The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points

TL;DR

This paper analyzes the distribution of exit points for a stochastic process in a domain, showing that at low temperatures, the exit points concentrate on the lowest energy saddle points on the boundary.

Contribution

It proves that the exit point distribution concentrates on minimal energy saddle points under general conditions, extending previous results to broader initial distributions.

Findings

Exit points concentrate on minimal energy saddle points at low temperature.

The support of the exit distribution localizes on points minimizing the potential on the boundary.

Results are extended to initial distributions beyond the quasi-stationary state.

Abstract

We consider the first exit point distribution from a bounded domain $\Omega$ of the stochastic process $(X_t)_{t\ge 0}$ solution to the overdamped Langevin dynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t$$ starting from the quasi-stationary distribution in $\Omega$. In the small temperature regime ($h\to 0$) and under rather general assumptions on $f$ (in particular, $f$ may have several critical points in $\Omega$), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of $f$ on $\partial \Omega$. The proof relies on tools to study tunnelling effects in semi-classical analysis. Extensions of the results to more general initial distributions than the quasi-stationary distribution are also presented.

Equations

$$d{X}_t=-\nabla f(X_t)dt+\sqrt{h}d{B}_t$$

$$d{X}_t=-\nabla f(X_t)dt+\sqrt{h}d{B}_t,$$

$${\tau }_{\Omega }=\inf \{t\ge 0|X_t\notin \Omega \}$$

$$\underset{h\to 0}{\lim }\mathbb{P}\left[X_{{\tau }_{\Omega }}\in {\mathcal{V}}_{\mathcal{Y}}\right]=1,$$

$$\underset{h\to 0}{\lim }\mathbb{P}\left[X_{{\tau }_{\Omega }}\in {\mathcal{V}}_x\right]>0.$$

$${\partial }_{n}f>0\text{ on }\partial \Omega ,$$

$$\{x\in \Omega ,|\nabla f(x)|=0\}=\{x_0\}\text{ with }f(x_0)=\underset{\overline{\Omega }}{\min }f\text{ and }\text{det Hess}f(x_0)>0,$$

$${\nu }_h(A)=\frac{{\int }_{\Omega }{\mathbb{P}}_x\left[X_t\in A,t<{\tau }_{\Omega }\right]{\nu }_h(dx)}{{\int }_{\Omega }{\mathbb{P}}_x\left[t<{\tau }_{\Omega }\right]{\nu }_h(dx)}.$$

$$\underset{t\to \infty }{\lim }\Vert \mathrm{Law}(X_t|t<{\tau }_{\Omega })-{\nu }_h{\Vert }_{TV}=0.$$

$$L_{f,h}^{(0)}=-\nabla f\cdot \nabla +\frac{h}{2}\Delta .$$

$$L_w^2(\Omega )=\left\{u:\Omega \to \mathbb{R},{\int }_{\Omega }{u}^2{e}^{-\frac{2}{h}f}<\infty \right\}$$

$${\int }_{\Omega }(L_{f,h}^{(0)}u)v{e}^{-\frac{2}{h}f}={\int }_{\Omega }(L_{f,h}^{(0)}v)u{e}^{-\frac{2}{h}f}=-\frac{h}{2}{\int }_{\Omega }\nabla u\cdot \nabla v{e}^{-\frac{2}{h}f}.$$

$$\varphi \in C_c^{\infty }(\Omega )\mapsto \frac{h}{2}{\int }_{\Omega }{\left|\nabla \varphi \right|}^2{e}^{-\frac{2}{h}f}$$

$$D\left(L_{f,h}^{D,(0)}\right)=H_{w,0}^1(\Omega )\cap H_w^2(\Omega ),$$

$$\lambda_{h}=\inf\sigma\big{(}-L^{D,(0)}_{f,h}\big{)}.$$

$$u_h>0\text{on}\Omega \text{and}{\int }_{\Omega }{u}_h^2{e}^{-\frac{2}{h}f}=1.$$

$$\{\begin{array}{rl}-L_{f,h}^{(0)}{u}_h& ={\lambda }_h{u}_h\mathrm{on}\Omega ,\\ u_h& =0\mathrm{on}\partial \Omega .\end{array}$$

$${\nu }_h(dx)=\frac{u_h(x)e^{-\frac{2}{h}f(x)}}{{\int }_{\Omega }{u}_h(y)e^{-\frac{2}{h}f(y)}dy}dx.$$

$$z\in \partial \Omega \mapsto -\frac{h}{2{\lambda }_h}\frac{{\partial }_n{u}_h(z)e^{-\frac{2}{h}f(z)}}{{\int }_{\Omega }{u}_h(y)e^{-\frac{2}{h}f(y)}dy}.$$

$$\{f<a\}=\{x\in \overline{\Omega },f(x)<a\},\{f\le a\}=\{x\in \overline{\Omega },f(x)\le a\},$$

$$\{f=a\}=\{x\in \overline{\Omega },f(x)=a\}.$$

$$\frac{d}{dt}{\phi }_t(x)=-\nabla f({\phi }_t(x))\text{ with }{\phi }_0(x)=x,$$

$$t_x=\inf \{t\ge 0,{\phi }_t(x)\notin \Omega \}>0.$$

$$\omega (x)=\{y\in \overline{\Omega },\exists (s_n{)}_{n\in \mathbb{N}}\in ({\mathbb{R}}_{+}{)}^{\mathbb{N}},\underset{n\to \infty }{\lim }{s}_n=+\infty ,\underset{n\to \infty }{\lim }{\phi }_{s_n}(x)=y\}.$$

$$\exists y\in \overline{\Omega },\omega (x)=\{y\}.$$

$$\mathcal{A}(D)=\{x\in \Omega ,t_x=+\infty \text{ and }\omega (x)\subset D\}.$$

$$\left.\begin{array}[]{ll}&\text{The function $f:\overline{\Omega}\to\mathbb{R}$ is a $C^{\infty}$ function.}\\ &\text{For all $x\in\partial\Omega$, $|\nabla f(x)|\neq 0$.}\\ &\text{The functions $f$ and $f|_{\partial\Omega}$ are Morse functions.}\\ &\text{Moreover,~{}$f$ has at least one local minimum in $\Omega$.}\end{array}\right\}$$

$$\mathsf{H}_{f}(x):=\,\inf_{\begin{subarray}{c}\gamma\in C^{0}([0,1],\overline{\Omega})\\ \gamma(0)=x\\ \gamma(1)\in\partial\Omega\end{subarray}}\ \,\max_{t\in[0,1]}\,f\big{(}\gamma(t)\big{)},$$

$$\exists!\mathsf{C}_{\mathsf{max}}\in\mathcal{C}\text{ such that }\max\limits_{\mathsf{C}\in\mathcal{C}}\,\Big{\{}\max_{\overline{\mathsf{C}}}f-\min_{\overline{\mathsf{C}}}f\Big{\}}=\max_{\overline{\mathsf{C}_{\mathsf{max}}}}f-\min_{\overline{\mathsf{C}_{\mathsf{max}}}}f$$

$$\displaystyle\mathcal{C}:=\big{\{}\mathsf{C}(x),\,x\text{ is a local minimum of $f$ in }\Omega\big{\}},$$

Full text

[TABLE]

starting from the quasi-stationary distribution in $\Omega$. In the small temperature regime ($h\to 0$) and under rather general assumptions on $f$ (in particular, $f$ may have several critical points in $\Omega$), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of $f$ on $\partial\Omega$. The proof relies on tools to study tunnelling effects in semi-classical analysis. Extensions of the results to more general initial distributions than the quasi-stationary distribution are also presented.

1. 1 Introduction and main results

2. 1.1 Setting and motivation

3. 1.1.1 Overdamped Langevin dynamics

4. 1.1.2 Exit point distribution and purpose of this work

5. 1.2 Metastability and the quasi-stationary distribution

6. 1.3 Hypotheses and main results

7. 1.3.1 Hypotheses and notation

8. 1.3.2 Notation for the local minima and saddle points of the function $f$

9. 1.3.3 Main results on the exit point distribution

10. 1.3.4 Intermediate results on the smallest eigenvalues and on the principal eigenfunction of $-L^{D,(0)}_{f,h}$

11. 1.4 Discussion of the hypotheses

12. 1.4.1 On the assumption (A0)

13. 1.4.2 On the assumption (A1)

14. 1.4.3 On the assumption (A2)

15. 1.4.4 On the assumption (A3)

16. 1.4.5 On the assumption (A4)

17. 1.5 Organization of the paper and outline of the proof

18. 2 Association of the local minima of $f$ with saddle points of $f$

19. 2.1 Connected components associated with the elements of $\mathsf{U}_{0}^{\Omega}$

20. 2.2 Separating saddle points

21. 2.3 A topological result under the assumption (A0)

22. 2.4 Constructions of the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$

23. 2.5 Rewriting the assumptions (A1)-(A4) in terms of the map $\mathbf{j}$

24. 3 Construction of the quasi-modes

25. 3.1 Notations and Witten Laplacian

26. 3.1.1 Notation for Sobolev spaces

27. 3.1.2 The Witten Laplacian and the infinitesimal generator of the diffusion (1)

28. 3.2 Construction of the quasi-modes for $-L_{f,h}^{D,(p)}$, $p\in\{0,1\}$

29. 3.2.1 Quasi-modes for the Witten Laplacian $\Delta_{f,h}^{D,(0)}$

30. 3.2.2 Quasi-modes for the Witten Laplacian $\Delta_{f,h}^{D,(1)}$

31. 3.2.3 Quasi-modes for $-L_{f,h}^{D,(p)}$, $p\in\{0,1\}$

32. 3.3 Bases of $\operatorname{Ran}\pi_{h}^{(0)}$ and $\operatorname{Ran}\pi_{h}^{(1)}$

33. 4 On the smallest eigenvalue of $-L_{f,h}^{D,(0)}$

34. 4.1 Estimates of interactions between quasi-modes

35. 4.1.1 Asymptotic estimates of boundary terms for $(\widetilde{\psi}_{i})_{i\in \{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$

36. 4.1.2 Quasi-modal estimates in $L^{2}_{w}(\Omega)$

37. 4.2 Restricted differential $\nabla:\,{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$

38. 4.2.1 Matrix of the restricted differential $\nabla:\,{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$

39. 4.2.2 On the $\mathsf{m}_{0}^{\Omega}$ smallest eigenvalues of $-L^{D,(0)}_{f,h}$

40. 4.2.3 Proof of Theorem 3

41. 5 On the principal eigenfunction of $-L_{f,h}^{D,(0)}$

42. 5.1 Proof of Proposition 59

43. 5.2 Proof of Theorem 6

44. 5.2.1 Gram-Schmidt orthonormalization

45. 5.2.2 Estimates of the interaction terms $\left(\langle\nabla u_{h},\psi_{j}\rangle_{L^{2}_{w}}\right)_{j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$

46. 5.2.3 Estimates of the boundary terms \Big{(}\displaystyle{\int_{\Sigma}\,F\,\psi_{j}\cdot n\,e^{-\frac{2}{h}f}}\Big{)}_{j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}}

47. 6 On the law of $X_{\tau_{\Omega}}$

48. 6.1 Proof of Theorem 1 when $X_{0}\sim\nu_{h}$

49. 6.2 Proof of Theorem 1 when $X_{0}=x\in\mathcal{A}(\mathsf{C}_{\mathsf{max}})$

50. 6.2.1 Leveling results

51. 6.2.2 End of the proof of Theorem 1

52. 6.3 Proof of Theorem 2

Introduction and main results

Setting and motivation

Overdamped Langevin dynamics

We are interested in the overdamped Langevin dynamics

[TABLE]

where $X_{t}\in\mathbb{R}^{d}$ is a vector in $\mathbb{R}^{d}$, $f:\mathbb{R}^{d}\to\mathbb{R}$ is a $C^{\infty}$ function, $h$ is a positive parameter and $(B_{t})_{t\geq 0}$ is a standard $d$-dimensional Brownian motion. Such a dynamics is prototypical of models used for example in computational statistical physics to simulate the evolution of a molecular system at a fixed temperature, in which case $f$ is the potential energy function and $h$ is proportional to the temperature. It admits as an invariant measure the Boltzmann-Gibbs measure (canonical ensemble) $Z^{-1}e^{-\frac{2}{h}f}$ where $Z=\int_{\mathbb{R}^{d}}e^{-\frac{2}{h}f}<\infty$. In the small temperature regime $h\to 0$, the stochastic process $(X_{t})_{t\geq 0}$ is typically metastable: it stays for a very long period of time in a subset of $\mathbb{R}^{d}$ (called a metastable state) before hopping to another metastable state. In the context of statistical physics, this behavior is expected since the molecular system typically jumps between various conformations, which are indeed these metastable states. For modelling purposes as well as for building efficient numerical methods, it is thus crucial to be able to precisely describe the exit event from a metastable state, namely the law of the first exit time and the first exit point.

The main objective of this work is to address the following question: given a metastable domain $\Omega\subset\mathbb{R}^{d}$, what are the exit points in the small temperature regime $h\to 0$? Compared to the work [10], we here only identify the support of the first exit point distribution, and the relative likelihood of the points in this support, whereas in [10], we also study the exit through points which occur with exponentially small probability in the limit $h\to 0$. The results here are thus less precise than in [10], but we also work under much more general assumptions on the function $f$ which are made precise in the next section.

Exit point distribution and purpose of this work

Let us consider a domain $\Omega\subset\mathbb{R}^{d}$ and the associated exit event from $\Omega$. More precisely, let us introduce

[TABLE]

the first exit time from $\Omega$. The concentration of the law of $X_{\tau_{\Omega}}$ on a subset of $\partial\Omega$ is defined as follows.

Definition 1**.**

Let $\mathcal{Y}\subset\partial\Omega$. The law of $X_{\tau_{\Omega}}$ concentrates on $\mathcal{Y}$ in the limit $h\to 0$ if for every neighborhood $\mathcal{V}_{\mathcal{Y}}$ of ${\mathcal{Y}}$ in $\partial\Omega$

[TABLE]

and if for all $x\in\mathcal{Y}$ and for all neighborhood $\mathcal{V}_{x}$ of $x$ in $\partial\Omega$

[TABLE]

In other words, $\mathcal{Y}$ is the support of the law of $X_{\tau_{\Omega}}$ in the limit $h\to 0$.

Previous results on the behaviour of the law of $X_{\tau_{\Omega}}$ when $h\to 0$. They are mainly three kinds of approaches to study where and how the law of $X_{\tau_{\Omega}}$ concentrates on $\partial\Omega$ when $h\to 0$. We refer to [5] for a comprehensive review of the literature.

The first approach one is based on formal computations: the concentration of the law of $X_{\tau_{\Omega}}$ on $\operatorname*{arg\,min}_{\partial\Omega}f$ in the small temperature regime ($h\to 0$) has been studied in [34] when $\partial_{n}f>0$ on $\partial\Omega$ and in [37, 42] when considering also the case when $\partial_{n}f=0$ on $\partial\Omega$.

The second approach is based on rigorous techniques developed for partial differential equations. When it holds,

[TABLE]

where $\partial_{n}f$ is the normal derivative of $f$ on $\partial\Omega$, and

[TABLE]

the concentration of the law of $X_{\tau_{\Omega}}$ in the limit $h\to 0$ on $\operatorname*{arg\,min}_{\partial\Omega}f$ has been obtained in [24, 23, 40, 6, 7], when $X_{0}=x\in\Omega$.

Finally, the last approach is based on techniques developed in large deviation theory. When (3) and (4) hold, and $f$ atteins its minimum on $\partial\Omega$ at one single point $y_{0}$, it is proved in [13, Theorem 2.1] that the law of $X_{\tau_{\Omega}}$ in the limit $h\to 0$ concentrates on $y_{0}$, when $X_{0}=x\in\Omega$. In [13, Theorem 5.1], under more general assumptions on $f$, for $\Sigma\subset\partial\Omega$, the limit of $h\ln\mathbb{P}\left[X_{\tau_{\Omega}}\in\Sigma\right]$ when $h\to 0$ is related to a minimization problem involving the quasipotential of the process (1). Let us mention two limitations when applying [13, Theorem 5.1] in order to obtain some information on the first exit point distribution. First, this theorem requires to be able to compute the quasipotential in order to get useful information: this is trivial under the assumptions (3) and (4) but more complicated under more general assumptions on $f$ (in particular when $f$ has several critical points in $\Omega$). Second, even when the quasi potential is analytically known, this result only gives the subset of $\partial\Omega$ where exit will not occur on an exponential scale in the limit $h\to 0$. It does not allow to exclude exit points with probability which goes to zero polynomially in $h$ (this indeed occurs, see Section 1.4.5), and it does not give the relative probability to exit through exit points with non-zero probability in the limit $h\to 0$.

Let us mention that [24, 23, 40, 6, 7, 13] also cover the case of non reversible diffusions.

Purpose of this work: the case when $f$ has several critical points in $\Omega$. As explained above, the concentration of the law of $X_{\tau_{\Omega}}$ on $\operatorname*{arg\,min}_{\partial\Omega}f$ was obtained when (3) and (4) hold (which imply in particular that $f$ has only one critical point in $\Omega$). Our work aims at generalizing in the reversible case the results [24, 23, 40, 6, 7] and [13, Theorem 2.1], when $f$ has several critical points in $\Omega$. In particular, we exhibit more general assumptions on $f$ in which the law of $X_{\tau_{\Omega}}$ concentrates on points belonging to $\operatorname*{arg\,min}_{\partial\Omega}f$ and we compute the relative probabilities to leave through each of them. For instance, we do not assume that $\partial_{n}f>0$ on $\partial\Omega$ (i.e. we drop the assumption (3)), we have no restriction on the number of critical points of $f$ in $\Omega$ (i.e. we do not assume (4)) and $f$ is allowed to have critical points in $\Omega$ (e.g. saddle points or local minima) with larger energies than $\min_{\partial\Omega}f$ (however we do not consider the case when $f$ has critical points on $\partial\Omega$). Here are examples of outputs of our work.

First, we show for example the following result: if $\{y\in\Omega,\,f(y)<\min_{\partial\Omega}f\}$ is connected and contains all the critical points of $f$ in $\Omega$ together with $\partial_{n}f>0$ on $\operatorname*{arg\,min}_{\partial\Omega}f$, then, the exit point distribution concentrates on $\operatorname*{arg\,min}_{\partial\Omega}f$ when $X_{0}$ is distributed according to the quasi-stationary distribution $\nu_{h}$ of the process (1) in $\Omega$ (see Definition 2 below) or $X_{0}=x\in\{y\in\Omega,\,f(y)<\min_{\partial\Omega}f\}$. This extends the results of [24, 23, 40, 6, 7] and [13, Theorem 2.1] to a more general geometric setting.

Second, we also study situations where critical points of $f$ in $\Omega$ are larger in energy than $\min_{\partial\Omega}f$. In such a case, again for $X_{0}$ distributed according to $\nu_{h}$ or $X_{0}=x\in K$ (where $K$ is a compact subset of $\Omega$ to be made precise below), the law of $X_{\tau_{\Omega}}$ concentrates when $h\to 0$ on a subset of $\partial\Omega$ which can be strictly included in $\operatorname*{arg\,min}_{\partial\Omega}f$. In particular, we show that the following phenomena can occur:

• (i)

There exist points $z\in\operatorname*{arg\,min}_{\partial\Omega}f$, $C>0$ and $c>0$, such that for all sufficiently small neighborhood $\Sigma_{z}$ of $z$ in $\partial\Omega$, in the limit $h\to 0$: $\mathbb{P}\left[X_{\tau_{\Omega}}\in\Sigma_{z}\right]\leq C\,e^{-\frac{c}{h}}$ (see (25) in Theorem 1 and the discussion after the statement of Theorem 1).

• (ii)

There exist points $z\in\operatorname*{arg\,min}_{\partial\Omega}f$ and $C>0$, for all sufficiently small neighborhood $\Sigma_{z}$ of $z$ in $\partial\Omega$, $\mathbb{P}\left[X_{\tau_{\Omega}}\in\Sigma_{z}\right]=C\sqrt{h}\,(1+o(1))$. This is explained in Section 1.4.5.

In particular, motivated by the desire to analyse the metastability of the exit event from $\Omega$, we exhibit explicit assumptions on $f$ which aim at ensuring the two following properties:

• [P1]

When $X_{0}$ is initially distributed according to the quasi-stationary distribution $\nu_{h}$ of the process (1) in $\Omega$ (see Definition 2 below), the law of $X_{\tau_{\Omega}}$ concentrates in the limit $h\to 0$ on some global minima of $f$ on $\partial\Omega$.

• [P2]

There exists a connected component of $\{f<\min_{\partial\Omega}f\}$ such that when $X_{0}=x$ and $x$ belongs to this connected component, the law of $X_{\tau_{\Omega}}$ concentrates in the limit $h\to 0$ on the same points of $\partial\Omega$ as it does when $X_{0}\sim\nu_{h}$.

Finally, we give sharp asymptotic estimates when $h\to 0$ on the principal eigenvalue and the principal eigenfunction of the infinitesimal generator of the diffusion (1) associated with Dirichlet boundary conditions on $\partial\Omega$, see Section 1.3.4. Let us mention that a simplified version of the results of this work is presented in [32].

Organization of the introduction. In Section 1.2, we introduce the quasi-stationary distribution associated with $\Omega$ and the process (1), and we explain why it is relevant to study the exit event from a metastable domain $\Omega$ assuming that the process (1) is initially distributed according to the quasi-stationary distribution. In Section 1.3, we introduce assumptions on $f$ which will be used throughout this paper and we state the main result of this work (see Theorem 1). Finally, in Section 1.4, we discuss the necessity of the assumptions related to obtain the results of Theorem 1.

Metastability and the quasi-stationary distribution

The quasi-stationary distribution is the cornerstone of our analysis. Here and in the following, we assume that $\Omega\subset\mathbb{R}^{d}$ is smooth, open, bounded and connected. Let us give the definition of the quasi-stationary distribution associated with the overdamped Langevin process (1) and $\Omega$:

Definition 2**.**

Let $\Omega\subset\mathbb{R}^{d}$ and consider the dynamics (1). A quasi-stationary distribution is a probability measure $\nu_{h}$ supported in $\Omega$ such that for all measurable sets $A\subset\Omega$ and for all $t\geq 0$

[TABLE]

Here and in the following, the subscript $x$ indicates that the stochastic process starts from $x\in\mathbb{R}^{d}$: $X_{0}=x$. In words, (5) means that if $X_{0}$ is distributed according to $\nu_{h}$, then for all $t>0$, $X_{t}$ is still distributed according to $\nu_{h}$ conditionally on $X_{s}\in\Omega$ for all $s\in(0,t)$. We have the following results from [28]:

Proposition 3**.**

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain and consider the dynamics (1). Then, there exists a probability measure $\nu_{h}$ with support in $\Omega$ such that, whatever the law of the initial condition $X_{0}$ with support in $\Omega$, it holds:

[TABLE]

Here, ${\rm Law}(X_{t}|t<\tau_{\Omega})$ denotes the law of $X_{t}$ conditional to the event $\{t<\tau_{\Omega}\}$. A corollary of this proposition is that the quasi-stationary distribution $\nu_{h}$ exists and is unique. For a given initial distribution of the process (1), if the convergence in (6) is much quicker than the exit from $\Omega$, the exit from the domain $\Omega$ is said to be metastable. When the exit from $\Omega$ is metastable, it is thus relevant to study the exit event from $\Omega$ assuming that the process (1) is initially distributed according to the quasi-stationary distribution $\nu_{h}$.

Let us introduce the infinitesimal generator of the dynamics (1), which is the differential operator

[TABLE]

In the notation $L^{(0)}_{f,h}$, the superscript $(0)$ indicates that we consider an operator on functions, namely [math]-forms. The basic observation to define our functional framework is that the operator $L^{(0)}_{f,h}$ is self-adjoint on the weighted $L^{2}$ space

[TABLE]

(the weighted Sobolev spaces $H^{k}_{w}(\Omega)$ are defined similarly). Indeed, for any smooth test functions $u$ and $v$ with compact supports in $\Omega$, one has

[TABLE]

This gives a proper framework to introduce the Dirichlet realization $L^{D,(0)}_{f,h}$ on $\Omega$ of the operator $L^{(0)}_{f,h}$:

Proposition 4**.**

The Friedrich’s extension associated with the quadratic form

[TABLE]

is denoted by $-L^{D,(0)}_{f,h}$. It is a non negative unbounded self adjoint operator on $L^{2}_{w}(\Omega)$ with domain

[TABLE]

where $H^{1}_{w,0}(\Omega)=\{u\in H^{1}_{w}(\Omega),\,u=0\text{ on }\partial\Omega\}$.

The compact injection $H^{1}_{w}(\Omega)\subset L^{2}_{w}(\Omega)$ implies that the operator $L^{D,(0)}_{f,h}$ has a compact resolvent and its spectrum is consequently purely discrete. Let us introduce $\lambda_{h}>0$ the smallest eigenvalue of $-L^{D,(0)}_{f,h}$:

[TABLE]

The eigenvalue $\lambda_{h}$ is called the principal eigenvalue of $-L^{D,(0)}_{f,h}$. From standard results on elliptic operator (see for example [15, 12]), $\lambda_{h}$ is non degenerate and its associated eigenfunction $u_{h}$ has a sign on $\Omega$. Moreover, $u_{h}\in C^{\infty}(\overline{\Omega})$. Without loss of generality, one can then assume that:

[TABLE]

The eigenvalue-eigenfunction pair $(\lambda_{h},u_{h})$ satisfies:

[TABLE]

The link between the quasi stationary distribution $\nu_{h}$ and the function $u_{h}$ is given by the following proposition (see for example [28]):

Proposition 5**.**

The unique quasi-stationary distribution $\nu_{h}$ associated with the dynamics (1) and the domain $\Omega$ is given by:

[TABLE]

The next proposition (which can also be found in [28]) characterizes the law of the exit event from $\Omega$.

Proposition 6**.**

Let us consider the dynamics (1) and the quasi stationary distribution $\nu_{h}$ associated with the domain $\Omega$. If $X_{0}$ is distributed according to $\nu_{h}$, the random variables $\tau_{\Omega}$ and $X_{\tau_{\Omega}}$ are independent. Furthermore $\tau_{\Omega}$ is exponentially distributed with parameter $\lambda_{h}$ and the law of $X_{\tau_{\Omega}}$ has a density with respect to the Lebesgue measure on $\partial\Omega$ given by

[TABLE]

Here and in the following, $\partial_{n}=n\cdot\nabla$ stands for the normal derivative and $n$ is the unit outward normal on $\partial\Omega$.

Hypotheses and main results

This section is dedicated to the statement of the main result of this work.

Hypotheses and notation

In the following, we consider a setting that is more general than the one of Section 1.2: $\overline{\Omega}$ is a $C^{\infty}$ oriented compact and connected Riemannian manifold of dimension $d$ with boundary $\partial\Omega$.

The following notation will be used: for $a\in\mathbb{R}$,

[TABLE]

and

[TABLE]

Let us recall the definition of the domain of attraction of a subset $D$ of $\Omega$ for the $-\nabla f$ dynamics. Let $f:\overline{\Omega}\to\mathbb{R}$ be a $C^{\infty}$ function. Let $x\in\Omega$ and denote by $\varphi_{t}(x)$ the solution to the ordinary differential equation

[TABLE]

on the interval $t\in[0,t_{x}]$, where

[TABLE]

Let $x\in\Omega$ be such that $t_{x}=+\infty$. The $\omega$-limit set of $x$, denoted by $\omega(x)$, is defined by

[TABLE]

Let us recall that the $\omega$-limit set $\omega(x)$ is included in the set of the critical points of $f$ in $\overline{\Omega}$. Moreover, when $f$ has a finite number of critical points in $\overline{\Omega}$,

[TABLE]

Let $D$ be a subset of $\Omega$. The domain of attraction of a subset $D$ of $\Omega$ is defined by

[TABLE]

Let us now introduce the basic assumption which is used throughout this work:

[TABLE]

A function $\phi:\overline{\Omega}\to\mathbb{R}$ is a Morse function if all its critical points are non degenerate (which implies in particular that $\phi$ has a finite number of critical points since $\overline{\Omega}$ is compact and a non degenerate critical point is isolated from the other critical points). Let us recall that a critical point $z\in\overline{\Omega}$ of $\phi$ is non degenerate if the hessian matrix of $\phi$ at $z$, denoted by ${\rm Hess\,}\phi(z)$, is invertible. We refer for example to [22, Definition 4.3.5] for a definition of the hessian matrix on a manifold. A non degenerate critical point $z\in\overline{\Omega}$ of $\phi$ is said to have index $p\in\{0,\dots,d\}$ if ${\rm Hess\,}\phi(z)$ has precisely $p$ negative eigenvalues (counted with multiplicity). In the case $p=1$, $z$ is called a saddle point.

For any local minimum $x$ of $f$ in $\Omega$, one defines

[TABLE]

where $C^{0}([0,1],\overline{\Omega})$ is the set of continuous paths from $[0,1]$ to $\overline{\Omega}$. In Section 2.1, another equivalent definition of $\mathsf{H}_{f}$ is given (see indeed (38) and (40)). Let us now define a set of assumptions which will ensure that [P1] and [P2] are satisfied (see indeed Theorem 1 and Section 1.4 for a discussion on these assumptions):

• •

(A0) holds and

[TABLE]

where

[TABLE]

with, for a local minimum $x$ of $f$ in $\Omega$,

[TABLE]

• •

(A1) holds and

[TABLE]

• •

(A1) holds and

[TABLE]

More precisely, the assumptions (A0), (A1), (A2), and (A3) ensure that when $X_{0}\sim\nu_{h}$ or $X_{0}=x\in\mathsf{C}_{\mathsf{max}}$, the law of $X_{\tau_{\Omega}}$ concentrates on the set $\partial\mathsf{C}_{\mathsf{max}}\cap\partial\Omega$, see items 1 and 2 in Theorem 1. Finally, let us introduce the following assumption:

• (A1) holds and

[TABLE]

where the definition of a separating saddle point of $f$ is introduced below in item 1 in Definition 18.

The assumption (A4) together with (A0), (A1), (A2), and (A3), ensures that the probability that the process (1) (starting from the quasi-stationary distribution $\nu_{h}$ or from $x\in\mathsf{C}_{\mathsf{max}}$) leaves $\Omega$ through any sufficiently small neighborhood of $z\in\partial\Omega\setminus\partial\mathsf{C}_{\mathsf{max}}$ in $\partial\Omega$ is exponentially small when $h\to 0$, see indeed item 3 in Theorem 1.

In Figure 1, one has represented a one-dimensional case where (A1), (A2), (A3) and (A4) are satisfied. In Section 1.4, the assumptions (A1), (A2), (A3), and (A4) are discussed. In particular, it is shown that if one of the assumptions among (A1), (A2), or (A3) does not hold, then there exists a function $f$ for which either [P1] or [P2] is not satisfied. Equivalent formulations of the assumptions (A1), (A2), (A3), and (A4) will be given in Section 2.5.

Remark 7**.**

It is proved in Proposition 20 that when (A0) holds, for all local minima $x$ of $f$ in $\Omega$, one has $\mathsf{C}(x)\subset\Omega$ (see (17)). This implies that for all $y\in\mathsf{C}(x)$, $t_{y}=+\infty$ and then, $\mathsf{C}(x)\subset\mathcal{A}(\mathsf{C}(x))$.

Notation for the local minima and saddle points of the function $f$

The main purpose of this section is to introduce the local minima and the generalized saddle points of $f$. These elements of $\overline{\Omega}$ are used extensively throughout this work and play a crucial role in our analysis. Roughly speaking, the generalized saddle points of $f$ are the saddle points $z\in\overline{\Omega}$ of the extension of $f$ by $-\infty$ outside $\overline{\Omega}$. Thus, when the function $f$ satisfies the assumption (A0), a generalized saddle point of $f$ (as introduced in [18]) is either a saddle point $z\in\Omega$ of $f$ or a local minimum $z\in\partial\Omega$ of $f|_{\partial\Omega}$ such that $\partial_{n}f(z)>0$.

Let us assume that the function $f$ satisfies the assumption (A0). Let us denote by

[TABLE]

the set of local minima of $f$ in $\Omega$ where $\mathsf{m}_{0}^{\Omega}\in\mathbb{N}$ is the number of local minima of $f$ in $\Omega$. Notice that since $f$ satisfies (A0), $\mathsf{m}_{0}^{\Omega}\geq 1$.

The set of saddle points of $f$ of index $1$ in $\Omega$ is denoted by $\mathsf{U}_{1}^{\Omega}$ and its cardinality by $\mathsf{m}_{1}^{\Omega}$. Let us define

[TABLE]

Notice that an equivalent definition of $\mathsf{U}_{1}^{\partial\Omega}$ is

[TABLE]

which follows from the fact that $\nabla f(x)\neq 0$ for all $x\in\partial\Omega$. Let us introduce

[TABLE]

In addition, one defines:

[TABLE]

The set $\mathsf{U}_{1}^{\overline{\Omega}}$ is the set of the generalized saddle points of $f$. If $\mathsf{U}_{1}^{\partial\Omega}$ is not empty, its elements are denoted by:

[TABLE]

and if $\mathsf{U}_{1}^{\Omega}$ is not empty, its elements are labeled as follows:

[TABLE]

Thus, one has:

[TABLE]

We assume that the elements of $\mathsf{U}_{1}^{\partial\Omega}$ are ordered such that:

[TABLE]

Notice that $\mathsf{k}_{1}^{\partial\Omega}\in\{0,\ldots,\mathsf{m}_{1}^{\partial\Omega}\}$.

Let us assume that the assumptions (A1), (A2), and (A3) are satisfied. In this case, let us recall that $\mathsf{C}_{\mathsf{max}}$ is defined by (A1). Moreover, in this case, one has $\mathsf{k}_{1}^{\partial\Omega}\geq 1$ and

[TABLE]

Indeed, by assumption $\partial\mathsf{C}_{\mathsf{max}}\cap\partial\Omega\subset\{f=\min_{\partial\Omega}f\}$ (see (A3)) and there is no local minima of $f$ in $\overline{\Omega}$ on $\partial\mathsf{C}_{\mathsf{max}}$ (since $\mathsf{C}_{\mathsf{max}}$ is a sublevel set of $f$). We assume lastly that the set $\{z_{1},\ldots,z_{\mathsf{k}_{1}^{\partial\Omega}}\}$ is ordered such that:

[TABLE]

Notice that $\mathsf{k}_{1}^{\partial\mathsf{C}_{\mathsf{max}}}\in\mathbb{N}^{*}$ and $\mathsf{k}_{1}^{\partial\mathsf{C}_{\mathsf{max}}}\leq\mathsf{k}_{1}^{\partial\Omega}$. We provide an example in Figure 2 to illustrate the notations introduced in this section.

As introduced in [18, Section 5.2], $\mathsf{U}_{0}^{\Omega}$ is the set of generalized critical points of $f$ of index [math] for the Witten Laplacian acting on functions with Dirichlet boundary conditions on $\partial\Omega$, and $\mathsf{U}_{1}^{\overline{\Omega}}$ is the set of generalized critical points of $f$ of index $1$ for the Witten Laplacian acting on $1$-forms with tangential Dirichlet boundary conditions on $\partial\Omega$. We refer to Section 3.1.2 for the definition of these Witten Laplacians.

Remark 8**.**

The assumption (A0) implies that $f$ does not have any saddle point (i.e critical point of index $1$) on $\partial\Omega$. Actually, under (A0), the points $(z_{i})_{i=1,\ldots,\mathsf{m}_{1}^{\partial\Omega}}$ play geometrically the role of saddle points. Indeed, zero Dirichlet boundary conditions are consistent with extending $f$ by $-\infty$ outside $\overline{\Omega}$, in which case the point $(z_{i})_{i=1,\ldots,\mathsf{m}_{1}^{\partial\Omega}}$ are geometrically saddle points of $f$ (i.e. $z_{i}$ is a local minimum of $f|_{\partial\Omega}$ and a local maximum of $f|_{D_{i}}$, where $D_{i}$ is the straight line passing through $z_{i}$ and orthogonal to $\partial\Omega$ at $z_{i}$).

Main results on the exit point distribution

The main result of this work is the following.

Theorem 1**.**

Let us assume that the assumptions (A0), (A1), (A2), and (A3) are satisfied. Let $F\in L^{\infty}(\partial\Omega,\mathbb{R})$ and $(\Sigma_{i})_{i\in\{1,\dots,\mathsf{k}_{1}^{\partial\Omega}\}}$ be a family of disjoint open subsets of $\partial\Omega$ such that

[TABLE]

where we recall that \big{\{}z_{1},\dots,z_{\mathsf{k}_{1}^{\partial\Omega}}\big{\}}=\mathsf{U}_{1}^{\partial\Omega}\cap\operatorname*{arg\,min}_{\partial\Omega}f (see (23)). Let $K$ be a compact subset of $\Omega$ such that $K\subset\mathcal{A}(\mathsf{C}_{\mathsf{max}})$ (see (A1) and (14)). Let $\mu_{0}$ be a probability distribution which is either supported in $K$ or equals to the quasi-stationary distribution $\nu_{h}$ of the process (1) in $\Omega$ (see Definition 2 and (11)). Then:

There exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

and

[TABLE]

where we recall that \big{\{}z_{1},\ldots,z_{\mathsf{k}_{1}^{\partial\mathsf{C}_{\mathsf{max}}}}\big{\}}=\partial\mathsf{C}_{\mathsf{max}}\cap\partial\Omega (see (24)). 2. 2.

When for some i\in\big{\{}1,\dots,\mathsf{k}_{1}^{\partial\mathsf{C}_{\mathsf{max}}}\big{\}} the function $F$ is $C^{\infty}$ in a neighborhood of $z_{i}$, one has when $h\to 0$:

[TABLE]

where

[TABLE] 3. 3.

When (A4) is satisfied the remainder term $O(h^{\frac{1}{4}})$ in (26) is of the order O\big{(}e^{-\frac{c}{h}}\big{)} for some $c>0$ and the remainder term O\big{(}h^{\frac{1}{4}}\big{)} in (27) is of the order $O(h)$ and admits a full asymptotic expansion in $h$ (as defined in Remark 9 below).

Finally, the constants involved in the remainder terms in (25), (26), and (27) are uniform with respect to the probability distribution $\mu_{0}$ supported in $K$.

Remark 9**.**

Let us recall that for $\alpha>0$, $(r(h))_{h>0}$ admits a full asymptotic expansion in $h^{\alpha}$ if there exists a sequence $(a_{k})_{k\geq 0}\in\mathbb{R}^{\mathbb{N}}$ such that for any $N\in\mathbb{N}$, it holds in the limit $h\to 0$:

[TABLE]

According to (25), when the function $F$ belongs to $C^{\infty}(\partial\Omega,\mathbb{R})$ and $x\in\mathcal{A}(\mathsf{C}_{\mathsf{max}})$, one has in the limit $h\to 0$:

[TABLE]

where the order in $h$ of the remainder term $o_{h}(1)$ depends on the support of $F$ and on whether or not the assumption (A4) is satisfied. This is reminiscent of previous results obtained in [24, 23, 40, 6, 7].

Theorem 1 implies that in the limit $h\to 0$, when $X_{0}\sim\nu_{h}$ or $X_{0}=x\in\mathcal{A}(\mathsf{C}_{\mathsf{max}})$, the law of $X_{\tau_{\Omega}}$ concentrates on the set $\{z_{1},\dots,z_{\mathsf{k}_{1}^{\mathsf{C}_{\mathsf{max}}}}\}=\partial\Omega\cap\partial\mathsf{C}_{\mathsf{max}}$ with explicit formulas for the probabilities to exit through each of the $z_{i}$’s. Therefore, [P1] and [P2] are satisfied when the assumptions (A1), (A2), and (A3) holds.

Another consequence of Theorem 1 is the following. The probability to exit through a global minimum $z$ of $f|_{\partial\Omega}$ which satisfies $\partial_{n}f(z)<0$ is exponentially small in the limit $h\to 0$ (see (25)) and when assuming (A4), the probability to exit through $z_{\mathsf{k}_{1}^{\mathsf{C}_{\mathsf{max}}}+1},\dots,z_{\mathsf{k}_{1}^{\partial\Omega}}$ is also exponentially small even though all these points belong to $\operatorname*{arg\,min}_{\partial\Omega}f$.

Let us now give two crucial results used in the proof of [P2] in Theorem 1. The first result shows that, when the assumptions (A0) and (A1) are satisfied, and $\min_{\mathsf{C}_{\mathsf{max}}}f=\min_{\overline{\Omega}}f$ (which is automatically the case when (A1), (A2), and (A3) hold, see Lemma 26), the quasi-stationary distribution $\nu_{h}$ (see Proposition 5) concentrates in neighborhoods of the global minima of $f$ in $\mathsf{C}_{\mathsf{max}}$. This is stated in the following proposition.

Proposition 10**.**

Assume that the assumptions (A0) and (A1) are satisfied. Furthermore, let us assume that

[TABLE]

where we recall that $\mathsf{C}_{\mathsf{max}}$ is introduced in (A1). Let $\mathsf{O}$ be an open subset of $\Omega$. Then, if $\mathsf{O}\cap\operatorname*{arg\,min}_{\mathsf{C}_{\mathsf{max}}}f\neq\emptyset$, one has in the limit $h\to 0$:

[TABLE]

When $\overline{\mathsf{O}}\cap\operatorname*{arg\,min}_{\mathsf{C}_{1}}f=\emptyset$, there exists $c>0$ such that when $h\to 0$:

[TABLE]

Proposition 10 is a direct consequence of (11) and Proposition 59 below (see the beginning of Section 5).

The second result used in the proof of Theorem 1 connects the law of $X_{\tau_{\Omega}}$ when $X_{0}\sim\nu_{h}$ and $X_{0}=x\in\mathcal{A}(\mathsf{C}_{\mathsf{max}})$ in the limit $h\to 0$. This is stated in the following proposition.

Proposition 11**.**

Assume that the assumptions (A0) and (A1) are satisfied. Let us moreover assume that

[TABLE]

where we recall that $\mathsf{C}_{\mathsf{max}}$ is introduced in (A1). Let $K$ be a compact subset of $\Omega$ such that $K\subset\mathcal{A}(\mathsf{C}_{\mathsf{max}})$ and let $F\in C^{\infty}(\partial\Omega,\mathbb{R})$. Then, there exists $c>0$ such that for all $x\in K$:

[TABLE]

in the limit $h\to 0$ and uniformly in $x\in K$.

Proposition 11 is a direct consequence of Lemma 69 below (see Section 6.2.2). It gives sufficient conditions to ensure that [P2] is satisfied.

Let us end this section with the following theorem dealing with the case when $X_{0}=x\in\mathcal{A}(\mathsf{C})$, when $\mathsf{C}\in\mathcal{C}$ (see (16)) is not necessarily $\mathsf{C}_{\mathsf{max}}$.

Theorem 2**.**

Let us assume that (A0) holds. Let $\mathsf{C}\in\mathcal{C}$ (see (16)). Let us assume that

[TABLE]

Recall that $\partial\mathsf{C}\cap\partial\Omega\subset\mathsf{U}_{1}^{\partial\Omega}$ (see (19) and (21)). For all $z\in\partial\mathsf{C}\cap\partial\Omega$, let $\Sigma_{z}$ be an open subset of $\partial\Omega$ such that $z\in\Sigma_{z}$. Let $K$ be a compact subset of $\Omega$ such that $K\subset\mathcal{A}(\mathsf{C})$. Then, there exists $c>0$ such that for $h$ small enough,

[TABLE]

Assume moreover that the sets $(\Sigma_{z})_{z\in\partial\mathsf{C}\cap\partial\Omega}$ are two by two disjoint. Let $z\in\partial\mathsf{C}\cap\partial\Omega$. Then, it holds for all $x\in K$,

[TABLE]

in the limit $h\to 0$ and uniformly in $x\in K$.

Theorem 2 implies that when $\mathsf{C}\in\mathcal{C}$ satisfies (29) (for instance, this is the case for $\mathsf{C}_{3}$ on Figures 1 and 2), the law of $X_{\tau_{\Omega}}$ when $X_{0}=x\in\mathcal{A}(\mathsf{C})$ concentrates when $h\to 0$ on $\partial\mathsf{C}\cap\partial\Omega$. Let us mention that the proof of Theorem 2 is based on the use of Theorem 1 with a suitable subdomain of $\Omega$ containing $\mathsf{C}$.

When ${\mathsf{C}}\in\mathcal{C}$ and does not satisfy (29), it is much harder to exhibit explicit assumptions on $\mathsf{C}$ to give the most probable places of exit from $\Omega$ of the process (1) when $h\to 0$ (as suggested by one dimensional-examples, see Appendix B).

Intermediate results on the smallest eigenvalues and on the principal eigenfunction of $-L^{D,(0)}_{f,h}$

Let us recall that from (12), one has:

[TABLE]

Therefore, to obtain the asymptotic estimates on $\mathbb{E}_{\nu_{h}}\left[F\left(X_{\tau_{\Omega}}\right)\right]$ stated in Theorem 1 when $h\to 0$, i is sufficient to study the asymptotic behaviour of the quantities

[TABLE]

The intermediate results we obtain are the following.

In Theorem 5, one gives for $h\to 0$ small enough, a lower and an upper bound for all the $\mathsf{m}_{0}^{\Omega}$ small eigenvalues of $-L^{D,(0)}_{f,h}$ when (A0) is satisfied. 2. 2.

In Theorems 3 and 4, one gives a sharp asymptotic equivalent in the limit $h\to 0$ of the smallest eigenvalue $\lambda_{h}$ of $-L^{D,(0)}_{f,h}$ when (A0) and (A1) are satisfied. 3. 3.

In Proposition 59, when (A0), (A1) and $\min_{\overline{\mathsf{C}_{\mathsf{max}}}}f=\min_{\overline{\Omega}}f$ hold, one shows that $u_{h}\,e^{-\frac{2}{h}f}$ concentrates in the $L^{1}(\Omega)$-norm on the global minima of $f$ in $\mathsf{C}_{\mathsf{max}}$ in the limit $h\to 0$. 4. 4.

In Theorem 6, one studies the concentration in the limit $h\to 0$ of the normal derivative of the principal eigenvalue $u_{h}$ of $-L^{D,(0)}_{f,h}$ on $\partial\Omega$ when (A0),(A1), (A2), and (A3) are satisfied. In particular, one computes sharp asymptotic equivalents of $\partial_{n}u_{h}$ in neighborhoods of $\partial\mathsf{C}_{\mathsf{max}}\cap\partial\Omega$ in $\partial\Omega$.

Discussion of the hypotheses

In this section, we discuss the necessity of the assumptions (A1), (A2), and (A3) to obtain [P1] and [P2]. We also discuss the necessity of the assumption (A4) in order to get item 3 in Theorem 1.

On the assumption (A0)

The results of this work actually still hold under a weaker assumption than (A0), namely by simply assuming that $f:\{x\in\partial\Omega,\partial_{n}f(x)>0\}\to\mathbb{R}$ is a Morse function instead of $f|_{\partial\Omega}$ is a Morse function. Indeed, as mentioned in [17, Section 7.1], the statement of Lemma 27 (which is the only place where we use the fact that $f$ is a Morse function on $\{x\in\partial\Omega,\partial_{n}f(x)\leq 0\}$, relying on [18, Section 3.4]) still holds under this weaker assumption, see Appendix A.

On the assumption (A1)

In this section, we discuss the necessity of the assumption (A1) in order to obtain the results of Theorem 1 (or equivalently [P1] and [P2]). More precisely, one first exhibits a case where (A1) and [P2] are not satisfied. Then, one shows that there are cases where [P1] and [P2] are satisfied but not (A1). Finally, one explains why it is more difficult to analyse [P1] and [P2] when (A1) does no hold.

An example where (A1) and [P2] are not satisfied.

Let us consider $z_{1}>0$, $z_{2}:=-z_{1}$, $z=0$ and $f\in C^{\infty}([z_{1},z_{2}],\mathbb{R})$ a Morse function such that

[TABLE]

where

[TABLE]

Notice that in this case $x_{1}=-x_{2}$, $x_{1}$ and $x_{2}$ are the two global minima of $f$ on $[z_{1},z_{2}]$, $z$ is the global maximum of $f$ on $[z_{1},z_{2}]$ and $\mathsf{H}_{f}(x_{1})=\mathsf{H}_{f}(x_{2})=f(z_{1})$. Such a function $f$ is represented in Figure 3. For such functions $f$, the assumption (A1) is not satisfied since \operatorname*{arg\,max}\,\big{\{}\mathsf{H}_{f}(x)-f(x),\ x\text{ is local minimum of }f\text{ in }\Omega\big{\}}=\{x_{1},x_{2}\} and $x_{1}$ belongs to a connected component of $\{f<\mathsf{H}_{f}(x_{1})\}$ which differs from the connected component of $\{f<\mathsf{H}_{f}(x_{1})\}$ which contains $x_{2}$.

Since for $x\in(z_{1},z_{2})$ and $h>0$, $\nu_{h}(x)=\nu_{h}(-x)$ and $\mathbb{P}_{x}[X_{\tau_{(z_{1},z_{2})}}=z_{1}]=\mathbb{P}_{-x}[X_{\tau_{(z_{1},z_{2})}}=z_{2}]$, one has for all $h>0$:

[TABLE]

However, from (282) below (see the Appendix) together with Laplace’s method, for $x\in(z_{1},z)$, there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

and for $x\in(z,z_{2})$, there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

Therefore, in this example, the assumption [P2] is not satisfied. The domain $\Omega$ is not metastable (see Section 1.2) for any deterministic initial conditions $X_{0}=x\in[z_{1},z_{2}]\setminus\{z\}$.

There are cases when [P1] and [P2] are satisfied but not (A1).

In the symmetric case depicted in Figure 3 the quasi-stationary distribution $\nu_{h}$ concentrates in the two wells $(z_{1},z)$ and $(z,z_{2})$ (see [31]): i.e. for any $a_{1}<b_{1}$ such that $(a_{1},b_{1})\subset(z_{1},z)$ and $x_{1}\in(a_{1},b_{1})$, and $a_{2}<b_{2}$ such that $(a_{2},b_{2})\subset(z,z_{2})$ and $x_{2}\in(a_{2},b_{2})$, it holds

[TABLE]

However, it is proved in [31], that this equal repartition of $\nu_{h}$ when $h\to 0$ is unstable with respect to perturbations. Indeed, changing a little bit the value of the determinant of the hessian matrix at $x_{1}$ or $x_{2}$, or the normal derivative at $z_{1}$ or $z_{2}$ of the symmetric potential $f$ depicted in Figure 3 (while keeping the fact that (A1) is not satisfied) makes $\nu_{h}$ concentrates in the limit $h\to 0$ in only one of the two wells $(z_{1},z)$ or $(z,z_{2})$, and [P1] and [P2] then also hold.

Remark 12**.**

The main goal of [31] is to study the repartition of $\nu_{h}$ when $h\to 0$ in the double-well case (where (A1) does not hold). In particular, in this case, it is shown that the asymptotic behaviour when $h\to 0$ of $\nu_{h}$ which generically happens is the following: $\nu_{h}$ concentrates in only one of the two wells in the limit $h\to 0$, and [P1] and [P2] hold.

On the analysis of [P1] and [P2] when (A1) does not hold.

To analyse whether [P1] or [P2] is satisfied when (A1) does not hold, one needs in particular to have access to the repartition of $\nu_{h}$ in neigborhoods of the local minima of $f$ in $\Omega$ when $h\to 0$.

When (A1) is not satisfied, the analysis of the repartition of $\nu_{h}$ is tricky. This can be explained as follows. When (A1) is not satisfied, one has from Theorem 5 below (see Section 4.2.2),

[TABLE]

where $\lambda_{2,h}$ is the second smallest eigenvalue of $-L^{D,(0)}_{f,h}$. It is difficult to measure the quality of the approximation of $u_{h}$ by projecting an ansatz on ${\rm Span}(u_{h})$, since the error is related to the ratio of $\lambda_{h}$ over $\lambda_{2,h}$ (see Lemma 28). Moreover, when (A1) is not satisfied, it is difficult to predict in which well $\nu_{h}$ concentrates when it does, as explained in [31]. This is again due to the fact that this prediction relies on a very accurate comparison between $\lambda_{h}$ and $\lambda_{2,h}$.

On the contrary, when the assumption (A1) is satisfied, one can more easily obtain an approximation of $u_{h}$ (see (212) below) since in that case $\lim_{h\to 0}h\ln\lambda_{h}<\lim_{h\to 0}h\ln\lambda_{2,h}$ and thus Lemma 28 provides a sufficiently accurate error estimate of the approximation of $u_{h}$ by a simple ansatz (namely a cut-off function), see indeed Proposition 10 above and Proposition 59 below.

On the assumption (A2)

In this section, we discuss the assumption (A2) to obtain the results stated in Theorem 1. To this end, let us consider the following one-dimensional example. Let $z_{1}<z_{2}$ and $f$: $[z_{1},z_{2}]\to\mathbb{R}$ be a $C^{\infty}$ Morse function. Let us assume that $\{x\in[z_{1},z_{2}],f^{\prime}(x)=0\}=\{x_{1},x_{2},c,d\}$ with $z_{1}<x_{1}<c<x_{2}<d<z_{2}$, $f(x_{2})<f(x_{1})<f(z_{1})<f(z_{2})<f(d)<f(c)$ (see Figure 4). This implies that $f^{\prime}(z_{2})<0$, $f(d)-f(x_{2})>f(z_{1})-f(x_{1})$. Moreover, it holds

[TABLE]

The assumption (A1) is satisfied but not (A2). From (284) below (see Appendix B), there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

Therefore, in the small temperature regime and starting from the quasi-stationary distribution, the process (1) leaves $\Omega=(z_{1},z_{2})$ through $z_{2}$ when $h\to 0$. Notice that $z_{2}$ is not the global minimum of $f|_{\partial\Omega}$ and is even not a generalized critical point of index $1$. Consequently, the condition [P1] is not satisfied.

On the assumption (A3)

In this section, we discuss the assumption (A3) to obtain the results of Theorem 1. To this end, let us consider the following one-dimensional case. Let $z_{1}<z_{2}$ and $f$: $[z_{1},z_{2}]\to\mathbb{R}$ be a $C^{\infty}$ Morse function. Let us assume that $\{x\in[z_{1},z_{2}],f^{\prime}(x)=0\}=\{x_{1},z,x_{2}\}$ where $z_{1}<x_{1}<z<x_{2}<z_{2}$ $f(x_{2})<f(x_{1})<f(z_{1})<f(z_{2})<f(z)$ (see Figure 5). This implies $f(z_{1})-f(x_{1})<f(z_{2})-f(x_{2})$, $f^{\prime}(z_{1})<0$, $f^{\prime}(z_{2})>0$, $x_{2}$ is the global minimum of $f$ in $[z_{1},z_{2}]$, $x_{1}$ is a local minimum of $f$ and $z$ the global maximum of $f$ in $[z_{1},z_{2}]$. Then it holds,

[TABLE]

and $\mathsf{C}_{\mathsf{max}}\subset(z,z_{2})$. The assumptions (A1) and (A2) are satisfied but not (A3). From (286) below (see Appendix B), there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

Therefore, when $X_{0}\sim\nu_{h}$, the law of $X_{\tau_{\Omega}}$ concentrates on $z_{2}$ in the limit $h\to 0$. Since $f(z_{2})>\min_{\partial\Omega}f$, the property [P1] is not satisfied.

On the assumption (A4)

In this section, one gives an example to show that when (A4) is not satisfied, the remainder term $O(h^{\frac{1}{4}})$ in (26) is not of the order $O(e^{-\frac{c}{h}})$ for some $c>0$. To this end, let us consider the following one-dimensional case. Let $z_{1}<z_{2}$ and $f$: $[z_{1},z_{2}]\to\mathbb{R}$ be a $C^{\infty}$ Morse function. Let us assume that $\{x\in[z_{1},z_{2}],f^{\prime}(x)=0\}=\{x_{1},z,x_{2}\}$ with $z_{1}<x_{1}<z<x_{2}<z_{2}$ and $f(x_{1})<f(x_{2})<f(z)=f(z_{1})=f(z_{2})$ (see Figure 6). This implies $f^{\prime}(z_{1})<0$, $f^{\prime}(z_{2})>0$, $x_{1}$ is the global minimum of $f$ in $[z_{1},z_{2}]$, $x_{2}$ is a local minimum of $f$ and $z$ is a local maximum of $f$. In this example, it holds:

[TABLE]

and

[TABLE]

where $\mathsf{C}\neq\mathsf{C}_{\mathsf{max}}$ is the other connected component of $\{f<\mathsf{H}_{f}(x_{1})\}$. The assumptions (A1), (A2), and (A3) are satisfied whereas, since $z\in\partial\mathsf{C}_{\mathsf{max}}\cap\Omega$ is a separating saddle point of $f$ (see item 1 in Definition 18 below), the hypothesis (A4) is not satisfied. From (282) (in Appendix B) together with Laplace’s method, for $x\in\mathsf{C}_{\mathsf{max}}$, one has in the limit $h\to 0$:

[TABLE]

Moreover, a similar result holds starting from $\nu_{h}$ (using Proposition 11 above): in the limit $h\to 0$:

[TABLE]

In this case, the exit through $z_{2}$ when $h\to 0$ is not exponentially small but is exactly of the order $\sqrt{h}$ even though $z_{2}$ is a generalized critical point of $f$ on $\partial\Omega$ (i.e $f(z_{2})\in\mathsf{U}_{1}^{\partial\Omega}$, see (19)) and $f(z_{2})=\min_{\partial\Omega}f$. In conclusion, the remainder term $O(h^{\frac{1}{4}})$ in (26) is in general not of the order $O(e^{-\frac{c}{h}})$ and is actually exactly of the order $O(\sqrt{h})$ in this example.

Remark 13**.**

This can be generalized to higher-dimensional settings. In [38, Proposition C.40, item 3], one shows with some higher-dimensional cases for which the assumption (A4) does not hold, that the remainder terms O\big{(}h^{\frac{1}{4}}\big{)} in (26) and (27) are of the order $O(\sqrt{h})$. We moreover expect that the reminder terms O\big{(}h^{\frac{1}{4}}\big{)} in (26) and (27) are of the order $O(\sqrt{h})$ in the setting considered in Theorem 1. Proving this fact would require some substantially finer analysis.

Organization of the paper and outline of the proof

The aim of this section is to give an overview of the strategy of the proof of Theorem 1. From (12) and in order to obtain an asymptotic estimate of $\mathbb{E}_{\nu_{h}}\left[F\left(X_{\tau_{\Omega}}\right)\right]$, we study the asymptotic behaviour when $h\to 0$ of the quantities

[TABLE]

where $\lambda_{h}$ is defined by (8) and $u_{h}$ by (10).

To study $\lambda_{h}$ and $\partial_{n}u_{h}$, the first key point is to notice that the gradient of any eigenfunction associated with an eigenvalue of $-L^{D,(0)}_{f,h}$ is also a solution to an eigenvalue problem for the same eigenvalue. Let us be more precise. Let $v$ be an eigenfunction associated with $\lambda\in\sigma(-L^{D,(0)}_{f,h})$. The eigenvalue-eigenfunction pair $(\lambda,v)$ satisfies:

[TABLE]

By differentiating this relation, we observe that $\nabla v$ satisfies

[TABLE]

where

[TABLE]

is an operator acting on $1$-forms (namely on vector fields). Therefore, the vector field $\nabla v$ is an eigen-$1$-form of the operator $-L^{D,(1)}_{f,h}$ which is the operator $-L^{(1)}_{f,h}$ with tangential Dirichlet boundary conditions (see (34)), associated with the eigenvalue $\lambda$.

The second key point (see for example [18]) is that, when (A0) holds, $-L^{D,(0)}_{f,h}$ admits exactly $\mathsf{m}_{0}^{\Omega}$ eigenvalues smaller than $\frac{\sqrt{h}}{2}$ (where we recall that $\mathsf{m}_{0}^{\Omega}$ is the number of local minima of $f$ in $\Omega$) and that $-L^{D,(1)}_{f,h}$ admits exactly $\mathsf{m}_{1}^{\overline{\Omega}}$ eigenvalues smaller than $\frac{\sqrt{h}}{2}$ (where, we recall that $\mathsf{m}_{1}^{\overline{\Omega}}$ is the number generalized saddle points of $f$ in $\overline{\Omega}$). Actually, all these small eigenvalues are exponentially small in the regime $h\to 0$ (namely they are bounded from above by $e^{-\frac{c}{h}}$ for some $c>0$), the other eigenvalues being bounded from below by a constant in this regime. This implies in particular that $\lambda_{h}$ is an exponentially small eigenvalue of $-L^{D,(1)}_{f,h}$. Let us denote by $\pi_{h}^{(0)}$ (resp. $\pi_{h}^{(1)}$) the projector onto the vector space spanned by the eigenfunctions (resp. eigenforms) associated with the $\mathsf{m}_{0}^{\Omega}$ (resp. $\mathsf{m}_{1}^{\overline{\Omega}}$) smallest eigenvalue of $-L^{D,(0)}_{f,h}$ (resp. of $-L^{D,(1)}_{f,h}$).

To obtain an asymptotic estimate on $\lambda_{h}$ when $h\to 0$, the strategy consists in studying the smallest singular values of the matrix of the gradient operator $\nabla$ which maps ${\rm Ran}\,\pi_{h}^{(0)}$, equipped with the scalar product of $L^{2}_{w}(\Omega)$, to ${\rm Ran}\,\pi_{h}^{(1)}$. Indeed, from Proposition 4, the squares of the smallest singular values of this matrix are the smallest eigenvalues of $-\frac{2}{h}L^{D,(0)}_{f,h}$. Working with the matrix of $\nabla|_{{\rm Ran}\,\pi_{h}^{(0)}}$ gives more flexibility than directly working with the matrix of $-L^{D,(0)}_{f,h}|_{{\rm Ran}\,\pi_{h}^{(0)}}$. To this end, the idea is then to construct an appropriate basis (with so called quasi-modes) of ${\rm Ran}\,\pi_{h}^{(0)}$ and ${\rm Ran}\,\pi_{h}^{(1)}$. Moreover, from (34), $\nabla u_{h}\in{\rm Ran}\,\pi_{h}^{(1)}$ and thus, to study the asymptotic behaviour of $\partial_{n}u_{h}$ on $\partial\Omega$ when $h\to 0$, one decomposes $\nabla u_{h}$ along the basis of ${\rm Ran}\,\pi_{h}^{(1)}$. The terms in the decomposition are approximated using quasi-modes.

The paper is organized as follows. In Section 2, one constructs two maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$ which will be extensively used in Section 3. These maps are useful in order to understand the different timescales of the process (1) in $\Omega$. Section 3 is dedicated to the construction of quasi-modes for $-L^{D,(0)}_{f,h}$ and $-L^{D,(1)}_{f,h}$. In Section 4, we study the asymptotic behaviors of the smallest eigenvalues of $-L^{D,(0)}_{f,h}$ (see Theorem 5) and we give an asymptotic estimate of $\lambda_{h}$ when $h\to 0$, see Theorem 3. In Section 5, we give asymptotic estimates for $\int_{\Omega}u_{h}e^{-\frac{2}{h}f}$ and for $\partial_{n}u_{h}$ on $\partial\Omega$ when $h\to 0$ (see Proposition 59 and Theorem 6). Finally, Section 6 is dedicated to the proof of Theorem 1.

For the ease of the reader, a list of the main notation used in this work is provided at the end of this work.

Association of the local minima of $f$ with saddle points of $f$

This section is dedicated to the construction of two maps: the map $\mathbf{j}$ which associates each local minimum of $f$ with an ensemble of saddle points of $f$ and the map $\mathbf{\widetilde{j}}$ which associates each local minimum of $f$ with a connected component of a sublevel set of $f$. These maps are useful to define the quasi-modes in Section 3.

This section is organized as follows. In Section 2.1, one introduces a set of connected components which play a crucial role in our analysis. The constructions of the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$ require two preliminary results: Propositions 20 and 22 which are respectively introduced in Section 2.2 and Section 2.3. Then, the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$ are defined in Section 2.4. Finally, in Section 2.5, one rewrites the assumptions (A1)-(A4) with the help of the map $\mathbf{j}$.

Connected components associated with the elements of $\mathsf{U}_{0}^{\Omega}$

The aim of this section is to define for each $x\in\mathsf{U}_{0}^{\Omega}$, the connected component of $\{f<\mathsf{H}_{f}(x)\}$ which contains $x$ (where $\mathsf{H}_{f}(x)$ is defined by (15)). For that purpose, let us introduce the following definitions.

Definition 14**.**

Let us assume that the assumption (A0) holds. For all $x\in\mathsf{U}_{0}^{\Omega}$ and $\lambda>f(x)$, one defines

[TABLE]

and

[TABLE]

Moreover, for all $x\in\mathsf{U}_{0}^{\Omega}$, one defines

[TABLE]

A direct consequence of Lemma 15 below is that for all $x\in\mathsf{U}_{0}^{\Omega}$, $\mathsf{C}(x)$ defined in (38) coincides with $\mathsf{C}(x)$ introduced in (17) and thus

[TABLE]

where $\mathcal{C}$ is defined by (16) and for $x\in\mathsf{U}_{0}^{\Omega}$, $\mathsf{C}(x)$ is defined by (38).

Notice that under (A0), for all $x\in\mathsf{U}_{0}^{\Omega}\subset\Omega$, $\lambda(x)$ is well defined. Indeed, for all $x\in\mathsf{U}_{0}^{\Omega}$, $\{\lambda>f(x)\ \text{s.t.}\ \mathsf{C}(\lambda,x)\cap\partial\Omega=\emptyset\}$ is bounded by $\sup_{\overline{\Omega}}f+1$ and nonempty because for $\beta>0$ small enough $\mathsf{C}(f(x)+\beta,x)$ is included in $\Omega$ (since $x\in\Omega$ and $f$ is Morse).

One has the following results which permits to give another definition of $\mathsf{H}_{f}$ (see (15)) which will be easier to handle in the sequel.

Lemma 15**.**

Let us assume that (A0) holds. Then, for all $x\in\mathsf{U}_{0}^{\Omega}$

[TABLE]

where $\mathsf{H}_{f}(x)$ is defined by (15) and $\lambda(x)$ is defined by (38).

Proof.

Let $x\in\mathsf{U}_{0}^{\Omega}$. By definition of $\mathsf{H}_{f}(x)$ (see (15)), for all $\varepsilon>0$, there exists $\gamma\in C^{0}([0,1],\overline{\Omega})$ such that $\gamma(0)=x$, $\gamma(1)\in\partial\Omega$ and

[TABLE]

Therefore, $\mathsf{C}(\mathsf{H}_{f}(x)+\varepsilon,x)\cap\partial\Omega\neq\emptyset$. Then, by definition of $\lambda(x)$ (see (38)) $\lambda(x)\leq\mathsf{H}_{f}(x)+\varepsilon$ which implies, letting $\varepsilon\to 0^{+}$, $\lambda(x)\leq\mathsf{H}_{f}(x)$. To prove that $\lambda(x)=\mathsf{H}_{f}(x)$, we argue by contradiction and we assume that $\lambda(x)<\mathsf{H}_{f}(x)$. Let us consider $\alpha\in(0,\mathsf{H}_{f}(x)-\lambda(x))$. By definition of $\lambda(x)$, $\mathsf{C}(\lambda(x)+\alpha,x)\cap\partial\Omega\neq\emptyset$. Thus, there exists $\gamma\in C^{0}([0,1],\overline{\Omega})$ such that $\gamma(0)=x$, $\gamma(1)\in\partial\Omega$ and for all $t\in[0,1]$, $f(\gamma(t))<\lambda(x)+\alpha$. This implies that, $\mathsf{H}_{f}(x)<\lambda(x)+\alpha$ which contradicts the definition of $\alpha$. Therefore $\lambda(x)=\mathsf{H}_{f}(x)$. This concludes the proof of Lemma 15.

Definition 16**.**

Let us assume that the assumption (A0) holds. The integer $\mathsf{N}_{1}$ is defined by:

[TABLE]

where we recall that $\mathsf{m}_{0}^{\Omega}={\rm Card}\,(\mathsf{U}_{0}^{\Omega})$ (see (18)), $\mathsf{C}(x)$ is defined by (38) and \mathcal{C}=\big{\{}\mathsf{C}(x),\,x\in\mathsf{U}_{0}^{\Omega}\big{\}} (see (16) and (39)). Moreover, the elements of $\mathcal{C}=\{\mathsf{C}(x),x\in\mathsf{U}_{0}^{\Omega}\}$ are denoted by $\mathsf{C}_{1},\dots,\mathsf{C}_{\mathsf{N}_{1}}$. Finally, for all $\ell\in\{1,\ldots,\mathsf{N}_{1}\}$, $\mathsf{C}_{k}$ is denoted by

[TABLE]

For example, on Figure 1, one has $\mathsf{m}_{0}^{\Omega}=4$ and $\mathsf{N}_{1}=3$. The notation (42) will be useful when constructing the maps $\mathbf{j}$ and $\widetilde{\mathbf{j}}$ in Section 2.4 below.

Separating saddle points

This section is devoted to the proof of Proposition 20 below which will be needed when constructing the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$ in Section 2.4. Let us first prove the following lemma which will be used in the proof of Proposition 20.

Lemma 17**.**

Let us assume that the function $f:\overline{\Omega}\to\mathbb{R}$ is a $C^{\infty}$ function. Let $x\in\mathsf{U}_{0}^{\Omega}$. For all $\mu>f(x)$, it holds:

[TABLE]

and

[TABLE]

where $\mathsf{C}(\mu,x)$ and $\mathsf{C}^{+}(\mu,x)$ are respectively defined in (36) and (37).

Proof.

The proof is divided into two steps.

Step 1. Proof of (43).

Since $\{f<\lambda\}$ is open in the locally connected space $\overline{\Omega}$, the set $\mathsf{C}(\lambda,x)\subset\overline{\Omega}$ is open for all $\lambda\in(f(x),\mu)$. Since moreover $\mathsf{C}(\lambda,x)\subset\mathsf{C}(\mu,x)$ for all $\lambda\in(f(x),\mu)$, the union $\cup_{\lambda\in(f(x),\mu)}\mathsf{C}(\lambda,x)$ is an open subset of $\mathsf{C}(\mu,x)$. Therefore, since $\mathsf{C}(\mu,x)$ is connected, to obtain (43), it is enough to prove that the set $\cup_{\lambda\in(f(x),\mu)}\mathsf{C}(\lambda,x)$ is closed in $\mathsf{C}(\mu,x)$. To this end, let us show that the complement of $\cup_{\lambda\in(f(x),\mu)}\mathsf{C}(\lambda,x)$ in $\mathsf{C}(\mu,x)$ is open. It is obviously the case if it is empty. If $\cup_{\lambda\in(f(x),\mu)}\mathsf{C}(\lambda,x)$ is not empty, let us choose

[TABLE]

Then, since $y\in\mathsf{C}(\mu,x)$, one has $f(y)<\mu$ and thus $y\in\mathsf{C}(\lambda,y)\cap\mathsf{C}(\mu,x)$ for all $\lambda\in(f(y),\mu)$. Therefore, it holds $\mathsf{C}(\lambda,y)\subset\mathsf{C}(\mu,x)$ and $\mathsf{C}(\lambda,y)\cap\mathsf{C}(\lambda,x)=\emptyset$ for all $\lambda\in(\max\{f(x),f(y)\},\mu)$. Hence, the open set $\mathsf{C}(\lambda,y)$ is included in $\mathsf{C}(\mu,x)$ and disjoint from the set $\cup_{\lambda\in(f(x),\mu)}\mathsf{C}(\lambda,x)$ for all $\lambda\in(f(y),\mu)$. This proves that $\cup_{\lambda\in(f(x),\mu)}\mathsf{C}(\lambda,x)$ is closed in $\mathsf{C}(\mu,x)$. This concludes the proof of (43).

Step 2. Proof of (44).

Since for all $\lambda$, $\mathsf{C}^{+}(\lambda,x)$ is a connected component of $\{f\leq\lambda\}$, it is closed in this closed set of $\overline{\Omega}$ and thus it is closed in $\overline{\Omega}$. It follows that the set $\cap_{\lambda>\mu}\mathsf{C}^{+}(\lambda,x)$ is connected as a decreasing intersection of compact connected sets. Since $\cap_{\lambda>\mu}\mathsf{C}^{+}(\lambda,x)$ is also obviously included in $\{f\leq\mu\}$ and contains $x$, it is then included in $\mathsf{C}^{+}(\mu,x)$ by definition of $\mathsf{C}^{+}(\mu,x)$. The reverse inclusion follows from the fact that $\mathsf{C}^{+}(\mu,x)\subset\mathsf{C}^{+}(\lambda,x)$ for all $\lambda>\mu$. This proves (44) and ends the proof of Lemma 17.

The constructions of the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$ made in Section 2.4 are based on the notions of separating saddle points and of critical components as introduced in [20, Section 4.1] for a case without boundary. Let us define and slightly adapt theses two notions to our setting. To this end, let us first recall that according to [18, Section 5.2], for any non critical point $z\in\Omega$, for $r>0$ small enough

[TABLE]

and for any critical point $z\in\Omega$ of index $p$ of the Morse function $f$, for $r>0$ small enough, one has the three possible cases:

[TABLE]

where $B(z,r):=\{x\in\overline{\Omega}\ \text{s.t.}\ |x-z|<r\}$. The separating saddle points of $f$ and the critical components of $f$ are defined as follows.

Definition 18**.**

Assume (A0). Let $\mathcal{C}=\{\mathsf{C}_{1},\dots,\mathsf{C}_{\mathsf{N}_{1}}\}$ be the set of connected sets introduced Definition 16.

A point $z\in\mathsf{U}_{1}^{\overline{\Omega}}$ is a separating saddle point if

• •

either $z\in\mathsf{U}_{1}^{\Omega}\cap\cup_{i=1}^{\mathsf{N}_{1}}\overline{\mathsf{C}_{i}}$ and for $r>0$ small enough, the two connected components of $\{f<f(z)\}\cap B(z,r)$ are contained in different connected components of $\{f<f(z)\}$,

• •

or $z\in\mathsf{U}_{1}^{\partial\Omega}\cap\cup_{i=1}^{\mathsf{N}_{1}}\partial\mathsf{C}_{i}$.

Notice that in the former case $z\in\Omega$ while in the latter case $z\in\partial\Omega$. The set of separating saddle points is denoted by $\mathsf{U}_{1}^{\mathsf{ssp}}$. 2. 2.

For any $\sigma\in\mathbb{R}$, a connected component $\mathsf{E}$ of the sublevel set $\{f<\sigma\}$ in $\overline{\Omega}$ is called a critical connected component if $\partial\mathsf{E}\cap\mathsf{U}_{1}^{\mathsf{ssp}}\neq\emptyset$. The family of critical connected components is denoted by $\mathcal{C}_{crit}$.

Remark 19**.**

It is natural to define generally a separating saddle point of a Morse function $f$ as follows: $z$ is a separating saddle point if for any sufficiently small connected neighborhood $\mathcal{V}_{z}$ of $z$, $\mathcal{V}_{z}\cap\{f<f(z)\}$ has two connected components included in two connected components of $\{f<f(z)\}$. Our definition of separating saddle point is consistent with this general definition when the function $f$ is extended by $-\infty$ outside $\overline{\Omega}$. To be more precise, let us introduce some new nonempty set $X^{\prime}$ and let us define the topological space $X$ as the disjoint union $X=\overline{\Omega}\cup X^{\prime}$ whose open sets are

[TABLE]

Note that it follows from this definition that $X^{\prime}$ is connected and that $\partial X^{\prime}=\partial\Omega$. We denote by $B_{X}(z,r)$ the ball in $X$: $B_{X}(z,r)=B(z,r)$ if $B(z,r)\cap\partial\Omega=\emptyset$ and $B_{X}(z,r)=B(z,r)\cup X^{\prime}$ if $B(z,r)\cap\partial\Omega\neq\emptyset$. Moreover, extending $f$ by $-\infty$ on $X^{\prime}$, the following holds for any $z\in\overline{\Omega}$ and $r>0$ small enough : $B_{X}(z,r)\cap\{f<f(z)\}$ has at least two connected components in $X$ iff $z\in\mathsf{U}_{1}^{\overline{\Omega}}$, in which case $B_{X}(z,r)\cap\{f<f(z)\}$ has precisely two connected components in $X$. Lastly, for $z\in\mathsf{U}_{1}^{\overline{\Omega}}$, the above two connected components of $B(z,r)\cap\{f<f(z)\}$ in $X$ are contained in different connected components of $\{f<f(z)\}$ in $X$ iff $z\in\mathsf{U}_{1}^{\mathsf{ssp}}$.

In Figure 7, one gives an example of a saddle point $z$ which is not a separating saddle point as introduced in Definition 18.

Let us now study the properties of $\mathsf{C}_{1},\dots,\mathsf{C}_{\mathsf{N}_{1}}$. The following proposition will be used in the first step of the construction of the map between points in $\mathsf{U}_{0}^{\Omega}$ and subsets of $\mathsf{U}_{1}^{\overline{\Omega}}$.

Proposition 20**.**

Let us assume that (A0) holds. Let $\mathcal{C}=\{\mathsf{C}_{1},\dots,\mathsf{C}_{\mathsf{N}_{1}}\}$ be the set of connected sets introduced Definition 16 and let $(k,\ell)\in\{1,\dots,\mathsf{N}_{1}\}^{2}$ with $k\neq\ell$. Then,

[TABLE]

In addition, one has

[TABLE]

where the set $\mathsf{U}_{1}^{\mathsf{ssp}}$ is introduced in item 1 in Definition 18. Finally, $\partial\mathsf{C}_{k}\cap\mathsf{U}_{1}^{\mathsf{ssp}}\neq\emptyset$.

Proof.

The proof of Proposition 20 is divided into $5$ steps.

Step 1. For $k\in\{1,\dots,\mathsf{N}_{1}\}$, let us show that $\mathsf{C}_{k}$ is an open subset of $\Omega$. To this end, let us first prove that $\mathsf{C}_{k}\subset\Omega$. From Definition 16 and (36), there exists $x_{k}\in\mathsf{U}_{0}^{\Omega}\cap\mathsf{C}_{k}$ such that $\mathsf{C}_{k}=\mathsf{C}(\lambda(x_{k}),x_{k})$ with $\lambda(x_{k})>f(x_{k})$. From Lemma 17, it holds

[TABLE]

Moreover, since $\lambda\mapsto\mathsf{C}(\lambda,x_{k})$ is increasing on $(f(x_{k}),+\infty)$, one has, by definition of $\lambda(x_{k})$ (see (36)), that $\mathsf{C}(\lambda,x_{k})\cap\partial\Omega=\emptyset$ for all $\lambda\in(f(x_{k}),\lambda(x_{k}))$. Therefore $\mathsf{C}(\lambda(x_{k}),x_{k})\subset\Omega$ and thus $\mathsf{C}_{k}\cap\partial\Omega=\emptyset$. Thus, $\mathsf{C}_{k}\subset\Omega$. Then, the fact that $\mathsf{C}_{k}$ is an open subset of $\Omega$ follows from the fact that $\mathsf{C}_{k}$ is open in $\overline{\Omega}$. Indeed, $\overline{\Omega}$ is locally connected and $\mathsf{C}_{k}$ is a connected component of the open set $\{f<\lambda(x_{k})\}$.

Step 2. Let us now show that the $\mathsf{C}_{k}$’s are two by two disjoint. To this end, let $(k,\ell)\in\{1,\dots,\mathsf{N}_{1}\}^{2}$ with $\ell\neq k$ and $\mathsf{C}_{k}\cap\mathsf{C}_{\ell}\neq\emptyset$. Therefore, since for $q\in\{k,\ell\}$, there exists $x_{q}\in\mathsf{U}_{0}^{\Omega}\cap\mathsf{C}_{q}$ such that $\mathsf{C}_{q}=\mathsf{C}(\lambda(x_{q}),x_{q})$ is a connected component of $\{f<\lambda(x_{q})\}$ (see Definition 16 and (36)), it holds $\mathsf{C}_{k}=\mathsf{C}_{\ell}$ if $\lambda(x_{k})=\lambda(x_{l})$. Let us prove that $\lambda(x_{k})=\lambda(x_{l})$ by contradiction and assume that, without loss of generality, $\lambda(x_{k})<\lambda(x_{l})$. Since $\mathsf{C}_{k}\cap\mathsf{C}_{\ell}\neq\emptyset$, this implies $\mathsf{C}_{k}\subset\mathsf{C}_{\ell}$. Therefore, for any $\varepsilon\in(0,\lambda(x_{l})-\lambda(x_{k}))$, $\mathsf{C}(\lambda(x_{k})+\varepsilon,x_{k})\subset\mathsf{C}_{\ell}$ and by definition of $\lambda(x_{k})$ (see (36)), $\mathsf{C}(\lambda(x_{k})+\varepsilon,x_{k})$ intersects $\partial\Omega$. This is in contradiction with the fact that $\mathsf{C}_{\ell}\subset\Omega$. Therefore, $\lambda(x_{k})=\lambda(x_{l})$ and thus $\mathsf{C}_{k}=\mathsf{C}_{\ell}$.

Step 3. Let us prove that for $k\in\{1,\dots,\mathsf{N}_{1}\}$, $\partial\mathsf{C}_{k}\cap\partial\Omega\subset\mathsf{U}_{1}^{\mathsf{ssp}}\cap\partial\Omega$ which is equivalent, according to Definition 18, to $\partial\mathsf{C}_{k}\cap\partial\Omega\subset\mathsf{U}_{1}^{\partial\Omega}$ (where $\mathsf{U}_{1}^{\partial\Omega}$ is defined in Section 1.3.2). If $\partial\mathsf{C}_{k}\cap\partial\Omega\neq\emptyset$, let us consider $y\in\partial\mathsf{C}_{k}\cap\partial\Omega$. According to [18, Section 5.2], if $y$ is not a critical point of $f|_{\partial\Omega}$, then the hypersurfaces $\{f=f(y)\}$ and $\partial\Omega$ interstects transversally in a neighborhood of $y$. This implies that for $r>0$ small enough, $\{f<f(y)\}\cap B(y,r)$ is connected and $\{f<f(y)\}\cap B(y,r)\cap\partial\Omega\neq\emptyset$. Therefore, since $\mathsf{C}_{k}$ is a connected component of $\{f<f(y)\}$, one has $\{f<f(y)\}\cap B(y,r)=\mathsf{C}_{k}\cap B(y,r)$ and thus, $\mathsf{C}_{k}\cap\partial\Omega\neq\emptyset$. This is impossible since $\mathsf{C}_{k}\subset\Omega$. Therefore $y$ is a critical point of $f|_{\partial\Omega}$ and according to [18, Section 5.2], there are three possible different cases:

either $y$ is local minimum of $f$ 2. 2.

or $y$ is a local minimum of $f|_{\partial\Omega}$ and $\partial_{n}f(y)>0$, 3. 3.

or for $r>0$ small enough, $\{f<f(y)\}\cap B(y,r)$ admits one or two connected components with nonempty intersection with $\partial\Omega$.

The first case is not possible in our setting since $y\in\overline{\mathsf{C}_{k}}$ implies that $y$ is not a local minimum of $f$. The third case is also not possible since $\mathsf{C}_{k}\subset\Omega$. Therefore $y$ is a local minimum of $f|_{\partial\Omega}$ and $\partial_{n}f(y)>0$. This proves that $\partial\mathsf{C}_{k}\cap\partial\Omega\subset\mathsf{U}_{1}^{\mathsf{ssp}}\cap\partial\Omega$.

Step 4. Let us prove that for all $(k,l)\in\{1,\dots,\mathsf{N}_{1}\}^{2}$ with $k\neq\ell$, $\partial\mathsf{C}_{k}\cap\partial\mathsf{C}_{\ell}\subset\mathsf{U}_{1}^{\mathsf{ssp}}\cap\Omega$ or equivalently (see item 1 in Definition 18) that $\partial\mathsf{C}_{k}\cap\partial\mathsf{C}_{\ell}\subset\mathsf{U}_{1}^{\Omega}$ (where $\mathsf{U}_{1}^{\Omega}$ is the set of saddle points of $f$ in $\Omega$, see Section 1.3.2). To this end, let us assume that $\partial\mathsf{C}_{k}\cap\partial\mathsf{C}_{\ell}\neq\emptyset$ for some $(k,l)\in\{1,\dots,\mathsf{N}_{1}\}^{2}$ with $k\neq\ell$. First, since for $q\in\{k,\ell\}$, there exists $x_{q}\in\mathsf{U}_{0}^{\Omega}\cap\mathsf{C}_{q}$ such that $\mathsf{C}_{q}$ is a connected component of $\{f<\lambda(x_{q})\}$, one has necessarily $\lambda(x_{\ell})=\lambda(x_{k})$. Moreover, it holds $\partial\mathsf{C}_{k}\cap\partial\mathsf{C}_{\ell}\subset\Omega$. Indeed, if there exists $z\in\partial\mathsf{C}_{k}\cap\partial\Omega$, we know from the analysis above that $z\in\mathsf{U}_{1}^{\partial\Omega}$. It follows that for $r>0$ small enough, $B(z,r)\cap\{f<\lambda(x_{k})\}$ is connected and therefore $B(z,r)\cap\{f<\lambda(x_{k})\}\subset\mathsf{C}_{k}$. This implies that $B(z,r)\cap\mathsf{C}_{\ell}=\emptyset$ (since we proved that for $k\neq l$, $\mathsf{C}_{k}\cap\mathsf{C}_{\ell}=\emptyset$) and hence $z\notin\overline{\mathsf{C}_{\ell}}$. Lastly, if there exists $z\in\partial\mathsf{C}_{k}\cap\partial\mathsf{C}_{\ell}\cap\Omega$, then one deduces from (46) that $z\in\mathsf{U}_{1}^{\Omega}$. Indeed, for all $r>0$ small enough, $B(z,r)\cap\{f<\lambda(x_{k})\}$ has two connected components respectively included in $\mathsf{C}_{k}$ and $\mathsf{C}_{\ell}$.

Step 5. To conclude the proof of Proposition 20, it remains to show that for all $k\in\{1,\dots,\mathsf{N}_{1}\}$, $\partial\mathsf{C}_{k}\cap\mathsf{U}_{1}^{\mathsf{ssp}}\neq\emptyset$. Let us argue by contradiction and assume that $\partial\mathsf{C}_{k}\cap\mathsf{U}_{1}^{\mathsf{ssp}}=\emptyset$. Since $(\partial\mathsf{C}_{k}\cap\partial\Omega)\cap\mathsf{U}_{1}^{\mathsf{ssp}}=\emptyset$, one has $\overline{\mathsf{C}_{k}}\subset\Omega$ (indeed we proved above that $\mathsf{C}_{k}\subset\Omega$ and $\partial\mathsf{C}_{k}\cap\partial\Omega\subset\mathsf{U}_{1}^{\mathsf{ssp}}$). Let us recall that $\mathsf{C}_{k}=\mathsf{C}(\lambda(x_{k}),x_{k})$ for some $x_{k}\in\mathsf{U}_{0}^{\Omega}\cap\mathsf{C}_{k}$ (see Definition 16, (36), and (38)). Then, using the fact that

[TABLE]

and the fact that the function $f$ is Morse, for all $z\in\partial\mathsf{C}_{k}$,

• •

either $z\notin\mathsf{U}_{1}^{\Omega}$ in which case there exists $r_{z}>0$ such that $B(z,r_{z})\cap\{f<\lambda(x_{k})\}$ is connected (see (45) together with the fact that $f(z)=\lambda(x_{k})$) and thus $B(z,r_{z})\cap\{f<\lambda(x_{k})\}$ is included in $\mathsf{C}_{k}$,

• •

or $z\in\mathsf{U}_{1}^{\Omega}$ in which case there exists $r_{z}>0$ such that $B(z,r_{z})\cap\{f<\lambda(x_{k})\}$ has two connected components both included in $\mathsf{C}_{k}$ (because we assume that there is no separating saddle point on $\partial\mathsf{C}_{k}$).

In all cases, one can assume in addition, choosing $r_{z}$ smaller, that $B(z,r_{z})\subset\Omega$ and $B(z,r_{z})\cap\mathsf{U}_{0}^{\Omega}=\emptyset$. Let us now consider

[TABLE]

Then, the set $\mathsf{V}_{k}$ is an open subset of $\Omega$ such that $\overline{\mathsf{C}_{k}}\subset\mathsf{V}_{k}$ and $\mathsf{V}_{k}\cap\{f\leq\lambda(x_{k})\}=\overline{\mathsf{C}_{k}}$. Therefore, the connected set $\overline{\mathsf{C}_{k}}$ is closed and open in $\{f\leq\lambda(x_{k})\}$, and thus

[TABLE]

Let us now denote by $\mathsf{C}^{+}_{k}(\mu)$, for $\mu\geq\lambda(x_{k})$, the connected component of $\{f\leq\mu\}$ containing $\overline{\mathsf{C}_{k}}$. It then holds, according to Lemma 17,

[TABLE]

Moreover, by definition of $\lambda(x_{k})$ (see (36)), $\mathsf{C}^{+}_{k}(\mu)$ meets $\partial\Omega$ for all $\mu>\lambda(x_{k})$. Hence, $\cap_{\mu>\lambda(x_{k})}\mathsf{C}^{+}_{k}(\mu)\cap\partial\Omega=\overline{\mathsf{C}_{k}}\cap\partial\Omega$ is nonempty as a decreasing intersection of nonempty compact sets. This contradicts the fact that $\overline{\mathsf{C}_{k}}\subset\Omega$. In conclusion $\partial\mathsf{C}_{k}\cap\mathsf{U}_{1}^{\mathsf{ssp}}\neq\emptyset$. This concludes the proof of Proposition 20.

We end this section with the following lemma which will be needed in the sequel.

Lemma 21**.**

Let us assume that (A0) is satisfied. Let $\mathcal{C}=\{\mathsf{C}_{1},\dots,\mathsf{C}_{\mathsf{N}_{1}}\}$ be the set of connected sets introduced Definition 16. Let us consider $\{j_{1},\ldots,j_{k}\}\subset\{1,\ldots,\mathsf{N}_{1}\}$ with $k\in \{1,\ldots,\mathsf{N}_{1}\}$ and $j_{1}<\ldots<j_{k}$ such that $\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}$ is connected and such that for all $q\in\{1,\ldots,\mathsf{N}_{1}\}\setminus\{j_{1},\ldots,j_{k}\}$, $\overline{\mathsf{C}_{q}}\cap\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}=\emptyset$. Then, there exists $z\in\mathsf{U}_{1}^{\mathsf{ssp}}$ and $\ell_{0}\in\{1,\ldots,k\}$ such that

[TABLE]

Proof.

There are two cases to consider: either $\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}\cap\partial\Omega\neq\emptyset$ or $\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}\cap\partial\Omega=\emptyset$. Let us consider the case $\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}\cap\partial\Omega\neq\emptyset$. Using (47) and (48), the result stated in Lemma 21 follows from the fact that for all $\ell\in\{1,\ldots,k\}$, the sets $\overline{\mathsf{C}_{j_{\ell}}}\cap\partial\Omega=\partial{\mathsf{C}_{j_{\ell}}}\cap\partial\Omega$ are two by two disjoint, together with the definition of $\mathsf{U}_{1}^{\mathsf{ssp}}$ (see the second point of item 1 in Definition 18). Let us now consider the case $\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}\cap\partial\Omega=\emptyset$. From (48), one has $\cup_{q,\ell=1,q\neq\ell}^{k}\partial{\mathsf{C}_{j_{\ell}}}\cap\partial{\mathsf{C}_{j_{q}}}\subset\mathsf{U}_{1}^{\mathsf{ssp}}\cap(\cup_{\ell=1}^{k}\partial{\mathsf{C}_{j_{\ell}}})$ and this inclusion is an equality if the statement of Lemma 21 is not satisfied. To prove Lemma 21, let us argue by contradiction, i.e. let us assume that

[TABLE]

Notice that there exists $x\in\mathsf{U}_{0}^{\Omega}$ such that for all $\ell\in\{1,\ldots,k\}$, $\mathsf{C}_{j_{\ell}}$ is a connected component of $\{f<\lambda(x)\}$. Let us prove, using the same arguments as those used to prove (50), that $\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}$ is a connected component of $\{f\leq\lambda(x)\}$. To this end, let us consider $z\in\cup_{\ell=1}^{k}\partial{\mathsf{C}_{j_{\ell}}}$. If $z$ is not a separating saddle point, there exists $r_{z}>0$ such that $B(z,r_{z})\subset\Omega$ and $B(z,r_{z})\cap\{f<\lambda(x)\}$ is included in $\cup_{\ell=1}^{k}{\mathsf{C}_{j_{\ell}}}$. Else, $z$ is a separating saddle point and thus, from (51), there exists $(\ell,q)\in\{1,\ldots,k\}^{2}$, $\ell\neq q$, such that $z\in\partial\mathsf{C}_{j_{\ell}}\cap\partial\mathsf{C}_{j_{q}}$. Thus, again, there exists $r_{z}>0$ such that $B(z,r_{z})\subset\Omega$ and $B(z,r_{z})\cap\{f<\lambda(x)\}$ is included in $\cup_{\ell=1}^{k}{\mathsf{C}_{j_{\ell}}}$. Therefore, the same arguments as those used to prove (50) imply that $\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}$ is a connected component of $\{f\leq\lambda(x)\}$. Using in addition (44) together with the fact that for all $\mu>\lambda(x)$, the connected component of $\{f\leq\mu\}$ which contains $\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}$ intersects $\partial\Omega$ (by definition of $\lambda(x)$), one obtains that $\cup_{\ell=1}^{k}\overline{\mathsf{C}_{j_{\ell}}}\cap\partial\Omega$ is not empty. This is a contradiction. Therefore, the set $\bigcup_{q,\ell=1,q\neq\ell}^{k}\partial{\mathsf{C}_{j_{\ell}}}\cap\partial{\mathsf{C}_{j_{q}}}$ is strictly included in $\mathsf{U}_{1}^{\mathsf{ssp}}\cap(\cup_{\ell=1}^{k}\partial{\mathsf{C}_{j_{\ell}}})$. This concludes the proof of Lemma 21.

A topological result under the assumption (A0)

This section is devoted to the proof of Proposition 22 which will be needed when constructing the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$ in Section 2.4.

Proposition 22**.**

Let us assume that the assumption (A0) is satisfied. Let us consider $\mathsf{C}_{q}$ for $q\in\{1,\ldots,\mathsf{N}_{1}\}$ (see Definition 16). From (36) and (38), there exists $x_{q}\in\mathsf{U}_{0}^{\Omega}\cap\mathsf{C}_{q}$ such that $\mathsf{C}_{q}=\mathsf{C}(x_{q},\lambda(x_{q}))$. Let $\lambda\in(\min_{\overline{\mathsf{C}_{q}}}f,\lambda(x_{q})]$ and $\mathsf{C}$ be a connected component of $\mathsf{C}_{q}\cap\{f<\lambda\}$. Then,

[TABLE]

Moreover, let us define

[TABLE]

with the convention $\sigma=\min_{\overline{\mathsf{C}}}f$ when $\mathsf{C}\cap\mathsf{U}_{1}^{\mathsf{ssp}}=\emptyset$. Then, the following assertions hold.

For all $\mu\in(\sigma,\lambda]$, the set $\mathsf{C}\cap\{f<\mu\}$ is a connected component of $\{f<\mu\}$. 2. 2.

If $\mathsf{C}\cap\mathsf{U}_{1}^{\mathsf{ssp}}\neq\emptyset$, one has $\mathsf{C}\cap\mathsf{U}_{0}^{\Omega}\subset\{f<\sigma\}$ and the connected components of $\mathsf{C}\cap\{f<\sigma\}$ belong to $\mathcal{C}_{crit}$.

Proof.

Notice that from (47), the set $\mathsf{C}$ is an open subset of $\Omega$. The proof of Proposition 22 is divided into three steps.

Step 1. Proof of (52).

The fact that

[TABLE]

is straightforward. Indeed, let $z\in\mathsf{C}\,\cap\,\mathsf{U}_{1}^{\mathsf{ssp}}\subset\Omega$. Then, $z\in\mathsf{U}_{1}^{\Omega}\cap\mathsf{C}_{q}$ and for $r>0$ small enough, the two connected components of $\{f<f(z)\}\cap B(z,r)$ are contained in different connected components of $\{f<f(z)\}$ (see item 1 in Definition 18). Then, since the set $\mathsf{C}$ is a connected component of $\{f<\lambda\}$, $\mathsf{C}$ contains at least two open connected components $\mathsf{A}_{1}$ and $\mathsf{A}_{2}$ of $\{f<f(z)\}$. Moreover, for $k\in\{1,2\}$, $\partial\mathsf{A}_{k}\subset\{f=f(z)\}$. Thus, for $k\in\{1,2\}$, the global minimum of $f$ on $\overline{A_{k}}$ is reached in $A_{k}$ and hence at some $y_{k}\in A_{k}\cap\mathsf{U}_{0}^{\Omega}$. This implies that $\mathsf{C}\cap\mathsf{U}_{0}^{\Omega}$ contains at least two elements, $y_{1}$ and $y_{2}$.

Let us now prove the reverse implication in (53). To this end, let us assume that there exist two points $x\neq y$ in $C\cap\mathsf{U}_{0}^{\Omega}$. One can assume without loss of generality that

[TABLE]

Let us recall that (see (36)) $\mathsf{C}(\mu,y)$ is the connected component of $\{f<\mu\}$ containing $y$. Let us define

[TABLE]

Let us show that

[TABLE]

Notice first that $\lambda_{x}(y)$ is well defined since $\{\mu>f(y)\ \text{s.t.}\ x\notin C(\mu,y)\}$ is nonempty and bounded. Indeed, since $y$ is a non degenerate local minimum of $f$, for $\beta>0$ sufficiently small, $f(w)>f(y)$ for all $w\in\mathsf{C}(f(y)+\beta,y)$, $w\neq y$. Therefore, $x\notin\mathsf{C}(f(y)+\beta,y)$ (because $x\neq y$ and $f(x)\leq f(y)$). Moreover for all $\eta\in\{\mu>f(y)\ \text{s.t.}\ x\notin\mathsf{C}(\mu,y)\}$, $\eta<\lambda$ (because $x,y\in\mathsf{C}$ implies $\mathsf{C}(\lambda,y)=\mathsf{C}$ since $\mathsf{C}(\lambda,y)$ and $\mathsf{C}$ are both connected components of $\{f<\lambda\}$). Therefore, $\lambda_{x}(y)$ is well defined and satisfies $f(y)<\lambda_{x}(y)\leq\lambda$ (which proves the first inequality in (55)). Since $\mu\mapsto\mathsf{C}(\mu,y)$ is increasing on $(f(y),+\infty)$, it holds $x\notin\mathsf{C}(\mu,y)$ for all $\mu\in(f(y),\lambda_{x}(y))$ by definition of $\lambda_{x}(y)$. Thus, since according to Lemma 17 (see (43)),

[TABLE]

the set $\mathsf{C}(\lambda_{x}(y),y)$ does not contain $x$ and hence $\lambda_{x}(y)<\lambda$. This proves (55). Notice that (55) implies

[TABLE]

Let us now prove that $\partial\mathsf{C}(\lambda_{x}(y),y)\cap\mathsf{U}_{1}^{\mathsf{ssp}}\neq\emptyset$ which will conclude the proof of (53). Let us prove it by contradiction and let us assume that $\partial\mathsf{C}(\lambda_{x}(y),y)\cap\mathsf{U}_{1}^{\mathsf{ssp}}=\emptyset$. Then, using in addition the fact the function $f$ is Morse and the fact that

[TABLE]

the same arguments as those used to prove (50) apply and lead to the fact that $\overline{\mathsf{C}(\lambda_{x}(y),y)}$ is a closed and open connected set in $\{f\leq\lambda_{x}(y)\}$. Thus,

[TABLE]

For $\mu\geq\lambda(y)$, let us now denote by $\mathsf{C}^{+}_{y}(\mu)$ the connected component of $\{f\leq\mu\}$ containing $\overline{\mathsf{C}(\lambda_{x}(y),y)}$. It then holds, according to Lemma 17 (see (44)),

[TABLE]

In addition, for all $\mu>\lambda(y)$, $x\in\mathsf{C}(\mu,y)$ by definition of $\lambda_{x}(y)$ (see (54)), and $\mathsf{C}(\mu,y)\subset\mathsf{C}^{+}_{y}(\mu)$. Thus, using (58), $x\in\overline{\mathsf{C}(\lambda_{x}(y),y)}$ and hence, since $x\notin\mathsf{C}(\lambda_{x}(y),y)$, it holds $f(x)=\lambda_{x}(y)>f(y)$. This contradicts the fact that that $x\in\operatorname*{arg\,min}_{\overline{C}}f$. Therefore, we have proven that

[TABLE]

Using (56), this implies that $\mathsf{C}\cap\mathsf{U}_{1}^{\mathsf{ssp}}\neq\emptyset$ which concludes the proof of the reverse implication in (53) and thus the proof of (52).

Step 2. Proof of item 1 in Proposition 22.

Let us first deal with the case $\mathsf{U}_{1}^{\mathsf{ssp}}\cap\mathsf{C}=\emptyset$. In that case, the set $\mathsf{C}\cap\mathsf{U}_{0}^{\Omega}$ is reduced to one element. This implies that for all $\lambda\in(\min_{\overline{\mathsf{C}}}f,\lambda]$, the set $\mathsf{C}\cap\{f<\lambda\}$ is connected since each of its connected components necessarily contains at least one element of $\mathsf{U}_{0}^{\Omega}$.

Let us now deal with the case $\mathsf{U}_{1}^{\mathsf{ssp}}\cap\mathsf{C}\neq\emptyset$. Let us then consider $x\in\operatorname*{arg\,min}_{\mathsf{C}}f\subset\mathsf{U}_{0}^{\Omega}$ and, for every $y\in\mathsf{C}\cap\mathsf{U}_{0}^{\Omega}\setminus\{x\}$, let $\lambda_{x}(y)$ be as defined in (54). Let us also define

[TABLE]

which is well defined since the set $\{y\in\mathsf{C}\cap\mathsf{U}_{0}^{\Omega}\setminus\{x\}\}$ is nonempty (by (52)) and contains a finite number of elements (since $f$ is Morse). Then, from (55), (59) and the first inclusion in (57), one has

[TABLE]

Then, since for all $\mu\in(\tilde{\sigma},\lambda]$ and for all y\in\mathsf{C}\cap\big{(}\mathsf{U}_{0}^{\Omega}\setminus\{x\}\big{)}, $x\in\mathsf{C}(\mu,y)$ (because $\mu>\lambda_{x}(y)$ and by definition of $\lambda_{x}(y)$, see (54)) and since the $\mathsf{C}(\mu,y)$’s are connected components of $\{f<\mu\}$, one obtains that $\mathsf{C}(\mu,y)=\mathsf{C}(\mu,w)$ for all y,w\in\mathsf{C}\cap\big{(}\mathsf{U}_{0}^{\Omega}\setminus\{x\}\big{)}. Thus, one has

[TABLE]

This implies that for any y\in\mathsf{C}\cap\big{(}\mathsf{U}_{0}^{\Omega}\setminus\{x\}\big{)}, $\{f<\mu\}\cap\mathsf{C}$ is equal to $\mathsf{C}(\mu,y)$ (since every connected component of $\{f<\mu\}\cap\mathsf{C}$ contains at least one element of $\mathsf{U}_{0}^{\Omega}$). Therefore, one has

[TABLE]

Moreover, it holds

[TABLE]

and thus $\tilde{\sigma}=\sigma$, where we recall that $\sigma$ is defined in Proposition 22. Indeed, if it is not the case, then from (61), one has $\tilde{\sigma}<\max_{x\in\mathsf{U}_{1}^{\mathsf{ssp}}\cap\mathsf{C}}f(x)$ and thus $\mathsf{C}$ contains at least two connected components of $\{f<\max_{\mathsf{U}_{1}^{\mathsf{ssp}}\cap\mathsf{C}}f\}$ with $\max_{\mathsf{U}_{1}^{\mathsf{ssp}}\cap\mathsf{C}}f>\tilde{\sigma}$, which contradicts (63). The fact that $\mathsf{C}\cap\{f<\mu\}$ is connected follows from (63) and (64).

Let us now prove that $\mathsf{C}\cap\{f<\mu\}$ is a connected component of $\{f<\mu\}$ for all $\mu\in(\sigma,\lambda]$. Since $\mathsf{C}\cap\{f<\mu\}$ is connected, one can consider the connected component $\mathsf{V}$ of $\{f<\mu\}$ which contains $\mathsf{C}\cap\{f<\mu\}$. Then, since $\mathsf{C}$ is a connected component of $\{f<\lambda\}$ and $\mu\leq\lambda$, it holds $\mathsf{V}\subset\mathsf{C}$ and thus, $\mathsf{V}\subset\mathsf{C}\cap\{f<\mu\}$. Therefore, one has $\mathsf{V}=\mathsf{C}\cap\{f<\mu\}$ is a connected component of $\{f<\mu\}$. This concludes the proof of item 1 in Proposition 22.

Step 3. Proof of item 2 in Proposition 22.

Let us assume that $\mathsf{U}_{1}^{\mathsf{ssp}}\cap\mathsf{C}\neq\emptyset$. Then, using (52), $\mathsf{C}\cap\mathsf{U}_{0}^{\Omega}$ contains at least two elements. Let $x\in\operatorname*{arg\,min}_{\mathsf{C}}f$ and $y\in\mathsf{C}\cap\mathsf{U}_{0}^{\Omega}\setminus\{x\}$. Then $f(x)\leq f(y)$ and according to (55), it holds $f(y)<\lambda_{x}(y)$. From (64) and (60), it holds moreover

[TABLE]

Therefore $f(x)\leq f(y)<\sigma$ and thus

[TABLE]

This proves the first statement of item 2 in Proposition 22.

Let us now prove that each connected component of $\mathsf{C}\cap\{f<\sigma\}$ is a critical connected component (as introduced in item 2 in Definition 18). Let us first notice that $\mathsf{C}\cap\mathsf{U}_{0}^{\Omega}\subset\{f<\sigma\}$ implies

[TABLE]

where $\mathsf{C}(\sigma,w)$ is defined in (36) (since every connected component of $\mathsf{C}\cap\{f<\sigma\}$ contains at least one element of $\mathsf{U}_{0}^{\Omega}$). Let us consider a connected component of $\mathsf{C}\cap\{f<\sigma\}$. From (66), this component has the form $\mathsf{C}(\sigma,y)$ for some $y\in\mathsf{U}_{0}^{\Omega}$. Since $\sigma\in f(\mathsf{U}_{1}^{\mathsf{ssp}})$ (see (61)), $\mathsf{C}\cap\{f<\sigma\}$ contains at least two connected components, and thus $(\mathsf{C}\cap\mathsf{U}_{0}^{\Omega})\setminus\mathsf{C}(\sigma,y)\neq\emptyset$. Let $w\in(\mathsf{C}\cap\mathsf{U}_{0}^{\Omega})\setminus\mathsf{C}(\sigma,y)$. Let us assume that $\mathsf{C}(\sigma,y)$ is not a critical connected component, i.e that $\partial\mathsf{C}(\sigma,y)\cap\mathsf{U}_{1}^{\mathsf{ssp}}=\emptyset$. Then, the arguments used to prove (50) imply that $\overline{\mathsf{C}(\sigma,y)}$ is a connected component of $\{f\leq\sigma\}$. Thus, using Lemma 17 (see (44)), it holds

[TABLE]

Moreover, since for all $\lambda>\sigma$, $w\in\mathsf{C}(\lambda,y)$ (see (62)) and $\mathsf{C}(\lambda,y)\subset\mathsf{C}^{+}(\lambda,y)$, it holds $w\in\overline{\mathsf{C}(\sigma,y)}$. Therefore, since $f(w)<\sigma$ (see (65)), one has $w\in\mathsf{C}(\sigma,y)$ which contradicts the fact that $w\in(\mathsf{C}\cap\mathsf{U}_{0}^{\Omega})\setminus\mathsf{C}(\sigma,y)$. This ends the proof of Proposition 22.

Constructions of the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$

In this section we construct, under (A0), two maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$. These maps are constructed using an association between the local minima of $f$ and the (generalized) saddle points $\mathsf{U}_{1}^{\overline{\Omega}}$. Such maps have been introduced in [1, 2] and [16, 20] in the boundaryless case in order to give sharp asymptotic estimates of the eigenvalues of the involved operators. This has been generalized in [18] to the boundary case (where the authors introduced the notion of generalized saddle points for $\Delta^{D,(0)}_{f,h}$).

Let us recall (see Lemma 27 below), that $L^{D,(0)}_{f,h}$ has exactly $\mathsf{m}_{0}^{\Omega}$ eigenvalues smaller than $\sqrt{h}$ for sufficiently small $h$. Actually, from [19, 18], it can be shown that these $\mathsf{m}_{0}^{\Omega}$ eigenvalues are exponentially small. The goal of the map $\mathbf{j}$ is to associate each local minimum $x$ of $f$ with a set of generalized saddle points $\mathbf{j}(x)\subset\mathsf{U}_{1}^{\overline{\Omega}}$ such that $f$ is constant over $\mathbf{j}(x)$ and such that, for sufficiently small $h$, there exists at least one eigenvalue of $-L^{D,(0)}_{f,h}$ whose exponential rate of decay is 2\big{(}f(\mathbf{j}(x))-f(x)\big{)} i.e.

[TABLE]

The map $\widetilde{\mathbf{j}}$ associates each local minimum $x$ of $f$ with the connected component of $\{f<f(\mathbf{j}(x))\}$ which contains $x$. To construct the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$, the procedure relies on the analysis made in Section 2.3 and on the analysis of the sublevel sets of $f$ following the general analysis of the sublevel sets of a Morse function on a manifold without boundary of [20, Section 4.1] which generalizes the procedure described in [16]. To build the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$, one considers the connected components of $\{f<\lambda\}\cap\mathsf{U}_{1}^{\mathsf{ssp}}$ appearing when $\lambda$ is decreasing from $\max_{\cup_{k=1}^{\mathsf{N}_{1}}\overline{\mathsf{C}_{k}}}f$ to $-\infty$. Each time a new connected component appears in $\cup_{k=1}^{\mathsf{N}_{1}}\mathsf{C}_{k}$, one picks arbitrarily a local minimum in it and then, one associates this local minimum with the separating saddle points on the boundary of this new connected component.

Let assume that the assumption (A0) holds. The constructions of the maps $\mathbf{j}$ and $\mathbf{\widetilde{j}}$ are made recursively as follows:

Initialization ($q=1$). We consider $\mathsf{E}_{1,\ell}=\mathsf{C}_{\ell}$ for $\ell\in\{1,\dots,\mathsf{N}_{1}\}$ (see (42)).

For each $\ell\in\{1,\dots,\mathsf{N}_{1}\}$, $x_{1,\ell}$ denotes one point in $\operatorname*{arg\,min}_{\overline{\mathsf{E}_{1,\ell}}}f=\operatorname*{arg\,min}_{\mathsf{E}_{1,\ell}}f$. Then we define, for all $k\in\{1,\dots,\mathsf{N}_{1}\}$,

[TABLE]

Notice that according to Proposition 20 and item 2 in Definition 18, it holds

[TABLE]

and

[TABLE]

Moreover, one has from Proposition 20 (and more precisely the second inclusion in (48)),

[TABLE] 2. 2.

First step ($q=2$). If $\mathsf{N}_{1}<\mathsf{m}_{0}^{\Omega}$, we consider \{f<\lambda\}\bigcap\Big{(}\cup_{\ell=1}^{\mathsf{N}_{1}}\mathsf{E}_{1,\ell}\Big{)} for $\lambda<\max_{\ell\in\{1,\dots,\mathsf{N}_{1}\}}\sigma_{1,\ell}$.

From Proposition 22, for each $\ell\in\{1,\dots,\mathsf{N}_{1}\}$, $\mathsf{E}_{1,\ell}\cap\mathsf{U}_{0}^{\Omega}\neq\{x_{1,\ell}\}$ if and only if $\mathsf{U}_{1}^{\mathsf{ssp}}\cap\mathsf{E}_{1,\ell}\neq~{}\emptyset$. As a consequence, one has:

[TABLE]

If \mathsf{U}_{1}^{\mathsf{ssp}}\bigcap\Big{(}\cup_{\ell=1}^{\mathsf{N}_{1}}\mathsf{E}_{1,\ell}\Big{)}=\emptyset (then $\mathsf{N}_{1}=\mathsf{m}_{0}^{\Omega}$), the constructions of the maps $\widetilde{\mathbf{j}}$ and ${\mathbf{j}}$ are finished and one goes to item 4 below. If \mathsf{U}_{1}^{\mathsf{ssp}}\bigcap\Big{(}\cup_{\ell=1}^{\mathsf{N}_{1}}\mathsf{E}_{1,\ell}\Big{)}\neq\emptyset, one defines

[TABLE]

The set

[TABLE]

is then the union of finitely many connected components. We denote by $\mathsf{E}_{2,1},\dots,\mathsf{E}_{2,\mathsf{N}_{2}}$ (with $\mathsf{N}_{2}\geq 1$) the connected components of \bigcup_{\ell=1}^{\mathsf{N}_{1}}\big{(}\mathsf{E}_{1,\ell}\cap\{f<\sigma_{2}\}\big{)} which do not contain any of the minima $\{x_{1,1},\dots,x_{1,\mathsf{N}_{1}}\}$. From items 1 and 2 in Proposition 22 (applied for each $\ell\in\{1,\ldots,\mathsf{N}_{1}\}$ with $\mathsf{C}=\mathsf{E}_{1,\ell}\cap\{f<\sigma_{2}\}$ there),

[TABLE]

Notice that the other connected components (i.e. those containing the $x_{1,\ell}$’s) may be not critical. Let us associate with each $\mathsf{E}_{2,\ell}$, $1\leq\ell\leq\mathsf{N}_{2}$, one point $x_{2,\ell}$ arbitrarily chosen in $\operatorname*{arg\,min}_{\overline{\mathsf{E}_{2,\ell}}}f=\operatorname*{arg\,min}_{\mathsf{E}_{2,\ell}}f$ (the last equality follows from the fact that $\partial\mathsf{E}_{2,\ell}\subset\{f=\sigma_{2}\}$). For $\ell\in\{1,\ldots,\mathsf{N}_{2}\}$, let us define:

[TABLE] 3. 3.

Recurrence ($q\geq 3$).

If all the local minima of $f$ in $\Omega$ have been labeled at the end of the previous step above ($q=2$), i.e. if $\cup_{j=1}^{2}\{x_{j,1},\ldots,x_{j,\mathsf{N}_{j}}\}=\mathsf{U}_{0}^{\Omega}$ (or equivalently if $\mathsf{N}_{1}+\mathsf{N}_{2}=\mathsf{m}_{0}^{\Omega}$), the constructions of the maps $\widetilde{\mathbf{j}}$ and ${\mathbf{j}}$ are finished, all the local minima of $f$ have been labeled and one goes to item 4 below. If it is not the case, from Proposition 22, there exists $m\in\mathbb{N}^{*}$ such that

[TABLE]

where one defines recursively the decreasing sequence $(\sigma_{q})_{q=3,\ldots,m+2}$ by

[TABLE]

for $q\in\{3,\ldots,m+2\}$. Let us now consider $m^{*}\in\mathbb{N}^{*}$ the larger integer among the integers $m\in\mathbb{N}^{*}$ such that (69) holds. Notice that $m^{*}$ is well defined since the cardinal of $\mathsf{U}_{0}^{\Omega}$ is finite. By definition of $m^{*}$, one has:

[TABLE]

Then, one repeats recursively $m^{*}$ times the procedure described above defining \big{(}\mathsf{E}_{2,\ell},\mathbf{j}(x_{2,\ell}),\widetilde{\mathbf{j}}(x_{2,\ell})\big{)}_{1\leq\ell\leq\mathsf{N}_{2}} : for $q\in\{2,\ldots,m^{*}+1\}$, one defines $(\mathsf{E}_{q+1,\ell})_{\ell\in\{1,\ldots,\mathsf{N}_{q+1}\}}$ as the set of connected components of

[TABLE]

which do not contains any of the local minima $\cup_{j=1}^{q}\{x_{j,1},\ldots,x_{j,\mathsf{N}_{j}}\}$ of $f$ in $\Omega$ which have been previously chosen. From items 1 and 2 in Proposition 22 (applied for each $\ell\in\{1,\ldots,\mathsf{N}_{1}\}$ with $\mathsf{C}=\mathsf{E}_{1,\ell}\cap\{f<\sigma_{q+1}\}$ there),

[TABLE]

For $\ell\in\{1,\ldots,\mathsf{N}_{q+1}\}$, we associate with each $\mathsf{E}_{q+1,\ell}$, one point $x_{q+1,\ell}$ arbitrarily chosen in $\operatorname*{arg\,min}_{\mathsf{E}_{q+1,\ell}}f$ . For $\ell\in\{1,\ldots,\mathsf{N}_{q+1}\}$, let us define:

[TABLE]

From (70) and Proposition 22, $\mathsf{U}_{0}^{\Omega}=\cup_{j=1}^{m^{*}+2}\{x_{j,1},\ldots,x_{j,\mathsf{N}_{j}}\}$ and thus, all the local minima of $f$ in $\Omega$ are labeled. The constructions of the maps of the maps $\widetilde{\mathbf{j}}$ and ${\mathbf{j}}$ are finished and one goes to item 4 below. 4. 4.

Properties of the maps $\widetilde{\mathbf{j}}$ and ${\mathbf{j}}$.

Let us now give important features of the map $\mathbf{j}$ which follows directly from its construction and which are used in the sequel. Two maps have been defined:

[TABLE]

which are clearly injective. Notice that the $\mathbf{j}(x)$, $x\in\mathsf{U}_{0}^{\Omega}$, are not disjoint in general. For all $x\in\mathsf{U}_{0}^{\Omega}$, $f(\mathbf{j}(x))$ contains exactly one value, which will be denoted by $f(\mathbf{j}(x))$. Moreover, since $\cup_{\ell=1}^{\mathsf{N}_{1}}\mathsf{E}_{1,\ell}\subset\Omega$ (see the first statement in (47)), one has for all $x\in\mathsf{U}_{0}^{\Omega}$,

[TABLE]

Moreover, it holds

[TABLE]

Finally, for all $x\in\mathsf{U}_{0}^{\Omega}$,

[TABLE]

and for all $x\in\mathsf{U}_{0}^{\Omega}\setminus\{x_{1,1},\dots,x_{1,\mathsf{N}_{1}}\}$,

[TABLE]

In Figure 8, one gives the constructions of the maps $\mathbf{j}$ and $\widetilde{\mathbf{j}}$ for a one-dimensional example. Since, one can pick a minimum or another in a critical connected component at each step of the construction of $\mathbf{j}$ and $\widetilde{\mathbf{j}}$, the maps are not uniquely defined if over one of the connected components $\mathsf{E}_{k,\ell}$ ($k\geq 1,\,\ell\in\{1,\ldots,\mathsf{N}_{k}\}$), $\operatorname*{arg\,min}f$ contains more than one point. As will be clear below, this non-uniqueness has no influence on the results proven hereafter (in particular Theorem 1). In Figure 9, we give an example for which two constructions of the maps $\mathbf{j}$ and $\widetilde{\mathbf{j}}$ are possible.

Remark 23**.**

In the case when for all local minima $x$ of $f$, $\mathbf{j}(x)$ is a single point, $\mathbf{j}(x)\cap\mathbf{j}(y)=\emptyset$ for all $x\neq y$ and when all the heights $(f(\mathbf{j}(x))-f(x))_{x\in\mathsf{U}_{1}^{\overline{\Omega}}}$ are distinct, the map $\mathbf{j}$ is exactly the one constructed in [18].

The next definition will be used in Section 3.2 to construct the quasi-modes.

Definition 24**.**

Let us assume that the assumption (A0) is satisfied. Let $\varepsilon$ be such that

[TABLE]

where the family $(\mathsf{E}_{k,\ell})_{k\geq 1,\,\ell\in\{1,\ldots,\mathsf{N}_{k}\}}$ is defined in the construction of the map $\mathbf{j}$ above. For $k\geq 1$ and $\ell\in\{1,\ldots,\mathsf{N}_{k}\}$, one defines

[TABLE]

which is a connected component of \big{\{}f<\max_{\overline{\mathsf{E}_{k,\ell}}}f-\varepsilon\big{\}} according to item 1 in Proposition 22.

Rewriting the assumptions (A1)-(A4) in terms of the map $\mathbf{j}$

In this section, one rewrites the assumptions (A1), (A2) (A3), and (A4) with the map $\mathbf{j}$ constructed in Section 2.4. To this end, let us prove the following lemma.

Lemma 25**.**

Let us assume that the hypothesis (A0) is satisfied. Then, the assumption (A1) is equivalent to the fact that there exists $\ell\in\{1,\ldots,\mathsf{N}_{1}\}$ such that for all $k\in\{1,\ldots,\mathsf{N}_{1}\}\setminus\{\ell\}$,

[TABLE]

Thus, when (A1) holds, the elements of $\mathcal{C}=\{\mathsf{C}_{1},\dots,\mathsf{C}_{\mathsf{N}_{1}}\}$ (see Definition 16) are ordered such that $\ell=1$, i.e for all $k\in\{2,\ldots,\mathsf{N}_{1}\}$:

[TABLE]

Moreover, under (A1) (or equivalently (76)), one has $\mathsf{C}_{1}=\mathsf{C}_{\mathsf{max}}$, where $\mathsf{C}_{\mathsf{max}}$ is defined by (A1).

Proof.

Assume that the hypothesis (A0) is satisfied. Let us recall that the set $\mathcal{C}$ (defined by (16)) satisfies from (39) and Definition 16:

[TABLE]

Let $\mathsf{C}\in\mathcal{C}$ and let $k\in\{1,\ldots,\mathsf{N}_{1}\}$, such that $\mathsf{C}=\mathsf{C}_{k}$. Then, from (40) and the first step of the construction of $\mathbf{j}$ in Section 2.4, one has for all $q\in\{1,\ldots,\mathsf{N}_{1}\}$: $\mathsf{H}_{f}(x_{1,q})=\lambda(x_{1,q})=f(\mathbf{j}(x_{1,q}))=\sup_{\mathsf{C}_{q}}f$ and $f(x_{1,q})=\min_{\overline{\mathsf{C}_{q}}}f$. Thus, it holds

[TABLE]

This implies the results stated in Lemma 25.

In view of Lemma 25 and by construction of the map $\mathbf{j}$ (see the first step in Section 2.4), one can rewrite the assumptions (A1), (A2) (A3), and (A4) with the map $\mathbf{j}$ as follows:

• •

The assumption (A1) is equivalent to the fact that, up to reordering the elements of $\mathcal{C}=\{\mathsf{C}_{1},\dots,\mathsf{C}_{\mathsf{N}_{1}}\}$ (see Definition 16) such that (76), it holds for all $x\in\{x_{1,2},\ldots,x_{1,\mathsf{N}_{1}}\}$:

[TABLE]

Furthermore, in this case, $\mathsf{C}_{1}=\mathsf{C}_{\mathsf{max}}$, where $\mathsf{C}_{\mathsf{max}}$ is defined by (A1).

• •

The assumption (A2) rewrites when (A1j) holds,

[TABLE]

• •

The assumption (A3) rewrites when (A1j) holds,

[TABLE]

• •

When (A1j) holds, the assumption (A4) is equivalent to

[TABLE]

This equivalence follows from (A1j) together with the fact that $\mathbf{j}(x_{1,1})=\partial\mathsf{C}_{1}\cap\mathsf{U}_{1}^{\mathsf{ssp}}$ (see (67)) and by definition of a separating saddle point.

From now on, we work with the formulations (A1j), (A2j), (A3j), and (A4j) of the assumptions (A1), (A2), (A3), and (A4).

Notice that under (A1j), it holds from (73), for all $x\in\mathsf{U}_{0}^{\Omega}\setminus\{x_{1,1}\}$,

[TABLE]

In Figure 10, ones gives an example when (A1j) holds but not (A2j), (A3j) and (A4j). In Figure 11, one gives an example when (A1j) and (A2j) hold but not (A3j) and (A4j). In Figure 12, one gives a case when (A1j), (A2j) and (A3j) hold but not (A4j). In Figure 13, one gives a case when (A1j), (A2j), (A3j), and (A4j) hold.

When (A1j) and (A2j) are satisfied, from Definition 18 and Proposition 20 (see the first inclusion in (48) and (67)), one has

[TABLE]

In that case, we assume that the elements $\{z_{1},\ldots,z_{\mathsf{m}_{1}^{\partial\Omega}}\}$ of $\mathsf{U}_{1}^{\partial\Omega}$ (see (21)) are ordered such that

[TABLE]

where $\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\in\mathbb{N}^{*}$ satisfies $\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\leq\mathsf{m}_{1}^{\partial\Omega}$ (see (21)). Notice that from Lemma 25, this labeling implies when (A3j) is satisfied:

[TABLE]

where $\mathsf{k}_{1}^{\partial\mathsf{C}_{\mathsf{max}}}$ is defined by (24). Let us finally prove the following result which will be used in the sequel.

Lemma 26**.**

Let us assume that the assumptions (A0), (A1j), (A2j) and (A3j) are satisfied. Then, one has

[TABLE]

and

[TABLE]

Proof.

The fact that $\min_{\overline{\Omega}}f<\min_{\partial\Omega}f$ is obvious. Let us prove (82). Let $k\in\{1,\ldots,\mathsf{N}_{1}\}$ and let us recall that from Definition 16, there exists $x\in\mathsf{U}_{0}^{\Omega}\cap\mathsf{C}_{k}$ such that $\mathsf{C}_{k}=\mathsf{C}(\lambda(x),x)$. Let us assume that $x\in\operatorname*{arg\,min}_{\Omega}f$. Then, by definition of the map $\mathbf{j}$ and by definition of $\lambda(x)$ (see (38)) together with the fact that (A1j), (A2j) and (A3j) hold, one has $f(\mathbf{j}(x_{1,k}))=\lambda(x)\geq\min_{\partial\Omega}f=f(\mathbf{j}(x_{1,1}))$. Thus, if $f(x_{1,k})=f(x)\leq f(x_{1,1})$, it holds

[TABLE]

This implies $\mathsf{C}_{k}=\mathsf{C}_{1}$ from the assumption (A1j). This concludes the proof of (82).

Construction of the quasi-modes

This section is dedicated to the construction of two families of quasi-modes: a family of functions which aims at approximating the vector space spanned by the eigenfunctions associated with the smallest eigenvalues of $-L^{D,(0)}_{f,h}$ and a family of $1$-forms which aims at approximating the vector space spanned by the eigenforms associated with the smallest eigenvalues of $-L^{D,(1)}_{f,h}$. This construction is made using the maps  $\mathbf{j}$ and $\widetilde{\mathbf{j}}$ previously constructed.

This section is organized as follows. In Section 3.1, we introduce the notations used throughout this paper for operators, and the properties of Witten Laplacians and of the operators $L_{f,h}^{D,(p)}$ needed in our analysis. The maps  $\mathbf{j}$ and $\widetilde{\mathbf{j}}$ constructed in the previous section are then used to build the quasi-modes in Section 3.2.

Notations and Witten Laplacian

In Section 3.1.1, one introduces the notations for the Sobolev spaces which are used in this paper. Section 3.1.2 is dedicated to the properties of Witten Laplacians and of the operators $L_{f,h}^{D,(p)}$ needed in our analysis.

Notation for Sobolev spaces

For $p\in\{0,\ldots,d\}$, one denotes by $\Lambda^{p}C^{\infty}(\overline{\Omega})$ the space of $C^{\infty}$ $p$-forms on $\overline{\Omega}$. Moreover, $\Lambda^{p}C^{\infty}_{T}(\overline{\Omega})$ is the set of $C^{\infty}$ $p$-forms $v$ such that $\mathbf{t}v=0$ on $\partial\Omega$, where $\mathbf{t}$ denotes the tangential trace on forms. For $p\in\{0,\ldots,d\}$ and $q\in\mathbb{N}$, one denotes by $\Lambda^{p}H^{q}_{w}(\Omega)$ the weighted Sobolev spaces of $p$-forms with regularity index $q$, for the weight function $e^{-\frac{2}{h}f(x)}$ on $\Omega$: $v\in\Lambda^{p}H^{q}_{w}(\Omega)$ if and only if for all multi-index $\alpha$ with $|\alpha|\leq q$, the $\alpha$ derivative of $v$ is in $\Lambda^{p}L^{2}_{w}(\Omega)$ where $\Lambda^{p}L^{2}_{w}(\Omega)$ is the completion of the space $\Lambda^{p}C^{\infty}(\overline{\Omega})$ for the norm

[TABLE]

The subscript $w$ in the notation $L^{2}_{w}(\Omega)$ refers to the fact that the weight function $x\in\Omega\mapsto e^{-\frac{2}{h}f(x)}$ appears in the inner product. See for example [43] for an introduction to Sobolev spaces on manifolds with boundaries. For $p\in\{0,\ldots,d\}$ and $q>\frac{1}{2}$, the set $\Lambda^{p}H^{q}_{w,T}(\Omega)$ is defined by

[TABLE]

Notice that $\Lambda^{p}L^{2}_{w}(\Omega)$ is the space $\Lambda^{p}H^{0}_{w}(\Omega)$, and that $\Lambda^{0}H^{1}_{w,T}(\Omega)$ is the space $H^{1}_{w,0}(\Omega)$ than we introduced in Proposition 4. We will denote by $\|.\|_{H^{q}_{w}}$ the norm on the weighted space $\Lambda^{p}H^{q}_{w}(\Omega)$. Moreover $\langle\cdot,\cdot\rangle_{L^{2}_{w}}$ denotes the scalar product in $\Lambda^{p}L^{2}_{w}(\Omega)$.

Finally, we will also use the same notation without the index $w$ to denote the standard Sobolev spaces defined with respect to the Lebesgue measure on $\Omega$.

The Witten Laplacian and the infinitesimal generator of the diffusion (1)

In this section, we recall some basic properties of Witten Laplacians, as well as the link between those and the operators $L_{f,h}^{(p)}$ introduced above (see (7) and (35)).

For $p\in\{0,\dots,n\}$, one defines the distorted exterior derivative à la Witten $d^{(p)}_{f,h}:\Lambda^{p}\,C^{\infty}(\Omega)\to\Lambda^{p+1}\,C^{\infty}(\Omega)$ and its formal adjoint: $d_{f,h}^{(p)*}:\Lambda^{p+1}\,C^{\infty}(\Omega)\to\Lambda^{p}\,C^{\infty}(\Omega)$ by

[TABLE]

The Witten Laplacian, firstly introduced in [46], is then defined similarly as the Hodge Laplacian $\Delta^{(p)}_{H}:=(d+d^{*})^{2}:\Lambda^{p}\,C^{\infty}(\Omega)\to\Lambda^{p}\,C^{\infty}(\Omega)$ by

[TABLE]

The Dirichlet realization of $\Delta_{f,h}^{(p)}$ on $\Lambda^{p}L^{2}(\Omega)$ is denoted by $\Delta_{f,h}^{D,(p)}$ and its domain is

[TABLE]

The operator $\Delta^{D,(p)}_{f,h}$ is self-adjoint, nonnegative, and its associated quadratic form is given by

[TABLE]

where

[TABLE]

We refer in particular to [18, Section 2.4] for a comprehensive definition of Witten Laplacians with Dirichlet tangential boundary conditions and statements on their properties. The link between the Witten Laplacian and the infinitesimal generator $L^{(0)}_{f,h}$ of the diffusion (1) is the following: since

[TABLE]

one has:

[TABLE]

where $U$ is the unitary operator

[TABLE]

In particular, the operator $L^{D,(0)}_{f,h}$ has a natural extension to $p$-forms defined by the relation

[TABLE]

For $p=1$, one recovers the operator $L^{(1)}_{f,h}$ with tangential Dirichlet boundary conditions defined by (34) and (35). The operator $L^{D,(p)}_{f,h}$ with domain

[TABLE]

is then self-adjoint on $\Lambda^{p}L^{2}_{w}\left(\Omega\right)$, non positive and its associated quadratic form is

[TABLE]

Let us also recall that $-L^{D,(p)}_{f,h}$ (and equivalently $\Delta^{D,(p)}_{f,h}$) has a compact resolvent. From general results on elliptic operators when $p=0$, $-L^{D,(0)}_{f,h}$ (and $\Delta^{D,(0)}_{f,h}$) admits a non degenerate smallest eigenvalue with an associated eigenfunction which has a sign on $\Omega$. Denoting moreover by $\pi_{E}(L^{D,(p)}_{f,h})$ the spectral projector associated with $L^{D,(p)}_{f,h}$ and some Borel set $E\subset\mathbb{R}$, the following commutation relations hold on $\Lambda^{p}H^{1}_{T}(\Omega)$:

[TABLE]

Let us recall that from the elliptic regularity of $L^{D,(p)}_{f,h}$, for any bounded Borel set $E\subset\mathbb{R}$, ${\rm Ran}\,\pi_{E}(L^{D,(d)}_{f,h})\subset\Lambda^{p}\,C^{\infty}_{T}(\overline{\Omega})$, the relation (86) then leads to the following complex structure:

[TABLE]

and

[TABLE]

For ease of notation, one defines:

[TABLE]

The following result, instrumental in our investigation of the smallest eigenvalue $\lambda_{h}$ of $-L_{f,h}^{D,(0)}$, is an immediate consequence of [18, Theorem 3.2.3] together with (85).

Lemma 27**.**

Under assumption (A0), there exists $h_{0}>0$ such that for all $h\in(0,h_{0})$,

[TABLE]

where ${\mathsf{m}_{0}^{\Omega}}={\rm Card}(\mathsf{U}_{0}^{\Omega})$ and $\mathsf{m}_{1}^{\overline{\Omega}}={\rm Card}(\mathsf{U}_{1}^{\overline{\Omega}})$ are defined in Section 1.3.2.

In the sequel, with a slight abuse of notation, one denotes the exterior differential $d$ acting on functions by $\nabla$. Note that it follows from the above considerations and Lemma 27 that under (A0), it holds

[TABLE]

Moreover, from (85), it is equivalent to study the spectrum of $L_{f,h}^{D,(0)}$ or the spectrum of $\Delta_{f,h}^{D,(0)}$. We end this section with the following lemma which will be frequently used throughout this work.

Lemma 28**.**

Let $(A,D\left(A\right))$ be a non negative self adjoint operator on a Hilbert space $\left(\mathcal{H},\|\cdot\|\right)$ with associated quadratic form $q_{A}(x)=(x,Ax)$ whose domain is $Q\left(A\right)$. It then holds, for any $u\in Q\left(A\right)$ and $b>0$,

[TABLE]

where, for a Borel set $E\subset\mathbb{R}$, $\pi_{E}(A)$ is the spectral projector associated with $A$ and $E$.

Construction of the quasi-modes for $-L_{f,h}^{D,(p)}$, $p\in\{0,1\}$

Let us recall that from Lemma 27, one has for any $h$ small enough

[TABLE]

where we recall that $\mathsf{m}_{0}^{\Omega}$ is the number of local minima of $f$ in $\Omega$ and $\mathsf{m}_{1}^{\overline{\Omega}}$ is the number generalized saddle points of $f$ in $\overline{\Omega}$, see Section 1.3.2. To prove Theorem 1, the strategy consists in constructing a family of $\mathsf{m}_{0}^{\Omega}$ quasi-modes in order to approximate ${\rm Ran}\,\pi_{h}^{(0)}$ and a family of $\mathsf{m}_{1}^{\overline{\Omega}}$ quasi-modes in order to approximate ${\rm Ran}\,\pi_{h}^{(1)}$, see (87).

Since the construction of the quasi-modes rely on the one made for Witten Laplacians in [16, 18, 20], we first construct quasi-modes for the Witten Laplacians $\Delta_{f,h}^{D,(0)}$ (Section 3.2.1) and $\Delta_{f,h}^{D,(1)}$ (Section 3.2.2). The quasi-modes for $-L_{f,h}^{D,(0)}$ and $-L_{f,h}^{D,(1)}$ are then obtained using (85) (Section 3.2.3).

Quasi-modes for the Witten Laplacian $\Delta_{f,h}^{D,(0)}$

Let us assume that the assumption (A0) is satisfied. Let us recall that from Lemma 27 and (85), there exists $h_{0}>0$ such that for any $h\in(0,h_{0})$:

[TABLE]

where we recall that $\mathsf{m}_{0}^{\Omega}$ is the number of local minima of $f$ in $\Omega$. In this section, one constructs using the maps $\mathbf{j}$ and $\widetilde{\mathbf{j}}$ constructed in Section 2.4, a family of $\mathsf{m}_{0}^{\Omega}$ functions whose span approximates {\rm Ran}\,\pi_{[0,h^{\frac{3}{2}})}\big{(}\Delta^{D,(0)}_{f,h}\big{)}. The properties of this family which are listed in this section will be useful to prove Proposition 46 below and Propositions 49 and 50 in the next section. Following [16, 18, 20], we associate each critical point $x\in\mathsf{U}_{0}^{\Omega}$ with a quasi-mode for $\Delta_{f,h}^{D,(0)}$. The notation follows the one introduced in Section 2.4.

Let us first introduce two parameters $\varepsilon_{1}>0$ and $\varepsilon>0$ which will be used to define the quasi-modes for $\Delta_{f,h}^{D,(0)}$. In the following, $d$ is the geodesic distance on $\overline{\Omega}$ for the initial metric. Let us consider $\varepsilon_{1}>0$ small enough such that

[TABLE]

and for all $z\in\mathsf{U}_{1}^{\overline{\Omega}}$,

[TABLE]

or

[TABLE]

The parameter $\varepsilon_{1}>0$ will be successively reduced a finite number of times in this section and in Section 3.2.2, and it will be kept fixed from the end of Section 3.2.2.

Let $\varepsilon>0$ be such that

[TABLE]

which ensures in particular that $\mathsf{E}_{k,\ell}(\varepsilon)$ and $\mathsf{E}_{k,\ell}(2\varepsilon)$ are connected for all $k\geq 1$ and $\ell\in\{1,\dots,\mathsf{N}_{1}\}$, see (75). The parameter $\varepsilon>0$ will be further reduced a finite number of times in the following sections so that $\partial\chi_{k,\ell}^{\varepsilon,\varepsilon_{1}}$ is as close as necessary to $\partial\mathsf{E}_{k,\ell}$ near $\mathbf{j}(x_{k,\ell})$, where $\chi_{k,\ell}^{\varepsilon,\varepsilon_{1}}$ is used to define the quasi-mode for $\Delta_{f,h}^{D,(0)}$ associated to $x_{k,\ell}$.

Definition 29**.**

Let us assume that the assumption (A0) holds. For $k\geq 1$ and $\ell\in\{1,\dots,\mathsf{N}_{k}\}$, the quasi-mode associated with $x_{k,\ell}$ is defined by:

[TABLE]

where the functions $\chi_{k,\ell}^{\varepsilon,\varepsilon_{1}}\in C_{c}^{\infty}(\Omega,\mathbb{R}^{+})$. There exists $\varepsilon_{1}^{0}>0$ such that for all $\varepsilon_{1}\in(0,\varepsilon_{1}^{0}]$, there exists $\varepsilon^{0}>0$ such that for all $\varepsilon\in(0,\varepsilon^{0}]$, the functions $\chi_{k,\ell}^{\varepsilon,\varepsilon_{1}}$ satisfy the following properties:

1. a)

It holds

[TABLE]

see (42) for the definition of $\mathsf{E}_{k,\ell}$ and (75) for the definition of $\mathsf{E}_{k,\ell}(2\varepsilon)$. 2. b)

For all $y\in\mathop{\rm supp}\nolimits\chi^{\varepsilon,\varepsilon_{1}}_{k,\ell}$,

[TABLE]

and hence, according to (93),

[TABLE] 3. c)

For all $z\in\mathbf{j}(x_{k,\ell})\cap\Omega$, it holds

[TABLE] 4. d)

For all $z\in\mathsf{U}_{1}^{\overline{\Omega}}\setminus\mathbf{j}(x_{k,\ell})$, it holds

[TABLE] 5. e)

For all $q\in\{1,\dots,\mathsf{N}_{k}\}\setminus\{\ell\}$, it holds $\mathop{\rm supp}\nolimits\chi_{k,q}^{\varepsilon,\varepsilon_{1}}\cap\mathop{\rm supp}\nolimits\chi_{k,\ell}^{\varepsilon,\varepsilon_{1}}=\emptyset$. 6. f)

For $k\geq 2$ and for any $(k^{\prime},\ell^{\prime})\in\{1,\dots,k-1\}\times\{1,\dots,\mathsf{N}_{k^{\prime}}\}$ such that $\mathsf{E}_{k,\ell}\subset\mathsf{E}_{k^{\prime},\ell^{\prime}}$, it holds

[TABLE]

Notice that by a conexity argument, it holds ${\mathsf{E}_{k,\ell}}\subset\mathsf{E}_{k^{\prime},\ell^{\prime}}$ or $\mathsf{E}_{k,\ell}\cap\mathsf{E}_{k^{\prime},\ell^{\prime}}=\emptyset$.

In Figures 14, 15 and 16, for $k\geq 1$, one gives a schematic representation of the cut-off function $\chi^{\varepsilon,\varepsilon_{1}}_{k,\ell}$ near $z\in\mathsf{U}_{1}^{\overline{\Omega}}$ respectively in the three situations:

• •

$k=1$, $\ell\in\{1,\ldots,\mathsf{N}_{1}\}$ and $z\in\mathbf{j}(x_{1,\ell})\cap\partial\Omega$.

• •

$\ell\in\{1,\dots,\mathsf{N}_{k}\}$ and $z\in\mathbf{j}(x_{k,\ell})\cap\Omega$.

• •

$\ell\in\{1,\ldots,\mathsf{N}_{k}\}$, z\in\big{(}\mathsf{U}_{1}^{\overline{\Omega}}\setminus\mathbf{j}(x_{k,\ell})\big{)}\cap\partial\mathsf{E}_{k,\ell}.

For the ease of notation, we do not indicate the dependance on the parameters $\varepsilon$ and $\varepsilon_{1}$ in the notation of the functions $\widetilde{v}_{k,\ell}$ for $k\geq 1$, $\ell\in\{1,\ldots,\mathsf{N}_{k}\}$, introduced in Definition 29.

The following lemma will be useful to estimate when $h\to 0$ the quantities

[TABLE]

for ${k\geq 1}$ and ${\ell\in\{1,\ldots,\mathsf{N}_{k}\}}$ (where the $\mathsf{m}_{0}^{\Omega}$ functions $(\widetilde{v}_{k,\ell})_{k\geq 1,\,\ell\in\{1,\ldots,\mathsf{N}_{k}\}}$ are introduced in Definition 29), see indeed item 2a in Proposition 46 below.

Lemma 30**.**

Let us assume that the assumption (A0) holds. Then, for $k\geq 1$ and $\ell\in\{1,\dots,\mathsf{N}_{k}\}$, there exist $c>0$, $C>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$,

[TABLE]

where the function $\widetilde{v}_{k,\ell}$ is introduced in Definition 29.

Proof.

This estimate follows from (92)–(94), and Laplace’s method applied to $\|\chi^{\varepsilon,\varepsilon_{1}}_{k,\ell}\,e^{-\frac{1}{h}f}\|_{L^{2}}$.

The following lemma ensures that the family $(\widetilde{v}_{k,\ell})_{k\geq 1,\,\ell\in\{1,\ldots,\mathsf{N}_{k}\}}$ is uniformly linearly independent for any $h$ small enough.

Lemma 31**.**

Let us assume that the assumption (A0) holds. The family $\mathcal{B}_{v}=(\widetilde{v}_{k,\ell})_{k\geq 1,\,\ell\in\{1,\ldots,\mathsf{N}_{k}\}}$ introduced in Definition 29, is linearly independent, uniformly with respect to $h$ small enough. This is equivalent to: for some (and hence for any) orthonormal (for the $L^{2}$-scalar product) family $\mathcal{B}_{o}$ spanning ${\rm Span}(\mathcal{B}_{v})$, for any matrix norm $\|\cdot\|$ on $\mathbb{R}^{\mathsf{m}_{0}^{\Omega}\times\mathsf{m}_{0}^{\Omega}}$, there exist $C>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$,

[TABLE]

Proof.

The proof of Lemma 31 is made in [20, Section 4.2].

Quasi-modes for the Witten Laplacian $\Delta_{f,h}^{D,(1)}$

Let us assume that the assumption (A0) is satisfied. Let us recall that from Lemma 27 and (85), there exists $h_{0}>0$ such that for any $h\in(0,h_{0})$:

[TABLE]

where we recall that $\mathsf{m}_{1}^{\overline{\Omega}}$ is the number of generalized saddle points of $f$ in $\overline{\Omega}$, see Section 1.3.2. In this section, one constructs a family of $1$-forms $(\widetilde{\phi}_{j})_{j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$ which aims at approximating {\rm Ran}\,\pi_{[0,h^{\frac{3}{2}})}\big{(}\Delta^{D,(1)}_{f,h}\big{)}. To this end, for each $z\in\mathsf{U}^{\overline{\Omega}}_{1}$, one constructs a $1$-form locally supported in a neighborhood of $z$ in $\overline{\Omega}$. More precisely, one proceeds as follows:

for each $z\in\mathsf{U}^{\Omega}_{1}$, the $1$-form associated with $z$ is constructed following the procedure in [16, 20] and, 2. 2.

for each $z\in\mathsf{U}^{\partial\Omega}_{1}$, the $1$-form associated with $z$ is constructed as in [18].

Let us recall these constructions and some estimates which will be used throughout this work.

Quasi-mode associated with $z\in\mathsf{U}^{\Omega}_{1}$.

Let us recall that from (22),

[TABLE]

is the set of saddle points of $f$ in $\Omega$. Let j\in\{\mathsf{m}_{1}^{\partial\Omega}+1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} and $z_{j}\in\mathsf{U}_{1}^{\Omega}$. Let $\mathcal{V}_{j}$ be some small smooth neighborhood of $z_{j}$ such that $\overline{\mathcal{V}_{j}}\cap\partial\Omega=\emptyset$ and for $x\in~{}\overline{\mathcal{V}_{j}}$, $|\nabla f(x)|=0$ if and only if $x=z_{j}$. Let us now consider the full Dirichlet realization $\Delta_{f,h}^{FD,(1)}(\mathcal{V}_{j})$ of the Witten Laplacian $\Delta_{f,h}^{(1)}$ in $\mathcal{V}_{j}$ whose domain is

[TABLE]

where the superscript $FD$ stands for full Dirichlet boundary conditions. Let us recall that according to [19, Section 2], there exists, choosing if necessary $\mathcal{V}_{j}$ smaller, a $C^{\infty}$ non negative solution $\Phi_{j}:\mathcal{V}_{j}\to\mathbb{R}^{+}$ to the eikonal equation

[TABLE]

Moreover, $\Phi_{j}$ is the unique non negative solution to (100) in the sense that if $\widetilde{\Phi}_{j}:\widetilde{\mathcal{V}_{j}}\to\mathbb{R}^{+}$ is another non negative $C^{\infty}$ solution to (100) on a neighborhood $\widetilde{\mathcal{V}_{j}}$ of $z_{j}$, then $\widetilde{\Phi}_{j}=\Phi_{j}$ on $\widetilde{\mathcal{V}_{j}}\cap\mathcal{V}_{j}$.

Remark 32**.**

The function $\Phi_{j}$ is actually the Agmon distance to $z_{j}$, i.e. $\Phi_{j}$ is the distance to $z_{j}$ in $\overline{\Omega}$ associated with the metric $|\nabla f|^{2}dx^{2}$, where $dx^{2}$ is the Riemannian metric on $\overline{\Omega}$ (see [19, Section 1]).

The next proposition, which follows from [19, Theorem 1.4 and Lemma 1.6], gathers all the estimates one needs in the following on the operator $\Delta_{f,h}^{FD,(1)}(\mathcal{V}_{j})$.

Proposition 33**.**

Let us assume that the assumption (A0) is satisfied. Then, the operator $\Delta_{f,h}^{FD,(1)}(\mathcal{V}_{j})$ is self-adjoint, has compact resolvent and is positive. Moreover:

• •

There exist $\varepsilon_{0}>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$:

[TABLE]

• •

The smallest eigenvalue $\lambda_{h}(\mathcal{V}_{j})$ of $\Delta_{f,h}^{FD,(1)}(\mathcal{V}_{j})$ is exponentially small: there exist $C>0$, $c>0$ and $h_{0}>0$ such that for any $h\in(0,h_{0})$:

[TABLE]

• •

Any $L^{2}$-normalized eigenform $w_{j}$ associated with the smallest eigenvalue $\lambda_{h}(\mathcal{V}_{j})$ of $\Delta_{f,h}^{FD,(1)}(\mathcal{V}_{j})$ satisfies the following Agmon estimates (see Remark 32): for all $\varepsilon>0$, there exist $C_{\varepsilon}>0$ and $h_{0}>0$ such that for any $h\in(0,h_{0})$, it holds:

[TABLE]

Choosing $\varepsilon_{1}$ smaller if necessary, one assumes that there exists $\alpha>0$ such that

[TABLE]

Let us now define the quasi-mode associated with $z_{j}\in\mathsf{U}_{1}^{\Omega}$.

Definition 34**.**

Let us assume that the assumption (A0) is satisfied. Let j\in\{\mathsf{m}_{1}^{\partial\Omega}+1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} and $z_{j}\in\mathsf{U}_{1}^{\Omega}$. The quasi-mode associated with $z_{j}$ is defined by

[TABLE]

where $w_{j}$ is a $L^{2}$-normalized eigenform associated with the smallest eigenvalue $\lambda_{h}(\mathcal{V}_{j})$ of $\Delta_{f,h}^{FD,(1)}(\mathcal{V}_{j})$ and $\theta_{j}$ is a smooth non negative cut-off function satisfying, $\mathop{\rm supp}\nolimits\theta_{j}\subset B(z_{j},2\varepsilon_{1})\subset\mathcal{V}_{j}$ and $\theta_{j}=1$ on $B(z_{j},\varepsilon_{1})$.

Notice that both $w_{j}$ and $-w_{j}$ can be used to build a quasi-mode and the choice of the sign is determined in Proposition 36. Moreover, using (103) together with the fact that for all j\in\{\mathsf{m}_{1}^{\partial\Omega}+1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, $\inf_{\text{supp }(1-\theta_{j})\cap\mathcal{V}_{j}}\Phi_{j}>0$ (see (100)), one has when $h\to 0$:

[TABLE]

for some $c>0$ independent of $h$.

Using Proposition 33 and (105), one deduces the following estimate on the quasi-mode $\widetilde{\phi}_{j}$ introduced in Definition 34.

Corollary 35**.**

Let us assume that the assumption (A0) holds. Let $\widetilde{\phi}_{j}$ be the quasi-mode associated with $z_{j}\in U_{1}^{\Omega}$ (j\in\{\mathsf{m}_{1}^{\partial\Omega}+1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}), see Definition 34. Then, there exist $C>0$, $c>0$ and $h_{0}>0$ such that for any $h\in(0,h_{0})$:

[TABLE]

Let us now recall the construction of a WKB approximation of $w_{j}$ made in [19] and which will be needed in the following. Let us denote by $\mathcal{W}_{+}(z_{j})$ and $\mathcal{W}_{-}(z_{j})$ respectively the stable and unstable manifolds of $z_{j}$ associated with the flow of $-\nabla f$ which are defined as follows. Denoting by $\varphi_{t}(y)$ the solution of $\frac{d}{dt}\varphi_{t}(y)=-\nabla f(\varphi_{t}(y))$ with initial condition $\varphi_{0}(y)=y$,

[TABLE]

It then holds (see indeed [19, Section 2] and [16, Section 4.2]): ${\rm\;dim\;}\mathcal{W}_{+}(z_{j})=d-1$, ${\rm\;dim\;}\mathcal{W}_{-}(z_{j})=1$, and for all $y\in\mathcal{V}_{j}$ (assuming $\mathcal{V}_{j}$ small enough),

[TABLE]

with moreover

[TABLE]

Additionally, there exists from [19, Proposition 1.3 and Section 2] a $C^{\infty}(\overline{\mathcal{V}_{j}})$ $1$-form $a_{j}(x,h)=\widetilde{a}_{j}(x)+O(h)$ such that $a_{j}(z_{j},h)=\widetilde{a}_{j}(z_{j})=n(z_{j})$, where $n(z_{j})$ is a unit normal to $\mathcal{W}_{+}(z_{j})$, and such that the $1$-form $u^{(1)}_{j,wkb}=a_{j}e^{-\frac{1}{h}\Phi_{j}}$ satisfies

[TABLE]

where $\mu(h)\sim h^{2}\sum_{k=0}^{\infty}\mu_{k}h^{k}$. Moreover, one has in the limit $h\to 0$ (see [19, Section 2]):

[TABLE]

where the remainder terms $O(h)$ admits a full asymptotic expansion in $h$. Using in addition the fact that $\Phi_{j}>0$ on ${\rm supp}\,\nabla\theta_{j}$, there exists $c>0$ such that for $h$ small enough:

[TABLE]

From (110), one then obtains that $\Delta_{f,h}^{FD,(1)}(\mathcal{V}_{j})$ admits an eigenvalue which equals $\mu(h)+O(h^{\infty})$. Since $\mu(h)=O(h^{2})$, from (101) and (102), one deduces that $\mu_{k}=0$ for all $k\geq 0$ and thus $\mu(h)=O(h^{\infty})$. Finally, one has:

[TABLE]

In the following proposition, $w_{j}$ and $u^{(1)}_{j,wkb}$ are compared.

Proposition 36**.**

Let us assume that the assumption (A0) is satisfied. Let $w_{j}$ be a $L^{2}$-normalized eigenform associated with the smallest eigenvalue $\lambda_{h}(\mathcal{V}_{j})$ of $\Delta_{f,h}^{FD,(1)}(\mathcal{V}_{j})$ (j\in\{\mathsf{m}_{1}^{\partial\Omega}+1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}). Then, there exists $h_{0}>0$ such that for all $h\in(0,h_{0})$ one has:

[TABLE]

where

[TABLE]

In addition, up to replacing $w_{j}$ by $-w_{j}$, one can assume that $c_{j}(h)\geq 0$ for $h$ small enough and then, in the limit $h\to 0$, one has:

[TABLE]

where the remainder terms $O(h)$ admits a full asymptotic expansion in $h$.

Proof.

Let us define k_{j}(h):=\big{\langle}w_{j},\theta_{j}u^{(1)}_{j,wkb}\big{\rangle}_{L^{2}}=c_{j}(h)^{-1}. If $k_{j}(h)<0$, then one changes $w_{j}$ to $-w_{j}$ so that one can suppose without loss of generality that $k_{j}(h)\geq 0$. For $h$ small enough, one has from (101):

[TABLE]

Let us define the following $1$-form

[TABLE]

Thus, the following identity holds for $h$ small enough

[TABLE]

Notice that, from (110), there exist $C>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$

[TABLE]

Therefore, using Lemma 28, (105), and (111), there exist $c>0$ and $C>0$ such that for $h$ small enough:

[TABLE]

Moreover, since $d_{f,h}=hd+df\wedge$ and $d_{f,h}^{*}=hd^{*}+\mathbf{i}_{\nabla f}$, one obtains using the Gaffney inequality (see [43]):

[TABLE]

Furthermore, from (106), it holds

[TABLE]

and from (111)

[TABLE]

Thus, there exists $C>0$ such that:

[TABLE]

This concludes the proof of (112). Finally, since $\|\theta_{j}w_{j}\|_{L^{2}}=1+O(e^{-\frac{c}{h}})$ (see (105)), by considering \|\theta_{j}(u^{(1)}_{j,wkb}-k_{j}(h)w_{j})\|^{2}_{L^{2}}=O\big{(}h^{\infty}\big{)}, one gets using (110):

[TABLE]

Since $k_{j}(h)\geq 0$, one has k_{j}(h)=\frac{(\pi h)^{\frac{d}{4}}}{|{\rm det\,{\rm Hess\,}}f(z_{j})|^{\frac{1}{4}}}\,\big{(}1+O(h)\big{)}. This concludes the proof of (113) since $c_{j}(h)=k_{j}(h)^{-1}$.

Quasi-mode associated with $z\in\mathsf{U}^{\partial\Omega}_{1}$.

Let us recall that from (21),

[TABLE]

Let j\in\{1,\dots,\mathsf{m}_{1}^{\partial\Omega}\big{\}} and $z_{j}\in\mathsf{U}_{1}^{\partial\Omega}$. To construct a $1$-form locally supported in a neighborhood of $z_{j}$ in $\overline{\Omega}$, one proceeds in the same way as in [18, Section 4.3]. Let $\mathcal{V}_{j}$ be a small neighborhood of $z_{j}$ in $\overline{\Omega}$ such that $\mathcal{V}_{j}$ satisfies: $|\nabla f|>0$ on $\overline{\mathcal{V}_{j}}$, for all $x\in\partial\mathcal{V}_{j}\cap\partial\Omega$, $|\nabla_{T}f(x)|=0$ if and only if $x=z_{j}$, and $\partial_{n}f>0$ on $\partial\Omega\cap\partial\mathcal{V}_{j}$. Let us now consider the mixed full Dirichlet–tangential Dirichlet realization $\Delta_{f,h}^{MD,(1)}(\mathcal{V}_{j})$ of the Witten Laplacian $\Delta_{f,h}^{(1)}$ in $\mathcal{V}_{j}$ whose domain is

[TABLE]

where the superscript $MD$ stands for mixed full Dirichlet–tangential Dirichlet boundary conditions (see [18, Remark 4.3.1] for the characterization of its domain). Since $\partial_{n}f>0$ on $\partial\Omega\cap\mathcal{V}_{j}$, from [18, Section 4.2], one has that, choosing $\mathcal{V}_{j}$ small enough, there exists a $C^{\infty}(\mathcal{V}_{j},\mathbb{R}^{+})$ non negative solution $\Phi_{j}$ to the eikonal equation

[TABLE]

Moreover, $\Phi_{j}$ is the unique non negative solution to (114) in the sense that if $\widetilde{\Phi}_{j}:\widetilde{\mathcal{V}_{j}}\to\mathbb{R}^{+}$ is another non negative $C^{\infty}$ solution to (114) on a neighborhood $\widetilde{\mathcal{V}_{j}}$ of $z_{j}$, then $\widetilde{\Phi}_{j}=\Phi_{j}$ on $\widetilde{\mathcal{V}_{j}}\cap\mathcal{V}_{j}$.

Remark 37**.**

The function $\Phi_{j}$ is actually the Agmon distance to $z_{j}$, see [10, Section 3] for a precise definition of the Agmon distance in a bounded domain.

Choosing $\varepsilon_{1}$ smaller if necessary, one can assume that there exists $\alpha>0$ such that

[TABLE]

The next proposition, which follows from [18, Proposition 4.3.2], gathers all the estimates one needs in the following on the operator $\Delta_{f,h}^{MD,(1)}(\mathcal{V}_{j})$.

Proposition 38**.**

Let us assume that the assumption (A0) is satisfied. Then, the operator $\Delta_{f,h}^{MD,(1)}(\mathcal{V}_{j})$ is self-adjoint, has compact resolvent and is positive. Moreover:

• •

There exists $h_{0}>0$ such that for all $h\in(0,h_{0})$:

[TABLE]

• •

The smallest eigenvalue $\lambda_{h}(\mathcal{V}_{j})$ of $\Delta_{f,h}^{MD,(1)}(\mathcal{V}_{j})$ is exponentially small: there exist $C>0$, $c>0$ and $h_{0}>0$ such that for any $h\in(0,h_{0})$:

[TABLE]

• •

Any $L^{2}$-normalized eigenform $w_{j}$ associated with the smallest eigenvalue $\lambda_{h}(\mathcal{V}_{j})$ of $\Delta_{f,h}^{MD,(1)}(\mathcal{V}_{j})$ satisfies the following Agmon estimates: there exist $C>0$, $n\in\mathbb{N}$ and $h_{0}>0$ such that for any $h\in(0,h_{0})$, it holds:

[TABLE]

Let us now define the quasi-mode associated with $z_{j}\in U_{1}^{\partial\Omega}$.

Definition 39**.**

Let us assume that the assumption (A0) holds. Let j\in\{1,\dots,\mathsf{m}_{1}^{\partial\Omega}\big{\}} and $z_{j}\in\mathsf{U}_{1}^{\partial\Omega}$. The quasi-mode associated with $z_{j}$ is defined by

[TABLE]

where $w_{j}$ is a $L^{2}$-normalized eigenform associated with the first eigenvalue $\lambda_{h}(\mathcal{V}_{j})$ of $\Delta_{f,h}^{(1),MD}(\mathcal{V}_{j})$ and $\theta_{j}$ is a smooth non negative cut-off function satisfying $\mathop{\rm supp}\nolimits\theta_{j}\subset B(z_{j},2\varepsilon_{1})\cap\overline{\Omega}\subset\mathcal{V}_{j}$, $\{z_{j}\}=\operatorname*{arg\,min}_{\mathop{\rm supp}\nolimits\theta_{j}\cap\partial\Omega}f$, and $\theta_{j}=1$ on $B(z_{j},\varepsilon_{1})\cap\overline{\Omega}$.

Notice again that both $w_{j}$ and $-w_{j}$ can be used to build a quasi-mode and the choice of the sign is determined in Proposition 41. Notice also that the fact that $\widetilde{\phi}_{j}\in\Lambda^{1}C^{\infty}\left(\overline{\Omega}\right)$ follows from standard elliptic regularity results. In addition, for all j\in\{1,\dots,\mathsf{m}_{1}^{\partial\Omega}\big{\}}, using (117) together with the fact that $\inf_{\text{supp }(1-\theta_{j})\cap\mathcal{V}_{j}}\Phi_{j}>0$ (see (114)), there exists $c>0$ such that when $h\to 0$:

[TABLE]

Using Proposition 38 and (119), one deduces the following estimate on the quasi-mode $\widetilde{\phi}_{j}$ introduced in Definition 34.

Corollary 40**.**

Let us assume that the assumption (A0) holds. Let $\widetilde{\phi}_{j}$ be the quasi-mode associated with $z_{j}\in U_{1}^{\partial\Omega}$ (j\in\{1,\dots,\mathsf{m}_{1}^{\partial\Omega}\big{\}}), see Definition 39. Then, there exist $C>0$, $c>0$ and $h_{0}>0$ such that for any $h\in(0,h_{0})$:

[TABLE]

Let us now give the corresponding versions of the WKB approximation estimates (111)–(113) for the quasi-mode $\widetilde{\phi}_{j}$ introduced in Definition 39. From [18, Section 4.2], there exists a $C^{\infty}(\overline{\mathcal{V}_{j}})$ function $a_{j}(x,h)=\widetilde{a}_{j}(x)+O(h)$ with $a_{j}\equiv\widetilde{a}_{j}\equiv 1$ on $\partial\Omega\cap\overline{\mathcal{V}_{j}}$ such that the $1$-form

[TABLE]

satisfies

[TABLE]

Moreover, one has in the limit $h\to 0$ (see [18, Section 4.2]):

[TABLE]

where the remainder terms $O(h)$ admits a full asymptotic expansion in $h$. In the following proposition, $w_{j}$ and $u^{(1)}_{j,wkb}$ are compared.

Proposition 41**.**

Let us assume that the assumption (A0) holds. Let $w_{j}$ be a $L^{2}$-normalized eigenform associated with the smallest eigenvalue $\lambda_{h}(\mathcal{V}_{j})$ of $\Delta_{f,h}^{FD,(1)}(\mathcal{V}_{j})$ (j\in\{1,\dots,\mathsf{m}_{1}^{\partial\Omega}\big{\}}). Then, there exists $h_{0}>0$ such that for all $h\in(0,h_{0})$ one has:

[TABLE]

where

[TABLE]

In addition, up to replacing $w_{j}$ by $-w_{j}$, one can assume that $c_{j}(h)\geq 0$ for $h$ small enough and then, in the limit $h\to 0$, one has:

[TABLE]

where the remainder terms $O(h)$ admits a full asymptotic expansion in $h$.

Proposition 41 is proved exactly as Proposition 36.

In conclusion, a family of $1$-forms $(\widetilde{\phi}_{j})_{j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$ has been constructed in this section. Since (89) guarantees that $\overline{B(z,2\varepsilon_{1})}\cap\overline{B(z^{\prime},2\varepsilon_{1})}=\emptyset$ for all $z\neq z^{\prime}\in\mathsf{U}_{1}^{\overline{\Omega}}$, the family $(\widetilde{\phi}_{j})_{j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$ is orthonormal in $L^{2}(\Omega)$. From now on, the parameter $\varepsilon_{1}$ is fixed and $\varepsilon>0$ will be successively reduced a finite number of times in the following.

WKB approximation of the quasi-modes $(\widetilde{\phi}_{j})_{j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$.

For upcoming computations, one needs the following definition.

Definition 42**.**

Let us assume that the assumption (A0) is satisfied. For all $j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}$, one defines:

[TABLE]

where for j\in\{1,\dots,\mathsf{m}_{1}^{\partial\Omega}\big{\}}, $u_{j,wkb}^{(1)}$ satisfies (122) and $\theta_{j}$ is introduced in Definition 39 and, for j\in\{\mathsf{m}_{1}^{\partial\Omega}+1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, $u_{j,wkb}^{(1)}$ satisfies (111) and $\theta_{j}$ is introduced in Definition 34.

From (105), Proposition 36, (119), and Proposition 41 one has the following lemma.

Lemma 43**.**

Let us assume that the assumption (A0) is satisfied. For j\in\{1,\dots,\mathsf{m}_{1}^{\partial\Omega}\big{\}}, let $\widetilde{\phi}_{j}$ be as defined in (118), and for j\in\{\mathsf{m}_{1}^{\partial\Omega}+1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, let $\widetilde{\phi}_{j}$ be as defined in (104). Moreover, for j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, let $\widetilde{\phi}_{j,wkb}$ be as defined in (125). Then, one has:

[TABLE]

Quasi-modes for $-L_{f,h}^{D,(p)}$, $p\in\{0,1\}$

Before defining the quasi-modes for $L^{D,(0)}_{f,h}$ and $L^{D,(1)}_{f,h}$, let us label the quasi-modes for $\Delta^{D,(0)}_{f,h}$ and the local minima of $f$ using the lexicographic order.

Definition 44**.**

Let us assume that the assumption (A0) is satisfied. Then, the family of critical connected components $(\mathsf{E}_{k,\ell})_{k\geq 1,\,\ell\in\{1,\ldots,N_{k}\}}$ introduced in Section 2.4, the local minima $(x_{k,\ell})_{k\geq 1,\,\ell\in\{1,\ldots,N_{k}\}}$ of $f$ labeled in Section 2.4, the family of cut-off functions $(\chi_{k,\ell}^{\varepsilon,\varepsilon_{1}})_{k,\ell})_{k\geq 1,\,\ell\in\{1,\ldots,N_{k}\}}$ introduced in Definition 29 and the family of quasi-modes $(\widetilde{v}_{k,\ell})_{k\geq 1,\,\ell\in\{1,\ldots,N_{k}\}}$ introduced in Definition 29 are labeled in the lexicographic order:

[TABLE]

Let us recall that the lexicographic order is defined by $(k^{\prime},\ell^{\prime})<(k,l)$ if and only if $k^{\prime}<k$ or if $k^{\prime}=k$, $\ell^{\prime}<\ell$. From now on, one uses the labeling introduced in Definition 44.

According to (85), the quasi-modes for $L^{D,(0)}_{f,h}$ and $L^{D,(1)}_{f,h}$ are obtained from those constructed previously for $\Delta^{D,(0)}_{f,h}$ and $\Delta^{D,(1)}_{f,h}$ using the unitary transformation $U$ defined in (84).

Definition 45**.**

Let us assume that the assumption (A0) is satisfied. Let $(\widetilde{v}_{k})_{k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}}$ be the family of quasi-modes for $\Delta^{D,(0)}_{f,h}$ introduced in Definition 29 (and labeled in the lexicographic order, see Definition 44) and let $(\widetilde{\phi}_{j})_{j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$ be the family of quasi-modes for $\Delta^{D,(1)}_{f,h}$ introduced in Definitions 34 and 39. The family of quasi-modes $(\widetilde{u}_{k})_{k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}}$ for $-L^{D,(0)}_{f,h}$ and the family of quasi-modes $(\widetilde{\psi}_{j})_{j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$ for $-L^{D,(1)}_{f,h}$ are defined by: for $k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}$, and for $j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$:

[TABLE]

Notice that, according to (92) and (126), for all $k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}$,

[TABLE]

and according to (104) and (118), for all $j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$,

[TABLE]

Bases of $\operatorname{Ran}\pi_{h}^{(0)}$ and $\operatorname{Ran}\pi_{h}^{(1)}$

Let us recall, that from (87),

[TABLE]

In this section, one proves that the quasi-modes introduced in Definition 45 form two bases of $\operatorname{Ran}\pi_{h}^{(0)}$ and $\operatorname{Ran}\pi_{h}^{(1)}$. In the following, the finite dimensional spaces ${\rm Ran}\,\pi^{(0)}_{h}$ and ${\rm Ran}\,\pi^{(1)}_{h}$ are endowed with the scalar product $\langle\cdot,\cdot\rangle_{L^{2}_{w}}$.

Proposition 46**.**

Let us assume that the assumption (A0) holds. Let $(\widetilde{u}_{k})_{k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}}$ be the family of quasi-modes for $-L^{D,(0)}_{f,h}$ and let $(\widetilde{\psi}_{j})_{j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$ be the family of quasi-modes for $-L^{D,(1)}_{f,h}$ introduced in Definition 45. Then,

For all k\in\big{\{}1,\dots,{\mathsf{m}_{0}^{\Omega}}\big{\}} and j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, $\widetilde{u}_{k}\in\Lambda^{0}H^{1}_{w,T}\left(\Omega\right)$, $\widetilde{\psi}_{j}\in\Lambda^{1}H^{1}_{w,T}\left(\Omega\right)$ and

[TABLE] 2. 2.

• a)

For any $\delta>0$, one can choose the parameter $\varepsilon$ in (92) (see also (126)) small enough such that for all $k\in\{1,\dots,{\mathsf{m}_{0}^{\Omega}}\}$, in the limit $h\to 0$:

[TABLE]

In particular, choosing the parameter $\varepsilon>0$ small enough in (92), there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

• b)

There exist $c>0$ such that for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, one has in the limit $h\to 0$:

[TABLE] 3. 3.

• a)

*The family $(\widetilde{u}_{k})_{k=1,\dots,{\mathsf{m}_{0}^{\Omega}}}$ is uniformly linearly independent (for the $L^{2}_{w}$-scalar product) for all $h$ sufficiently small (as defined in Lemma 31). *

• b)

For all (i,j)\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}^{2},

[TABLE]

In particular, there exists $h_{0}>0$ such that for all $h\in(0,h_{0})$:

[TABLE]

and

[TABLE]

Proof.

The proof of Proposition 46 is divided into two steps.

Step 1. Proofs of items 1 and 2.

The first item is immediate according to the definition of the families $(\widetilde{u}_{k})_{k=1,\dots,{\mathsf{m}_{0}^{\Omega}}}$ and $(\widetilde{\psi}_{i})_{i=1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}}$ introduced in Definition 45.

The first inequality appearing in 2a is a direct consequence of Lemma 28 applied to $A=-L_{f,h}^{D,(0)}$ whose associated quadratic form is given by $\frac{h}{2}\langle\nabla\cdot,\nabla\cdot\rangle_{L^{2}_{w}}$ on $H^{1}_{w,T}(\Omega)$. The second inequality in 2a follows from Laplace’s methods and from the properties of the cut-off functions used to define the quasi-modes $\widetilde{u}_{k}$ (see Definition 45 and Lemma 30). Indeed, it is just a rewriting of (98) using Definition 45 and the labeling introduced in Definition 44.

Let us now deal with 2b. First, Lemma 28 together with (106) and (120) implies the existence of some $c>0$ such that for all $i\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$ and $h$ small enough,

[TABLE]

Consequently, using again (106) and (120), and owing to the following relations on $\Lambda^{1}H^{1}_{T}(\Omega)$,

[TABLE]

[TABLE]

$d^{*}_{f,h}=hd^{*}+\mathbf{i}_{\nabla f}$, and $d_{f,h}=hd+\nabla f\wedge$, one obtains the existence of $c>0$ such that in the limit $h\to 0$:

[TABLE]

Since $\widetilde{\phi}_{i}\in\Lambda^{1}H^{1}_{T}(\Omega)$, the estimates (127) and (128) then lead, owing to Gaffney’s inequality (see [43, Corollary 2.1.6]), to

[TABLE]

Therefore, we deduce from the relation $\|u\|_{H^{1}_{w}}\leq\frac{C}{h}\|u\,e^{-\frac{1}{h}f}\|_{H^{1}}$, valid for all $u\in\Lambda^{p}H^{1}(\Omega)$ and $h>0$, and from

[TABLE]

resulting from (85) and (87), that there exists $c>0$ such that for all $i\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$ and $h$ small enough,

[TABLE]

This ends the proof of 2b.

Step 2. Proof of item 3.

The fact that the family $(\widetilde{u}_{k})_{k=1,\dots,{\mathsf{m}_{0}^{\Omega}}}$ is uniformly linearly independent is a consequence of Lemma 31 together with (126). Item 3b follows from items 1 and 2.b together with the relation

[TABLE]

holding for $f,g$ in $\Lambda^{\ell}L^{2}_{w}(\Omega)$ and $\ell\in\{0,1\}$. Finally, the fact that for $h$ small enough, \operatorname{Ran}\pi_{h}^{(0)}={\rm Span}\big{(}\pi_{h}^{(0)}\widetilde{u}_{k},\ k=1,\dots,{\mathsf{m}_{0}^{\Omega}}\big{)} and \operatorname{Ran}\pi_{h}^{(1)}={\rm Span}\big{(}\pi_{h}^{(1)}\widetilde{\psi}_{i},\ i=1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{)} are consequences of items 2a, 3a and 3b together with Lemma 27.

On the smallest eigenvalue of $-L_{f,h}^{D,(0)}$

This section is dedicated to the proof of the following theorem.

Theorem 3**.**

Assume that the assumptions (A0) and (A1j) are satisfied. Let $\lambda_{h}$ be the principal eigenvalue of $-L_{f,h}^{D,(0)}$ (see (8)). Then, denoting by $\lambda_{2,h}$ the second smallest eigenvalue of $-L_{f,h}^{D,(0)}$, there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

Moreover, when (A2j) is satisfied, one has in the limit $h\to 0$:

[TABLE]

where we recall that $x_{1}\in\operatorname*{arg\,min}_{\mathsf{C}_{1}}f$. Finally, when (A4j) holds, the remainder term $O(\sqrt{h})$ in (131) is actually of order $O(h)$ and admits a full asymptotic expansion in $h$.

Remark 47**.**

Without the assumption (A4j), we are not able to prove an asymptotic expansion in $\sqrt{h}$ of the remainder term $O(\sqrt{h})$ in (131) except in some specific cases, see Theorem 4 below or [38, Proposition C.40].

Let us mention that sharp asymptotic estimates when $h\to 0$ of the principal eigenvalue of $-L_{f,h}^{D,(0)}$ have been obtained in [18, 33, 10] in the Dirichlet case and in [30] in the Neumann case. However, these results do not apply under the assumptions considered in Theorem 3. Let us also mention that when $\Omega=\mathbb{R}^{d}$ or when $\Omega$ is a compact Riemannian manifold, sharp asymptotic estimates of the second smallest eigenvalue of $-L_{f,h}^{(0)}$ have been obtained in [2, 16, 35, 26, 20, 21, 36].

The analysis led in this section will also allow us to give a lower and an upper bound for all the $\mathsf{m}_{0}^{\Omega}$ small eigenvalues of $-L^{D,(0)}_{f,h}$ (and not only $\lambda_{h}$). This is the purpose of Theorem 5 below.

Remark 48**.**

Combining Theorem 3 and Proposition 6, under the assumptions (A0), (A1j) and (A2j), one obtains that in the limit $h\to 0$:

[TABLE]

In some specific cases, one can drop the assumption (A2j) in Theorem 3 and still obtain a sharp asymptotic equivalent of $\lambda_{h}$ when $h\to 0$. Indeed, in view of the proof of Theorem 3, one has the following result.

Theorem 4**.**

Assume that the assumptions (A0) and (A1j) are satisfied. Assume moreover that for all $j\in\{2,\ldots,\mathsf{N}_{1}\}$, $\partial\mathsf{C}_{1}\cap\partial\mathsf{C}_{j}=\emptyset$ (this last assumption is for instance satisfied when $\mathsf{N}_{1}=1$). Let us define,

[TABLE]

and

[TABLE]

where $\lambda_{-}(z_{j})$ is the negative eigenvalue of ${\rm Hess\,}f(z)$ (notice that $a_{1}$ and $a_{2}$ cannot be both equal to [math] since from Proposition 20, $\partial\mathsf{C}_{1}\cap\mathsf{U}_{1}^{\mathsf{ssp}}\neq\emptyset$). Then, one has when $h\to 0$:

[TABLE]

where the two remainder terms $O(h)$ admit a full asymptotic expansion in $h$.

This section is organized as follows. In Section 4.1, one gives the quasi-modal estimates which are used to prove Theorem 3. Section 4.2 is then dedicated to the proof of Theorem 3.

Estimates of interactions between quasi-modes

The main result of this section is Proposition 50 which gives the quasi-modal estimates in $L^{2}_{w}(\Omega)$ needed to prove Theorem 3. This section is divided into two parts. In Section 4.1.1, one gives the asymptotic estimates of the boundary terms

[TABLE]

which are then used in the proof of Proposition 50. In Section 4.1.2, one states and proves Proposition 50.

For all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, let us define the constant

[TABLE]

where $\lambda_{-}(z_{j})$ is the negative eigenvalue of ${\rm Hess\,}f(z_{j})$. These constants will appear in the upcoming computations.

Asymptotic estimates of boundary terms for $(\widetilde{\psi}_{i})_{i\in \{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$

The following boundary estimates will be used several times in the sequel.

Proposition 49**.**

Let us assume that the assumption (A0) is satisfied. Let us consider $j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$, an open set $\Sigma$ of $\partial\Omega$, and $F\in L^{\infty}(\partial\Omega,\mathbb{R})$. Then, there exists $c>0$ such that one has in the limit $h\to 0$:

[TABLE]

where $\widetilde{\psi}_{j}$ is introduced in (126) and $m_{1}^{\partial\Omega}$ is defined in (20). Moreover, when j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\partial\Omega}\big{\}}, $z_{j}\in\Sigma$, and $F$ is $C^{\infty}$ in a neighborhood of $z_{j}$, it holds

[TABLE]

where the remainder terms $O(h)$ admits a full asymptotic expansion in $h$ (as defined in Remark 9), and $B_{j}$ is defined by (132)

Proof.

Let $F\in L^{\infty}(\partial\Omega,\mathbb{R})$. From (126) and (104), the quasi-mode $\widetilde{\psi}_{j}$ is supported in $\Omega$ for $j\in\{\mathsf{m}_{1}^{\partial\Omega}+1\dots\mathsf{m}_{1}^{\overline{\Omega}}\}$ and thus:

[TABLE]

Let us now consider the case $j\in\{1,\dots,\mathsf{m}_{1}^{\partial\Omega}\}$. Notice that one has for all $h$ small enough, from the trace theorem, (126), (118), and (117),

[TABLE]

where $p$ is independent of $h$. Therefore, since $z_{j}$ is the only minimum of $f$ on ${\rm supp}\,\theta_{j}\cap\partial\Omega$, if $z_{j}\notin\overline{\Sigma}$, one has in the limit $h\to 0$:

[TABLE]

for some $c>0$ independent of $h$.

Let us now now consider the case $j\in\{1,\dots,\mathsf{m}_{1}^{\partial\Omega}\}$ and $z_{j}\in\Sigma$. One has:

[TABLE]

where $\widetilde{\phi}_{j,wkb}=c_{j}(h)\theta_{j}u_{j,wkb}^{(1)}$ is defined in (125). From (121), let us recall that in the limit $h\to 0$:

[TABLE]

with $\widetilde{a}_{j}=1$ on $\partial\Omega\cap\mathop{\rm supp}\nolimits\theta_{j}$. Thus on $\partial\Omega\cap\mathop{\rm supp}\nolimits\theta_{j}$, using also (114),

[TABLE]

Thus, the term $\displaystyle\int_{\Sigma}F\,\widetilde{\phi}_{j,wkb}\cdot n\ e^{-\frac{1}{h}f}$ appearing in the right-hand side of (136) satisfies in the limit $h\to 0$:

[TABLE]

where the last line follows from $\{z_{j}\}=\operatorname*{arg\,min}_{\partial\Omega\cap\mathop{\rm supp}\nolimits\theta_{j}}f$, $\theta_{j}(z_{j})=1$, Laplace’s method, and $c_{j}(h)=O(h^{-\frac{d+1}{4}})$ according to (124). When $z_{j}\in\Sigma$ and $F$ is $C^{\infty}$ in a neighborhood of $z_{j}$, the same arguments applied to (137) yield, in the limit $h\to 0$:

[TABLE]

where the remainder terms $O(h)$ admits a full asymptotic expansion in $h$ (which follows from Laplace’s method).

Besides, from the trace theorem and Lemma 43, the second term in the right-hand side of (136) satisfies in the limit $h\to 0$:

[TABLE]

The first part of Proposition 49 then results from (133)–(136), (138), and (140), and its second part from (136), (139)–(140), and from the asymptotic estimate of $c_{j}(h)$ given in (124) which yields, when $h\to 0$

[TABLE]

This ends the proof of Proposition 49.

Quasi-modal estimates in $L^{2}_{w}(\Omega)$

We are now in position to prove Proposition 50 which will be crucial to prove Theorem 3. This proposition allows indeed to study accurately the small singular values of the restricted differential $\nabla:\,{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$. The square of the smallest singular values is indeed  $\frac{2}{h}\lambda_{h}$, where $\lambda_{h}$ is the principal eigenvalue of $-L^{D,(0)}_{f,h}$ (see (8) and Proposition 4).

Proposition 50**.**

Let us assume that the assumption (A0) holds. Let $(\widetilde{u}_{k})_{k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}}$ be the family of quasi-modes for $-L^{D,(0)}_{f,h}$ and let $(\widetilde{\psi}_{j})_{j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$ be the family of quasi-modes for $-L^{D,(1)}_{f,h}$ introduced in Definition 45. Then, there exists $\varepsilon_{0}>0$ such that for for all $\varepsilon\in(0,\varepsilon_{0})$, for all k\in\big{\{}1,\dots,{\mathsf{m}_{0}^{\Omega}}\big{\}} and j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, there exists $\varepsilon_{j,k}\in\{-1,1\}$ independent of $h$ such that in the limit $h\to 0$,

[TABLE]

where all the remainder terms $O(h)$ admits a full asymptotic expansions in $h$ (as defined in Remark 9),

[TABLE]

and

[TABLE]

where the constant $B_{j}$ is defined in (132), and $(\mathsf{E}_{k})_{k\in\{1,\dots,\mathsf{m}_{0}^{\Omega}\}}$ is defined in Section 2.4 and labeled in Definition 44. Finally, if $z_{j}\in\mathbf{j}(x_{k})\cap\partial\Omega$ (and thus, it holds necessarily $k\in\{1,\ldots,\mathsf{N}_{1}\}$, see (72) where $\mathsf{N}_{1}$ is defined by (41)), one has

[TABLE]

Proof.

The proof of Proposition 50 is divided into three steps.

Step 1.

Let k\in\big{\{}1,\ldots,\mathsf{m}_{0}^{\Omega}\big{\}} and j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}. Let us consider the case $z_{j}\notin\mathbf{j}(x_{k})$. According to Definitions 34, 39 and 45, one has that for all $j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$, the quasi-mode $\widetilde{\psi}_{j}$ is supported in $B(z_{j},2\varepsilon_{1})\cap\overline{\Omega}$. Therefore, from (92), (96), and Definition 45, one has $\mathop{\rm supp}\nolimits\nabla\chi_{k}^{\varepsilon,\varepsilon_{1}}\cap\overline{B(z_{j},2\varepsilon_{1})}=\emptyset$ and thus

[TABLE]

Step 2.

Let us now deal with the computation of the terms $\langle\nabla\widetilde{u}_{k},\widetilde{\psi}_{j}\rangle_{L^{2}_{w}}$ for k\in\big{\{}1,\ldots,\mathsf{m}_{0}^{\Omega}\big{\}} and j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} such that $z_{j}\in\mathbf{j}(x_{k})\cap\Omega$. In this case, these computations follow from the analysis led in the proof of [16, Proposition 6.4], the only difference arising from the fact $\mathbf{j}(x_{k})$ and $\operatorname*{arg\,min}_{\mathop{\rm supp}\nolimits\chi^{\varepsilon,\varepsilon_{1}}_{{k}}}f$ were both reduced to one single point there. Let us give a proof for the sake of completeness. One has:

[TABLE]

From (93)–(94), it holds $\operatorname*{arg\,min}_{\mathop{\rm supp}\nolimits\chi^{\varepsilon,\varepsilon_{1}}_{{k}}}f=\operatorname*{arg\,min}_{\mathsf{E}_{k}}f$. Thus, using Laplace’s method together with the fact that $\min_{\overline{\mathsf{E}_{k}}}f=f(x_{k})$, one has in the limit $h\to 0$:

[TABLE]

Let us now give the estimate of the numerator of the right-hand side of (142).

To prepare this computation, let us first recall that the set $B(z_{j},2\varepsilon_{1})\cap\{f<f(z_{j})\}$ has, according to (90), two connected components. Since $z_{j}\in\mathsf{U}_{1}^{\mathsf{ssp}}$ (see Definition 18), exactly one of these two connected components intersects – and is then included in – the critical connected component $\widetilde{\mathbf{j}}(x_{k})=\mathsf{E}_{k}$ associated with $x_{k}$ (see Definition 18 and (71)). Moreover, the set $B(z_{j},2\varepsilon_{1})\setminus\mathcal{W}_{+}(z_{j})$, where the stable manifold $\mathcal{W}_{+}(z_{j})$ has been defined in (107), has also two connected components and one of them contains the connected component of $B(z_{j},2\varepsilon_{1})\cap\{f<f(z_{j})\}$ which intersects $\mathsf{E}_{k}$, namely $B(z_{j},2\varepsilon_{1})\cap\mathsf{E}_{k}$. Let us denote by

[TABLE]

Since $\mathop{\rm supp}\nolimits\widetilde{\phi}_{j}\subset B(z_{j},2\varepsilon_{1})$, one has using in addition (95):

[TABLE]

Therefore, using an integration by parts, it holds:

[TABLE]

since $\widetilde{\phi}_{j}=0$ on $\partial B(z_{j},2\varepsilon_{1})$. From (106), it holds for $h$ small enough,

[TABLE]

where $c>0$ is independent of $h$. Since moreover $f\geq f(z_{j})-2\varepsilon$ on $\mathsf{B}_{j,k}\cap\mathop{\rm supp}\nolimits(1-\chi_{k}^{\varepsilon,\varepsilon_{1}})$ by (93) and (94), there exist $c^{\prime}>0$ and $\varepsilon_{0}>0$ such that for $\varepsilon\in(0,\varepsilon_{0})$, in the limit $h\to 0$:

[TABLE]

Lastly, using Lemma 43 and the trace theorem, one obtains in the limit $h\to 0$:

[TABLE]

where $\lambda_{-}(z_{j})$ denotes the negative eigenvalue of ${\rm Hess\,}f(z_{j})$. The second equality follows from Laplace’s method and from the fact that

[TABLE]

and $\widetilde{a}_{j}(z_{j})\cdot n=\pm 1$, see indeed the lines between (109) and (110). The last line follows from (113). The asymptotic estimate of the term $\langle\nabla\widetilde{u}_{k},\widetilde{\psi}_{j}\rangle_{L^{2}_{w}}$is a consequence of the latter estimate together with (142)–(145) which gives in the limit $h\to 0$:

[TABLE]

where the remainder terms $O(h)$ admits a full asymptotic expansion in $h$ (which follows from Laplace’s method). This proves Proposition 50 for all k\in\big{\{}\mathsf{1},\ldots,\mathsf{m}_{0}^{\Omega}\big{\}} and j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} such that $z_{j}\in\mathbf{j}(x_{k})\cap\Omega$.

Step 3.

Let us now deal with the computation of the terms $\langle\nabla\widetilde{u}_{k},\widetilde{\psi}_{j}\rangle_{L^{2}_{w}}$ for k\in\big{\{}\mathsf{1},\ldots,\mathsf{m}_{0}^{\Omega}\big{\}} and j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} when $z_{j}\in\mathbf{j}(x_{k})\cap\partial\Omega$. Notice that according to Definition 45 and (92) and by definition of lexicographic labeling introduced in Definition 44, this situation can only occur when

[TABLE]

One has

[TABLE]

Notice that (143) also holds here for $\int_{\Omega}(\chi_{k}^{\varepsilon,\varepsilon_{1}})^{2}e^{-\frac{2}{h}f}$. Since $\mathop{\rm supp}\nolimits\widetilde{\phi}_{j}\subset B(z_{j},2\varepsilon_{1})\cap\overline{\Omega}$, the numerator of the right-hand side of (146) can be rewritten as

[TABLE]

From (120), there exists $c>0$ such that for $h$ small enough, $d_{f,h}^{*}\widetilde{\phi}_{j}=O(e^{-\frac{c}{h}})$ in $L^{2}(\Omega)$. Since $f\geq f(z_{j})-2\varepsilon$ on $B(z_{j},2\varepsilon_{1})\cap\mathop{\rm supp}\nolimits(1-\chi_{k}^{\varepsilon,\varepsilon_{1}})$ by (91) and (93), there exist $c^{\prime}>0$ and $\varepsilon_{0}>0$ such that for $\varepsilon\in(0,\varepsilon_{0})$, in the limit $h\to 0$:

[TABLE]

Furthermore, applying Proposition 49 with $\Sigma=\partial\Omega$ and $F=1_{\partial B(z_{j},2\varepsilon_{1})\cap\partial\Omega}$, one has in the limit $h\to 0$:

[TABLE]

Therefore, injecting the estimates (143), (148) and (149) into (146), one obtains in the limit $h\to 0$:

[TABLE]

where the remainder terms $O(h)$ admits a full asymptotic expansion in $h$ (which follows from Laplace’s method). This is the desired estimate according to (141)–(132). This ends the proof of Proposition 50.

Restricted differential $\nabla:\,{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$

This section is devoted to the proof of Theorem 3. In this section, one also gives lower and upper bounds on the $\mathsf{m}_{0}^{\Omega}$ first eigenvalues of $-L^{D,(0)}_{f,h}$, see Theorem 5 below. According to Lemma 27 and Proposition 4, the square of the $\mathsf{m}_{0}^{\Omega}$ singular values of the restricted differential $\nabla:\,{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$ (${\rm Ran}\,\pi^{(0)}_{h}$ and ${\rm Ran}\,\pi^{(1)}_{h}$ being endowed with the $L^{2}_{w}$ inner product) are the first $\mathsf{m}_{0}^{\Omega}$ first eigenvalues of $-L^{D,(0)}_{f,h}$ times $\frac{2}{h}$. Therefore, the strategy consists in estimating in the limit $h\to 0$ the $\mathsf{m}_{0}^{\Omega}$ singular values of the restricted differential $\nabla:\,{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$.

This section is organized as follows. Section 4.2.1 is dedicated to the definition of the matrix of the restricted differential $\nabla:\,{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$ and preliminary asymptotic estimates on its coefficients. In Section 4.2.2, lower and upper bounds for the $\mathsf{m}_{0}^{\Omega}$ smallest eigenvalues of $-L^{D,(0)}_{f,h}$ are obtained. Finally, one proves Theorem 3 in Section 4.2.3.

Matrix of the restricted differential $\nabla:\,{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$

Let us introduce the matrix of the restricted differential $\nabla:\,{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$ in a basis of projected quasi-modes.

Definition 51**.**

Let us assume that the assumption (A0) is satisfied. Let $(\widetilde{u}_{k})_{k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}}$ be the family of quasi-modes for $-L^{D,(0)}_{f,h}$ and let $(\widetilde{\psi}_{j})_{j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$ be the family of quasi-modes for $-L^{D,(1)}_{f,h}$, both introduced in Definition 45. Let us denote by $S=(S_{j,k})_{j,k}$ the $\mathsf{m}_{1}^{\overline{\Omega}}\times{\mathsf{m}_{0}^{\Omega}}$ matrix defined by: for all $k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}$ and for all $j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$

[TABLE]

Notice that from (86), it holds for all $k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}$ and for all $j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}$:

[TABLE]

Then, using the identity

[TABLE]

together with item 2 in Proposition 46 and Proposition 50, one gets the following estimates of the coefficients of $S$:

Proposition 52**.**

Let us assume that the assumption (A0) is satisfied. Let $S$ be the matrix introduced in Definition 51. Let $k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}$ and $j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$. Then, there exists $\varepsilon_{0}>0$ such that for $\varepsilon\in(0,\varepsilon_{0})$ (where $\varepsilon$ is introduced in (92)), there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

and

[TABLE]

where we recall that, from Proposition 50, $\varepsilon_{j,k}\in\{-1,1\}$,

[TABLE]

and $C_{j,k}$ is defined in (141). Moreover, when $z_{j}\in\mathbf{j}(x_{k})\cap\partial\Omega$ (and thus $k\in\{1,\ldots,\mathsf{N}_{1}\}$), one has $\varepsilon_{j,k}=-1$.

To study the singular values of the matrix $S$, one defines the following matrices:

• •

let \widetilde{S}=\big{(}\widetilde{S}_{j,k}\big{)}_{j,k} be the real value $\mathsf{m}_{1}^{\overline{\Omega}}\times{\mathsf{m}_{0}^{\Omega}}$ matrix defined by

[TABLE]

• •

let $D$ be the diagonal ${\mathsf{m}_{0}^{\Omega}}\times{\mathsf{m}_{0}^{\Omega}}$ matrix defined by

[TABLE]

and where $p_{j,k}$ is defined in (151).

Notice that when the assumptions (A1j) and (A2j) are satisfied, one has in the limit $h\to 0$ (see (78) together with the fact $q_{1}=-\frac{3}{4}$ since $\mathbf{j}(x_{1})\cap\partial\Omega=\partial\mathsf{C}_{1}\cap\partial\Omega\neq\emptyset$ which follows from (67) and (A2j)):

[TABLE]

and for some $c>0$ independent of $h$, it holds:

[TABLE]

• •

let $\widetilde{C}$ be the real value $\mathsf{m}_{1}^{\overline{\Omega}}\times\mathsf{m}_{0}^{\Omega}$ matrix defined by

[TABLE]

The matrix $\widetilde{C}$ is the $\mathsf{m}_{1}^{\overline{\Omega}}\times{\mathsf{m}_{0}^{\Omega}}$ matrix whose coefficients are given by:

[TABLE]

for all (j,k)\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}\times\big{\{}1,\dots,{\mathsf{m}_{0}^{\Omega}}\big{\}}. It satisfies, according to Proposition 52 and (153), in the limit $h\to 0$:

[TABLE]

where $p_{j,k}$ is defined by (151), $\varepsilon_{j,k}\in\{-1,1\}$ and $C_{j,k}$ is defined in (141). From (157), one has when $h\to 0$, $\widetilde{C}=O(1)$ which means that there exist $K>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$:

[TABLE]

Under (A0), by definition of the matrices $S$, $\widetilde{S}$, $D$ and $\widetilde{C}$ (see Definition 51 and Equations (152), (153), (156)), there exists $c>0$ such that the matrix \big{(}S-\widetilde{S}\big{)}D^{-1} satisfies in the limit $h\to 0$:

[TABLE]

The following Lemma will be needed in the sequel.

Lemma 53**.**

Let us assume that the assumption (A0) holds. Let $\widetilde{C}$ be the matrix defined in (156). Then, there exist $c>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$:

[TABLE]

where $\|.\|_{2}$ denotes the usual Euclidean norm on $\mathbb{R}^{K}$, $K\in\mathbb{N}^{*}$.

Proof.

The proof of Lemma 53 is divided into two steps.

Step 1. Block-diagonal decomposition of $\widetilde{C}$.

According to (156), (153) and (152), $\widetilde{C}$ has the form, up to reordering the $z_{i}$ for $i\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$:

[TABLE]

where:

• •

the block matrix $(0,0)$ on the first line corresponds to the rows of $\widetilde{C}$ associated with $j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}$ such that $z_{j}\notin\cup_{k=1}^{\mathsf{m}_{0}^{\Omega}}\mathbf{j}(x_{k})$.

• •

$[\widetilde{C}]_{a}$ is a matrix of size {\rm Card}\big{(}\cup_{k=1}^{\mathsf{N}_{1}}\mathbf{j}(x_{k})\big{)}\times\mathsf{N}_{1} (where $\mathsf{N}_{1}$ is defined in (41)). The coefficients $([\widetilde{C}]_{a})_{j,k}=\widetilde{C}_{j,k}$ are associated with 0-forms $\widetilde{u}_{k}$, for $k\in\{1,\dots,\mathsf{N}_{1}\}$ (see Definition 45 and (92)) and with 1-forms $\widetilde{\psi}_{j}$ for $j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}$ such that $z_{j}\in\cup_{k=1}^{\mathsf{N}_{1}}\mathbf{j}(x_{k})$.

• •

$[\widetilde{C}]_{c}$ is a matrix of size {\rm Card}\big{(}\cup_{k=\mathsf{N}_{1}+1}^{\mathsf{m}_{0}^{\Omega}}\mathbf{j}(x_{k})\big{)}\times(\mathsf{m}_{0}^{\Omega}-\mathsf{N}_{1}), which has the following block diagonal form:

[TABLE]

where for \ell\in\big{\{}1,\ldots,\mathsf{N}_{1}\big{\}}, $[\widetilde{C}]_{c,\ell}$ is a matrix of size

[TABLE]

with the convention that $[\widetilde{C}]_{c,\ell}$ does not exist if $\operatorname*{arg\,min}_{\overline{\mathsf{C}_{\ell}}}f=\{x_{\ell}\}$. Let us recall that for \ell\in\big{\{}1,\ldots,\mathsf{N}_{1}\big{\}}, $\mathsf{C}_{\ell}$ is introduced in Definition 16. For all $\ell\in\{1,\ldots,\mathsf{N}_{1}\}$, $[\widetilde{C}]_{c,\ell}$ contains the non zero terms of $\widetilde{C}$ associated with 0-forms $\widetilde{u}_{k}$ and 1-forms $\widetilde{\psi}_{j}$ with:

$\widetilde{u}_{k}$ such that $\mathop{\rm supp}\nolimits(\widetilde{u}_{k})\subset\{\widetilde{\chi}_{\ell}=1\}$ (according to (97)), 2. 2.

for those $\widetilde{u}_{k}$, $j$ is such that $z_{j}\in\mathbf{j}(x_{k})\subset\mathsf{C}_{\ell}$.

This explains in particular the block structure of $[\widetilde{C}]_{c}$ since by construction $\widetilde{S}_{j,k}=0$ if $z_{j}\not\in\mathbf{j}(x_{k})$ (see (156) and (152)).

From [20, Section 7.3 and Equation (7.4) in Section 7.2], for all \ell\in\big{\{}1,\ldots,\mathsf{N}_{1}\big{\}} there exist $c_{\ell}>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$ and for all z\in\mathbb{R}^{{\rm Card}\big{(}\operatorname*{arg\,min}_{\overline{\mathsf{C}_{\ell}}}f\big{)}~{}-1},

[TABLE]

Thus, there exist $\alpha>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$ and for all $z\in\mathbb{R}^{\mathsf{m}_{0}^{\Omega}-\mathsf{N}_{1}}$,

[TABLE]

For any $x=\,(y,z)^{T}\in\mathbb{R}^{\mathsf{m}_{0}^{\Omega}}$ ($y\in\mathbb{R}^{\mathsf{N}_{1}}$ and $z\in\mathbb{R}^{\mathsf{m}_{0}^{\Omega}-\mathsf{N}_{1}}$), it holds

[TABLE]

Therefore, to prove (160), let show that there exist $\beta>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$ and for all $y\in\mathbb{R}^{\mathsf{N}_{1}}$,

[TABLE]

Step 2. Proof of (162).

Let us divide the family $(\mathsf{C}_{k})_{k\in\{1,\ldots,\mathsf{N}_{1}\}}$ into $\mathsf{K}$ groups ($\mathsf{K}\leq\mathsf{N}_{1}$):

[TABLE]

which are such that for all $\ell\in\{1,\ldots,\mathsf{K}\}$,

[TABLE]

and for all $q\in\{1,\ldots,\mathsf{N}_{1}\}$ such that $\mathsf{C}_{q}\notin\{\mathsf{C}_{1}^{\ell},\ldots,\mathsf{C}^{\ell}_{\mathsf{k}_{\ell}}\}$,

[TABLE]

Then, by definition of the matrix $[\widetilde{C}]_{a}$ (see Step 1 above), up to a reordering, $[\widetilde{C}]_{a}$ has the block-diagonal form

[TABLE]

where for \ell\in\big{\{}1,\ldots,\mathsf{K}\big{\}}, $[\widetilde{C}]_{a,\ell}$ is a matrix of size {\rm Card}\Big{(}\bigcup_{k=1,\,x_{k}\in\cup_{j=1}^{k_{\ell}}\mathsf{C}_{j}^{\ell}}^{\mathsf{N}_{1}}\mathbf{j}(x_{k})\Big{)}\times k_{\ell}. For \ell\in\big{\{}1,\ldots,\mathsf{K}\big{\}}, the coefficients $([\widetilde{C}]_{a,\ell})_{j,k}=\widetilde{C}_{j,k}$ are associated with 0-forms $\widetilde{u}_{k}$, for $k\in\{1,\dots,\mathsf{N}_{1}\}$ such that $x_{k}\in\cup_{j=1}^{k_{\ell}}\mathsf{C}_{j}^{\ell}$ and with 1-forms $\widetilde{\psi}_{j}$ for $j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}$ such that $z_{j}\in\bigcup_{k=1,\,x_{k}\in\cup_{j=1}^{k_{\ell}}\mathsf{C}_{j}^{\ell}}^{\mathsf{N}_{1}}\mathbf{j}(x_{k})$.

Therefore, to prove (162), let us show that for \ell\in\big{\{}1,\ldots,\mathsf{K}\big{\}}, there exist $\beta_{\ell}>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$ and for all $y\in\mathbb{R}^{k_{\ell}}$,

[TABLE]

In view of the block structure of $[\widetilde{C}]_{a}$, to prove it, one can assume, without loss of generality, that $\mathsf{K}=1$ which is equivalent to the fact that the set $\bigcup_{j=1}^{\mathsf{N}_{1}}\overline{\mathsf{C}_{j}}$ is connected. Let us thus assume that $\bigcup_{j=1}^{\mathsf{N}_{1}}\overline{\mathsf{C}_{j}}$ is connected and let us then write

[TABLE]

where $A$ is a matrix which has the same size as $[\widetilde{C}]_{a}$, and which satisfies, from (157), for all $k\in\{1,\ldots,\mathsf{N}_{1}\}$ and all $j$ such that $z_{j}\in\bigcup_{k=1}^{\mathsf{N}_{1}}\mathbf{j}(x_{k})$,

[TABLE]

and

[TABLE]

where $\varepsilon_{j,k}\in\{-1,1\}$ and $\,C_{j,k}>0$ for all $k\in\{1,\ldots,\mathsf{N}_{1}\}$ and $j$ such that $z_{j}\in\mathbf{j}(x_{k})$. To prove (162), it sufficient to show that (162) holds for $A$ instead of $[\widetilde{C}]_{a}$, i.e. that there exist $\beta>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$ and for all $y\in\mathbb{R}^{\mathsf{N}_{1}}$,

[TABLE]

Before proving (166), let us label the family $(\mathsf{C}_{k})_{k\in\{1,\ldots,\mathsf{N}_{1}\}}$ as follows. According to Lemma 21, one can assume without loss of generality that $\mathsf{C}_{1}$ is such that there exists $z\in\mathsf{U}_{1}^{\mathsf{ssp}}$ such that

[TABLE]

Let us now label $(\mathsf{C}_{j})_{j\in\{2,\ldots,\mathsf{N}_{1}\}}$ such that for all $k\in\{1,\ldots,\mathsf{N}_{1}\}$, $\bigcup_{j=1}^{k}\overline{\mathsf{C}_{j}}$ is connected. Let us prove (166) by induction on $k\in\{1,\ldots,\mathsf{N}_{1}\}$ (the proof is similar to the proof made in [20, Section 7.3] in a different context). For $k\in\{1,\ldots,\mathsf{N}_{1}\}$, one denotes by $\mathcal{P}_{k}$ the following property: there exists $\alpha_{k}>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$ and for all $y\in\mathbb{R}^{\mathsf{N}_{1}}$,

[TABLE]

Let us prove $\mathcal{P}_{1}$. Using (167) together with (164) and (165), the $j$-th row of $A$ equals

[TABLE]

where $j$ is such that z_{j}\in\partial\mathsf{C}_{1}\setminus\Big{(}\cup_{\ell=2}^{\mathsf{N}_{1}}\partial{\mathsf{C}_{\ell}}\Big{)}. Thus, one has for $y\in\mathbb{R}^{\mathsf{N}_{1}}$, \big{\|}A\,y\big{\|}_{2}\geq|\varepsilon_{j,1}\,C_{j,1}y_{1}|. Therefore, $\mathcal{P}_{1}$ is satisfied. Let us now assume that $\mathcal{P}_{k}$ is satisfied for some $k\in\{1,\ldots,\mathsf{N}_{1}-1\}$ and let us prove $\mathcal{P}_{k+1}$. If $\partial\mathsf{C}_{k+1}\cap\partial\Omega\neq\emptyset$, there exists $j\in \{1,\ldots,\mathsf{N}_{1}\}$, such that $z_{j}\in\partial\mathsf{C}_{k+1}\cap\partial\Omega$. Thus, using (164), one has

[TABLE]

Therefore, one obtains y_{k+1}^{2}C_{j,k+1}^{2}\leq\big{|}(Ay)_{j}\big{|}^{2}\leq\big{\|}Ay\big{\|}^{2}_{2}. Applying the property $\mathcal{P}_{k}$, one gets

[TABLE]

This implies that the property $\mathcal{P}_{k+1}$ is satisfied. Let us now consider the case $\partial\mathsf{C}_{k+1}\cap\partial\Omega=\emptyset$. Using (165) together with the fact that the set $\bigcup_{j=1}^{k+1}\overline{\mathsf{C}_{j}}$ is connected, there exist $\ell\in\{1,\ldots,k\}$ and $j$ such that $z_{j}\in\partial\mathsf{C}_{\ell}\cap\partial\mathsf{C}_{k+1}$. Thus,

[TABLE]

Thus, since the exists $M>0$ such that $|A_{j\ell}|\leq M$ for all $h$ small enough, one obtains using the triangular inequality and the property $\mathcal{P}_{k}$,

[TABLE]

Using the property $\mathcal{P}_{k}$, one gets that \min\Big{(}\frac{C_{j,k+1}^{2}}{2(1+M^{2})},\alpha_{k}\Big{)}\sum_{\ell=1}^{k+1}y_{\ell}^{2}\leq 2\big{\|}Ay\big{\|}^{2}_{2}. Therefore, the property $\mathcal{P}_{k+1}$ is satisfied. This ends the proof of (166) by induction. Together with (161) and (162), one then obtains (160). This concludes the proof of Lemma 53.

As a consequence of (160), the rectangular matrix $\widetilde{C}$ admits a left inverse $\widetilde{C}^{-1}$ which satisfies

[TABLE]

for some $M>0$ independent of $h$. This implies that, using (159) and (156):

[TABLE]

On the $\mathsf{m}_{0}^{\Omega}$ smallest eigenvalues of $-L^{D,(0)}_{f,h}$

This section is dedicated to the proof of the following proposition which aims at giving lower bound and an upper bounds for the $\mathsf{m}_{0}^{\Omega}$ smallest eigenvalues of $-L^{D,(0)}_{f,h}$.

Theorem 5**.**

Let us assume that the assumption (A0) holds. Let $\mathbf{j}$ be the map constructed in Section 2.4. Let us reorder the set $\{x_{1},\ldots,x_{\mathsf{m}_{0}^{\Omega}}\}$ such that the sequence

[TABLE]

is decreasing, and, on any $\mathcal{I}\subset\{1,\dots,{\mathsf{m}_{0}^{\Omega}}\}$ such that \big{(}\mathsf{S}_{k}\big{)}_{k\in\mathcal{I}} is constant, the sequence $(q_{k})_{k\in\mathcal{I}}$ is decreasing (where the $q_{k}$’s are introduced in (153)). Finally let us denote by $\lambda_{k,h}$, for $k\in\mathbb{N}^{*}$, the $k$-th eigenvalue of $-L_{f,h}^{D,(0)}$ counted with multiplicity (with these notations, $\lambda_{1,h}=\lambda_{h}$, see (8)). Then, there exist $C>0$ and $h_{0}>0$ such that for all k\in\big{\{}1,\dots,{\mathsf{m}_{0}^{\Omega}}\big{\}} and for all $h\in(0,h_{0})$,

[TABLE]

The reordering of $\{x_{1},\ldots,x_{\mathsf{m}_{0}^{\Omega}}\}$ introduced in (170) is only used in Theorem 5. One recalls that, except in Theorem 5, the labelling of $\{x_{1},\ldots,x_{\mathsf{m}_{0}^{\Omega}}\}$ is the one introduced in Definition 44.

A direct consequence of Theorem 5 is the following.

Corollary 54**.**

Let us assume that the assumptions (A0) and (A1j) are satisfied. Then, the estimate (130) is satisfied.

Before starting the proof of Theorem 5, let us recall the Fan inequalities, which is the purpose of Lemma 55 below (see for instance [44, Theorem 1.6] or [29]) and its consequences, see Lemma 56 below.

Lemma 55**.**

Let $A\in M_{m,m}(\mathbb{C})$, $B\in M_{m,n}(\mathbb{C})$ and $C\in M_{n,n}(\mathbb{C})$. Then, it holds

[TABLE]

where, for any matrix $T\in M_{m,n}(\mathbb{C})$, $\|T\|=\eta_{1}(T)\geq\cdots\geq\eta_{n}(T)$ denote the singular values of the matrix $T$ and where \big{\|}T\big{\|}:=\sqrt{\max\sigma(\,^{t}TT)} is the spectral norm of $T$.

Let us recall that from item 3 in Proposition 46, there exists $h_{0}>0$ such that for all $h\in(0,h_{0})$,

[TABLE]

and

[TABLE]

where the projectors $\pi^{(0)}_{h}$ and $\pi^{(1)}_{h}$ are defined in (87). Let us define \widetilde{\Upsilon}:=\big{(}\pi^{(0)}_{h}\widetilde{u}_{k}\big{)}_{1\leq k\leq{\mathsf{m}_{0}^{\Omega}}} and \widetilde{\Psi}:=\big{(}\pi^{(1)}_{h}\widetilde{\psi}_{j}\big{)}_{1\leq j\leq\mathsf{m}_{1}^{\overline{\Omega}}}. For $i\in\{0,1\}$, let $\mathcal{B}_{i}$ be an orthonormal basis of ${\rm Ran}\,\pi^{(i)}_{h}$ and let us define the matrices

[TABLE]

Notice that from item 3 in Proposition 46, there exist $K>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$:

[TABLE]

and

[TABLE]

A consequence of the Fan inequalities is the following.

Lemma 56**.**

Let us assume that the assumption (A0) holds. Let us denote by $\lambda_{k,h}$, for $k\in\mathbb{N}^{*}$, the $k$-th eigenvalue of $-L_{f,h}^{D,(0)}$ counted with multiplicity and let $\tilde{S}$ be the matrix defined in (152). Then, there exists $c>0$ such that for all $k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}$, one has in the limit $h\to 0$:

[TABLE]

where $\|\widetilde{S}C_{0}\|=\eta_{1}(\widetilde{S}C_{0})\geq\cdots\geq\eta_{\mathsf{m}_{0}^{\Omega}}(\widetilde{S}C_{0})$ denote the singular values of $\widetilde{S}C_{0}$.

Proof.

The ${\mathsf{m}_{0}^{\Omega}}$ smallest eigenvalues of $-L^{D,(0)}_{f,h}$ are the eigenvalues of

[TABLE]

Moreover, since the $L^{2}_{w}$-adjoint of $d|_{{\rm Ran}\,\pi^{(0)}_{h}}:{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$ is $d^{*}_{\frac{2f}{h},1}|_{{\rm Ran}\,\pi^{(1)}_{h}}:{\rm Ran}\,\pi^{(1)}_{h}\to{\rm Ran}\,\pi^{(0)}_{h}$, one has that the ${\mathsf{m}_{0}^{\Omega}}$ smallest eigenvalues of $-L^{D,(0)}_{f,h}$ are given by $\frac{h}{2}$ times the squares of the singular values of $d|_{{\rm Ran}\,\pi^{(0)}_{h}}:{\rm Ran}\,\pi^{(0)}_{h}\to{\rm Ran}\,\pi^{(1)}_{h}$. Thus, the eigenvalues of $-L^{D,(0)}_{f,h}|_{{\rm Ran}\,\pi^{(0)}_{h}}$ are given by $\frac{h}{2}$ times the squares of the singular values of the matrix $Q$ defined by

[TABLE]

In addition, by definition of the matrices $C_{0}$ and $C_{1}$ (see (171)), one has

[TABLE]

By (169), it holds

[TABLE]

Furthermore, from (169), there exists $c>0$ such that in the limit $h\to 0$

[TABLE]

and from item 3b in Proposition 46,

[TABLE]

where we recall that \big{\|}T\big{\|}:=\sqrt{\max\sigma(\,^{t}TT)} is the spectral norm of a matrix $T$. Therefore, it follows from the Fan inequalities (see Lemma 55) that the singular values of $Q$ are, up to multiplication by $1+O(e^{-\frac{c}{h}})$, the singular values of $\widetilde{S}\,C_{0}$. This concludes the proof of Lemma 56.

Remark 57**.**

Notice that in general, the spectral norm of the matrix $C_{0}$ defined in (171) does not equal $1+O(e^{-\frac{c}{h}})$ when $h\to 0$. For instance, in the case when $f$ is a one-dimensional symmetric double-well potential with the saddle point lower than $\min_{\partial\Omega}f$, it can be checked that the Gramian matrix of the functions $\widetilde{u}_{1}$ and $\widetilde{u}_{2}$ introduced in Definition 45 converges when $h\to 0$ towards the matrix

[TABLE]

where $0<c_{1}<1$.

Let us now prove Theorem 5.

Proof.

Theorem 5 is equivalent, according Lemma 56, to the existence of $C>0$ and $h_{0}>0$ such that for all k\in\big{\{}1,\dots,{\mathsf{m}_{0}^{\Omega}}\big{\}} and for all $h\in(0,h_{0})$

[TABLE]

Let us prove (175). According to (153) and to the ordering of $k\in\{1,\dots,{\mathsf{m}_{0}^{\Omega}}\}$ introduced in the statement of Theorem 5, the singular values of $D$ satisfy for $h$ small enough (see (153) and (170)):

[TABLE]

Using the fact that $\widetilde{S}C_{0}=\widetilde{C}DC_{0}$ (see (156)) together with (158), (172) and Lemma 55, one obtains that for all $k\in\{1,\dots,{\mathsf{m}_{0}^{\Omega}}\}$

[TABLE]

which provide the required upper bound in (175). To obtain a lower bound on the singular values of $\widetilde{S}C_{0}$, we write

[TABLE]

Using (168), (173) and Lemma 55, one has for all $k\in\{1,\dots,{\mathsf{m}_{0}^{\Omega}}\}$

[TABLE]

Then, (175) follows from (176), (178) and (177). This concludes the proof of Theorem 5.

To prove Theorem 3 and to ease upcoming computations, we replace in the Fan inequalities (174) the matrix $C_{0}$ by another matrix which has a simpler form than $C_{0}$: this is the purpose of Lemma 58. Before stating Lemma 58, let us choose a particular orthonormal basis $\mathcal{B}_{0}$ of ${\rm Ran}\,\pi_{h}^{(0)}$ to define $C_{0}$ in (171). Recall that the when the assumption (A1j) is satisfied the well $\mathsf{C}_{1}$ (see Definition 16) satisfies: for all $x\in\mathsf{U}_{0}^{\Omega}\setminus\{x_{1}\}$,

[TABLE]

Let us define

[TABLE]

According to item 2a in Proposition 46, there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

Then, let us choose $\{e_{2},\dots,e_{{\mathsf{m}_{0}^{\Omega}}}\}$ such that

[TABLE]

is an orthonormal basis of ${\rm Ran}\,\pi^{(0)}_{h}$. In that case, the matrix $C_{0}$ defined in (171), satisfies in the limit $h\to 0$:

[TABLE]

Let us now define the $\mathsf{m}_{0}^{\Omega}\times\mathsf{m}_{0}^{\Omega}$ matrix $\widetilde{C}_{0}$ by:

[TABLE]

and

[TABLE]

Lemma 58**.**

Let us assume that the assumptions (A0) and (A1j) are satisfied. Let us denote by $\lambda_{k,h}$, for $k\in\mathbb{N}^{*}$, the $k$-th eigenvalue of $-L_{f,h}^{D,(0)}$ counted with multiplicity and let $\tilde{S}$ be the matrix defined in (152). Then, there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

where $\|\widetilde{S}\widetilde{C}_{0}\|=\eta_{1}(\widetilde{S}C_{0})\geq\cdots\geq\eta_{\mathsf{m}_{0}^{\Omega}}(\widetilde{S}\widetilde{C}_{0})$ denote the singular values of $\widetilde{S}\,\widetilde{C}_{0}$ and $\lambda_{1,h}=\lambda_{h}$ is the principal eigenvalue of $-L^{D,(0)}_{f,h}$ (see (8)).

Proof.

Let us prove that there exists $c>0$ such that in the limit $h\to 0$,

[TABLE]

From (179), the $\mathsf{m}_{0}^{\Omega}\times\mathsf{m}_{0}^{\Omega}$ matrix $C_{0}$ has the form

[TABLE]

for some $c>0$. Moreover, according to (180) and (181), the $\mathsf{m}_{0}^{\Omega}\times\mathsf{m}_{0}^{\Omega}$ matrix $\widetilde{C}_{0}$ has the form

[TABLE]

Let us recall that by definition of $C_{0}$ (see (171)) and from item 3a in Proposition 46, $C_{0}$ is invertible and thus $[C_{0}]_{2}$ is invertible. Therefore, one has

[TABLE]

and thus,

[TABLE]

This proves (182). Lemma 58 is then a consequence of (182) together with Lemma 56 and Lemma 55.

Proof of Theorem 3

This section is dedicated to the proof of Theorem 3 which gives the asymptotic estimate of the principal eigenvalue of $-L^{D,(0)}_{f,h}$ under the assumptions (A1j) and (A2j).

Proof of Theorem 3.

Let us assume that the assumptions (A0) and (A1j) hold. The spectral gap (130) has already been proved, see Corollary 54. It thus remains to prove (131). The proof of (131) is partly inspired by the analysis led in [20, Section 7.4]. According to Lemma 58, there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

where $\widetilde{C}_{0}$ is defined in (180) and (181). Therefore, the analysis of the estimate of $\lambda_{h}$ is then reduced to precisely computing $\eta_{{\mathsf{m}_{0}^{\Omega}}}(\widetilde{S}\widetilde{C}_{0})$. One has:

[TABLE]

Let us assume in addition to (A0) and (A1j) that (A2j) holds. Recall that (A1j) and (A2j) consists in assuming that for all $x\in\mathsf{U}_{0}^{\Omega}\setminus\{x_{1}\}$,

[TABLE]

and

[TABLE]

Then, it holds

[TABLE]

Thus, using in addition Proposition 52 and (152), one has in the limit $h\to 0$:

[TABLE]

for some $c>0$ independent of $h$ and where

[TABLE]

where the constants $C_{j,1}$’s are defined in (141) and where all the remainder terms $O(h)$ admits a full expansion in $h$.

Let us first obtain an upper bound on $\eta_{{\mathsf{m}_{0}^{\Omega}}}(\widetilde{S}\widetilde{C}_{0})$. Let us denote by $y_{0}$ the vector $\,{}^{t}(1,0,\dots,0)$. Then, it holds from (184),

[TABLE]

Using in addition the fact from (180), one has $\widetilde{C}_{0}y_{0}=y_{0}$, one obtains

[TABLE]

This provides the required upper bound. Notice that (187), (185), and (186) imply that in the limit $h\to 0$

[TABLE]

Let us now give a lower bound on $\eta_{{\mathsf{m}_{0}^{\Omega}}}(\widetilde{S}\widetilde{C}_{0})$. To this end let us consider $y^{*}\in\mathbb{R}^{\mathsf{m}_{0}^{\Omega}}$ with $\|y^{*}\|_{2}=1$, realizing the minimum in (184). Let us write $y^{*}=\,^{t}(y^{*}_{\alpha},y^{*}_{\beta})$, where $y^{*}_{\alpha}\in\mathbb{R}$ and $y^{*}_{\beta}$ is a row vector of size ${\mathsf{m}_{0}^{\Omega}}-1$. We claim that there exists $\mu>0$ such that for $h$ small enough,

[TABLE]

Let us prove (189). By definition of $y^{*}$ and according to (156), one has

[TABLE]

To prove (189), we use the block structure of the matrices $\widetilde{C}$, $\widetilde{C}_{0}$ and $D$. Let us recall that from (79) and (80), since (A2j) holds,

[TABLE]

Then, according to (156), (153) and (152), the $\mathsf{m}_{1}^{\overline{\Omega}}\times\mathsf{m}_{0}^{\Omega}$ matrix $\widetilde{C}$ has the form, up to reordering the $z_{i}$, $i\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$,

[TABLE]

where:

• •

$[\widetilde{C}]_{1}$ is a matrix of size $\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\times 1$ where we recall that $\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}$ is defined in (80). The coefficients $([\widetilde{C}]_{1})_{j,1}=\widetilde{C}_{j,1}$ are associated with the function $\widetilde{u}_{1}$ (see Definition 45 and (92)) and with 1-forms $\widetilde{\psi}_{j}$ for $j\in\{1,\ldots,\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\}$ (or equivalently, $j$ such that $z_{j}\in\mathbf{j}(x_{1})\cap\partial\Omega$).

• •

$[\widetilde{C}]_{3}$ is a matrix of size \big{(}\mathsf{m}_{1}^{\overline{\Omega}}-\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\big{)}\times 1 . The coefficients $([\widetilde{C}]_{3})_{j,k}=\widetilde{C}_{j,k}$ are associated with the function $\widetilde{u}_{1}$ and with 1-forms $\widetilde{\psi}_{j}$ for $j\in\{\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}+1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}$ (or equivalently, $j$ such that $z_{j}\notin\mathbf{j}(x_{1})\cap\partial\Omega$).

• •

$[\widetilde{C}]_{2}$ is a matrix of size \big{(}\mathsf{m}_{1}^{\overline{\Omega}}-\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\big{)}\times\big{(}\mathsf{m}_{0}^{\Omega}-1\big{)}. The coefficients $([\widetilde{C}]_{2})_{j,k}=\widetilde{C}_{j,k}$ are associated with 0-forms $\widetilde{u}_{k}$, for $k\in\{2,\ldots,\mathsf{m}_{0}^{\Omega}\}$ and with 1-forms $\widetilde{\psi}_{j}$ for $j\in\{\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}+1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}$ (or equivalently, $j$ such that $z_{j}\notin\mathbf{j}(x_{1})\cap\partial\Omega$).

From (160) and (190), $[\widetilde{C}]_{2}$ is injective and satisfies, for some constant $c>0$ and for all $h>0$ small enough,

[TABLE]

This is indeed obvious by applying (160) to the vector $\,{}^{t}(0,x)$. Let us now decompose the square matrices $D$ and $\widetilde{C}_{0}$ in blocks which are compatible with the decomposition of $\widetilde{C}$ made in (190). According to (153), (180), and (181), one has

[TABLE]

where for a square matrix $U$ of size $\mathsf{m}_{0}^{\Omega}$:

[TABLE]

Notice that from (154), it holds

[TABLE]

and from (173), there exists $M>0$ such that when $h>0$

[TABLE]

We are now in position to prove (189). Le us recall that by definition of $y^{*}$ , one has

[TABLE]

Therefore, since $\widetilde{C}_{0}\,^{t}(y^{*}_{\alpha},0)=\,^{t}(y^{*}_{\alpha},0)$ (see (180)) and $\widetilde{S}=\widetilde{C}D$ (see (156)), one has using (188), (190), and (193) together with the fact that $|y^{*}_{\alpha}|\leq 1$ and $[\widetilde{C}]_{1}=O(1)$ (see (190) and (158)),

[TABLE]

Moreover, using (193) and since $[\widetilde{C}]_{3}=O(1)$ (see (190) and (158)) and $[\widetilde{C}_{0}]_{\gamma}=O(1)$ (since $\widetilde{C}_{0}=C_{0}+O(e^{-\frac{c}{h}})$ and $C_{0}=O(1)$ see (180), (181), and (172)), one has

[TABLE]

Therefore, one deduces from the latter inequality and from (195) and (191) that

[TABLE]

In addition, since $[\widetilde{C}_{0}]_{\beta}^{-1}=O(1)$ (which follows from $\widetilde{C}_{0}=C_{0}+O(e^{-\frac{c}{h}})$, see indeed (180), (181), (173), and (192)) and since there exists $c>0$ such that it holds:

[TABLE]

which follows from (153) and (155), one obtains from (196) that there exists $\mu>0$ such that for $h$ small enough,

[TABLE]

This ends the proof of (189). We are now in position to give a lower bound on $\eta_{{\mathsf{m}_{0}^{\Omega}}}(\widetilde{S}\widetilde{C}_{0})$. Notice that from (189) together with the fact that $\|y^{*}\|_{2}=1$, one has

[TABLE]

Using (190) and (192), there exists $c>0$ such that

[TABLE]

where we recall that $\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}$ is defined by (80). Using in addition (189) and (197) together with the fact that $\widetilde{C}_{0}=O(1)$, there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

By definition of $\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}$ (see (80)) it holds

[TABLE]

where the last equality follows from (156). Thus, one obtains the following lower bound:

[TABLE]

In conclusion, from (187) and (198), one has for some $c>0$, in the limit $h\to 0$:

[TABLE]

Using (185) and (186), one gets

[TABLE]

Thus, since $\lambda_{1,h}=\lambda_{h}$, using in addition Proposition 52, (183), and (199), it holds int the limit $h\to 0$:

[TABLE]

Then, (200) together with Proposition 50 and the fact that

[TABLE]

imply when $h\to 0$:

[TABLE]

Recall that (A4j) consists in assuming that $\mathbf{j}(x_{1})\cap\Omega=\emptyset$. This concludes the proof of Theorem 3.

On the principal eigenfunction of $-L_{f,h}^{D,(0)}$

This section is dedicated to the proof of Proposition 59 and Theorem 6 stated below which gives respectively the asymptotic behaviour in the limit $h\to 0$ of $\displaystyle\int_{\Omega}u_{h}\ e^{-\frac{2}{h}f}$ and $\partial_{n}u_{h}$ on $\partial\Omega$.

Proposition 59 gives a sufficient condition to obtain that $u_{h}\,e^{-\frac{2}{h}f}$ (and thus the quasi-stationnary distribution $\nu_{h}$, see Proposition 5) concentrates in only one of the wells $(\mathsf{C}_{k})_{k\in\{1,\ldots,\mathsf{N}_{1}\}}$ when $h\to 0$ in the $L^{1}(\Omega)$-norm.

Proposition 59**.**

Assume that the assumptions (A0) and (A1j) are satisfied. Let us moreover assume that

[TABLE]

Let $u_{h}$ be the eigenfunction associated with the principal eigenvalue $\lambda_{h}$ of $-L_{f,h}^{D,(0)}$ (see (8)) which satisfies (9). Let $\mathsf{O}$ be an open subset of $\Omega$. On the one hand, if

[TABLE]

one has in the limit $h\to 0$:

[TABLE]

On the other hand, if

[TABLE]

then, there exists $c>0$ such that when $h\to 0$:

[TABLE]

When (A0) and (A1j) are satisfied and when $\min_{\overline{\mathsf{C}_{1}}}f=\min_{\overline{\Omega}}f$ holds, Proposition 59 implies that when $h\to 0$, $u_{h}\,e^{-\frac{2}{h}f}$ concentrates in the $L^{1}$-norm on the global minima of $f$ in $\mathsf{C}_{\mathsf{max}}$. Proposition 59 together with (11) and the fact that $\mathsf{C}_{1}=\mathsf{C}_{\mathsf{max}}$ when (A1j) holds (or equivalently (A1), see Lemma 25), imply Proposition 10. Notice that when $\mathsf{O}=\Omega$ in Proposition 59, one has from (201), when $h\to 0$:

[TABLE]

The following theorem shows that, under the hypotheses (A1j), (A2j), and (A3j), the $L^{1}_{w}(\partial\Omega)$-norm of the normal derivative of the principal eigenfunction of $-L_{f,h}^{D,(0)}$ concentrates when $h\to 0$ on $\partial\Omega\cap\partial\mathsf{C}_{1}$.

Theorem 6**.**

Let us assume that the assumptions (A0), (A1j), (A2j) and (A3j) are satisfied. Let $u_{h}$ be the eigenfunction associated with the principal eigenvalue $\lambda_{h}$ of $-L_{f,h}^{D,(0)}$ which satisfies (9). Let $F\in L^{\infty}(\partial\Omega,\mathbb{R})$ and $\Sigma$ be an open subset of $\partial\Omega$.

• (i)

When $\overline{\Sigma}\cap\{z_{1},\dots,z_{\mathsf{k}_{1}^{\partial\Omega}}\}=\emptyset$, one has in the limit $h\to 0$:

[TABLE]

where $c>0$ is independent of $h$.

• (ii)

When $\overline{\Sigma}\cap\{z_{1},\dots,z_{\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}}\}=\emptyset$, one has in the limit $h\to 0$:

[TABLE]

where, for some $c>0$ independent of $h$,

[TABLE]

• (iii)

When, for some $i\in\{1,\dots,\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\}$, $\overline{\Sigma}\cap\{z_{1},\dots,z_{\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}}\}=\{z_{i}\}$, $z_{i}\in{\Sigma}$, and $F$ is $C^{\infty}$ in a neighborhood of $z_{i}$, one has in the limit $h\to 0$:

[TABLE]

where $\varepsilon_{h}$ satisfies (204) and the constants $B_{i}$ and $C_{i,1}$ are defined in (141)–(132).

The following rewriting of Theorem 6 will be useful to prove Theorem 1. Assume that the assumptions (A0), (A1j), (A2j), and (A3j) are satisfied. Let $F\in L^{\infty}(\partial\Omega,\mathbb{R})$ and $(\Sigma_{i})_{i\in\{1,\dots,\mathsf{k}_{1}^{\partial\Omega}\}}$ be a family of disjoint open subsets of $\partial\Omega$ such that

[TABLE]

where we recall that $\{z_{1},\dots,z_{\mathsf{k}_{1}^{\partial\Omega}}\}=\mathsf{U}_{1}^{\partial\Omega}\cap\operatorname*{arg\,min}_{\partial\Omega}f$ (see (23)). Then:

There exists $c>0$ such that in the limit $h\to 0$,

[TABLE]

and

[TABLE]

with the convention $\sum\limits_{i=n}^{m}=0$ if $n>m$ and where we recall that (see (79), (80) and (A3j)),

[TABLE]

The asymptotic estimate (205) follows from item $(i)$ in Theorem 6 taking $\Sigma=\partial\Omega\setminus\cup_{i=1}^{\mathsf{k}_{1}^{\partial\Omega}}\Sigma_{i}$, while (206) follows from item $(ii)$ in Theorem 6 taking $\Sigma=\cup_{i=\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}+1}^{\mathsf{k}_{1}^{\partial\Omega}}\Sigma_{i}$. 2. 2.

Moreover, when, for some $i\in\{1,\dots,\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\}$, $F$ is $C^{\infty}$ in a neighborhood of $z_{i}$, one has in the limit $h\to 0$:

[TABLE]

where

[TABLE]

This asymptotic equivalent follows from item $(iii)$ in Theorem 6 taking $\Sigma=\Sigma_{i}$ for some $i\in\{1,\dots,\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\}$. 3. 3.

Lastly, when (A4j) (i.e when $\mathbf{j}(x_{1})\subset\partial\Omega$), the remainder term O\big{(}h^{\frac{d-5}{4}}e^{-\frac{1}{h}(2f(z_{1})-f(x_{1}))}\big{)} in (206) is of the order O\big{(}e^{-\frac{1}{h}\big{(}2\min_{\partial\Omega}f-\min_{\overline{\Omega}}f+c\big{)}}\big{)} for some $c>0$ and the remainder term O\big{(}h^{\frac{1}{4}}\big{)} in (207) is of the order $O(h)$ and admits a full asymptotic expansion in $h$.

According to Theorem 6, when the function $F$ belongs to $C^{\infty}(\partial\Omega,\mathbb{R})$, one has the following equivalent of (205) in the limit $h\to 0$:

[TABLE]

Remark 60**.**

When the assumption (A4j) is not satisfied, the remainder terms in (206) and (207) may not be optimal. In [38, Section C.4.2.2], it is proved with a one-dimensional example, that when the assumption (A4j) is not satisfied, the optimal remainder term in (206) is O\big{(}h^{\frac{d-4}{4}}e^{-\frac{1}{h}\big{(}2\min_{\partial\Omega}f-\min_{\overline{\Omega}}f\big{)}}\big{)} and the optimal remainder term in (207) is $O(\sqrt{h})$. In higher-dimension, these optimal remainder terms can be obtained in some specific cases, see [38, Proposition C.40].

This section is organized as follows. In Section 5.1, one proves Proposition 59. Section 5.2 is then dedicated to the proof of Theorem 6.

Proof of Proposition 59

This section is dedicated to the proof Proposition 59. Let us first give a corollary of Theorem 5 which is used in the proof of Proposition 59.

Corollary 61**.**

Let us assume that the assumptions (A0) and (A1j) are satisfied. Then, there exists $\beta_{0}>0$ such that for all $\beta\in(0,\beta_{0})$, there exists $h_{0}>0$ such that for all $h\in(0,h_{0})$, the orthogonal projector

[TABLE]

Moreover, choosing the parameter $\varepsilon>0$ appearing in (92) small enough, there exists $h_{0}>0$ such that for all $h\in(0,h_{0})$, one has:

[TABLE]

where the function $\widetilde{u}_{1}$ is introduced in Definition 45.

Proof.

The fact that ${\rm\;dim\;}{\rm Ran}\,\widetilde{\pi}^{(0)}_{h}=1$ is a direct consequence of Corollary 54. Let us now prove (209). Using Lemma 28, Proposition 4 and using item 2a of Proposition 46, for any $\delta>0$, there exist $\varepsilon>0$ (see (92)), $C>0$ and $h_{0}>0$ such that one has for all $h\in(0,h_{0})$,

[TABLE]

Therefore, choosing $\varepsilon>0$ small enough such that $\delta\in(0,\beta)$, there exists $c>0$ and $h_{0}>0$ such that one has for all $h\in(0,h_{0})$,

[TABLE]

This concludes the proof of (209) and thus the proof of Corollary 61.

Let us now prove Proposition 59.

Proof of Proposition 59.

Let us first assume that only the assumptions (A0) and (A1j) are satisfied. Let us recall that $u_{h}$ is the eigenfunction associated with the principal eigenvalue $\lambda_{h}$ of $-L_{f,h}^{D,(0)}$ (see (8)) which satisfies (9). As a direct consequence of Corollary 61 and (211), one has since the functions $u_{h}$ and $\widetilde{u}_{1}$ are non negative,

[TABLE]

Let $\mathsf{O}$ be an open subset of $\Omega$. Using (212) and thanks to the Cauchy-Schwarz inequality, one obtains in the limit $h\to 0$:

[TABLE]

Let us recall that by construction (see Definition 45 and (92)),

[TABLE]

Then, from the definition of $\chi_{1}^{\varepsilon,\varepsilon_{1}}$ (see (92) and the lines below) and using Laplace’s method, one has in the limit $h\to 0$,

[TABLE]

Let us assume that

[TABLE]

Then, using Laplace’s method, one has when $h\to 0$,

[TABLE]

where we recall that $x_{1}\in\operatorname*{arg\,min}_{\mathsf{C}_{1}}f$. Thus, from (213), (214), and (215), one has when $h\to 0$:

[TABLE]

Let us assume moreover that

[TABLE]

Then, (201) in Proposition 59 is a consequence of (216). Let us now consider the case where

[TABLE]

Then, it holds

[TABLE]

Since in the limit $h\to 0$:

[TABLE]

one obtains using (217), (213), and (214), that there exist $c>0$ and $\tilde{c}>0$ such that when $h\to 0$:

[TABLE]

This proves (202) and concludes the proof of proposition 59.

Proof of Theorem 6

Let us briefly explain the strategy for the proof of Theorem 6. The basic idea is to notice that, since $\nabla u_{h}$ belongs to ${\rm Ran}\,\pi_{h}^{(1)}$ (according to (88)), one has for any open set $\Sigma$ of $\partial\Omega$ and for any $L_{w}^{2}$-orthonormal basis $(\psi_{1},\dots,\psi_{\mathsf{m}_{1}^{\overline{\Omega}}})$ of ${\rm Ran}\,\pi_{h}^{(1)}$,

[TABLE]

Notice that this decomposition of $\nabla u_{h}$ is valid on $\partial\Omega$. Indeed, for all $i\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}$, $\psi_{i}$ has a smooth trace on $\partial\Omega$ since $\psi_{i}\in\Lambda^{1}C^{\infty}(\overline{\Omega})$ (due to the fact that the eigenforms of $L^{D,(1)}_{f,h}$ belongs to $C^{\infty}(\overline{\Omega})$ and $\pi_{h}^{(1)}$ is a projector onto a finite number of eigenforms of $-L^{D,(1)}_{f,h}$). In the rest of this section, one first introduces such a family $\{\psi_{1},\dots,\psi_{\mathsf{m}_{1}^{\overline{\Omega}}}\}$ using a Gram-Schmidt orthonormalization of the family \big{\{}\pi_{h}^{(1)}\widetilde{\psi}_{1},\dots,\pi_{h}^{(1)}\widetilde{\psi}_{\mathsf{m}_{1}^{\overline{\Omega}}}\big{\}}. Then, one gives estimates of the terms $\langle\nabla u_{h},\psi_{i}\rangle_{L^{2}_{w}}$ appearing in (219). Finally, one concludes the proof of Theorem 6 in Section 5.2.3, with estimations of the boundary terms $\int_{\Sigma}F\,\psi_{i}\cdot n\ e^{-\frac{2}{h}f}$.

Gram-Schmidt orthonormalization

Let us assume that the hypothesis (A0) holds, and assume $h>0$ small enough such that the family $\left\{\pi_{h}^{(1)}\widetilde{\psi}_{i},i=1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\right\}$ is independent (which is guaranteed for small $h$ by item 3b in Proposition 46). Using a Gram-Schmidt procedure, there exists, for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, a family $(\kappa_{ji})_{i=1,\dots,j-1}\subset\mathbb{R}^{j-1}$ such that the $1$-forms

[TABLE]

satisfy:

• (i)

for all $k\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$, {\rm Span}\big{(}\{f_{i},i=1,\dots,k\}\big{)}={\rm Span}\big{(}\{\pi_{h}^{(1)}\widetilde{\psi}_{i},i=1,\dots,k\}\big{)},

• (ii)

for all $i\neq j$, $\langle f_{i},f_{j}\rangle_{L^{2}_{w}}=0$.

One defines moreover, for $j\in\left\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\right\}$,

[TABLE]

so that (\psi_{j})_{j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}} is a $L^{2}_{w}$-orthonormal basis of ${\rm Ran}\,\pi_{h}^{(1)}$. By reasoning by induction (see [10, Section 2] for a similar proof), Proposition 46 easily leads to the following estimates showing in particular that the family $(\pi_{h}^{(1)}\widetilde{\psi}_{i})_{i\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$ is close to the family $(\psi_{i})_{i\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$.

Lemma 62**.**

Let us assume that the assumption (A0) is satisfied. Then, there exists $c>0$ such that for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, $i\in\{1,\dots,j-1\}$ and $h>0$ small enough,

[TABLE]

Estimates of the interaction terms $\left(\langle\nabla u_{h},\psi_{j}\rangle_{L^{2}_{w}}\right)_{j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$

Let us begin with the estimates of the terms $\langle\nabla\pi_{h}^{(0)}\widetilde{u}_{k},\psi_{j}\rangle_{L^{2}_{w}}$, where j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} and $k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}$.

Lemma 63**.**

Let us assume that the assumption (A0) holds. Then, there exists $c>0$ such that for all for all $k\in\{1,\ldots,\mathsf{m}_{0}^{\Omega}\}$, j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} and $h>0$ small enough, it holds:

[TABLE]

where we recall that the asymptotic expansion of the term $\langle\nabla\widetilde{u}_{k},\widetilde{\psi}_{j}\rangle_{L^{2}_{w}}$ is given in Proposition 52.

Proof.

Using (220), (221), and Lemma 62, one has for some $c>0$ and for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} and $h>0$ small enough,

[TABLE]

Using Proposition 52, the statement of Lemma 63 follows immediately.

In view of (219), we need to give an asymptotic estimate of the terms

[TABLE]

To do that, we would like to prove that $\langle\nabla u_{h},\psi_{j}\rangle_{L^{2}_{w}}$ is well approximated by $\langle\nabla\pi_{h}^{(0)}\widetilde{u}_{1},\psi_{j}\rangle_{L^{2}_{w}}$. To this end, let us show that $\pi_{h}^{(0)}\widetilde{u}_{1}$ is an accurate approximation of $u_{h}$ in $H^{1}_{w}(\Omega)$.

Before, let us recall that when (A0) and (A1j) hold, Corollary 61 implies that there exists $\beta_{0}>0$ such that for all $\beta\in(0,\beta_{0})$, there exists $h_{0}>0$ such that for all $h\in(0,h_{0})$, the orthogonal projector

[TABLE]

Therefore, $\widetilde{\pi}^{(0)}_{h}$ is the orthogonal projector onto ${\rm Span}\,u_{h}$. Moreover, from the second equality in (212) and item 3a in Proposition 46, one has

[TABLE]

Therefore, $\pi^{(0)}_{h}\widetilde{u}_{1}$ is an accurate approximation of $u_{h}$ in $L^{2}_{w}(\Omega)$. The following result extends this result in $H^{1}_{w}(\Omega)$ when assuming (A2j) in addition to (A0) and (A1j).

Lemma 64**.**

Assume that (A0), (A1j) and (A2j) hold. Then, it holds in the limit $h\to 0$:

[TABLE]

and

[TABLE]

where in the limit $h\to 0$, $\varepsilon_{h}$ satisfies (204)

Proof.

Applying the Parseval identity to $\nabla\pi_{h}^{(0)}\widetilde{u}_{1}\in{\rm Ran}\,\pi_{h}^{(1)}$ (see (86)), one gets

[TABLE]

Using Lemma 63, there exists consequently $c>0$ such that for all $h>0$ small enough,

[TABLE]

Using in addition (200), one then obtains the first part of Lemma 64:

[TABLE]

where, in the limit $h\to 0$, $\varepsilon_{h}$ satisfies (204).

Let us recall that from (212) and (211), one has for any $h$ small enough

[TABLE]

Now, since the projectors $\pi_{h}^{(0)}$ and $\widetilde{\pi}_{h}^{(0)}$ commute with $L^{D,(0)}_{f,h}$, and $\widetilde{\pi}_{h}^{(0)}\pi_{h}^{(0)}=\widetilde{\pi}_{h}^{(0)}$, one has

[TABLE]

where the last line follows from (223). Using in addition (222), one obtains in the limit $h\to 0$:

[TABLE]

which proves Lemma 64, using also the asymptotic estimate of $\lambda_{h}$ given in Theorem 3, see (131).

We are now in position to estimate the interaction terms $\left(\langle\nabla u_{h},\psi_{j}\rangle_{L^{2}_{w}}\right)_{j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}}$.

Corollary 65**.**

Let us assume that the assumptions (A0), (A1j) and (A2j) hold. Let $u_{h}$ be the eigenfunction associated with the principal eigenvalue $\lambda_{h}$ of $-L_{f,h}^{D,(0)}$ (see (8)) which satisfies (9). Then, in the limit $h\to 0$:

1. (i)

for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} such that $z_{j}\in\mathbf{j}(x_{1})\cap\partial\Omega$ (i.e. for all j\in\big{\{}1,\dots,\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\big{\}}, see (79) and (80)),

[TABLE] 2. (ii)

for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} such that $z_{j}\in\mathbf{j}(x_{1})\cap\Omega$,

[TABLE] 3. (iii)

and for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} such that $z_{j}\notin\mathbf{j}(x_{1})$,

[TABLE]

where in the limit $h\to 0$, $\varepsilon_{h}$ satisfies (204).

Proof.

Using (223), there exists $c>0$ such that for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, in the limit $h\to 0$:

[TABLE]

In addition, using the Cauchy-Schwarz inequality and the second statement in Lemma 64, it holds for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, in the limit $h\to 0$:

[TABLE]

where $\varepsilon_{h}$ is of the order given by (204). Then, the statement of Corollary 65 follows by injecting (225) into (224) and by using the estimates of the terms \big{\langle}\nabla\pi_{h}^{(0)}\widetilde{u}_{1},\psi_{j}\big{\rangle}_{L^{2}_{w}} ($j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$) given in Lemma 63.

Estimates of the boundary terms \Big{(}\displaystyle{\int_{\Sigma}\,F\,\psi_{j}\cdot n\,e^{-\frac{2}{h}f}}\Big{)}_{j\in\{1,\ldots,\mathsf{m}_{1}^{\overline{\Omega}}\}}

Proposition 66**.**

Let us assume that the assumption (A0) is satisfied. Let us consider $i\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$, an open set $\Sigma$ of $\partial\Omega$, and $F\in L^{\infty}(\partial\Omega,\mathbb{R})$. Then, there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

where we recall that $\{z_{1},\ldots,z_{\mathsf{k}_{1}^{\partial\Omega}}\}=\mathsf{U}_{1}^{\partial\Omega}\cap\operatorname*{arg\,min}_{\partial\Omega}f$ (see (23)). Moreover, when i\in\big{\{}1,\dots,\mathsf{k}_{1}^{\partial\Omega}\big{\}}, $z_{i}\in{\Sigma}$, and $F$ is $C^{\infty}$ in a neighborhood of $z_{i}$, it holds in the limit $h\to 0$:

[TABLE]

where the constant $B_{i}$ is defined in (132).

Proof.

Let $F\in L^{\infty}(\partial\Omega,\mathbb{R})$. Using (220), (221), the trace theorem, and the Cauchy-Schwarz inequality, one has for all $j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$,

[TABLE]

From Lemma 62 and item 2b in Proposition 46, there exists $c>0$ such that for all j\in\big{\{}1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}}, $i\in\{1,\dots,j-1\}$, in the limit $h\to 0$:

[TABLE]

Therefore, using Proposition 49, there exists $c>0$ such that for all $j\in\{1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\}$, in the limit $h\to 0$:

[TABLE]

The statement of Proposition 66 is then a straightforward consequence of Proposition 49.

We are now in position to prove Theorem 6.

Proof of Theorem 6.

Let us assume that the assumptions (A0), (A1j), (A2j) and (A3j) hold. Recall that in this case, for all $x\in\mathsf{U}_{0}^{\Omega}\setminus\{x_{1}\}$, one has

[TABLE]

and

[TABLE]

Moreover, from (82), it holds

[TABLE]

Thus, one has

[TABLE]

Let us now consider $F\in L^{\infty}(\partial\Omega,\mathbb{R})$ and $\Sigma$ an open subset of $\partial\Omega$. First, since \big{\{}\psi_{j},\,j=1,\dots,\mathsf{m}_{1}^{\overline{\Omega}}\big{\}} is an orthonormal basis of ${\rm Ran}\,\pi^{(1)}_{h}$ and $\nabla u_{h}\in{\rm Ran}\,\pi^{(1)}_{h}$, one has the following decomposition:

[TABLE]

Using in addition Corollary 65, Proposition 66, and (226), there exists $c>0$ such that for all $h>0$ small enough,

[TABLE]

Hence, when $\overline{\Sigma}$ does not contain any of the $z_{i}$, $i\in\{1,\dots,\mathsf{k}_{1}^{\partial\Omega}\}$, from (227), Corollary 65, Proposition 66, and (226), one deduces the following relation for some $c>0$ independent of $h$ and every $h>0$ small enough:

[TABLE]

This proves item $(i)$ in Theorem 6.

Assume now that $\overline{\Sigma}$ does not contain any of the $z_{i}$, $i\in\{1,\dots,\mathsf{k}^{\partial\mathsf{C}_{1}}_{1}\}$. Then , from (227), Corollary 65, Proposition 66, and (226), one deduces that in the limit $h\to 0$:

[TABLE]

where the constant $c>0$ is independent of $h$ and $\varepsilon_{h}$ satisfies (204). This proves item $(ii)$ in Theorem 6.

Assume lastly that $\overline{\Sigma}\cap\{z_{1},\dots,z_{\mathsf{k}^{\partial\mathsf{C}_{1}}_{1}}\}=\{z_{i}\}$, $F$ is $C^{\infty}$ in a neighborhood of $z_{i}$ and $z_{i}\in{\Sigma}$. From (227), Corollary 65, Proposition 66, and (226), one then deduces that in the limit $h\to 0$, it holds for some $c>0$ and $\varepsilon_{h}$ which satisfies (204),

[TABLE]

where the constants $B_{i}$ and $C_{i,1}$ are defined in (141)–(132). This concludes the proof of item $(iii)$ in Theorem 6.

On the law of $X_{\tau_{\Omega}}$

The main goal of this section is to prove Theorem 1. In Section 6.1, one proves Theorem 1 when $X_{0}\sim\nu_{h}$ (where $\nu_{h}$ is the quasi-stationary distribution of the process (1) in $\Omega$, see Definition 2). In Section 6.2, one proves Theorem 1 when $X_{0}=x\in\mathcal{A}(\mathsf{C}_{\mathsf{max}})$.

Proof of Theorem 1 when $X_{0}\sim\nu_{h}$

The proof of Theorem 1 when $X_{0}\sim\nu_{h}$ is a straightforward consequence of Theorem 3, Proposition 59 and Theorem 6. Indeed, let us recall that from (12), one has:

[TABLE]

Moreover, recall that (A1), (A2), and (A3) (see Section 2.5 and more precisely Lemma 25) are equivalent to the assumptions (A1j), (A2j), and (A3j) . In addition, under (A1j), one has $\mathsf{C}_{1}=\mathsf{C}_{\mathsf{max}}$ (see Lemma 25), $\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}=\mathsf{k}_{1}^{\partial\mathsf{C}_{\mathsf{max}}}$ (see (81)) and

[TABLE]

and

[TABLE]

Thus, injecting the results of Theorem 3 (and more precisely (131)), Proposition 59 (applied to $\mathsf{O}=\Omega$, see (203)) and Theorem 6 in (228), one obtains the statement of Theorem 1 when $X_{0}\sim\nu_{h}$.

Proof of Theorem 1 when $X_{0}=x\in\mathcal{A}(\mathsf{C}_{\mathsf{max}})$

Recall that, under (A1), $\mathsf{C}_{\mathsf{max}}=\mathsf{C}_{1}$ (see Lemma 25). To prove Theorem 1 when $X_{0}=x\in\mathcal{A}(\mathsf{C}_{1})$, one first proves that a sufficiently accurate leveling property (as introduced in [3]) holds in $\mathsf{C}_{1}$ for $x\mapsto\mathbb{E}_{x}[F(X_{\tau_{\Omega}})]$, see Proposition 68 in Section 6.2.1. Then, combining Theorem 1 when $X_{0}\sim\nu_{h}$ with Proposition 68, one proves Theorem 1 when $X_{0}=x\in\mathcal{A}(\mathsf{C}_{1})$ in Section 6.2.2.

Leveling results

The leveling property for $x\mapsto\mathbb{E}_{x}[F(X_{\tau_{\Omega}})]$ is defined as follows.

Definition 67**.**

Let $K$ be a compact subset of $\Omega$ and $F\in C^{\infty}(\partial\Omega,\mathbb{R})$. We say that $x\mapsto\mathbb{E}_{x}[F(X_{\tau_{\Omega}})]$ satisfies a leveling property on $K$ if

[TABLE]

and this limit holds uniformly with respect to $(x,y)\in K\times K$.

The leveling property (229) has been widely studied in the literature in various geometrical settings, see for example [40, 23, 9, 3, 13, 11]. We prove the following proposition which is a leveling property in our framework.

Proposition 68**.**

Let us assume that the assumption (A0) holds. Let $\lambda\in\mathbb{R}$ and $\mathsf{C}$ be a connected component of $\{f<\lambda\}$ such that $\mathsf{C}\subset\Omega$. Then, for any path-connected compact set $K\subset\mathsf{C}$ and for any $F\in C^{\infty}(\partial\Omega,\mathbb{R})$, there exist $c>0$ and $M>0$, such that for all $(x,y)\in K\times K$,

[TABLE]

Proof.

The proof is inspired from techniques used in [9]. The proof of Proposition 68 is divided into two steps. In the following $C>0$ is a constant which can change from one occurrence to another and which does not depend on $h$.

Step 1. Let $F\in C^{\infty}(\partial\Omega,\mathbb{R})$. Let us denote by $v_{h}\in H^{1}(\Omega)$ the unique weak solution to the elliptic boundary value problem

[TABLE]

Then, $v_{h}$ belongs to $C^{\infty}(\overline{\Omega},\mathbb{R})$ and for all $k\in\mathbb{N}$, there exist $C>0$, $n\in\mathbb{N}$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$, it holds

[TABLE]

Moreover, the Dynkin’s formula implies that

[TABLE]

Let us prove that $v_{h}$ belongs to $C^{\infty}(\overline{\Omega},\mathbb{R})$ and (232). Since $F$ is $C^{\infty}$, for all $k\geq 1$, the exists $\widetilde{F}\in H^{k}(\Omega)$ such that $\widetilde{F}=F$ on $\partial\Omega$ and $\|\widetilde{F}\|_{H^{k}}\leq C\|F\|_{H^{k-\frac{1}{2}}(\partial\Omega)}$. From (231), the function $\widetilde{v}_{h}=v_{h}-\widetilde{F}\in H^{1}(\Omega)$ and is the weak solution to

[TABLE]

Thus, using [12, Theorem 5, Section 6.3], $\widetilde{v}_{h}\in H^{2}(\Omega)$ (and thus $v_{h}\in H^{2}(\Omega)$) and there exist $C>0$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$

[TABLE]

and thus

[TABLE]

This proves (232) for $k=0$ The inequality (232) is then obtained by a bootstrap argument (by induction on $k$). This implies by Sobolev embeddings that $v_{h}$ belongs to $C^{\infty}(\overline{\Omega},\mathbb{R})$. Let us now prove that there exist $\alpha>0$ and $C>0$ such that

[TABLE]

Notice that from (233), one has that for all $h>0$, $\|v_{h}\|_{L^{\infty}(\Omega)}\leq\|F\|_{L^{\infty}(\partial\Omega)}$. From (231) and (235) there exists $C>0$ such that for any $\varepsilon>0$ and $\varepsilon^{\prime}>0$,

[TABLE]

Choosing $\varepsilon=\frac{h^{2}}{4(CC_{1}+1)}$ and $\varepsilon^{\prime}=\frac{h}{4(C+1)}$ we get

[TABLE]

Therefore, using (232), one obtains that for all $k\geq 0$, there exist $C>0$, $n\in\mathbb{N}$ and $h_{0}>0$ such that for all $h\in(0,h_{0})$

[TABLE]

Let $k\geq 0$ such that $k-\frac{d}{2}>1$. Then, one obtains (236) from the continuous Sobolev injection $H^{k}(\Omega)\subset W^{1,\infty}(\Omega)$.

Step 2. Let us assume that the assumption (A0) holds. Let $\lambda\in\mathbb{R}$ and $\mathsf{C}$ be a connected component of $\{f<\lambda\}$ such that $\mathsf{C}\subset\Omega$. To prove Proposition 68, we will prove that for any compact subset $K$ of $\mathsf{C}$, there exists $c_{K}>0$ such that

[TABLE]

Indeed, since $K$ is path-connected, the following inequality

[TABLE]

where $C_{K}>0$ depends on $K$, will then conclude the proof of (230).

Let us now define the set $\mathsf{C}_{r}$ by

[TABLE]

which is not empty and $C^{\infty}$ for all $r\in(0,r_{1})$, for some $r_{1}>0$. Indeed, the boundary of $\mathsf{C}_{r}$ is the set $\{f=\lambda-r\}\cap\mathsf{C}$ (since $\mathsf{C}\subset\Omega$) which contains no critical points of $f$ for $r\in(0,r_{1})$, with $r_{1}>0$ small enough (since there is a finite number of critical points under the assumption (A0)). We now prove that for all $r_{0}\in(0,r_{1})$ there exists $\alpha_{0}>0$ such that

[TABLE]

Let $r$ be such that $2^{n}r=r_{0}$ where $n\in\mathbb{N}$ will be fixed later (since $r\leq r_{0}$, $r\in(0,r_{1})$). Equation (231) rewrites

[TABLE]

Using (236), there exist $C>0$ and $\alpha>0$ such that,

[TABLE]

where we used the Green formula (valid since $\mathsf{C}_{r}$ is $C^{\infty}$ for all $r\in(0,r_{1})$) and the inclusion $\partial\mathsf{C}_{r/2}\subset\{f=\lambda-\frac{r}{2}\}$. In addition, since $\mathsf{C}_{r}\subset\mathsf{C}_{r/2}$ it holds,

[TABLE]

Therefore, there exists $\beta>0$ such that for $h$ small enough,

[TABLE]

and from (231), we then have $\|\Delta v_{h}\|_{L^{2}(\mathsf{C}_{r})}\leq C\,e^{-\frac{\beta}{h}}$ for some constant $\beta>0$ which has been reduced. In the following, $\beta>0$ is a constant which may change from one occurrence to another and does not depend on $h$. Let $\chi_{1}\in C_{c}^{\infty}(\mathsf{C}_{r})$ be such that $\chi_{1}\equiv 1$ on $\mathsf{C}_{2r}$. Since $\Delta(\chi_{1}v_{h})=\chi_{1}\,\Delta v_{h}+v_{h}\,\Delta\chi_{1}+2\nabla\chi_{1}\cdot\nabla v_{h}$, there exists $C$, such that $\|\Delta(\chi_{1}v_{h})\|_{L^{2}(\mathsf{C}_{r})}\leq C$ for $h$ small enough. By elliptic regularity (see [12, Theorem 5, Section 6.3]) it comes

[TABLE]

Let $\alpha\in(0,1)$ be such that $p_{1}=\frac{2d}{d-2\alpha}>0$. From the Gagliardo-Nirenberg interpolation inequality (see [39, Lecture II]), the following inequality holds

[TABLE]

From (231), $\|\Delta v_{h}\|_{L^{p_{1}}(\mathsf{C}_{2r})}\leq C\,e^{-\frac{\beta}{h}}$. Using a cutoff function $\chi_{2}\in C_{c}^{\infty}(\mathsf{C}_{2r})$ such that $\chi_{2}\equiv 1$ on $\mathsf{C}_{4r}$, we get, as previously, from the elliptic regularity $\|v_{h}\|_{W^{2,p_{1}}(\mathsf{C}_{4r})}\leq C$. Let $p_{2}=\frac{2d}{d-4\alpha}$ (i.e. $1/p_{2}=1/p_{1}-\alpha/d$). If $p_{2}<0$, then [39, Lecture II] implies

[TABLE]

Thus, (239) is proved (and one then chooses $n=2$, i.e. $2^{2}r=r_{0}$). Otherwise, we prove (239) by induction as follows. From the Gagliardo-Nirenberg interpolation inequality (see [39, Lecture II]), we get

[TABLE]

We repeat this procedure $n$ times where $n$ is the first integer such that $d-2n\alpha<0$ and the Gagliardo-Nirenberg interpolation inequality implies that $\|\nabla v_{h}\|_{L^{\infty}(\mathsf{C}_{2^{n}r})}\leq C\,e^{-\frac{\beta}{h}}$ which ends the proof of (239). Since for any compact $K\subset\mathsf{C}$, there exists $r_{0}\in(0,r_{1})$ such that $K\subset\mathsf{C}_{r_{0}}$, the inequality (237) is proved. This concludes the proof of Proposition 68.

End of the proof of Theorem 1

To prove Theorem 1 when $X_{0}=x\in\mathcal{A}(\mathsf{C}_{1})$, the following lemma will be needed.

Lemma 69**.**

Assume that the assumptions (A0) and (A1j) are satisfied. Let $\mathsf{C}_{1}$ be as in (A1j). Let us moreover assume that

[TABLE]

Let $K$ be a compact subset of $\Omega$ such that $K\subset\mathcal{A}(\mathsf{C}_{1})$ and let $F\in C^{\infty}(\partial\Omega,\mathbb{R})$. Then, there exists $c>0$ such that for all $y\in K$, one has in the limit $h\to 0$:

[TABLE]

uniformly in $y\in K$.

Lemma 69 implies Proposition 11 since we recall that when (A0) and (A1j) are satisfied, $\mathsf{C}_{1}=\mathsf{C}_{\mathsf{max}}$ (see Lemma 25).

Proof.

Assume that the assumptions (A0) and (A1j) are satisfied. Let us assume that

[TABLE]

Step 1. For $\alpha>0$ small enough, let $\mathsf{C}_{1}(\alpha)$ be as introduced in Definition 24 (see (75)):

[TABLE]

Let $F\in C^{\infty}(\partial\Omega,\mathbb{R})$. In this first step, we will prove that  $\exists\alpha_{0}>0$, $\forall\alpha\in(0,\alpha_{0})$, $\exists c>0$, $\forall y\in\overline{\mathsf{C}_{1}(\alpha)}$:

[TABLE]

in the limit $h\to 0$ and uniformly in $y\in\overline{\mathsf{C}_{1}(\alpha)}$.

Let us recall that from the notation of Proposition 68, for all $x\in\overline{\Omega}$:

[TABLE]

From (11), one has:

[TABLE]

where $Z_{h}(\Omega):=\displaystyle\int_{\Omega}u_{h}\ e^{-\frac{2}{h}f}$ and $u_{h}$ is the eigenfunction associated with the principal eigenvalue $\lambda_{h}$ of $-L^{D,(0)}_{f,h}$ (see (8)) which satisfies (9). Let us first deal with the second term in (242). Since (A0) and (A1j) hold, and because it is assumed that $\min_{\overline{\mathsf{C}_{1}}}f=\min_{\overline{\Omega}}f$, one obtains from Proposition 59 (applied to $\mathsf{O}=\Omega$, see (203)) that there exists $C>0$ such that for $h$ small enough:

[TABLE]

For all $\alpha>0$ small enough, one has \big{(}\overline{\Omega}\setminus\mathsf{C}_{1}(\alpha)\big{)}\cap\operatorname*{arg\,min}_{\mathsf{C}_{1}}f=\emptyset. Therefore, using (202) (applied for $\alpha$ small enough with $\mathsf{O}=\Omega\setminus\overline{\mathsf{C}_{1}(\alpha)}$), one has from (240), that for all $\alpha>0$ small enough, there exists $c>0$ such that when $h\to 0$:

[TABLE]

Thus, there exists $\alpha_{0}>0$ such that for all $\alpha\in(0,\alpha_{0})$ there exists $c>0$ such that when $h\to 0$:

[TABLE]

Then, since $\|v_{h}\|_{L^{\infty}(\overline{\Omega})}\leq\|F\|_{L^{\infty}(\partial\Omega)}$, one obtains that

[TABLE]

Let us now deal with the first term in (242). Let us recall that $\mathsf{C}_{1}\subset\Omega$ is a connected component of $\mathsf{\{}f<\max_{\overline{\mathsf{C}_{1}}}f\}$. Moreover, for $\alpha\in(0,\alpha_{0})$ ($\alpha_{0}>0$ small enough), the compact set $\overline{\mathsf{C}_{1}(\alpha)}$ is connected and $\overline{\mathsf{C}_{1}(\alpha)}\subset\mathsf{C}_{1}$. Therefore, from Proposition 68 applied to $K=\overline{\mathsf{C}_{1}(\alpha)}$ for $\alpha\in(0,\alpha_{0})$, one obtains that there exists $\delta_{\alpha}>0$ such that for all $y\in\overline{\mathsf{C}_{1}(\alpha)}$,

[TABLE]

in the limit $h\to 0$ and uniformly with respect to $y\in\overline{\mathsf{C}_{1}(\alpha)}$. Moreover, for all $\alpha\in(0,\alpha_{0})$ there exists $c>0$ such that in the limit $h\to 0$:

[TABLE]

which follows from the fact that

[TABLE]

together with (243). Let us now fix $\alpha\in(0,\alpha_{0})$. Then, using (245) and (246), $\exists c>0$, $\exists\delta_{\alpha}>0$, $\forall y\in\overline{\mathsf{C}_{1}(\alpha)}$:

[TABLE]

in the limit $h\to 0$ and uniformly with respect to $y\in\overline{\mathsf{C}_{1}(\alpha)}$. Therefore, using (242), (244) and (247), $\exists\alpha_{0}>0$, $\forall\alpha\in(0,\alpha_{0})$, $\exists c>0$, $\forall y\in\overline{\mathsf{C}_{1}(\alpha)}$:

[TABLE]

in the limit $h\to 0$ and uniformly with respect to $y\in\overline{\mathsf{C}_{1}(\alpha)}$. This concludes the proof of (241).

Step 2. Let us now conclude the proof of Lemma 69 by considering a compact subset $K$ of $\Omega$ such that $K\subset\mathcal{A}(\mathsf{C}_{1})$. Let us recall that (see (14)):

[TABLE]

Since $\mathsf{C}_{1}$ is open and stable by the flow $\varphi_{t}(\cdot)$ (see (13)), the continuity of $\varphi_{t}(\cdot)$ implies that there exists $T_{K}\geq 0$ such that for all $x\in K$,

[TABLE]

Moreover, since $K$ is compact and for all $x\in K$, $t_{x}=+\infty$ (i.e. $\varphi_{t}(x)\in\Omega$ for all $t\geq 0$), there exists $\delta>0$ such that all continuous curves $\gamma:[0,T_{K}]\to\overline{\Omega}$ such that

[TABLE]

satisfy:

[TABLE]

Furthermore, up to choosing $\delta>0$ smaller, there exists $\alpha_{K}>0$ such that

[TABLE]

Let us now recall the following estimate of Freidlin and Wentzell (see [13, Theorems 2.2 and 2.3 in Chapter 3, and Theorem 1.1 in Chapter 4], [4], [8, Theorem 3.5] and [14, Theorem 5.6.3]). For all $x\in K$, it holds:

[TABLE]

where

[TABLE]

and $H^{1}_{x,T_{K}}(\delta)$ is the set of curves $\gamma:[0,T_{K}]\to\Omega$ of regularity $H^{1}$ such that $\gamma(0)=x$ and \sup_{t\in[0,T_{K}]}\big{|}\gamma(t)-\varphi_{t}(x)\big{|}\geq\delta. Since $K$ is compact, there exists $\eta_{K}>0$ such that for $h$ small enough, it holds:

[TABLE]

Notice that when $X_{0}=x\in K$ and \sup_{t\in[0,T_{K}]}\big{|}X_{t}-\varphi_{t}(x)\big{|}\leq\delta, it holds from (248) and (249):

[TABLE]

Let us now consider $F\in C^{\infty}(\partial\Omega,\mathbb{R})$. Let $x\in K$. Then,

[TABLE]

Using (251), it holds for $h$ small enough:

[TABLE]

Using (252), (241) (with $\alpha=\alpha_{K}$), (251), and the Markov property of the process (1), there exists $c>0$ such that for all $x\in K$, one has when $h\to 0$:

[TABLE]

uniformly in $x\in K$. This concludes the proof of Lemma 69.

Let us now end the proof of Theorem 1 when $X_{0}=x\in K$, where $K$ is a compact subset of $\Omega$ such that $K\subset\mathcal{A}(\mathsf{C}_{1})$. Recall that, when (A1) holds, which is equivalent to (A1j), one has $\mathsf{C}_{1}=\mathsf{C}_{\mathsf{max}}$, see Lemma 25.

Proof.

Let $K$ be a compact subset of $\Omega$ such that

[TABLE]

and let us consider that the process starts from $X_{0}=x\in K$. Let $F\in C^{\infty}(\partial\Omega,\mathbb{R})$. Let us notice that the proof is not a direct consequence of Lemma 69 since in Theorem 1, less regular test functions $F$ are considered. The proof of Theorem 1 is divided into three steps. In the following we assume that (A0), (A1j), (A2j) and (A3j) are satisfied.

Step 1. Proof of (25) and (26) when $X_{0}=x\in K$.

Let us first show that if $\Sigma\subset\partial\Omega$ is open and there exists $\beta>0$ such that $\Sigma\cap\bigcup_{i=1}^{k_{1}^{\partial\Omega}}B_{\partial\Omega}(z_{i},\beta)=\emptyset$ (where $B_{\partial\Omega}(z_{i},\beta)$ is the open ball in $\partial\Omega$ of radius $\beta$ centered at $z_{i}$), then, for all $x\in K$,

[TABLE]

in the limit $h\to 0$ and uniformly in $x\in K$. To this end, let us consider $\tilde{F}\in C^{\infty}(\partial\Omega,[0,1])$ be such that

[TABLE]

Using Lemma 69, there exists $c>0$ such that for all $x\in K$:

[TABLE]

in the limit $h\to 0$ and uniformly in $x\in K$. Then, (253) follows from (25) applied to $\tilde{F}$ and the family of sets $\Sigma_{i}=B_{\partial\Omega}(z_{i},\frac{\beta}{2})$ for $i\in\{1,\ldots,\mathsf{k}_{1}^{\partial\Omega}\}$ when $X_{0}\sim\nu_{h}$.

Let us now prove (25) and (26). Let $F\in L^{\infty}(\partial\Omega,\mathbb{R})$ and for all $i\in\{1,\dots,\mathsf{k}_{1}^{\partial\Omega}\}$, let $\Sigma_{i}\subset\partial\Omega$ be an open set which contains $z_{i}$. Let us assume in addition that $\Sigma_{i}\cap\Sigma_{j}=\emptyset$ if $i\neq j$. One has for any $x\in K$

[TABLE]

Moreover, one has:

[TABLE]

Using (253) with $\Sigma=\partial\Omega\setminus\bigcup_{i=1}^{\mathsf{k}_{1}^{\partial\Omega}}\Sigma_{i}$, one gets (25).

Let us now prove (26). Let $j\in\{\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}+1,\ldots,\mathsf{k}_{1}^{\partial\Omega}\}$. Let us introduce $\mathsf{k}_{1}^{\partial\Omega}$ disjoint open sets $\tilde{\Sigma}_{i}\subset\partial\Omega$ ($i\in\{1,\ldots,\mathsf{k}_{1}^{\partial\Omega}\}$) and a smooth function $G$ supported in $\tilde{\Sigma}_{j}$ such that $\mathbf{1}_{\Sigma_{j}}F\leq\|F\|_{L^{\infty}}\,G$ (in order to apply Lemma 69 with $G$). To this end, let $\delta>0$ be such that for any $k\in\{1,\ldots,\mathsf{k}_{1}^{\partial\Omega}\}$ with $k\neq j$, the sets $\tilde{\Sigma}_{k}:=B_{\partial\Omega}(z_{k},\delta)$ and $\tilde{\Sigma}_{j}:=\cup_{z\in\Sigma_{j}}B_{\partial\Omega}(z,\delta)$ are disjoint. Let us consider

[TABLE]

Using Lemma 69, there exists $c>0$ such that for all $x\in K$

[TABLE]

in the limit $h\to 0$ and uniformly in $x\in K$. Then, using (26) and item 3 in Theorem 1 applied with $X_{0}\sim\nu_{h}$, $(\tilde{\Sigma}_{i})_{i\in\{1,\ldots,\mathsf{k}_{1}^{\partial\Omega}\}}$, and $G$, it holds when $h\to 0$: \mathbb{E}_{x}\left[\mathbf{1}_{\Sigma_{j}}F(X_{\tau_{\Omega}})\right]=O\big{(}h^{\frac{1}{4}}\big{)} and when (A4j) holds, one has when $h\to 0$:

[TABLE]

for some $c>0$. This concludes the proof of (26).

Step 2. Proof of (27) when $X_{0}=x\in K$.

For all $j\in\{1,\dots,\mathsf{k}_{1}^{\partial\Omega}\}$, let $\Sigma_{j}$ be open subset of $\partial\Omega$ such that $z_{j}\in\Sigma_{j}$. Let us assume that $\Sigma_{k}\cap\Sigma_{j}=\emptyset$ if $k\neq j$. Let $F\in L^{\infty}(\partial\Omega,\mathbb{R})$ be $C^{\infty}$ in a neighborhood of $z_{i}$ for some $i\in\{1,\dots,\mathsf{k}_{1}^{\partial\mathsf{C}_{1}}\}$. Let $\beta>0$ be such that $F$ is $C^{\infty}$ on $B_{\partial\Omega}(z_{i},2\beta)\subset\Sigma_{i}$ and let $\chi_{i}\in C^{\infty}(\partial\Omega,[0,1])$ be such that ${\rm supp}\,\chi_{i}\subset B_{\partial\Omega}(z_{i},\beta)$, $\chi_{i}=1$ on $B_{\partial\Omega}(z_{i},\frac{\beta}{2})$. One has:

[TABLE]

Using Lemma 69 with $\chi_{i}F\in C^{\infty}$ and (25)-(26) with $X_{0}\sim\nu_{h}$, $F\chi_{i}$ and the family of disjoint open sets $\{\Sigma_{j},j=1,\ldots,\mathsf{k}_{1}^{\partial\Omega},j\neq i\}\cup\{B_{\partial\Omega}(z_{i},\frac{\beta}{2})\}$, there exists $c>0$ such that for all $x\in K$:

[TABLE]

in the limit $h\to 0$ and uniformly in $x\in K$, and where $a_{i}$ is defined in (28). In addition, using Theorem 1 when $X_{0}\sim\nu_{h}$, when (A4j) holds, one can replace O\big{(}h^{\frac{1}{4}}\big{)} in the last computation by $O(h)$. Moreover, using (253) with $\Sigma=\Sigma_{i}\setminus B_{\partial\Omega}(z_{i},\frac{\beta}{2})$: there exists $c>0$ such that for all $x\in K$:

[TABLE]

in the limit $h\to 0$ and uniformly in $x\in K$. Thus, one has when $h\to 0$ and uniformly with respect to $x\in K$:

[TABLE]

and when (A4j) holds, one has:

[TABLE]

This concludes the proof of (27) when $X_{0}=x\in K$ and the proof of Theorem 1.

Proof of Theorem 2

In this section, one proves Theorem 2.

Proof of Theorem 2..

Let us assume that (A0) holds. Let $\mathsf{C}\in\mathcal{C}$. Assume that (see (29))

[TABLE]

To prove Theorem 2, the strategy consists in using Theorem 1 with a subdomain $\Omega_{\mathsf{C}}$ of $\Omega$ containing $\mathsf{C}$ such that in the limit $h\to 0$, the most probable places of exit of the process (1) from $\Omega_{\mathsf{C}}$ when $X_{0}=x\in\mathsf{C}$ are the elements of $\partial\mathsf{C}\cap\partial\Omega$. This will imply (since the trajectories of the process (1) are continuous) that the most probable places of exit of the process (1) from $\Omega$ when $X_{0}=x\in\mathsf{C}$ are the elements of $\partial\mathsf{C}\cap\partial\Omega$, which is the statement of Theorem 2.

The proof of Theorem 2 is divided into two steps.

Step 1: Construction of a domain $\Omega_{\mathsf{C}}$ containing $\mathsf{C}$.

In this step, one constructs a subset $\Omega_{\mathsf{C}}$ of $\Omega$ such that

[TABLE]

and

[TABLE]

To construct a domain $\Omega_{\mathsf{C}}\subset\Omega$ which satisfies (254) and (255), we first briefly recall the local structure of $f$ near $\partial\mathsf{C}$ to then build a neighborhood $\mathsf{V}_{\mathsf{C}}$ of $\overline{\mathsf{C}}$ in $\overline{\Omega}$ (this construction is similar to the the construction of $\mathsf{V}_{k}$ made in Step 5 in the proof of Proposition 20). The set $\mathsf{V}_{\mathsf{C}}$ is then used to justify the existence of a domain $\Omega_{\mathsf{C}}$ satisfying (254) and (255). Let $\lambda\in\mathbb{R}$ be such that $\mathsf{C}$ is a connected component of $\{f<\lambda\}$ (see (17)). Then, for $z\in\partial\mathsf{C}$, we introduce a ball of radius $\varepsilon_{z}>0$ centred at $z$ in $\overline{\Omega}$ as follows.

If $z\in\partial\mathsf{C}\cap\Omega$: Since $z\in\Omega$ and $|\nabla f(z)|\neq 0$, there exists $\varepsilon_{z}>0$ such that $\overline{B(z,\varepsilon_{z})}\subset\Omega$, $|\nabla f(z)|\neq 0$ on $\overline{B(z,\varepsilon_{z})}$, and, according to [18, Section 5.2], ${B(z,\varepsilon_{z})}\cap\{f<\lambda\}$ is connected and ${B(z,\varepsilon_{z})}\cap\partial\{f<\lambda\}={B(z,\varepsilon_{z})}\cap\{f=\lambda\}$ (where we recall that $B(z,\varepsilon_{z})=\{x\in\overline{\Omega}\ \text{s.t.}\ |x-z|<\varepsilon_{z}\}$). 2. 2.

If $z\in\partial\mathsf{C}\cap\partial\Omega$: Recall that $z\in\mathsf{U}_{1}^{\partial\Omega}$ (see (21)) and thus, $\partial_{n}f(z)>0$ and $z$ is a non degenerate local minimum of $f|_{\partial\Omega}$. Thus, there exists $\varepsilon_{z}>0$, such that $|\nabla f(z)|\neq 0$ on $\overline{B(z,\varepsilon_{z})}$ and such that, according to [18, Section 5.2], ${B(z,\varepsilon_{z})}\cap\{f<\lambda\}$ is connected and included in $\Omega$. In addition, it holds ${B(z,\varepsilon_{z})}\cap\partial\{f<\lambda\}={B(z,\varepsilon_{z})}\cap\{f=\lambda\}$ is included in $\overline{\Omega}$. Finally, up to choosing $\varepsilon_{z}>0$ smaller, one has:

[TABLE]

where we recall that $B_{\partial\Omega}(z,\varepsilon_{z})$ is the open ball of radius $\varepsilon_{z}$ centred in $z$ in $\partial\Omega$, and,

[TABLE]

Items 1 an 2 above imply that for all $z\in\partial\mathsf{C}$, by definition of $\mathsf{C}$ (see Theorem 2 and Definition 16),

[TABLE]

One then defines:

[TABLE]

The set $\mathsf{V}_{\mathsf{C}}$ is an open neighborhood of $\overline{\mathsf{C}}$ in $\overline{\Omega}$. Moreover, according to items 1 and 2 above,

[TABLE]

and using in addition (258),

[TABLE]

The second statement in (260) implies that $\overline{\mathsf{C}}$ is a connected component of $\{f\leq\lambda\}$. Thus, for $r>0$ small enough $\overline{{\mathsf{C}}(\lambda+r)}\subset\mathsf{V}_{\mathsf{C}}$, where ${\mathsf{C}}(\lambda+r)$ is the connected component of $\{f<\lambda+r\}$ which contains $\mathsf{C}$. This suggests that a natural candidate to satisfy (254) and (255) is the domain ${\mathsf{C}}(\lambda+r)$. However, for $r>0$ small enough, the boundary of ${\mathsf{C}}(\lambda+r)$ is not $C^{\infty}$, it is composed of two smooth pieces $\overline{\partial\mathsf{C}(\lambda+r)\cap\Omega}=\{x\in\partial\mathsf{C}(\lambda+r),\,f(x)=\lambda+r\,\}$ and $\partial\mathsf{C}(\lambda+r)\cap\partial\Omega$. The union of this two sets gives rise to ”corners”. Moreover, the function $f|_{\partial\mathsf{C}(\lambda+r)\cap\Omega}$ is not a Morse function since $f\equiv\lambda+r$ on $\overline{\partial\mathsf{C}(\lambda+r)\cap\Omega}$. To justify the existence of a domain $\Omega_{\mathsf{C}}$ which satisfies (254) and (255), we proceed in two steps, as follows.

• •

Domain $D_{\mathsf{C}}$ containing $\mathsf{C}$ which satisfies (254) and $\partial_{n}f>0$ on $\partial D_{\mathsf{C}}$. The subdomain $D_{\mathsf{C}}$ of $\Omega$ is constructed as a smooth regularization of the set $\mathsf{C}(\lambda+r)$ with $r>0$ such that $\overline{{\mathsf{C}}(\lambda+r)}\subset\mathsf{V}_{\mathsf{C}}$ by modifying ${\mathsf{C}}(\lambda+r)$ in a neighborhood of ${\{x\in\partial\mathsf{C}(\lambda+r),\,f(x)=\lambda+r\,\}}\cap\partial\Omega$ (where the two smooth pieces of $\partial\mathsf{C}(\lambda+r)$ intersect each other). Moreover, $\partial_{n}f>0$ on $\overline{\partial\mathsf{C}(\lambda+r)\cap\Omega}$ (since there is no critical point of $f$ on $\overline{\partial\mathsf{C}(\lambda+r)\cap\Omega}=\{x\in\partial\mathsf{C}(\lambda+r),\,f(x)=\lambda+r\,\}$) and on $\partial\mathsf{C}(\lambda+r)\cap\partial\Omega$ (since $\overline{{\mathsf{C}}(\lambda+r)}\subset\mathsf{V}_{\mathsf{C}}$ and $\partial_{n}f>0$ on $\mathsf{V}_{\mathsf{C}}\cap\partial\Omega$, see the second inequality in (257)). Thus, using in addition (260) together with the fact that $\mathsf{V}_{\mathsf{C}}$ is an open neighborhood of $\overline{\mathsf{C}}$ in $\overline{\Omega}$, there exists a $C^{\infty}$ connected open subset $D_{\mathsf{C}}$ of $\Omega$ such that

[TABLE]

and

[TABLE]

which satisfies, for some $\beta>0$ and $\Sigma_{\mathsf{C}}\subset\Omega$,

[TABLE]

Finally, according to the first statement in (257), there exists $\delta_{0}>0$ such that for any open $\delta$-neighborhood $U_{\partial\Omega}^{\delta}$ of $\partial\Omega$ in $\overline{\Omega}$, with $\delta\in(0,\delta_{0})$, one has

[TABLE]

where $\nabla_{T}f$ is the tangential gradient of $f$ on $\partial D_{\mathsf{C}}$.

• •

Domain $\Omega_{\mathsf{C}}$ containing $\mathsf{C}$ which satisfies (254) and (255). To construct such a domain, we use Proposition 72 (see Appendix C below) with $D=D_{\mathsf{C}}$, $\mathcal{V}_{-}=\mathsf{C}$, $\mathcal{V}_{+}=\mathsf{V}_{\mathsf{C}}$,

• (i)

$S_{1}=\partial D_{\mathsf{C}}\cap U^{\delta/2}_{\partial\Omega}$ on which $f$ is a Morse function with no critical point on $\partial S_{1}$ (see (264) together with the fact that $\partial\mathsf{C}\cap\partial\Omega$ is composed of non degenerate critical points of $f|_{\partial\Omega}$),

• (ii)

$S_{1}^{\prime}=\partial D_{\mathsf{C}}\cap U^{\delta/4}_{\partial\Omega}$ which satisfies, according to (264), $|\nabla_{T}f|\neq 0$ on $\overline{S_{1}\setminus S_{1}^{\prime}}$.

Therefore, using in addition the fact that $D_{\mathsf{C}}$ satisfies (261)–(263), there exists a $C^{\infty}$ connected open subset $\Omega_{\mathsf{C}}$ of $\Omega$ such that $\mathsf{C}\subset\Omega_{\mathsf{C}}$, $\overline{\Omega_{\mathsf{C}}}\subset\mathsf{V}_{\mathsf{C}}$,

[TABLE]

and for some $r>0$ and $\Gamma_{\mathsf{C}}\subset\Omega$,

[TABLE]

It then remains to check that $\Omega_{\mathsf{C}}$ satisfies (254). From (265) and (256), $\Omega_{\mathsf{C}}$ satisfies the two first statements in (254) and $\min_{\partial\Omega_{\mathsf{C}}}f=\lambda$. Since $\mathsf{C}\subset\Omega_{\mathsf{C}}$ and $\overline{\Omega_{\mathsf{C}}}\subset\mathsf{V}_{\mathsf{C}}$, one deduces from the first statement in (260), that

[TABLE]

and from (259),

[TABLE]

This proves that $\Omega_{\mathsf{C}}$ satisfies the two last statements in (254). This concludes the construction of a domain $\Omega_{\mathsf{C}}$ which satisfies (254) and (255). A schematic representation of such a domain ${\Omega_{\mathsf{C}}}$ is given in Figure 17.

Step 2: End of the proof of Theorem 2.

For all $z\in\partial\mathsf{C}\cap\partial\Omega$, let $\Sigma_{z}$ be an open subset of $\partial\Omega$ such that $z\in\Sigma_{z}$. Let $K$ be a compact subset of $\Omega$ such that $K\subset\mathcal{A}(\mathsf{C})$. Let us first consider the case when $K\subset\mathsf{C}$.

Let $\Omega_{\mathsf{C}}$ be the $C^{\infty}$ subdomain of $\Omega$ constructed in the previous step and which, we recall, contains $\mathsf{C}$ and satisfies (254) and (255). Then, one easily deduces that when $\Omega$ is replaced by $\Omega_{\mathsf{C}}$, the function $f:\overline{\Omega_{\mathsf{C}}}\to\mathbb{R}$ satisfies (A0) and $\mathcal{C}=\{\mathsf{C}\}$ (see (16) for the definition of $\mathcal{C}$). Thus, in this case $\mathsf{C}_{\mathsf{max}}=\mathsf{C}$. Moreover, using in addition the second and third statements in (254), one obtains that the assumptions (A1), (A2), (A3) and (A4) are satisfied for the function $f:\overline{\Omega_{\mathsf{C}}}\to\mathbb{R}$. Thus, according Theorem 1 applied to the function $f:\overline{\Omega_{\mathsf{C}}}\to\mathbb{R}$, the most probable places of exit of the process (1) from $\Omega_{\mathsf{C}}$ when $X_{0}=x\in\mathsf{C}$, are $\partial\mathsf{C}\cap\partial\Omega$ (and the relative asymptotic probabilities are given by item 2 in Theorem 1). In particular, from items 1 and 3 in Theorem 1, for any open subset $\Sigma$ of $\partial\Omega_{\mathsf{C}}$ such that

[TABLE]

there exists $c>0$ such that for $h$ small enough:

[TABLE]

where $\tau_{\Omega_{\mathsf{C}}}$ is the first exit time from $\Omega_{\mathsf{C}}$ of the process (1).

Step 2a: Proof of the first asymptotic estimate in Theorem 2 when $K\subset\mathsf{C}$.

Writing $\partial\Omega=(\partial\Omega\cap\partial\Omega_{\mathsf{C}})\cup(\partial\Omega\setminus\partial\Omega_{\mathsf{C}})$, it holds:

[TABLE]

To prove the first asymptotic estimate in Theorem 2, let us prove that when $X_{0}=x\in K$, the probability that $X_{\tau_{\Omega}}$ belongs to each of the two sets in the right-hand side of (267), is exponentially small when $h\to 0$. Let us recall that $\tau_{\Omega_{\mathsf{C}}}$ is the first exit time from $\Omega_{\mathsf{C}}$ of the process (1) and thus, when $X_{0}=x\in\Omega_{\mathsf{C}}$, $\tau_{\Omega_{\mathsf{C}}}\leq\tau_{\Omega}$, and

[TABLE]

Thus, from (268), when $X_{0}=x\in\Omega_{\mathsf{C}}$, it holds:

[TABLE]

Using (266), there exists $c>0$ such that for $h$ small enough:

[TABLE]

Thus, there exists $c>0$ such that for $h$ small enough:

[TABLE]

Let us now consider the case when $X_{\tau_{\Omega}}\in\partial\Omega\setminus\partial\Omega_{\mathsf{C}}$ and $X_{0}=x\in K$. When $X_{0}=x\in K$, it holds from (268):

[TABLE]

From (266), there exists $c>0$ such that for $h$ small enough:

[TABLE]

Therefore, there exists $c>0$ such that for $h$ small enough:

[TABLE]

In conclusion, from (267), (269) and (270), one obtains that there exists $c>0$ such that for $h$ small enough:

[TABLE]

This proves the first asymptotic estimate in Theorem 2 when $K\subset\mathsf{C}$.

Step 2b: Proof of the second asymptotic estimate in Theorem 2 when $K\subset\mathsf{C}$.

Let us assume that the open sets $(\Sigma_{z})_{z\in\partial\mathsf{C}\cap\partial\Omega}$ are two by two disjoint. Let us consider $z\in\partial\mathsf{C}\cap\partial\Omega$ and $\beta>0$ such that (see indeed the first statement in (254)),

[TABLE]

Then, one writes:

[TABLE]

Let us first deal with the second term in the right-hand side (273). It holds (since the sets $(\Sigma_{y})_{y\in\partial\mathsf{C}\cap\partial\Omega}$ are two by two disjoint and $B_{\partial\Omega}(z,\beta)\subset\Sigma_{z}$, see (272)), when $X_{0}=x\in\Omega$:

[TABLE]

Thus, from (271) (applied with $B_{\partial\Omega}(z,\beta)$ instead of $\Sigma_{z}$), one obtains that there exists $c>0$ such that for $h$ small enough:

[TABLE]

Let us now deal with the first term in the right-hand side (273). It holds from (272) and (268), when $X_{0}=x\in K$:

[TABLE]

Applying item 2 in Theorem 1 with the function $f:\overline{\Omega_{\mathsf{C}}}\to\mathbb{R}$ and $F=\mathbf{1}_{B_{\partial\Omega}(z,\beta)}$, one has:

[TABLE]

in the limit $h\to 0$ and uniformly in $x\in K$. Together with (273), (274), and (275), this concludes the proof of the second asymptotic estimate in Theorem 2 when $X_{0}=x\in K\subset\mathsf{C}$.

The case when $X_{0}=x\in K\subset\mathcal{A}(\mathsf{C})$ is proved using the estimate of Freidlin and Wentzell (250) (see the second step of the proof of Lemma 69). This concludes the proof of Theorem 2.

\addappheadtotoc

**A. On the assumption (A0) and Lemma 27 **

In this appendix and as stated in Section 1.4.1, one explains why the conclusion of Lemma 27, proved in [18, Section 3.4] (and made for $\Delta^{D,(p)}_{f,h}$, $p\in\{0,1\}$), is still valid when assuming in (A0) that $f:\{x\in\partial\Omega,\,\partial_{n}f(x)>0\}\to\mathbb{R}$ is a Morse function instead of $f|_{\partial\Omega}$ is a Morse function.

For that purpose, let us assume that $f:\overline{\Omega}\to\mathbb{R}$ is $C^{\infty}$, $|\nabla f|\neq 0$ on $\partial\Omega$ and that, the functions $f:\overline{\Omega}\to\mathbb{R}$ and $f:\{x\in\partial\Omega,\,\partial_{n}f(x)>0\}\to\mathbb{R}$ are Morse functions. From (85), we are going to prove that in this case, for $h$ small enough, one still has:

[TABLE]

Let us notice that there exists an open neighborhood $\mathcal{U}\subset\partial\Omega$ of $\{x\in\partial\Omega,\,\partial_{n}f(x)\geq 0\}$ such that $f|_{\mathcal{U}}$ is a Morse function. Therefore, in view of [18, Section 3.4] (and more precisely of the IMS formula used there to prove an upper bound on the number of small eigenvalues), to prove (276), it is sufficient to show that for all $z\in\{x\in\partial\Omega,\,\partial_{n}f(x)<0\}$, there exists a neighbourhood $\mathcal{V}_{z}$ of $z$ in $\overline{\Omega}$ such that for any $w\in\Lambda^{p}H^{1}_{T}(\Omega)$ (for $p\in\{0,1\}$) supported in $\mathcal{V}_{z}$,

[TABLE]

for some $C>0$ independent of $h$ and $w$. Let us recall the two following Green formulas [18, Lemma 2.3.2]. For all $w\in\Lambda^{0}H^{1}_{T}(\Omega)$, one has:

[TABLE]

For all $w\in\Lambda^{1}H^{1}_{T}(\Omega)$, one has:

[TABLE]

where $\mathcal{L}_{\nabla f}$ is the Lie derivative with respect to the vector field $\nabla f$ and $\mathcal{L}_{\nabla f}^{*}$ its formal adjoint in $L^{2}(\Omega)$. Let us recall that the operator $\mathcal{L}_{\nabla f}+\mathcal{L}_{\nabla f}^{*}$ is a zeroth order operator (see for instance [19, Appendice 1]).

Since there is no boundary term in (278), the first equality in (276) is just a consequence of $|\nabla f|\neq 0$ on $\partial\Omega$. Indeed, there exist a neighbourhood $\mathcal{V}_{\partial\Omega}$ of $\partial\Omega$ in $\overline{\Omega}$ and a constant $c>0$ such that $\inf_{\overline{\mathcal{V}_{\partial\Omega}}}|\nabla f|\geq c$. Then, from (278), for $h$ small enough, one has for all $w\in\Lambda^{0}H^{1}_{T}(\Omega)$ supported in $\mathcal{V}_{\partial\Omega}$:

[TABLE]

Thus, (277) is satisfied and for $h$ small enough, it holds:

[TABLE]

Let us now prove the second equality in (276). To this end, let $z\in\partial\Omega$ such that $\partial_{n}f(z)<0$. Then, there exists a neighborhood $\mathcal{V}_{z}\subset\mathcal{V}_{\partial\Omega}$ of $z$ in $\overline{\Omega}$ such that $\partial_{n}f<0$ on $\partial\Omega\cap\mathcal{V}_{z}$. Therefore, using (279), for $h$ small enough, one has for all $w\in\Lambda^{1}H^{0}_{T}(\Omega)$ supported in $\mathcal{V}_{z}$:

[TABLE]

The estimate (281) implies, according again to (277), that for $h$ small enough, it holds

[TABLE]

This ends the proof of (276).

B. Proofs of the results of Sections 1.4.3, 1.4.4, and 1.4.5

In this section, one proves the asymptotic estimates (30), (31), (32) and (33) stated in Section 1.4. Let us start with the following result. Let $z_{1}<z_{2}$ and $f$: $[z_{1},z_{2}]\to\mathbb{R}$ be a $C^{\infty}$ function. Then, for all $x\in[z_{1},z_{2}]$, one has:

[TABLE]

Indeed, let $v\in C^{\infty}([z_{1},z_{2}],\mathbb{R})$ be the unique solution to the elliptic boundary value problem on $(z_{1},z_{2})$:

[TABLE]

Clearly, one has for all $x\in[z_{1},z_{2}]$,

[TABLE]

Finally, using the Dynkin’s formula [25, Theorem 11.2], one has for all $x\in[z_{1},z_{2}]$,

[TABLE]

This proves (282).

Let us now explain how to prove (30), (31), (32) and (33). The asymptotic estimates (30) and (33) follow directly from (282) together with Laplace’s method.

Let us now prove (31). In the example depicted in Figure 4, the assumption (A1) is satisfied. Therefore, using (212), there exits $\chi\in C^{\infty}_{c}\left((c,d),[0,1]\right)$ such that $\chi=1$ on a neighborhood of $x_{1}$ and:

[TABLE]

where $r\in L^{2}_{w}(z_{1},z_{2})$ satisfies $\|r\|_{L^{2}_{w}}=O(e^{-\frac{c}{h}})$ and $c>0$ is independent of $h$. Moreover, one has (see Figure 4)

[TABLE]

Thus, from Proposition 59 (applied to $\mathsf{O}=(z_{1},z_{2})$, see (203)), one has in the limit $h\to 0$:

[TABLE]

Moreover, if we denote by $g(x)=\mathbb{P}_{x}[X_{\tau_{(z_{1},z_{2})}}=z_{1}]$ for $x\in[z_{1},z_{2}]$, since $\chi\in C^{\infty}_{c}\left((c,d),[0,1]\right)$ and $\|g\|_{L^{\infty}}\leq 1$, one has in the limit $h\to 0$ (using  (283) and (282) in the third equality):

[TABLE]

for some $c>0$ independent of $h$. This proves (31).

Let us now prove (32). In the example depicted in Figure 5, the assumption (A1) is satisfied. Therefore, using (212), there exits $\chi\in C^{\infty}_{c}\left((z,z_{2}),[0,1]\right)$ such that $\chi=1$ on a neighborhood of $x_{2}$ and:

[TABLE]

where $r\in L^{2}_{w}(z_{1},z_{2})$ satisfies $\|r\|_{L^{2}_{w}}=O(e^{-\frac{c}{h}})$ and $c>0$ is independent of $h$. Moreover, one has (see Figure 5)

[TABLE]

Thus, from Proposition 59 (applied to $\mathsf{O}=(z_{1},z_{2})$, see (203)), one has in the limit $h\to 0$:

[TABLE]

Then, the same computations as those made to obtain (284), imply that when $h\to 0$:

[TABLE]

for some $c>0$ independent of $h$. This proves (32).

**C. On the proof of (255). **

In this section, we prove the existence of a domain $\Omega_{\mathsf{C}}$ which satisfies (255) in addition to (254). To this end, we first give in Proposition 70 a simple perturbation result to present the main idea of the proof. Then, we extend this result to the setting we are interested in to prove the existence of such a domain $\Omega_{\mathsf{C}}$ in Proposition 72.

Proposition 70**.**

Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be a $C^{\infty}$ function and $D$ be a $C^{\infty}$ open bounded and connected subset of $\mathbb{R}^{d}$. Let us assume that

[TABLE]

Then, for any open sets $\mathcal{V}_{-}$ and $\mathcal{V}_{+}$ such that $\overline{\mathcal{V}_{-}}\subset D\text{ and }\overline{D}\subset\mathcal{V}_{+}$, there exists a $C^{\infty}$ open bounded and connected subset $D^{\prime}$ of $\mathbb{R}^{d}$ such that

[TABLE]

Remark 71**.**

We would like to thank François Laudenbach who gave us the main ingredient of the proof of Proposition 70. The proof is inspired by a method due to René Thom [45] based on Sard’s theorem [41], see [27, Section 5.6].

Proof.

Let $\mathcal{V}_{-}$ and $\mathcal{V}_{+}$ be two open subsets of $\mathbb{R}^{d}$ such that $\overline{\mathcal{V}_{-}}\subset D\text{ and }\overline{D}\subset\mathcal{V}_{+}$. Let us denote by $S$ the boundary of $D$ which is a smooth compact hypersurface of $\mathbb{R}^{d}$. For $r>0$, one denotes by $B(0,r)$ the ball of radius $r$ centred at [math] in $\mathbb{R}^{d}$. Let $\mathcal{V}$ be a neighborhood of $S$ in $\mathbb{R}^{d}$. By assumption on $S$, there exist $\varepsilon_{0}>0$ and $\varepsilon_{1}>0$ such that the map

[TABLE]

is well defined and is a diffeomorphism onto its image, and, for all $(x,v)\in S\times B(0,\varepsilon_{1})$, there exists a unique $\lambda(x,v)\in(-\varepsilon_{0},\varepsilon_{0})$ such that

[TABLE]

Moreover, for every $v\in B(0,\varepsilon_{1})$, according to the implicit function theorem, the map $x\in S\mapsto\lambda(x,v)\in(-\varepsilon_{0},\varepsilon_{0})$ is smooth and then also is $x\in S\mapsto x+\lambda(x,v)\nabla f(x)\in\mathbb{R}^{d}$. The latter application is then an injective immersion and hence, since $S$ is compact, it follows that $S_{v}:=\{x+\lambda(x,v)\nabla f(x)\}$ is a smooth compact hypersurface. Up to choosing $\varepsilon_{1}>0$ smaller, for any $v\in B(0,\varepsilon_{1})$, $S_{v}$ is the boundary of a $C^{\infty}$ open bounded and connected subset $D_{v}$ of $\mathbb{R}^{d}$ such that

[TABLE]

To prove Proposition 70, it remains to show that there exists $v\in B(0,\varepsilon_{1})$ such that $f|_{S_{v}}$ is a Morse function. Let us introduce the function

[TABLE]

For all $x\in S$ and for all $v\in B(0,\varepsilon_{1})$, let $v^{T}_{x}\in T_{x}S$ and $v_{x}^{N}\in\mathbb{R}$ be such that

[TABLE]

where we recall that $n(x)$ is a unit outward normal to $S$. At $(x,v)\in S\times B(0,\varepsilon_{1})$, it holds $\partial_{x}F(x,v):z\in T_{x}S\mapsto d_{x}f(x)z+v_{x}^{T}\cdot z$, where $\partial_{x}F(x,v)$ is the $x$-derivative of $F$ at $(x,v)$. The function $G:S\times B(0,\varepsilon_{1})\to T_{x}^{*}S$ defined by

[TABLE]

is a submersion onto a small tube around the zero section of $T^{*}S$. This is obvious by considering the $v$-derivative of $G$. Hence, $G$ is transverse to the zero section $0_{T^{*}S}$ of $T^{*}S$ (see [27, Chapitre 5.1] for the definition of transversality). Using the parametric transversality theorem (which is a consequence of Sard’s theorem, see for instance [27, Chapitre 5.3.1]), one obtains that for almost every $v\in B(0,\varepsilon_{1})$, $\partial_{x}(F|_{S\times\{v\}})=d(f|_{S_{v}})$ is transverse to $0_{T^{*}S}$, which is equivalent to $f|_{S_{v}}$ is a Morse function. This concludes the proof of Proposition 70.

The next proposition gives sufficient conditions on $D$ and $f$ to modify the result of Proposition 70 so that the perturbed domain $D′$ has the same boundary as $D$ on a prescribed subset $S_{1}^{\prime}$ of $\partial D$ on which $f$ is already a Morse function.

Proposition 72**.**

Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be a $C^{\infty}$ function and $D$ be a $C^{\infty}$ open bounded and connected subset of $\mathbb{R}^{d}$. Let us assume that

[TABLE]

Furthermore, let us assume that there exists an open subset $S_{1}$ of $\partial D$ such that $f:\overline{S_{1}}\to\mathbb{R}$ is a Morse function with no critical point on $\partial S_{1}$. Let us now consider an open set $S_{1}^{\prime}$ such that $\overline{S_{1}^{\prime}}\subset S_{1}$ and $f|_{\partial D}$ has no critical point on $\overline{S_{1}\setminus S_{1}^{\prime}}$. Then, for any open sets $\mathcal{V}_{-}$ and $\mathcal{V}_{+}$ such that $\overline{\mathcal{V}_{-}}\subset D\cup S_{1}^{\prime}\text{ and }\overline{D}\setminus S_{1}^{\prime}\subset\mathcal{V}_{+}$, there exists a $C^{\infty}$ open bounded and connected subset $D^{\prime}$ of $\mathbb{R}^{d}$ such that $S_{1}^{\prime}\subset\partial D^{\prime}$,

[TABLE]

Proof.

Let $\mathcal{V}_{-}$ and $\mathcal{V}_{+}$ be two open subsets of $\mathbb{R}^{d}$ such that $\overline{\mathcal{V}_{-}}\subset D\cup S_{1}^{\prime}\text{ and }\overline{D}\setminus S_{1}^{\prime}\subset\mathcal{V}_{+}$. Let us denote by $S$ the boundary of $D$ which is a smooth compact hypersurface of $\mathbb{R}^{d}$. Let us introduce a function $\chi\in C^{\infty}(S)$ such that $\chi(x)=1$ for all $x\in S\setminus S_{1}$ and $\chi(x)=0$ for all $x\in\mathcal{V}_{S_{1}^{\prime}}$ where $\mathcal{V}_{S_{1}^{\prime}}$ is an open neighborhood of $\overline{S_{1}^{\prime}}$ in $S$ such that $\overline{\mathcal{V}_{S_{1}^{\prime}}}\subset S_{1}$. To prove Proposition 72, one uses the cutoff function $\chi$ in the definition of $\lambda(x,t)$ to ensure that $S_{1}^{\prime}\subset S_{v}$ (see the proof of Proposition 70 for the notation $S_{v}$). This is made as follows. Let us first consider $\varepsilon_{0}>0$ and $\varepsilon_{1}>0$ such that the map

[TABLE]

is well defined and is a diffeomorphism onto its image, and, for all $(x,v)\in S\times B(0,\varepsilon_{1})$, there exists a unique $\lambda(x,v)\in(-\varepsilon_{0},\varepsilon_{0})$ such that

[TABLE]

Notice that $\lambda(x,v)=0$ for all $x\in\mathcal{V}_{S_{1}^{\prime}}$ and $v\in B(0,\varepsilon_{1})$ (since $\chi=0$ on $\mathcal{V}_{S_{1}^{\prime}}$). Thus, for all $v\in B(0,\varepsilon_{1})$, $\mathcal{V}_{S_{1}^{\prime}}\subset S_{v}$ which implies that $S_{1}^{\prime}\subset S_{v}$. Again, $S_{v}:=\{x+\lambda(x,v)\nabla f(x)\}$ is a smooth compact hypersurface. A schematic representation of the function $\chi$ and the hypersurface $S_{v}$ are given in Figure 18. Up to choosing $\varepsilon_{1}>0$ smaller, for any $v\in B(0,\varepsilon_{1})$, $S_{v}$ is the boundary of a $C^{\infty}$ open bounded and connected subset $D_{v}$ of $\mathbb{R}^{d}$ such that, since $\mathcal{V}_{S_{1}^{\prime}}\subset S_{v}$,

[TABLE]

Let us now show that there exists $v\in B(0,\varepsilon_{1})$ such that $f|_{S_{v}}$ is a Morse function. For that purpose, we consider the function

[TABLE]

and the function $G:S\times B(0,\varepsilon_{1})\to T_{x}^{*}S$ defined by $G:(x,v)\mapsto(x,\partial_{x}F(x,v))$. Notice that for all $v\in B(0,\varepsilon_{1})$, $x\in\overline{S_{1}^{\prime}}\mapsto F(x,v)=f(x)$ is already, by assumption, a Morse function (with no critical point on $\partial S_{1}^{\prime}$). This implies that $G$ is transverse to the zero section $0_{T^{*}S}$ of $T^{*}S$ along $S_{1}^{\prime}\times B(0,\varepsilon_{1})$. Thus, to prove Proposition 72, it remains to study the function $x\in S\setminus S^{\prime}_{1}\mapsto F(x,v)$, for $v\in B(0,\varepsilon_{1})$. For $(x,v)\in\overline{S_{1}\setminus S^{\prime}_{1}}\times B(0,\varepsilon_{1})$ and for all $z\in T_{x}S$, it holds:

[TABLE]

Since by assumption $d_{x}f(x)\neq 0_{T^{*}_{x}S}$ for all $x$ belonging to the compact set $\overline{S_{1}\setminus S_{1}^{\prime}}$, one has up to choosing $\varepsilon_{1}>0$ smaller, for all $x\in\overline{S_{1}\setminus S_{1}^{\prime}}$ and $v\in B(0,\varepsilon_{1})$, $\partial_{x}F(x,v)\neq 0_{T^{*}_{x}S}$. Finally, for $(x,v)\in S\setminus S_{1}\times B(0,\varepsilon_{1})$ and for all $z\in T_{x}S$, it holds:

[TABLE]

Thus, the function $G:S\setminus S_{1}\times B(0,\varepsilon_{1})\to T_{x}^{*}S$ is a submersion onto a small tube around the zero section $0_{T^{*}S}$ of $T^{*}S$. This implies that $G$ is transverse to the zero section of $T^{*}S$ along $S\setminus S_{1}\times B(0,\varepsilon_{1})$. In conclusion, the function $G:S\times B(0,\varepsilon_{1})\to T_{x}^{*}S$ is transverse to the zero section of $T^{*}S$. The parametric transversality theorem implies that for almost every $v\in B(0,\varepsilon_{1})$, $\partial_{x}(F|_{S\times\{v\}})=d(f|_{S_{v}})$ is transverse to $0_{T^{*}S}$, which is equivalent to $f|_{S_{v}}$ is a Morse function. This concludes the proof of Proposition 72.

Acknowledgements

This work is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492. The authors thank Laurent Michel and François Laudenbach for fruitful discussions.

Main notation used in this work

• $\bullet$

$(\tau_{\Omega},X_{\tau_{\Omega}})$, p. 2

• $\bullet$

$L^{(0)}_{f,h}$, p. 7

• $\bullet$

$L^{D,(0)}_{f,h}$, p. 4 (see also p. 3.1.2)

• $\bullet$

$\lambda_{h}$, p. 8

• $\bullet$

$u_{h}$, p. 9

• $\bullet$

$\nu_{h}$, p. 5

• $\bullet$

$\mathbb{P}_{x}$, p. 5

• $\bullet$

Assumption (A0), p. 1.3.1.

• $\bullet$

$\mathcal{A}(D)$, p. 1.3.1

• $\bullet$

$\{f<a\}$, $\{f\leq a\}$, and $\{f=a\}$, p. 1.3.1

• $\bullet$

$\mathsf{H}_{f}(x)$, p. 1.3.1,

• $\bullet$

$\mathsf{C}_{\mathsf{max}}$, p. 1.3.1

• $\bullet$

$\mathcal{C}$, p. • ‣ 1.3.1 (see also p. 14)

• $\bullet$

$\mathsf{C}(x)$, p. 17

• $\bullet$

Assumptions (A1), (A2), (A3), and (A4), p. • ‣ 1.3.1

• $\bullet$

$\mathsf{U}_{0}^{\Omega}=\{x_{1},\dots,x_{\mathsf{m}_{0}^{\Omega}}\}$ and $\mathsf{m}_{0}^{\Omega}$, p. 1.3.2

• $\bullet$

$\mathsf{U}_{1}^{\Omega}=\{z_{\mathsf{m}_{1}^{\partial\Omega}+1},\ldots,z_{\mathsf{m}_{1}^{\overline{\Omega}}}\}$ and $\mathsf{m}_{1}^{\Omega}$, p. 1.3.2

• $\bullet$

$\mathsf{U}_{1}^{\partial\Omega}=\{z_{1},\ldots,z_{\mathsf{m}_{1}^{\partial\Omega}}\}$ and $\mathsf{m}_{1}^{\partial\Omega}$, p. 1.3.2

• $\bullet$

$\mathsf{U}_{1}^{\overline{\Omega}}=\{z_{1},\ldots,z_{\mathsf{m}_{1}^{\partial\Omega}},z_{\mathsf{m}_{1}^{\partial\Omega}+1},\ldots,z_{\mathsf{m}_{1}^{\overline{\Omega}}}\}$ and $\mathsf{m}_{1}^{\overline{\Omega}}$, p. 1.3.2

• $\bullet$

$\{z_{1},\ldots,z_{\mathsf{k}_{1}^{\partial\Omega}}\}$ and $\mathsf{k}_{1}^{\partial\Omega}$, p. 1.3.2

• $\bullet$

$\{z_{1},\ldots,z_{\mathsf{k}_{1}^{\partial\mathsf{C}_{\mathsf{max}}}}\}$, and $\mathsf{k}_{1}^{\partial\mathsf{C}_{\mathsf{max}}}$, p. 1.3.2

• $\bullet$

$a_{i}$, p. 27

• $\bullet$

$\mathsf{C}(\lambda,x)$, p. 14

• $\bullet$

$\mathsf{C}^{+}(\lambda,x)$, p. 14

• $\bullet$

$\lambda(x)$, p. 14

• $\bullet$

$\mathsf{N}_{1}$ and $\{\mathsf{C}_{1},\dots,\mathsf{C}_{N_{1}}\}$, p. 16

• $\bullet$

$\mathsf{E}_{1,\ell}$ ($\ell\in\{1,\ldots,\mathsf{N}_{1}\}$), p. 16 (labeled with the lexicographic order in p. 3.2.3)

• $\bullet$

$\mathsf{U}_{1}^{\mathsf{ssp}}$, page 1

• $\bullet$

$\mathcal{C}_{crit}$, page 2

• $\bullet$

$\mathbf{j}$ and $\mathbf{\widetilde{j}}$, page 2.4

• $\bullet$

$x_{k,l}$ ($k\geq 1$, $\ell\in\{1,\ldots,\mathsf{N}_{k}\}$), page 1 (labeled with the lexicographic order in p. 3.2.3)

• $\bullet$

$\mathsf{E}_{k,l}$ ($k\geq 2$, $\ell\in\{1,\ldots,\mathsf{N}_{k}\}$), page 2 (labeled with the lexicographic order in p. 3.2.3)

• $\bullet$

$\mathsf{N}_{2},\mathsf{N}_{3},\mathsf{N}_{4},\ldots$, page 2

• $\bullet$

$\Lambda^{p}C^{\infty}(\overline{\Omega})$, p. 3.1.1

• $\bullet$

$\Lambda^{p}C^{\infty}_{T}(\overline{\Omega})$, p. 3.1.1

• $\bullet$

$\Lambda^{p}L^{2}_{w}(\Omega)$ and $\Lambda^{p}H^{q}_{w}(\Omega)$, p. 3.1.1

• $\bullet$

$\Lambda^{p}H^{q}_{w,T}(\Omega)$, p. 3.1.1

• $\bullet$

$\Lambda^{p}L^{2}(\Omega)$ and $\Lambda^{p}H^{q}(\Omega)$, p. 3.1.1

• $\bullet$

$\Lambda^{p}H^{q}_{T}(\Omega)$ and $\Lambda^{p}H^{q}_{N}(\Omega)$, p. 3.1.1

• $\bullet$

$\|.\|_{H^{q}_{w}}$ and $\langle\,,\,\rangle_{L^{2}_{w}}$, p. 3.1.1

• $\bullet$

$\|.\|_{H^{q}}$ and $\langle\,,\,\rangle_{L^{2}}$, p. 3.1.1

• $\bullet$

$\Delta^{(p)}_{f,h}$, p. 3.1.2

• $\bullet$

$\Delta^{D,(p)}_{f,h}(\Omega)$, p. 3.1.2

• $\bullet$

$L^{D,(p)}_{f,h}(\Omega)$, p. 3.1.2

• $\bullet$

$\pi_{E}$, p. 3.1.2

• $\bullet$

For $p\in\{0,\dots,d\}$ and $\pi^{(p)}_{h}$, p. 3.1.2

• $\bullet$

$\widetilde{v}_{k,\ell}$ and $\chi_{k,\ell}$, p. 29 (labeled with the lexicographic order in p. 3.2.3)

• $\bullet$

$\Phi_{j}$, p. 3.2.2 and 3.2.2

• $\bullet$

$w_{j}$, p. • ‣ 33 and • ‣ 38

• $\bullet$

$\theta_{j}$ and $\tilde{\phi}_{j}$, p. 34 and 39

• $\bullet$

$u^{(1)}_{j,wkb}$ p. 3.2.2 and 40

• $\bullet$

$c_{j}(h)$, p. 36 and 41

• $\bullet$

$\widetilde{\phi}_{j,wkb}$, p. 42

• $\bullet$

$\widetilde{u}_{k}$ and $\widetilde{\psi}_{j}$, page 45

• $\bullet$

$\lambda_{2,h}$, p. 3

• $\bullet$

$B_{j}$, p. 4.1

• $\bullet$

$p_{j,k}$, $C_{j,k}$, and $\varepsilon_{j,k}$, p. 50

• $\bullet$

$S$ and $S_{j,k}$, p. 51

• $\bullet$

$\widetilde{S}$ and $\widetilde{S}_{j,k}$, p. • ‣ 4.2.1

• $\bullet$

$D$, $D_{k,k}$, and $q_{k}$ p. • ‣ 4.2.1

• $\bullet$

$\widetilde{C}$ and $\widetilde{C}_{j,k}$, p. • ‣ 4.2.1

• $\bullet$

$\mathsf{S}_{k}$, p. 5

• $\bullet$

$\mu_{i}(T)$, p. 54

• $\bullet$

$C_{0}$, $C_{1}$, p. 55

• $\bullet$

$\lambda_{k,h}$, p. 56

• $\bullet$

$\varepsilon_{h}$, p. (ii)

• $\bullet$

$\widetilde{\pi}^{(0)}_{h}$, p. 61

• $\bullet$

$\kappa_{ji}$ p. 5.2.1

• $\bullet$

$Z_{j}$ and $\psi_{j}$, p. 220

• $\bullet$

$v_{h}$, p. 232

• $\bullet$

$\mathsf{C}_{r}$ ($r>0$), p. 6.2.1