Nondegeneracy and stability in the limit of a one-phase singular perturbation problem

TL;DR

This paper investigates the stability and nondegeneracy of solutions to a one-phase singular perturbation problem relevant in combustion theory, establishing conditions for flatness of solutions in low dimensions.

Contribution

It introduces a density condition ensuring nondegeneracy preservation in the limit and classifies stable solutions with flat level sets in dimensions up to four.

Findings

Density condition guarantees nondegeneracy in the limit

Stable solutions have flat level sets in dimensions ≤ 4

New formulas for first and second inner variations derived

Abstract

We study solutions to a one-phase singular perturbation problem that arises in combustion theory and that formally approximates the classical one-phase free boundary problem. We introduce a natural density condition on the transition layers themselves that guarantees that the key nondegeneracy growth property of solutions is satisfied and preserved in the limit. We then apply our result to the problem of classifying global stable solutions of the underlying semilinear problem and we show that those have flat level sets in dimensions $n\leq 4$, provided the density condition is fulfilled. The notion of stability that we use is the one with respect to inner domain deformations and in the process, we derive succinct new formulas for the first and second inner variations of general functionals of the form $I(v) = \int |\nabla v|^2 + \mathcal{F}(v)$ that hold in a Riemannian manifold setting.

Equations

$$I_{\epsilon }(v,\Omega ):={\int }_{\Omega }\left(|\nabla v{|}^2+{\mathcal{F}}_{\epsilon }(v)\right)dx,$$

$${\mathcal{F}}_0(t):=1_{(0,\infty )}(t),$$

$${\mathcal{F}}_{\epsilon }(t):=\{\begin{array}{ll}{\int }_0^{t}2f_{\epsilon }(s)ds,& \text{for }t\ge 0,\\ 0& \text{for }t<0,\end{array}$$

$$f\ge 0,\text{supp }f=[0,T],{\int }_0^{T}2f(s)ds=1,$$

$$c_{0}s\le f(s)\le c_0^{-1}s\text{when }s\in [0,\tau ],$$

$$\{\begin{array}{rlr}{u}_{\epsilon }& \ge 0& \text{mbox}in\Omega ,\\ \Delta u_{\epsilon }& =f_{\epsilon }(u_{\epsilon })& \text{mbox}in\Omega ,\end{array}$$

$$I_0(v,\Omega ):={\int }_{\Omega }\left(|\nabla v{|}^2+{\mathcal{F}}_0(v)\right)dx,v:\Omega \to [0,\infty ),$$

$$\{\begin{array}{rl}u\ge 0& \text{mbox}in\Omega ,\\ \Delta u=0& \text{mbox}in{\Omega }_0^{+}(u):=\{x\in \Omega :u(x)>0\},\\ |\nabla u|=1& \text{mbox}on{F}_0(u):=\partial {\Omega }_0^{+}(u)\cap \Omega ,\end{array}$$

$$\underset{B_r(x)}{\sup }{u}_{\epsilon }\ge cr,$$

$$\underset{B_r(x)}{\sup }u\ge cr,\text{for all }x\in \overline{{\Omega }_0^{+}(u)}\text{and all }{B}_r(x)\subset \Omega ,$$

$$|Z_0(u)\cap B_r(x)|\ge \kappa |B_r|\text{for all }x\in Z_0(u),\text{ and all }{B}_r(x)\subset \Omega ,$$

$$(V_{\epsilon }^s{)}^{\prime \prime }=f_{\epsilon }(V_{\epsilon }^s)\text{in }\mathbb{R},\text{with}(V_{\epsilon }^s{)}^{\prime }(0)=0\text{and}\underset{t\to \pm \infty }{\lim }(V_{\epsilon }^s{)}^{\prime }(t)=\pm s.$$

$$Z_{\epsilon }^{\theta }(u_{\epsilon }):=\{x\in \Omega :u_{\epsilon }(x)\le \theta \epsilon \}\text{and}{F}_{\epsilon }^{\theta }(u_{\epsilon }):=\{x\in \Omega :\theta \epsilon \le u_{\epsilon }(x)\le T\epsilon \}$$

$$|Z_{\epsilon }^{\tau /4}(u_{\epsilon })\cap B_{r/2}(x)|\ge \kappa |B_{r/2}|\text{for all }x\in F_{\epsilon }^{\tau }(u_{\epsilon })\text{ and all }r\ge L\epsilon ,\text{ such that }{B}_r(x)\subseteq \Omega .$$

$$\underset{B_r(p)}{\sup }{u}_{\epsilon }\ge cr.$$

$$\{\begin{array}{rlr}u& \ge 0& \text{mbox}in{\mathbb{R}}^n,\\ \Delta u& =f(u)& \text{mbox}in{\mathbb{R}}^n,\end{array}$$

$$V^{\prime \prime }=f(V)\text{in }\mathbb{R},\text{with}V(0)=T\text{and}{V}^{\prime }(0)=1.$$

$$\underset{x_n\to -\infty }{\lim }u(x^{\prime },x_n)=0\text{and}\underset{x_n\to \infty }{\lim }u(x^{\prime },x_n)=\infty ,$$

$$I_1^{\prime \prime }(u,\Omega )[\varphi ]:={\frac{d^2}{d{t}^2}|}_{t=0}{I}_1(u+t\varphi ,\Omega )=2{\int }_{\Omega }\left(|\nabla \varphi {|}^2+f^{\prime }(u){\varphi }^2\right)dx\ge 0\text{for all }\varphi \in C_c^{\infty }(\Omega ).$$

$${\partial }_t{\varphi }_t(x)=X({\varphi }_t(x)),{\varphi }_0(x)=x.$$

$$\delta I_{\epsilon }(u,\Omega )[X]:={\frac{d}{dt}|}_{t=0}{I}_{\epsilon }(u({\varphi }_t^{-1}(x)),\Omega )\text{and}{\delta }^2{I}_{\epsilon }(u,\Omega )[X]:={\frac{d^2}{d{t}^2}|}_{t=0}{I}_{\epsilon }(u({\varphi }_t^{-1}(x)),\Omega ).$$

$${\delta }^2{I}_{\epsilon }(u,\Omega )[X]\ge 0\text{for all }X\in C_c^{\infty }(\Omega ;{\mathbb{R}}^n).$$

$${\delta }^2{I}_1(u,\Omega )[X]=I_1^{\prime \prime }(u,\Omega )[⟨\nabla u,X⟩]\forall X\in C_c^{\infty }(\Omega ;{\mathbb{R}}^n).$$

$$|B_R(x)\cap \{u\le \tau /4\}|\ge \kappa |B_R(x)|\text{for all }x\in \{\tau \le u\le T\}\text{ and all large }R\ge L.$$

$$5\le n^{\ast }\le n_e^{\ast }\le 7.$$

$${\mathcal{F}}_{\epsilon }(u_{\epsilon })\to {\mathcal{F}}_0(u_0)\text{in }{L}_{\text{loc}}^1(\Omega )\text{as }\epsilon \downarrow 0.$$

$${\int }_{F_0(u_0)}H{\varphi }^{2}d{\mathcal{H}}^{n-1}\le {\int }_{{\Omega }_0^{+}(u)}|\nabla \varphi {|}^2\text{for all }\varphi \in C_c^{\infty }({\mathbb{R}}^n\setminus \{0\});$$

$$\frac{1}{2}{\delta }^2{I}_0(u_0,\Omega )[X]={\int }_{{\Omega }_0^{+}(u_0)}|\nabla (L_X{u}_0){|}^{2}dx-{\int }_{F_0(u_0)}H(L_X{u}_0{)}^{2}d{\mathcal{H}}^{n-1},$$

$$\Vert \nabla u_{\epsilon }{\Vert }_{L^{\infty }(B_1)}\le C$$

$$Z_{\epsilon }^{\theta }(u_{\epsilon }):=\{x\in \Omega :u_{\epsilon }(x)\le \theta \epsilon \}\text{and}{F}_{\epsilon }^{\theta }(u_{\epsilon }):=\{x\in \Omega :\theta \epsilon \le u_{\epsilon }(x)\le T\epsilon \}$$

$$u_{\epsilon }(0)\ge \theta \epsilon \text{for some }\theta \in (0,\tau ).$$

Full text

The author was partially supported by Proyecto FONDECYT Regular No. 1201087.

1. Introduction

The present paper aims to contribute to the understanding of the limit behaviour of nonnegative critical points of the energy functional

[TABLE]

in which $\Omega\subseteq\mathbb{R}^{n}$ is a domain and the potential $\mathcal{F}_{\varepsilon}(t)$ approximates the characteristic function

[TABLE]

as $\varepsilon\downarrow 0$. Specifically, we will be interested in potentials $\mathcal{F}_{\varepsilon}$ of the form

[TABLE]

where $f_{\varepsilon}(t):=\varepsilon^{-1}f(t/\varepsilon)$ for a given nonnegative function $f\in C^{\infty}_{c}([0,\infty))$, satisfying

[TABLE]

for some constants $0<\tau<T<\infty$ and $c_{0}>0$. Note that hypothesis (1.4) is simply a quantitative way of expressing $f^{\prime}(0)>0$ (which can be relaxed – see the discussion after Theorem 1.2 below).

For any $\varepsilon>0$, nonnegative critical points $u_{\varepsilon}\in H^{1}_{\text{loc}}(\Omega)$ of $I_{\varepsilon}$ solve the semilinear elliptic PDE

[TABLE]

in a weak sense. As $f_{\varepsilon}(u_{\varepsilon})\in L^{\infty}(\Omega),$ the Harnack inequality implies that a solution $u_{\varepsilon}$ of (1.5) must be locally bounded, while because of (1.4), the strong maximum principle yields that either $u_{\varepsilon}>0$ a.e. or $u_{\varepsilon}\equiv 0$ in $\Omega$. Semilinear elliptic regularity theory then tells us that $u_{\varepsilon}$ is actually a smooth, classical solution of (1.5), that is either identically zero, or strictly positive.

The functionals $I_{\varepsilon}(v,\Omega)$ formally converge as $\varepsilon\downarrow 0$ to the Alt-Caffarelli energy functional

[TABLE]

whose associated Euler-Lagrange equations form the classical one-phase free boundary problem (FBP)

[TABLE]

in which the set $\Omega_{0}^{+}(u)$ is the positive phase of $u$, its complement $Z_{0}(u):=\{x\in\Omega:u(x)=0\}$ is its zero phase, while the abrupt interface $F_{0}(u)$, caused by the discontinuity of $\mathcal{F}_{0}$, is known as the free boundary.

The energy functional $I_{\varepsilon}$ appears in models of flame propagation ([BL08]) and there has been substantial literature devoted to understanding the underlying singular perturbation problem (1.5) and its parabolic counterparts (we refer the reader to [BCN90, CV95, CLW97, Wei03, LW06b, Kar20] and references therein). Of particular interest has been exploring the sense in which critical points $u_{\varepsilon}$ of $I_{\varepsilon}$ and their transition layers $\{\theta\varepsilon\leq u_{\varepsilon}\leq T\varepsilon\}$, $\theta\in(0,T]$, converge to solutions $u$ of (1.7) and their free boundaries $F_{0}(u)$, respectively, and how regular the latter are.

The case of nonnegative critical points $u_{\varepsilon}$ that locally mimimize the energy $I_{\varepsilon}$ was studied in detail in the book by Caffarelli and Salsa [CS05]. The analysis of the interface convergence as well as the preliminary, measure-theoretic regularity of the resulting free boundary rests on the two fundamental estimates of uniform Lipschitz continuity (see Proposition 2.1 below) and uniform nondegeneracy. The latter precisely states (see [CS05, Theorem 1.8] or [AS22, Lemma 4.2]) that at a distance $r\geq\lambda\varepsilon$ away from points $x\in\Omega$, where $u_{\varepsilon}(x)\geq\theta\varepsilon$ for a fixed $\theta\in(0,T]$, the solution grows to be at least a multiple of $r$:

[TABLE]

for some constants $c,\lambda>0$. The nondegeneracy property underpins the local Hausdorff distance convergence of the superlevel sets $\{u_{\varepsilon}\geq\theta\varepsilon\}$ to the positive phase $\Omega^{+}_{0}(u)$ of the limit $u$. Being passed down to $u$,

[TABLE]

it is then instrumental in the blow-up analysis that explores the regularity of the free boundary $F_{0}(u)$ and the sense in which $u$ solves (1.7). At this stage, there is another key basic estimate at play: the positive density of the zero phase $Z_{0}(u)$, which states that

[TABLE]

for some constant $\kappa>0$. The positive density estimate (1.10) is essential in ruling out the possibility of a blowup limit $u_{0}$ of $u$ that is of wedge type: $u_{0}(x)=s|x_{n}|$ for some $s>0$, which is a vestige of a singularity in $F_{0}(u)$.

Whereas the uniform Lipschitz continuity continues to hold for solutions of (1.5) that are not necessarily energy minimizing, the nondegeneracy property does not and neither is valid the positive zero-phase density in the limit. This is illustrated by the family of one-dimensional, wedge-like, solutions $\{V_{\varepsilon}^{s}(t)\}_{\varepsilon>0,s\in(0,1)}$, given by the unique solutions to the ODE problem:

[TABLE]

which blow down to the wedge $s|t|$ for slopes $s\in(0,1)$ (see [LW06a, Proposition 3.1] or [FRRO19, Section 2.3]). As $V_{\varepsilon}^{s}(0)\in(0,T\varepsilon)$, these solutions have a nontrivial interface region, and the 1D solutions of (1.5) in $\mathbb{R}^{n}$, given by $u_{\varepsilon}(x)=V_{\varepsilon}^{\varepsilon}(x_{n}),$ certainly fail the uniform nondegeneracy estimate (1.8), since they tend to [math], as $\varepsilon\downarrow 0$.

The case of general critical points of $I_{\varepsilon}$ (nonnegative as well as sign-changing) was studied in depth in a series of papers by Lederman and Wolanski [LW98, LW06a, LW06b]. For the one-phase singular perturbation scenario, the authors showed that $u_{\varepsilon}$ converge locally uniformly to a limit $u$ which is harmonic in $\Omega_{0}^{+}(u)$ and which satisfies the free boundary gradient condition in viscosity sense (see Definition 4.6) as well as pointwise at regular points of $F_{0}(u)$, provided the limit $u$ satisfies the nondegeneracy condition (1.8). Assuming additionally the positive density condition (1.10) on the zero phase of $u$, they then obtained that the free boundary $F_{0}(u)$ is a smooth hypersurface, except on a relatively closed subset of $(n-1)$ Hausdorff measure zero.

We would like to emphasize that in the cited results above, the additional hypotheses leading to a good regularity theory are made on the limit $u$, and not on the critical points $u_{\varepsilon}$ of $I_{\varepsilon}$. The first objective of our paper is to identify a natural condition on the solutions $u_{\varepsilon}$ of (1.5) themselves that guarantees that the limit $u$ will inherit both key properties (1.9)-(1.10). We achieve it by introducing the notion of $\mathcal{D}(\kappa,L)$ density property of the interface of $u_{\varepsilon}$. Denote by

[TABLE]

the two parts of the transition region $\{u_{\varepsilon}\leq T\varepsilon\}$, divided by the level set $\{u_{\varepsilon}=\theta\varepsilon\}$, for $\theta\in(0,T]$.

Definition 1.1**.**

We will say that (the interface of) $u_{\varepsilon}$ satisfies the density property $\mathcal{D}(\kappa,L)$ in $\Omega$ for some $\kappa\in(0,1]$ and $L>0$ if

[TABLE]

Here $\tau$ refers to the constant in (1.4).

The condition (1.13) is a natural one that minimizers of $I_{\varepsilon}$, in particular, fulfill for universal positive constants $\kappa,L$ (see Proposition 3.4). It is not difficult to envision why the limit $u$ of solutions $u_{\varepsilon}$ of (1.5), which satisfy a $\mathcal{D}(\kappa,L)$ density property uniformly as $\varepsilon\downarrow 0$, will inherit the positive density (1.10) of the zero phase $Z_{0}(u)$. What is less obvious is that this property actually guarantees that the $u_{\varepsilon}$ satisfy the uniform nondegeneracy bound (1.8). This is the content of our first main result.

Theorem 1.2**.**

Let $\kappa\in(0,1]$, $L>0$ and $\theta\in(0,\tau]$. There exist positive constants $c$ and $\varepsilon_{0}$, depending on $\kappa,L,n$ and $f$, and a constant $M>0$, depending on $\theta,n$ and $f$, such that if $\varepsilon\leq\varepsilon_{0}$ and $u_{\varepsilon}\in C^{2}(B_{2})$ is a solution of (1.5) in $B_{2}$, for which

1. (1)

$u_{\varepsilon}(0)\leq T\varepsilon$, and 2. (2)

the interface of $u_{\varepsilon}$ satisfies the density property $\mathcal{D}(\kappa,L)$ in $B_{1}$,

then for all $p\in\{x\in B_{1/4}:u_{\varepsilon}(x)\geq\theta\varepsilon\}$ and all $r\geq 2\max(L,M)\varepsilon$ such that $B_{r}(p)\subset B_{1}$,

[TABLE]

Assumption (1) above is made to ensure that $u_{\varepsilon}$ satisfies the universal Lipschitz bound in $B_{1}$. The proof of Theorem 1.2 is achieved in several stages over Section 2, in which the $\mathcal{D}(\kappa,L)$ density property hypothesis (2) is first crucially utilized in a Poincaré-Sobolev type estimate (see key Lemma 2.4) to get the nondegeneracy growth away from points $p$ in the transition layer $F_{\varepsilon}^{\tau}$, and later in a limiting argument to extend it for points $p\in\{u_{\varepsilon}>T\varepsilon\}$, for small enough $\varepsilon>0$. The condition (1.4) that we impose on the nonlinearity $f$, allows us to handle points in the remaining layer $\{\theta\varepsilon\leq u_{\varepsilon}\leq\tau\varepsilon\}\subset Z_{\varepsilon}^{\tau}$, since it entails that $u_{\varepsilon}$ experiences exponential growth in $Z_{\varepsilon}^{\tau}$ (see Lemma 2.2). Just as in [AS22, Remark 2.3], one can relax (1.4) to the assumption $\liminf_{s\downarrow 0}f(s)s^{-p}>0$ for some $p\geq 1$, with virtually no effect on the proof of the theorem (with the only change being that the exponential growth of $u_{\varepsilon}$ in $Z_{\varepsilon}^{\tau}$ is replaced by a polynomial one, leading to a slightly different constant $M$).

Our motivation to find conditions under which solutions of the singular perturbation problem (1.5) enjoy the uniform nondegeneracy property sprang from the recent progress in classifying global nonnegative solutions of the semilinear equation (1.5) for $\varepsilon=1$:

[TABLE]

and, in particular, the solutions that locally minimize the energy $I_{1}$. Taking into consideration that their blow-downs $u_{\varepsilon}(x):=\varepsilon u(x/\varepsilon)$ are local minimizers of $I_{\varepsilon}$ that converge to globally defined, energy minimizing, homogeneous solutions of the one-phase FBP (1.7), Fernández-Real and Ros-Oton formulated a natural conjecture, akin to the celebrated De Giorgi conjecture [DG79] for the Allen-Cahn equation.

Conjecture 1.3** ([FRRO19]).**

Suppose that $u:\mathbb{R}^{n}\to(0,\infty)$ minimizes the energy $I_{1}$ locally. Then for $n\leq n_{e}^{*}-1$, $u$ has to be one-dimensional, i.e. $u(x)=V(x_{n})$ in a suitable Euclidean coordinate system, where $V(t)$ is the unique (positive) solution to the ODE problem

[TABLE]

Here $n^{*}_{e}$ denotes the lowest dimension in which there exists a global singular homogeneous minimizer of $I_{0}$. By the works of Caffarelli-Jerison-Kenig [CJK04], Jerison-Savin [JS15] and De Silva-Jerison [DSJ09], it is currently known that $5\leq n^{*}_{e}\leq 7.$

The conjecture was recently established by Audrito and Serra [AS22] who devised for the context an “improvement of flatness” technique inspired by Savin’s proof [Sav09] of the De Giorgi conjecture, by bulding upon De Silva’s regularity theory method [DS11] for the one-phase FBP. Their result holds more generally for any critical point $u$ of $I_{1}$ in any dimension, provided $u$ has an asymptotically flat interface and blows down to $x_{n}^{+}$. The Audrito-Serra theorem has since been used by Engelstein, Fernández-Real and Yu [EFRY22] in proving that global solutions of (1.5) that are monotone in $x_{n}$ and satisfy

[TABLE]

have to be one-dimensional in dimensions $n\leq n^{*}_{e}$.

There is another, stronger version of Conjecture 1.3 that concerns more broadly global stable critical points of $I_{1}$.

Conjecture 1.4** ([FRRO19]).**

Suppose that $u:\mathbb{R}^{n}\to(0,\infty)$ is a stable critical point of $I_{1}$ in $\Omega=\mathbb{R}^{n}$, i.e. the second variation of $I_{1}$ at $u$

[TABLE]

Then for $n\leq 4$, $u(x)=V(x_{n})$ in an appropriate Euclidean coordinate system, where $V$ is the solution of (1.16).

The rigidity statement in Conjecture 1.4 is currently known to be true only for $n=2$ [FV09]. What seems to make this version more challenging (if one is to employ the blow-down strategy) is a lack of understanding if blow-down limits of $u$ even solve the one-phase FBP (1.7) in certain weak sense, let alone what notion of stability is preserved in the limit. To start, it is not known if the strong nondegeneracy property (1.8) holds for stable solutions of (1.15). The implementation of the strategy is further impeded by the possibility of wedge blow-down limits $s|x_{n}|$, $s\in(0,1]$, which afflict the study of the rigidity problem for global stable solutions of the one-phase FBP (1.7) itself (see [KW23]).

In our second main result we prove that if blow-downs of $u$ satisfy a $\mathcal{D}(\kappa,L)$ density property uniformly, then a weaker notion of stability is preserved in the limit thanks to the nondegeneracy Theorem 1.2. This enables the blow-down strategy to be executed, yielding the rigidity result in Conjecture 1.4. The precise notion of stability that we employ is the one with respect to compact domain deformations.

Definition 1.5**.**

Let $\Omega\subseteq\mathbb{R}^{n}$ be a domain and let $X\in C^{\infty}_{c}(\Omega;\mathbb{R}^{n})$ be a smooth, compactly supported vector field. Denote by $\phi:\mathbb{R}\times\Omega\to\Omega$ its associated flow in $\Omega$:

[TABLE]

The first and second inner variations of the functional $I_{\varepsilon}(\cdot,\Omega)$, $\varepsilon\geq 0$, at $u\in H^{1}_{\text{loc}}(\Omega)$, along the vector field $X$ are given respectively by

[TABLE]

A critical point $u_{\varepsilon}$ of $I_{\varepsilon}(\cdot,\Omega)$ is stable with respect to compact domain deformations if

[TABLE]

Note that if $u$ is a positive, stable critical point of $I_{1}(\cdot,\Omega)$ (in the sense of (1.18)), then it is also stable with respect to compact domain deformations, since (see Proposition A.5)

[TABLE]

The grace of the stability notion in Definition 1.5 is that it behaves very well under taking limits. This becomes manifest from the succinct formulas (3.1)-(3.2) that we derive for the first and second inner variations of $I_{\varepsilon}$, which also hold for $\varepsilon=0$ (and, in fact, apply to general potentials $\mathcal{F}_{\varepsilon}$ inside the functional $I_{\varepsilon}$). The formulas appear in a different (albeit longer) form already in [Le11] in the context of the Allen-Cahn equation, but here we derive them with the apparatus of differential geometry which, we insist, provides the right conceptual framework for the calculations (see Appendix A). In this way, we produce formulas (Proposition A.1) that are valid for general Riemannian manifolds.

We can now state our second main result.

Theorem 1.6**.**

Let $u\in C^{\infty}(\mathbb{R}^{n})$ be a global positive solution of (1.15) that is stable with respect to compact domain deformations. Assume further that there exist constants $\kappa\in(0,1]$ and $L>0$ such that

[TABLE]

Let $n^{*}$ be the critical dimension from Definition 4.10 (explained also below), which satisfies

[TABLE]

If $n\leq n^{*}-1$, then in appropriate Euclidean coordinates, $u(x)=V(x_{n})$, where $V(t)$ is given by the solution of (1.16).

We establish Theorem 1.6 by showing that $u$ has asymptotically flat interface in dimensions $n\leq n^{*}-1$ and then invoking the result of Audrito and Serra [AS22, Theorem 1.4]. In order to prove the asymptotic flatness of the interface, we build a general theory of convergence of solutions $u_{\varepsilon}$ to (1.5) in a domain $\Omega\subseteq\mathbb{R}^{n}$, which are stable with respect to compact domain deformations and satisfy the density property $\mathcal{D}(\kappa,L)$, as $\varepsilon\downarrow 0$. The nondegeneracy result in our first Theorem 2.4 ensures the Hausdorff distance convergence of the interface of $u_{\varepsilon}$ to the free boundary of the limit $u_{0}$, as well as the key convergence

[TABLE]

The latter, along with the known $H^{1}_{\text{loc}}$-convergence of $u_{\varepsilon}$ to $u_{0}$, permits stability (in the sense of Definition 1.5) to be preserved in the limit.

To encapsulate all the good properties of the limiting function $u_{0}$, we employ the notion of inner-stable solution to the one-phase FBP (see Definition 4.1 below), introduced recently in [BMM*+*22]. This type of weak stable solution of (1.7) shares much of the same regularity theory as minimizers of the Alt-Caffarelli functional $I_{0}$. In particular, the free boundary $F_{0}(u_{0})$ is a smooth hypersurface, except possibly on a closed singular subset of Hausdorff dimension at most $n-n^{*}$, where $n^{*}$ is precisely the lowest dimension which admits a singular homogeneous inner-stable solution. Since local minimizers of $I_{0}$ are inner-stable solutions themselves (see Remark 4.2), one trivially has $n^{*}\leq n^{*}_{e}$. The lower bound $n^{*}\geq 5$ was proved in [BMM*+*22] by showing that the nonnegative second inner variation condition implies the stability inequality of Caffarelli-Jerison-Kenig [CJK04] for free boundary cones $u_{0}$ with an isolated singularity at the origin:

[TABLE]

here $H$ denotes the mean curvature of $F_{0}(u_{0})$ with respect to the outer unit normal to $\partial\Omega_{0}^{+}(u_{0})$. It is only the partial information that energy minimizing cones with an isolated singularity satisfy (1.20), in conjunction with the dimension reduction argument of Weiss [Wei98], that is used in [JS15] to obtain the bound $n^{*}_{e}\geq 5$. Given that the inner-stable solution class also admits a dimension reduction principle, the same lower bound holds for $n^{*}$.

In Section 4 of our paper we present the regularity theory of inner-stable solutions to the one-phase FBP, developed by [BMM*+*22], almost in its entirety. The reason is two-fold. First, we do it for the reader’s convenience, and second – because we are naturally guided to use our elementary formula (3.2) for the second inner variation $\delta^{2}I_{0}$, which is aligned with the (formal) convergence of $\delta^{2}I_{\varepsilon}$ to $\delta^{2}I_{0}$, in lieu of their more sophisticated formula [BMM*+*22, (7.8)-(7.9)]. The tools of differential geometry allow us to perform the computations leading to the stability inequality (1.20) in a transparent, methodical fashion, and in fact, we show that for any test vector field $X\in C^{\infty}_{c}(\Omega;\mathbb{R}^{n})$ that avoids the singular part of the free boundary $F_{0}(u_{0})$ of a one-phase FBP solution $u_{0}$, the second inner variation of the Alt-Caffarelli energy $I_{0}$ at $u_{0}$ along $X$ has the representation formula (see Proposition B.3):

[TABLE]

where $L_{X}$ denotes the Lie derivative along $X$ (which coincides with the directional derivative when applied to functions). Since $\nabla u\neq 0$ in $\Omega^{+}_{0}(u)$ for homogeneous one-phase FBP solutions $u$, the condition $\delta^{2}I_{0}(u)[X]\geq 0$ is equivalent to (1.20).

The paper is organized as follows. In Section 2 we prove several nondegenerecy estimates for solutions of (1.5), which ultimately lead to the proof of Theorem 1.2. In Section 3 we state the formulas (3.1)-(3.2) for the first and second inner variations of the energies $I_{\varepsilon}$, $\varepsilon\geq 0$, in the Euclidean setting. We then build a convergence theory for solutions $u_{\varepsilon}$ of (1.5), which satisfy the strong nondegeneracy property (1.8) for all $\theta\in(0,T]$. In particular, we show that their limits have trivial first inner variation $\delta I_{0}$. Section 4 describes the regularity theory of inner-stable solutions to the one-phase FBP, developed by [BMM*+*22]. Finally, in Section 5 we show that solutions $u_{\varepsilon}$ of (1.5) that are stable with respect to compact domain deformations and satisfy a $\mathcal{D}(\kappa,L)$ property uniformly for all small $\varepsilon>0$, converge to inner-stable solutions of (1.7) as $\varepsilon\downarrow 0$. As a corollary, we obtain the proof of Theorem 1.6.

In the two appendices A and B to the article, we provide the key technical results related to the first and second inner variation for the functionals $I_{\varepsilon}$, $\varepsilon\geq 0$, which support the exposition in Sections 3-4. Appendix A is devoted to the computation of $\delta I_{\varepsilon}$ and $\delta^{2}I_{\varepsilon}$ in the setting of a general Riemannian manifold (see Proposition A.1); its reading requires a very basic acquaintance with tensor calculus. In it we also expand on the divergence structure of the integrands appearing in the integral formulas for the inner variations (see Lemmas A.2 and A.4). This latter information is then exploited in Appendix B, in which we simplify the formulas for $\delta I_{0}$ and $\delta^{2}I_{0}$ in the Euclidean setting and establish the formula (1.21) for $\delta^{2}I_{0}(u_{0})$ at a critical point $u_{0}$ of the Alt-Caffarelli energy.

With great pleasure I dedicate this paper to David Jerison on the occasion of his 70th birthday. I am profoundly grateful for all the math that I have learned and continue to learn from him, for his generosity, guidance and friendship.

2. Nondegeneracy estimates

The goal of this section is to establish Theorem 1.2, which we do in a sequence of nondeneracy estimates. Before we start with these, we record the uniform interior Lipschitz bound that solutions of (1.5) satisfy.

Proposition 2.1** (Uniform Lipschitz continuity; see Theorem 1.2 of [CS05]).**

Let $u_{\varepsilon}\in C^{2}(B_{2})$ be a solution of (1.5) in $B_{2}$ and assume that $0\in\{u_{\varepsilon}\leq T\varepsilon\}$. Then

[TABLE]

for some constant $C=C(n,f)>0$.

We also recall the notation set earlier in (1.12):

[TABLE]

from which we will often drop the reference to $u_{\varepsilon}$, whenever it is implicit.

The first nondegeneracy lemma can be viewed as the statement that solutions $u_{\varepsilon}$ of (1.5) experience (exponential) growth inside the set $Z_{\varepsilon}^{\tau}(u_{\varepsilon})$.

Lemma 2.2**.**

Let $u_{\varepsilon}\in C^{2}(B_{1})$ be a solution of (1.5) in $B_{1}\subset\mathbb{R}^{n}$ and assume that

[TABLE]

Then there exists a constant $c_{1}=c_{1}(n)$ such that for $M:=2\log(c_{1}\tau/\theta)/\sqrt{c_{0}}=M(\theta,n,c_{0})$

[TABLE]

Proof.

Let $B_{R}(0)$ be the largest ball, centered at the origin, such that $B_{R}\subseteq Z_{\varepsilon}^{\tau}$. We would like to show that $R\leq M\varepsilon$. We notice that

[TABLE]

and thus $v(x):=u_{\varepsilon}(\varepsilon x/\sqrt{c_{0}})/\varepsilon$ solves

[TABLE]

where $R_{1}:=R\sqrt{c_{0}}/\varepsilon$. Defining $w\in C^{2}(B_{R_{1}})\cap C(\overline{B_{R_{1}}})$ to be the solution of $-\Delta w+w=0$ in $B_{R_{1}}$ with boundary values given by $v$, the maximum principle tells us that $v\leq w$ in $B_{R_{1}}$. Now, it is known (see [CC06, pp. 214]) that $w$ satisfies the weighted mean-value formula

[TABLE]

for some $c_{1}=c_{1}(n)$. Therefore, we have

[TABLE]

and we can conclude the desired bound

[TABLE]

∎

For the next nondegeneracy result we will need the following Poincaré-Sobolev inequality, whose proof can be adapted from [Eva10, Theorem 1 on pp. 290]:

Lemma 2.3**.**

Assume that $g\in W^{1,1}(B_{R})$ satisfies

[TABLE]

Then there exists a constant $C=C(\kappa)$ such that

[TABLE]

We now present our key uniform nondegeneracy lemma.

Lemma 2.4**.**

Let $u_{\varepsilon}\in C^{2}(B_{1})\cap C(\overline{B_{1}})$ be a solution of (1.5) in $B_{1}\subset\mathbb{R}^{n}$, $\varepsilon>0$. Suppose that

[TABLE]

[TABLE]

and that $u_{\varepsilon}$ satisfies the universal Lipschitz bound (2.1) in $B_{1}$. Then

[TABLE]

for some constant $\mu=\mu(n,\kappa,f)>0$.

Proof of Theorem 2.4.

Denote by

[TABLE]

We will carry out the proof in several steps. In what follows, the letters $C,c$ (possibly with indices and primes) will denote positive constants which depend only on $n$, $\kappa$, and $f$. Take $\rho\in(0,1/4]$.

Step 1. We start with the simple estimate

[TABLE]

Indeed, taking a standard, nonnegative cut-off function $\phi\in C^{2}_{c}(B_{1})$ such that $\phi\equiv 1$ in $B_{1-\rho}$ and $\|\phi\|_{C^{2}(B_{1})}\leq c/\rho^{2}$, we have

[TABLE]

Step 2. We will next show that

[TABLE]

by exploiting the observation that

[TABLE]

with constants depending only on $f$. For the purpose, consider $g:=(\mathcal{F}_{\varepsilon}(u_{\varepsilon})-\mathcal{F}_{\varepsilon}(\tau\varepsilon/4))^{+}$ and observe that

[TABLE]

because of the assumed universal Lipschitz bound of $u_{\varepsilon}$ in $B_{1}$. Furthermore, because of (2.5), $g$ vanishes inside $B_{1-\rho}$ on a set of measure at least $\kappa B_{1/2}$, so that we may apply the Poincaré-Sobolev inequality (2.3) to $g$ in $\Omega=B_{1-\rho}$, obtaining

[TABLE]

where the last inequality is a consequence of the bound (2.7) from Step 1. Now, we get

[TABLE]

Step 3. At this stage, we will obtain an $L^{1}(B_{1-\rho})$ bound on $u_{\varepsilon}$ in terms of the square of $\sigma=\sup_{B_{1}}u_{\varepsilon}$:

[TABLE]

First, we claim that

[TABLE]

Indeed, we have

[TABLE]

where the last inequality follows from the mean-value property, enjoyed by the subharmonic $u_{\varepsilon}$. As $u_{\varepsilon}(0)\geq\tau\varepsilon$, we confirm the validity of (2.11):

[TABLE]

Now, (2.10) follows after combining (2.11) with (2.8)

[TABLE]

Step 4. The $L^{1}$-estimate of the subharmonic $u_{\varepsilon}$ in $B_{1-\rho}$ entails a bound on the supremum of $u_{\varepsilon}$ on a slightly smaller scale. Indeed, since $u_{\varepsilon}$ is subharmonic, the function

[TABLE]

is increasing in $r$, so that for $\rho\in(0,1/4]$

[TABLE]

Furthermore, if $h$ is the harmonic function in $B_{1-3\rho/2}$ whose boundary values on $\partial B_{1-3\rho/2}$ are given by $u_{\varepsilon}$, we can estimate $\sup_{B_{1-2\rho}}u_{\varepsilon}$ via the maximum principle and the Poisson representation formula in $B_{1-3\rho/2}$:

[TABLE]

Now, the combination of (2.14), (2.13) and the estimate (2.10) from Step 3 yields

[TABLE]

Step 5. In this ultimate step we perform a standard iteration that produces a contradiction if $\sigma=\sup_{B_{1}}u_{\varepsilon}$ is too small. Denote

[TABLE]

The final estimate of Step 4 implies that for $\rho\in(0,1/4]$

[TABLE]

Since for $r\in(0,1)$ the blow-up

[TABLE]

is a nonnegative solution of $\Delta u=f_{\varepsilon/r}(u)$ in $B_{1}$ and satisfies the hypotheses (2.4)-(2.5), under which (2.16) was derived, we obtain, after rescaling, that for $0<r<R\leq 1$,

[TABLE]

In particular, we have that

[TABLE]

Setting $r_{0}=1$ and defining $r_{m}=r_{m-1}-2^{-m-1}$ iteratively for $m\in\mathbb{N}$, we see that $1/2<r_{m}<r_{m-1}\leq 1$, hence we are allowed to iterate (2.17):

[TABLE]

We claim that (2.18) implies that if $\sigma_{1}\leq\mu=\mu(n,\kappa,f)$ is small enough, then

[TABLE]

for some constant $\gamma=\gamma(n)>1$. Obviously, (2.19) is true for $m=0$, and assume it is true for index $m-1$. Using (2.18), we get that

[TABLE]

provided we choose $\gamma=2^{(n+1)}>1$ and $\sigma_{1}\leq\mu=\mu(n,\kappa,f)$ where $\mu C\gamma^{2}=1$. However, (2.19) leads to a contradiction, because for sufficiently large $m$

[TABLE]

We conclude that $\sigma_{1}>\mu$.

∎

As a corollary to Lemma 2.4, we get that solutions $u_{\varepsilon}$ of (1.5) grow linearly away from points of the interface $F_{\varepsilon}^{\theta}(u_{\varepsilon})$, whenever $u_{\varepsilon}$ possesses the density property $\mathcal{D}(\kappa,L)$ of Definition 1.1.

Corollary 2.5**.**

Let $u_{\varepsilon}\in C^{2}(B_{1})\cap C(\overline{B_{1}})$ be a nonnegative solution of (1.5) in $B_{1}$ that satisfies the Lipschitz estimate (2.1). Let $\kappa\in(0,1)$, $L>0$ and $\theta\in(0,\tau]$. If the interface of $u_{\varepsilon}$ has the density property $\mathcal{D}(\kappa,L)$, then for some positive constants $c=c(n,\kappa,f)$ and $M=M(n,\theta,f)$,

[TABLE]

Proof.

Assume that $B_{r}(p)\subset B_{1}$ with a radius

[TABLE]

where $L_{0}\geq L$. We will analyze the following two cases, in the process of which we will determine the size of $L_{0}$.

Case 1. Assume first that we are located at a point $p\in F_{\varepsilon}^{\tau}(u_{\varepsilon})$. After rescaling at $p$,

[TABLE]

we see that $\tilde{u}_{\varepsilon/r}$ satisfies all the hypotheses of Theorem 2.4. Therefore,

[TABLE]

and thus $\sup_{B_{r}(p)}u_{\varepsilon}\geq\mu r$.

Case 2. Suppose now that $p\in B_{1}$ is such that $\theta\varepsilon\leq u_{\varepsilon}(p)<\tau\varepsilon$. Lemma 2.2 informs us that for some $M=M(\theta,n,f)$

[TABLE]

Noting that (2.22) says

[TABLE]

let us choose $L_{0}=\max(M,L)$. In this way, (2.22) implies the existence of a point $\tilde{p}\in\overline{B_{r/2}(p)}$, where $u_{\varepsilon}(\tilde{p})\geq\tau\varepsilon$. Applying now the rescaling argument from Case 1 to $u_{\varepsilon}$ in $B_{r/2}(\tilde{p})\subset B_{r}(p)$ (which is permitted, as $r/2\geq L\varepsilon$ implies that $\tilde{u}_{2\varepsilon/r}(x):=u_{\varepsilon}(\tilde{p}+xr/2)/(r/2)$ satisfies the hypotheses of Theorem 2.4), we obtain

[TABLE]

∎

In the next lemma, we will use the $\mathcal{D}(\kappa,L)$ property to obtain an important distance nondegeneracy estimate in the spirit of [LW98, Lemma 5.1]. We introduce the notation

[TABLE]

to denote the $T\varepsilon$-superlevel set of $u_{\varepsilon}$ in a domain $\Omega$.

Lemma 2.6**.**

Let $u_{\varepsilon}\in C^{2}(B_{1})$ be a solution of (1.5) in $B_{1}\subset\mathbb{R}^{n}$ and assume that for some positive constants $C_{1},C_{2},\kappa$ and $L$, it satisfies

• •

the uniform Lipschitz estimate: $\|\nabla u_{\varepsilon}\|_{L^{\infty}(B_{1})}\leq C_{1}$;

• •

the uniform nondegeneracy condition: $\sup_{B_{r}(p)}u_{\varepsilon}\geq C_{2}r$ whenever p\in\partial\big{(}B_{1}^{+}(u_{\varepsilon})\big{)}\cap B_{1}, $B_{r}(p)\subseteq B_{1}$ and $r\geq 2L\varepsilon$;

• •

the density property $\mathcal{D}(\kappa,L)$ of Definition 1.1.

Then there exist positive constants $\varepsilon_{0},\mu_{0}$, such that for every $\varepsilon\leq\varepsilon_{0}$, we have

[TABLE]

Proof.

We will argue by contradiction. Assume that the statement of the proposition is false and we have a sequence of counterexamples $u_{\varepsilon_{k}}$ with $\varepsilon_{k}\downarrow 0$, for each of which there exists a point $y_{k}\in B_{1/2}^{+}(u_{\varepsilon_{k}})$ with $d_{k}:=d(y_{k},F_{\varepsilon_{k}}^{\tau})\leq 1/4$, where

[TABLE]

Let $z_{k}\in F_{\varepsilon}(u_{\varepsilon})\cap\partial B_{d_{k}}(y_{k})$ realize the distance $d_{k}$ between $y_{k}$ and $F_{\varepsilon_{k}}^{\tau}$. Taking into account that $B_{2d_{k}}(z_{k})\subset B_{1}$, we may define the rescaled solutions

[TABLE]

Then $v_{k}:=\tilde{u}_{\varepsilon_{k}/d_{k}}$ are uniformly Lipschitz continuous in $\overline{B_{2}}$ and fulfill:

[TABLE]

Hence, up to taking subsequences, we can assume that the points $q_{k}\in\partial B_{1}$ converge to some $q_{\infty}\in\partial B_{1}$ and that $v_{k}$ converges uniformly in $\overline{B_{2}}$ to a Lipschitz continuous function $v_{\infty}\in C(B_{1})$ that is harmonic in its positive phase $\Omega:=\{x\in B_{2}:v_{\infty}(x)>0\}$. Furthermore, as

[TABLE]

we see by (2.25) that $v_{\infty}(0)=0$ and that $v_{\infty}$ is nondegenerate at all scales at [math]:

[TABLE]

so that $0\in\partial\Omega$. Because of (2.26), we deduce that $v_{\infty}\geq 0$ is harmonic in $B_{1}(q_{\infty})$. However, (2.27) means that $v_{\infty}(q_{\infty})=0$ and the maximum principle yields $v_{\infty}\equiv 0$ in $B_{1}(q_{\infty})$. Hence, $B_{1}(q_{\infty})\subseteq\Omega^{c}$ is a ball touching $0\in\partial\Omega$ from the zero phase of $v_{\infty}$. Hence, the asymptotic development result [CS05, Lemma 11.17] for positive harmonic functions at (left) regular points, in combination with the nondegeneracy of $v_{\infty}$, entails that for some $\beta>0$,

[TABLE]

In particular, this means that

[TABLE]

On the other hand, since for every $\delta>0$, the uniform convergence of $v_{k}$ to $v_{\infty}$ in $B_{2}$ implies

[TABLE]

we obtain from the monotone convergence theorem and (2.28) that

[TABLE]

The latter contradicts (2.29) for $\kappa>0$. ∎

We are now in a position to establish Theorem 1.2.

Proof of Theorem 1.2.

As $u_{\varepsilon}(0)\leq T\varepsilon$, Proposition 2.1 tells us that the uniform Lipschitz bound (2.1) holds in $B_{1}$. For points $p\in F_{\varepsilon}^{\theta}(u_{\varepsilon})\cap B_{1/4}$, the nondegeneracy bound (1.14) thus follows from (2.20) of Corollary 2.5.

Assume now that $p\in B_{1/4}^{+}(u_{\varepsilon})=\{x\in B_{1/4}:u_{\varepsilon}(x)>T\varepsilon\}$, $r\geq 2L\varepsilon$ and $B_{r}(p)\subset B_{1}$. Since

[TABLE]

Lemma 2.6 states that as long as $\varepsilon\leq\varepsilon_{0}=\varepsilon_{0}(n,\kappa,f,L)$ is small enough,

[TABLE]

for some constant $\mu_{0}=\mu_{0}(n,\kappa,L,f)$. If $r\leq 2d(p,F_{\varepsilon}^{\tau})$, then (2.30) directly implies that

[TABLE]

In case that $r>2d(p,F_{\varepsilon}^{\tau})$, let $\tilde{p}$ be the point in $F_{\varepsilon}^{\tau}\cap B_{1/2}$ that realizes the distance $d(p,F_{\varepsilon}^{\tau})$. Because we have assumed that $r/2\geq L\varepsilon$, we can rescale $u_{\varepsilon}$ in $B_{r/2}(\tilde{p})$ as in the proof of Corollary 2.5 and apply Theorem 2.4 to get $\sup_{B_{r/2}(\tilde{p})}u_{\varepsilon}\geq\mu r/2.$ Now, the fact that $r>2|p-\tilde{p}|$ implies $B_{r}(p)\supseteq B_{r/2}(\tilde{p})$, so that

[TABLE]

∎

3. Limits of solutions of (1.5) as $\varepsilon\downarrow 0$

We begin this section by recalling the notion of inner-stationary solutions of (1.5), resp. (1.7), which are defined as the critical points of $I_{\varepsilon}$ (resp. $I_{0}$) with respect to inner domain deformations.

Definition 3.1**.**

A function $u=u_{\varepsilon}\in H^{1}_{\text{loc}}(\Omega)$ is an inner-stationary solution of (1.5) (resp. (1.7) when $\varepsilon=0$) in a domain $\Omega\subseteq\mathbb{R}^{n}$ if the first inner variation

[TABLE]

In Proposition A.1 of Appendix A we will compute explicit formulas for the first and second inner variations of $I_{\varepsilon}$, $\varepsilon\geq 0$, that hold in the general setting of an oriented Riemannnian manifold. For our Euclidean case they read

[TABLE]

Here $\bar{\delta}$ is the contravariant $(2,0)$-tensor $\bar{\delta}=\sum_{ij}\delta^{ij}\partial_{x_{i}}\otimes\partial_{x_{j}}$, which gives the Euclidean induced inner product on covectors, and $L_{X}$ denotes the Lie derivative. In standard coordinates, the tensors $L_{X}\bar{\delta}$ and $L_{X}^{2}\bar{\delta}$ have components (see the calculations preceding (A.3)-(A.4)):

[TABLE]

where we have adopted the standard summation convention over repeated indices.

It is worth mentioning the well known fact that if $u_{\varepsilon}$ is a classical solution to (1.5), then it is also an inner-stationary solution of (1.5) (see Proposition A.3). The benefit of working with these weak solutions is that they behave well under taking limits. The main goal of this section is to establish the convergence result for solutions to the one-phase singular perturbation problem (1.5), presented next. Its proof uses classical, well known arguments, with the only novelty being the argument behind the important $L^{1}_{\text{loc}}$ convergence $\mathcal{F}_{\varepsilon}(u_{\varepsilon})\to\mathcal{F}_{0}(u)$.

Proposition 3.2**.**

Let $\{u_{\varepsilon}\}_{\varepsilon}$ be a family of solutions of (1.5) in a domain $\Omega\subset\mathbb{R}^{n}$, that satisfy

• •

(Uniform Lipschitz continuity) There exists a constant $C$, such that $\|\nabla u_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq C$;

• •

(Uniform nondegeneracy) For every $\theta\in(0,T]$, there exist positive constants $\varepsilon_{0}$, $c$ and $\lambda$, such that if $\varepsilon\leq\varepsilon_{0}$, then $\sup_{B_{r}(x)}u_{\varepsilon}\geq cr$ for every $B_{r}(x)\subseteq\Omega$, centered at a point $x\in\{u_{\varepsilon}\geq\theta\varepsilon\}$, with $r\geq\lambda\varepsilon$.

Then any limit $u\in H^{1}_{\text{loc}}(\Omega)\cap C(\Omega)$ of a uniformly convergent on compacts sequence $u_{k}:=u_{\varepsilon_{k}}\to u$, as $\varepsilon_{k}\to 0$, satisfies

1. (a)

$u$* is harmonic in $\Omega^{+}_{0}(u)$;* 2. (b)

$\{u_{k}\geq\theta\varepsilon_{k}\}\to\overline{\{u>0\}}$* locally in the Hausdorff distance, for all $\theta>0$;* 3. (c)

$F_{\varepsilon_{k}}^{\theta}(u_{k})\to F_{0}(u)$* locally in the Hausdorff distance, for all $\theta\in(0,\tau]$;* 4. (d)

$\nabla u_{k}\to\nabla u$* in $L^{2}_{\text{loc}}(\Omega)$.* 5. (e)

$\mathcal{F}_{\varepsilon_{k}}(u_{k})\to\mathcal{F}_{0}(u)=1_{\{u>0\}}$* in $L^{1}_{\text{loc}}(\Omega)$;*

Moreover, $u$ is a Lipschitz continuous, inner-stationary solution of (1.7) that is nondegenerate:

[TABLE]

for some constant $\bar{c}>0$.

Proof.

The uniform limit $u$ is clearly harmonic in its positive phase $\Omega_{0}^{+}(u)$ and satisfies the same Lipschitz bound: $\|\nabla u\|_{L^{\infty}(\Omega)}\leq C$. Let us show that $u$ possesses the nondegeneracy property (3.3). Fix $x_{0}\in\Omega^{+}_{0}(u)$ and $r>0$ such that $B_{r}(x_{0})\subset\Omega$. Since $u_{k}(x_{0})\to u(x_{0})>0$, we have for all $k$ large enough $u_{k}(x_{0})\geq T\varepsilon_{k}$. Because $r\geq\lambda(T)\varepsilon_{k}$ for large $k$ as well, the nondegeneracy property of $u_{k}$ gives us that $\sup_{B_{r}(x_{0})}u_{k}\geq\bar{c}r$ for all $k$ large enough, with $\bar{c}=c(T)$. Thus, the uniform convergence yields $\sup_{B_{r}(x_{0})}u\geq\bar{c}r$ and we can conclude by continuity that (3.3) is valid for all points $x$ in the closure $\overline{\Omega^{+}_{0}(u)}$. In particular, for every $p\in F_{0}(u)$ and $B_{r}(p)\subset\Omega$, there exists a point $q\in\overline{B_{r/2}(p)}$ such that $u(q)\geq(\bar{c}/2)r$, so that by the Lipschitz continuity of $u$, the ball $B_{\nu r}(q)\subseteq\Omega^{+}_{0}(u)\cap B_{r}(p)$ for $\nu:=\min(\bar{c}/(2C),1/2)$. Hence,

[TABLE]

implying that the set of Lebesgue density points of $F_{0}(u)$ is empty. Therefore,

[TABLE]

The proofs of b) and d) are standard and their proofs can be found in [AS22, Lemma 5.3] and [CLW97, Lemma 3.1], respectively, so here we will focus only on proving c) and e). Fix $\delta>0$ and choose a compact subset $K\Subset\Omega$ such that $d(K,\Omega^{c})>\delta$. Denote

[TABLE]

In order to establish that $F_{0}\subseteq F_{k}^{\delta}$, it suffices to show that for every $x\in F_{0}$, $B_{\delta}(x)\cap F_{k}\neq\emptyset$ for all $k$ large enough. Assume not: then for all large $k$ (after possibly taking a subsequence) either $B_{\delta}(x)\subset\{u_{k}<\theta\varepsilon_{k}\}$, or $B_{\delta}(x)\subset\{u_{k}>T\varepsilon_{k}\}$. In the first case, we will get by uniform convergence that $u\equiv 0$ in $B_{\delta}(x)$, which is impossible as $x\in F_{0}$. In the second case, Harnack’s inequality implies

[TABLE]

so that $u\equiv 0$ in $B_{\delta/2}(x)$, which is impossible again.

To prove that $F_{k}\subseteq F_{0}^{\delta}$ for all large $k$, assume by contradiction that there is a sequence of points $\{x_{k}\in F_{k}\}$, such that $B_{\delta}(x_{k})\cap F_{0}=\emptyset$. By taking a subsequence, we may assume that $x_{k}\to x_{\infty}\in K$ and

[TABLE]

By possibly taking a further subsequence, it must be the case that either $B_{\delta}(x_{k})\subset\{u=0\}$, or $B_{\delta}(x_{k})\subset\{u>0\}$ for all $k$ large enough. The first scenario is impossible, since by the uniform nondegeneracy property of $u_{k}$, we have

[TABLE]

so that by uniform convergence, $\sup_{B_{\delta/2}(x_{\infty})}u\geq c(\theta)\delta/4>0$, contradicting the fact that $B_{\delta/2}(x_{\infty})\subset\{u=0\}.$ The second scenario doesn’t occur either, because by the continuity of $u_{k}$ and the uniform convergence of $u_{k}\to u$, we would have

[TABLE]

which would contradict the fact that $B_{\delta/2}(x_{\infty})\subset\{u>0\}$. The proof of c) is completed.

Let us now treat the claim in e). For the purpose, we will need the following lemma about the relation between Hausdorff convergence and convergence in measure of compact sets.

Lemma 3.3**.**

Let $\{K_{k}\}$ be a sequence of compact subsets of $\mathbb{R}^{n}$ that converge in the Hausdorff distance to the compact $K_{\infty}\subset\mathbb{R}^{n}$. Then

[TABLE]

Proof.

Fix $\epsilon>0$ and let $O\supset K_{\infty}$ be an open set, such that $|O|\leq|K_{\infty}|+\varepsilon$. Because $K_{\infty}\cap O^{c}=\emptyset$, the separation between the compact $K_{\infty}$ and the closed $O^{c}$, $d(K_{\infty},O^{c})\geq\delta>0$, for some $\delta>0$. Hence, $K_{\infty}^{\delta}:=\{x\in\mathbb{R}^{n}:d(x,K_{\infty})<\delta\}$ is disjoint from $O^{c}$, and by the Hausdorff convergence of $K_{k}\to K_{\infty}$, we have that $K_{k}\cap O^{c}\subset K_{\infty}^{\delta}\cap O^{c}=\emptyset$, i.e. $K_{k}\subseteq O$ for all large $k$. Thus,

[TABLE]

Taking the limit superior as $k\to\infty$, and noting that $\epsilon>0$ is arbitrary, we arrive at (3.5). ∎

Going back to proving e), we first write

[TABLE]

Take an arbitrary compact $K\Subset\Omega$. Claim that

[TABLE]

i.e. that $\mathrm{1}_{\{u_{k}\geq T\varepsilon_{k}\}}\to\mathrm{1}_{\{u>0\}}$ in $L^{1}_{\text{loc}}$. Note that if $x\in\{u>0\}$, then $x\in\{u_{k}\geq T\varepsilon_{k}\}$ for all large $k$, so we must have

[TABLE]

Fatou’s lemma then tells us that

[TABLE]

with equality if and only if (3.7) is valid. Furthermore, equality in (3.9) does hold, because the result of Lemma 3.3 yields that

[TABLE]

on account of the Hausdorff convergence of $K\cap\{u_{k}\geq T\varepsilon_{k}\}\to K\cap\overline{\{u>0\}}$ from a) plus the fact (3.4) that $|F_{0}(u)|=0$.

To show that the integrals over $K$ of $|A_{2}+A_{3}|$ in (3.6) go to [math] as $k\to\infty$, fix $\delta>0$ arbitrary and choose $\theta>0$ small enough such that $\mathcal{F}(u)\leq\delta$ for $u\leq\theta$. In this way,

[TABLE]

Now, to bound the integral of $|A_{2}|$ over $K$, we will use the fact from b) that $\{\theta\varepsilon_{k}\leq u_{k}<T\varepsilon_{k}\}\}\to F_{0}(u)\cap K$ in the Hausdorff distance. As a result, Lemma 3.3 implies that for all large enough $k$,

[TABLE]

since $|F_{0}(u)\cap K|=0$. Therefore, for all large $k$, we have

[TABLE]

Combining (3.7), (3.12) and (3.10) and taking $\delta\to 0$, we can complete the proof of $e)$.

Finally, that $u=\lim_{k\to\infty}u_{k}$ is an inner-stationary solution of the one-phase FBP (1.7), is a result of the strong convergences in d) and e):

[TABLE]

coupled with the fact that $\delta I_{\varepsilon}(u_{k})=0$, since classical solutions $u_{k}=u_{\varepsilon_{k}}\in C^{2}(\Omega)$ of (1.5) are inner-stationary by default. ∎

We end this section by showing that the interface of a minimizer of $I_{\varepsilon}(\cdot,\Omega)$ actually satisfies a $\mathcal{D}(\kappa,L)$ density property in $\Omega$ (cf. Definition 1.1) for some universal constants $\kappa,L>0$. We place the result here because its proof requires some of the ideas present in the convergence result above.

Proposition 3.4**.**

Let $u_{\varepsilon}\in H^{1}(\Omega)$ be a positive minimizer of $I_{\varepsilon}$ in $\Omega$, $\varepsilon>0$. Then there exist positive constants $\kappa$ and $L$, depending only on $n$ and $f$, such that the interface of $u_{\varepsilon}$ satisfies the density property $\mathcal{D}(\kappa,L)$ in $\Omega$.

Proof.

Let $p\in F_{\varepsilon}^{\tau}(u_{\varepsilon})$ and assume $B_{r}(p)\subset\Omega$. By recentering and rescaling,

[TABLE]

it suffices to prove the following statement: there exist absolute constants $\varepsilon_{0}=:1/L$ and $\kappa>0$ such that if $\varepsilon\leq\varepsilon_{0}$ and $u_{\varepsilon}$ is a minimizer of $I_{\varepsilon}$ in $B_{2}$ with $u_{\varepsilon}(0)\in(\tau\varepsilon,T\varepsilon)$, then

[TABLE]

We remark that $u_{\varepsilon}$ satisfies the universal Lipschitz bound (2.1) in $B_{1}$: $\|\nabla u_{\varepsilon}\|_{L^{\infty}(B_{1})}\leq C$.

Denote by $h$ the harmonic function in $B_{1}$ with $h=u_{\varepsilon}$ on $\partial B_{1}$. Since $h$ is a competitor to $u_{\varepsilon}$ in $B_{1}$, we have $I_{\varepsilon}(u_{\varepsilon},B_{1})\leq I_{\varepsilon}(h,B_{1})$, so that

[TABLE]

where the first equality follows from the harmonicity of $h$. By the Poincare-Sobolev inequality, we then get

[TABLE]

for a dimensional constant $c>0$. Taking into consideration that minimizers of $I_{\varepsilon}$ satisfy the nondegeneracy property (1.8) (see [AS22, Lemma 4.2]), we have $\max_{\partial B_{1}}u_{\varepsilon}\geq c_{1}$ for some absolute positive constant $c_{1}=c_{1}(n,f)$. In combination with the Lipschitz bound, this implies that for some positive constant $c_{2}=c_{2}(n,f)$

[TABLE]

so that the mean-value property and the Harnack inequality for harmonic functions entail

[TABLE]

for some $c_{3}=c_{3}(n,f).$ On the other hand, from the Lipschitz bound we know that $u_{\varepsilon}\leq T\varepsilon+Cr$ in $B_{r}$ for all $r\in(0,1)$, so that for $\varepsilon\leq\varepsilon_{1}:=c_{3}/(4T)$ and $r_{0}=\min(c_{3}/(4C),1/2)$, we have $u_{\varepsilon}\leq c_{3}/2$ in $B_{\rho}$. Hence, $h-u_{\varepsilon}\geq c_{3}-c_{3}/2=c_{3}/2$ in $B_{r_{0}}$, and (3.15) gives

[TABLE]

for some $c_{4}=c_{4}(n,f)>0$ and all small $\varepsilon\leq\varepsilon_{1}$. Furthermore, we get from (3.16) and the Harnack inequality that

[TABLE]

Now, if $\rho=\rho(n,f)$ is small enough, (3.18) plus the fact that $\mathcal{F}_{\varepsilon}(h)-\mathcal{F}_{\varepsilon}(u_{\varepsilon})\leq 1$ yield for $\varepsilon\leq\varepsilon_{1}$

[TABLE]

Hence, if $\varepsilon\leq\min(\varepsilon_{1},\varepsilon_{2})$, where $\varepsilon_{2}=\varepsilon_{2}(n,f)$ is defined by $T\varepsilon_{2}=c_{3}(1-\rho)$, we obtain from (3.19) that $h\geq T\varepsilon$ in $B_{1-\rho}$, so that (3.20) becomes

[TABLE]

Writing the integral on the left-hand side of (3.21) as

[TABLE]

we see that the claim (3.13) will be established for $\kappa:=c_{4}/[2(1-\mathcal{F}(\tau/4))]$ and some $\varepsilon_{0}=\varepsilon_{0}(n,f)\leq\min\{\varepsilon_{1},\varepsilon_{2}\}$, once we show that

[TABLE]

Now, the “thinning out” of the interface $F_{\varepsilon}^{\tau/4}$, expressed in (3.22), is a consequence of the uniform nondegeneracy property and can be established in the same way as in the proof of Proposition 3.2 (see (3.11) above). ∎

4. Inner-stable solutions of the one-phase FBP

In this section we present the regularity theory, developed by Buttazo et al. [BMM*+*22], for a class of weak solutions of (1.7), which carries a notion of stability and which is closed under taking locally uniform limits.

Definition 4.1**.**

Let $C,c,\kappa$ be positive real numbers, and let $\Omega\subseteq\mathbb{R}^{n}$ be a domain. We will say that a nonnegative function $u\in H^{1}_{\text{loc}}(\Omega)\cap C(\overline{\Omega})$ belongs to the class $\mathcal{S}(C,c,\kappa;\Omega)$ if the following are satisfied:

1. (1)

$u$* is an inner-stationary solution of (1.7) in $\Omega$:*

[TABLE]

that is harmonic in its positive phase $\Omega^{+}_{0}(u)$; 2. (2)

the second inner variation of $I_{0}$ at $u$ is nonnegative:

[TABLE] 3. (3)

$u$* is Lipschitz continuous in $\Omega$ with a Lipschitz constant bounded by $C$:*

[TABLE] 4. (4)

$u$* is nondegenerate in $\Omega$ with a nondegeneracy constant $c$:*

[TABLE] 5. (5)

the zero phase has positive density at least $\kappa$:

[TABLE]

We will say that $u$ is a inner-stable solution to the one-phase FBP (1.7) in $\Omega$ if $u\in\mathcal{S}(C,c,\kappa;\Omega)$ for some positive constants $C,c,$ and $\kappa$.

Remark 4.2**.**

Local minimizers $u\in\mathcal{H}^{1}_{\text{loc}}(\tilde{\Omega})$ of the Alt-Caffarelli functional $I_{0}(\cdot,\tilde{\Omega})$ with $0\in F_{0}(u)$ are inner-stable solutions in any domain $\Omega\Subset\tilde{\Omega}$. They are known to satisfy properties (3)-(5) (see [AC81]). To check that they satisfy (1)-(2) as well, we simply note that if $\phi_{t}$ denotes the flow along a test vector field $X\in C^{\infty}_{c}(\Omega;\mathbb{R}^{n})$, then $u_{t}(x):=u(\phi_{t}^{-1}(x))$ is a competitor to $u$ in $\Omega$ for all $t\in\mathbb{R}$, so that $I_{0}(u_{t},\Omega)\geq I_{0}(u,\Omega)$. As $u_{0}=u$, we have

[TABLE]

The goal of this section is to show that inner-stable solutions to the one-phase FBP enjoy virtually the same regularity theory as local minimizers of the Alt-Caffarelli functional. Namely, we will present the proof of the following theorem.

Theorem 4.3** ([BMM*+*22]).**

Let $u$ be an inner-stable solution of (1.7) in a domain $\Omega\subseteq\mathbb{R}^{n}$. Then its free boundary $F_{0}(u)$ is a smooth hypersurface, except possibly on a closed singular subset of Hausdorff dimension at most $n-n^{*}$, where the critical dimension $n^{*}$ is given in Definition 4.10 below, and satisfies $5\leq n^{*}\leq 7$.

We will first collect some basic results necessary for the blow-up analysis behind Theorem 4.3. We start with the fact that the class $\mathcal{S}(C,c,\kappa;\Omega)$ is compact in the uniform (on compacts) topology.

Proposition 4.4**.**

Let $\{u_{k}\}_{k}$ be a sequence in $\mathcal{S}(C,c,\kappa;\Omega)$ with $0\in F_{0}(u_{k})$ for every $k\in\mathbb{N}$. Then, up to taking a subsequence, $u_{k}$ converges uniformly on compact subsets to some $u\in\mathcal{S}(C,c,\kappa;\Omega)$. Moreover, the subsequence can be taken so that

[TABLE]

Proof.

The uniform Lipschitz continuity, in combination with $u_{k}(0)=0$, implies the uniform local boundedness of the sequence. Thus, by Arzela-Ascoli, $\{u_{k}\}$ subconverges on compacts to a continuous function $u$ that satisfies $u(0)=0$ and the same Lipschitz bound $\|\nabla u\|_{L^{\infty}(\Omega)}\leq C$. Moreover, $u\lfloor_{\Omega^{+}_{0}(u)}$ is harmonic as the uniform limit of the harmonic functions $u_{k}\lfloor_{\Omega^{+}_{0}(u)}$. That $u$ inherits (4)-(5) is straightforward to verify.

Now, it is well known classically (see [CS05, Lemma 1.21]) that the uniform Lipschitz continuity and the uniform nondegeneracy of the sequence imply the Hausdorff distance convergence (4.1), as well as the convergences:

[TABLE]

These, in turn, entail that for any test vector field $X\in C^{\infty}_{c}(\Omega;\mathbb{R}^{n})$,

[TABLE]

i.e. $u$ inherits the variational properties (1) and (2), as well. ∎

Note that all the properties (1)-(5) of Definition 4.1 are scale invariant. Thus, if $u\in\mathcal{S}(C,c,\kappa;\Omega)$, then its rescale $u_{r}(x):=r^{-1}u(rx)$ belongs to $\mathcal{S}(C,c,\kappa,r^{-1}\Omega)$. As a corollary to Proposition 4.4, we see that both blow-up and blow-down limits of solutions in the class $\mathcal{S}$ remain inner-stable solutions.

Corollary 4.5**.**

Let $\Omega\subseteq\mathbb{R}^{n}$ be a domain and let $u\in\mathcal{S}(C,c,\kappa;\Omega)$ for some positive constants $C,c,\kappa$. Assuming that $x_{0}\in F_{0}(u)$, then

1. (a)

For every sequence $r_{k}\downarrow 0$, the blow-ups $u_{r_{k}}(x):=(r_{k})^{-1}u(x_{0}+r_{k}x)$ subconverge on compact subsets of $\mathbb{R}^{n}$ to some $u_{0}\in\mathcal{S}(C,c,\kappa;\mathbb{R}^{n}).$ 2. (b)

If $\Omega=\mathbb{R}^{n}$, then for every sequence $r_{k}\uparrow\infty$, the blow-downs $u_{r_{k}}$ subconverge uniformly on compact subsets of $\mathbb{R}^{n}$ to some $u_{\infty}\in\mathcal{S}(C,c,\kappa;\mathbb{R}^{n})$.

Moreover, the blow-up limit $u_{0}$ and the blow-down limit $u_{\infty}$ are homogeneous functions of degree $1$.

Proof.

The claims in (a) and (b) follow from Proposition 4.4. That the limits $u_{0}$ and $u_{\infty}$ are homogeneous of degree one is a consequence of the the Weiss Monotonicity Formula ([Wei98]), which applies to inner-stationary solutions of (1.7). ∎

Next, we state the notion of viscosity solution to the one-phase FBP ([Caf87], [CS05]) and show that, in fact, inner-stable solutions are viscosity solutions, as well.

Definition 4.6**.**

A nonnegative function $u\in C(\Omega)$ is a viscosity solution of (1.7) if $u$ is harmonic in $\Omega_{0}^{+}(u)$ and

1. (1)

(supersolution property) for every $x_{0}\in F(u)$ with a tangent ball $B$ from the positive side ($x_{0}\in\partial B$* and $B\subset\Omega^{+}_{0}(u)$), there is $\alpha\leq 1$ such that*

[TABLE]

as $x\to x_{0}$ non-tangentially in $B$, with $\nu$ the inner normal to $\partial B$ at $x_{0}$; 2. (2)

(subsolution property) for every $x_{0}\in F(u)$ with a tangent ball $B$ from the zero side ($x_{0}\in\partial B$* and $B\subset Z_{0}(u)$), there is $\beta\geq 1$ such that*

[TABLE]

as $x\to x_{0}$ non-tangentially in $B^{c}$, with $\nu$ the outer normal to $\partial B$ at $x_{0}$.

Lemma 4.7**.**

Let $u$ be an inner-stable solution of (1.7) in a domain $\Omega\subseteq\mathbb{R}^{n}$. Then $u$ is a viscosity solution of (1.7) in $\Omega$.

Proof.

We will provide the proof of the supersolution property of $u$; the proof of the subsolution property is analogous.

If $F(u)$ has a tangent ball $B$ from the positive side at $x_{0}$, then by [CS05, Lemma 11.17] (4.3) is satisfied from some $\alpha>0$. According to Corollary 4.5, any blow-up limit $u_{0}$ of $u$ at $x_{0}$, is an inner-stable solution which is homogeneous of degree $1$. Therefore,

[TABLE]

If $\tilde{\Omega}:=\Omega^{+}(u_{0})\setminus P^{+}=\emptyset$, then $u_{0}(x)=\alpha\langle x,\nu\rangle^{+}$ in all of $\mathbb{R}^{n}$, so that $F_{0}(u_{0})$ is regular everywhere. By Proposition B.2, we then get that $\alpha=1$.

If $\Omega_{1}\neq\emptyset$, we notice that in the spherical section $\tilde{\Omega}_{\mathbb{S}}:=\tilde{\Omega}\cap\mathbb{S}^{n-1}$

[TABLE]

i.e. $u\lfloor_{\tilde{\Omega}_{\mathbb{S}}}$ is the first Dirichlet eigenfunction of $-\Delta_{S^{n-1}}$ in $\tilde{\Omega}_{\mathbb{S}}$, with associated eigenvalue $(n-1)$. Since, the half-sphere has the same first Dirichlet eigenvalue and contains $\tilde{\Omega}\cap\mathbb{S}^{n-1}$, then $\tilde{\Omega}\cap\mathbb{S}^{n-1}$ is a half-sphere, and $u_{0}(x)=\alpha\langle x,\nu\rangle^{+}+\tilde{\alpha}\langle x,\nu\rangle^{-}$ for some $\tilde{\alpha}>0$. This, however, is inconsistent with the positive density of $Z(u_{0})$. ∎

Definition 4.8**.**

Let $u$ be an inner-stable solution of (1.7) in $\Omega$. A point $x_{0}\in F_{0}(u)$ is called regular if $u$ has a blow-up limit at $x_{0}$ of the form $u_{0}(x)=\langle x,\nu\rangle^{+}$ for some unit vector $\nu\in\mathbb{R}^{n}$. Otherwise, the point is called singular. We will denote by $\text{Reg}(u)$ the subset of all regular points of $F_{0}(u)$ and by $\text{Sing}(u):=F_{0}(u)\setminus\text{Reg}(u)$ – the subset of its singular points.

Remark 4.9**.**

Let $x_{0}$ be a regular point of $F_{0}(u)$ of an inner-stable solution $u$ and let $u_{k}(x):=r_{k}^{-1}u(x_{0}+r_{k}x)$ be a sequence of blow-ups converging to $u_{0}(x)=\langle x,\nu\rangle$, where we may assume $\nu=e_{n}$. We note that $u_{k}$ are viscosity solutions of (1.7) by the previous Lemma 4.7. Since by (4.1) we have $\Omega^{+}_{0}(u_{k})\cap B_{1}\to\Omega^{+}_{0}(u_{0})\cap B_{1}$ in the Hausdorff distance, then for every $\delta>0$ there is $k$ large enough such that

[TABLE]

i.e. the free boundary $F_{0}(u_{k})$ is $\delta$-flat. When $\delta$ is sufficiently small, the classical regularity result “Flat $\Rightarrow$ Smooth” of Caffarelli (see [Caf87, Caf89]) kicks in and yields that $F_{0}(u_{k})$ is a smooth graph in $B_{1/2}$. Therefore, in a neighbourhood $U$ of every regular point, $F_{0}(u)\cap U$ is a smooth hypersurface, separating positive from zero phase, and $u$ is a classical solution of (1.7) in $U$.

Definition 4.10**.**

Define the critical dimension $n^{*}$ for inner-stable solutions to the one-phase FBP to be the lowest dimension $n$ for which there exists a global inner-stable solution $u:\mathbb{R}^{n}\to\mathbb{R}$ that is homogeneous of degree one, with $0\in\text{Sing}(u)$.

Remark 4.11**.**

Note that if $u\in\mathcal{S}(C,c,\kappa;\mathbb{R}^{n})$ is a global inner-stable solution, then by Corollary 4.5 any blow-down limit $u_{\infty}=\lim_{k\to\infty}u_{R_{k}}$, $R_{k}\to\infty$, belongs to $\mathcal{S}(C,c,\kappa;\mathbb{R}^{n})$ and is homogeneous of degree $1$. Therefore, when $n\leq n^{*}-1$, the fact that $F_{0}(u_{\infty})$ is a smooth hypersurface implies that $u_{\infty}=x_{n}^{+}$ in some Euclidean coordinate system. Now, since $u_{R_{k}}\to x_{n}^{+}$ locally uniformly, the free boundary $F(u_{\infty})$ is asymptotically flat, i.e.

[TABLE]

with the aspect ratio $\delta_{k}/R_{k}\to 0$ as $k\to\infty$. As $u$ is a viscosity solution of (1.7) as well, we conclude from Caffarelli’s theorem that $u(x)=x_{n}^{+}$.

The existence of a singular entire minimizer of (1.7) in $\mathbb{R}^{7}$ that is homogeneous of degree $1$, constructed by De Silva and Jerison ([DSJ09]), and the observation in Remark 4.2 suggest that $n^{*}\leq 7$. Due to works by Caffarelli, Jerison and Kenig [CJK04], and Jerison and Savin [JS15], it is currently known that the lower bound for the critical dimension $n^{*}_{\text{e}}$, in the case of energy minimizing solutions is $n^{*}_{\text{e}}\geq 5$. This was achieved by proving the following slightly more general result.

Theorem 4.12** ([JS15]).**

Let $u$ be a homogeneous solution of (1.7) in $\Omega=\mathbb{R}^{n}$, such that $0\in F_{0}(u)$ and $F_{0}(u)\setminus\{0\}$ is a smooth cone separating positive from zero phase. Assume further that $u$ satisfies the stability inequality

[TABLE]

where $H$ denotes the mean curvature of $F_{0}(u)$ with respect to the outer unit normal to $\partial\Omega^{+}_{0}(u)$. Then, for $n=2,3,4$, $F_{0}(u)$ is a hyperplane and $u(x)=\langle x,\nu\rangle^{+}$ for some unit vector $\nu\in\mathbb{R}^{n}$.

To obtain that the critical dimension for inner-stable solutions enjoys the same lower bound $n^{*}\geq 5$, [BMM*+*22] prove

Proposition 4.13** (Proposition 7.12 of [BMM*+*22]).**

Let $u$ be an inner-stable solution of (1.7) in $\Omega=\mathbb{R}^{n}$ that is homogeneous of degree one, with $\text{Sing}(u)=\{0\}$. Then $u$ satisfies (4.5). In particular, $n^{*}\geq 5$.

Here we give a different proof of this proposition, which is based on the formula (1.21) for the second inner variation $\delta^{2}I_{0}(u)$ that we derive in Proposition B.3 of Appendix B.

Proof.

Since $u$ is homogeneous of degree one, we have

[TABLE]

In particular, $\nabla u\neq 0$ in $\{u>0\}$ and for every test function $\phi\in C^{\infty}_{c}(\mathbb{R}^{n}\setminus\{0\})$, we can define a test vector field $X\in C^{\infty}_{c}(\mathbb{R}^{n}\setminus\{0\})$ by letting

[TABLE]

and extending it across the smooth hypersurface $F_{0}(u)\setminus\{0\}$ as a smooth vector field, compactly supported away from [math]. In this way, $L_{X}u=\langle X,\nabla u\rangle=\phi$ in $\Omega^{+}_{0}(u)$. Since $u$ is harmonic in $\Omega^{+}_{0}(u)$, smooth up to $\text{Reg}(F_{0}(u))=F_{0}(u)\setminus\{0\}$ and an inner-stable solution to (1.7), Proposition B.3 informs us that

[TABLE]

i.e. the stability inequality of Caffarelli-Jerison-Kenig (4.5) is satisfied. ∎

Proof of Theorem 4.3.

Given the bounds $5\leq n^{*}\leq 7$ for the critical dimension $n^{*}$ established in Proposition 4.13, the proof of the regularity statement now follows from Federer’s classical technique of dimension reduction, introduced in the free boundary context by Weiss [Wei98]. See [Vel19, Section 10] for details. ∎

5. Proof of Theorem 1.6

In this last section we will provide the proof of our second main result, Theorem 1.6, which will be a consequence of the nondegeneracy Theorem 1.2, the convergence result Proposition 3.2, the regularity Theorem 4.3 for inner-stable solutions, and ultimately, the Audrito-Serra theorem [AS22].

We begin by showing that a sequence of solutions $u_{\varepsilon_{k}}$ of (1.5), $k\in\mathbb{N}$, that fulfill a $\mathcal{D}(\kappa,L)$ property uniformly as $\varepsilon_{k}\downarrow 0$ and are stable with respect to compact domain deformations, subconverges to an inner-stable solution of the one-phase FBP (Definition 4.1).

Proposition 5.1**.**

Let $\{u_{\varepsilon_{k}}\}_{k}$ be a sequence of solutions of (1.5) in $B_{2R}$, with $\varepsilon_{k}\downarrow 0$, such that

• •

$u_{\varepsilon}(0)\leq T\varepsilon$,

• •

the interface of each $u_{\varepsilon_{k}}$ satisfies the density property $\mathcal{D}(\kappa,L)$ in $B_{2R}$ for some positive constants $\kappa$ and $L$;

• •

$u_{\varepsilon_{k}}$* has a non-negative second inner variation with respect to $I_{\varepsilon}(\cdot,B_{2R})$: $\delta^{2}I_{\varepsilon}(u_{\varepsilon_{k}},B_{2R})\geq 0$.*

Then, up to taking a subsequence, $u_{\varepsilon_{k}}$ converge uniformly in $B_{R/4}$ to a function $u$ that is an inner-stable solution to the one-phase FBP in $B_{R/4}$.

Proof.

After, rescaling we may assume that $R=1$. Since $u_{k}:=u_{\varepsilon_{k}}(0)\leq T\varepsilon$, we know by Proposition 2.1, that $u_{k}$ are uniformly Lipschitz continuous in $B_{1}$. Furthermore, the nondegeneracy result of Theorem 1.2 tells us that for each $\theta\in(0,\tau]$, there are positive constants $\varepsilon_{0}$, $c$ and $\lambda:=2\max(L,M)$ such that if $\varepsilon_{k}\leq\varepsilon_{0}$, then

[TABLE]

The hypotheses of Proposition 3.2 are therefore met in $\Omega:=B_{1/4}$, so we can infer that the sequence $\{u_{k}\}$ subconverges on $B_{1/4}$ to a nonnegative Lipschitz continuous function $u$ that is harmonic in its positive phase $\Omega_{0}^{+}(u)$ and is a non-degenerate inner-stationary solution of (1.7). Because the same proposition gives us that $\nabla u_{k}\to\nabla u$ in $L^{2}(B_{1/4})$ and $\mathcal{F}_{\varepsilon_{k}}(u_{k})\to\mathcal{F}_{0}(u)$ in $L^{1}(B_{1/4})$, we get that for any fixed test fector field $X\in C^{\infty}(B_{1/4},\mathbb{R}^{n})$,

[TABLE]

i.e. $u$ is a stable critical point of $I_{0}$ with respect to compactly supported deformations of $B_{1/4}$.

We have thus confirmed that $u$ satisfies properties (1)-(4) of Definition 4.1 in $B_{1/4}$. To conclude that $u$ is an inner-stable solution of (1.7) in $B_{1/4}$, it remains to check the positive density of the zero phase along the the free boundary $F_{0}(u)$. Let $x\in F_{0}(u)\cap B_{1/4}$ and $B_{r}(x)\subseteq B_{1/4}$. By the Hausdorff convergence of the interface of $u_{k}$ to $F_{0}(u)$ (statement (c) of Proposition 3.2), we know that there exists a point $p\in B_{r/2}(x)$ that belongs to $\{\tau\varepsilon_{k}\leq u_{k}\leq T\varepsilon_{k}\}\cap B_{1/4}$ for all large $k$. Now, as $u_{k}$ satisfies the density property $\mathcal{D}(\kappa,L)$, we have

[TABLE]

Moreover, as $\mathrm{1}_{B_{r/4}(p)\cap\{u>0\}}\leq\liminf_{k\to\infty}\mathrm{1}_{B_{r/4}(p)\cap\{u_{k}>T\varepsilon_{k}\}}$, we get by Fatou’s lemma that

[TABLE]

Combining (5.1) and (5.2), we obtain the desired density bound:

[TABLE]

∎

We are now finally in a position to prove Theorem 1.6.

Proof of Theorem 1.6.

Without loss of generality, assume that $u(0)=T$. Let $R_{k}\uparrow\infty$ and let $\varepsilon_{k}:=1/R_{k}$. Consider the blow-downs of $u$ at [math],

[TABLE]

which are solutions of (1.5) in $\mathbb{R}^{n}$, that are stable with respect to compact domain deformations. Furthermore, condition (1.6) says that the interface of each $u_{\varepsilon_{k}}$ satisfies the density property $\mathcal{D}(\kappa,L)$ in $\mathbb{R}^{n}$. Invoking Proposition 5.1, we see that $u_{\varepsilon_{k}}$ subconverge uniformly on compact subsets of $\mathbb{R}^{n}$ to a global inner-stable solution $u_{0}$ of the one-phase FBP (1.7). Given that $n<n^{*}$, Remark 4.11 informs us that $u_{0}$ actually has a flat free boundary and equals $x_{n}^{+}$, in an appropriate Euclidean coordinate system. From the Hausdorff distance convergence result of Proposition 3.2, we see that

[TABLE]

with the aspect ratio $\delta_{k}/R_{k}\to$ as $k\to\infty$. We may thus invoke the rigidity result [AS22, Theorem 1.4] of Audrito and Serra and conlude that $u(x)=V(x_{n})$, where $V$ is the solution of (1.16). ∎

Appendix A First and second inner variations of $I_{\varepsilon}$ in an oriented Riemannian manifold

Let $(M,g)$ be an oriented Riemannian manifold with induced volume form $\mathrm{vol}_{g}$. In this section we will compute expressions for the first and the second inner variations of the functional

[TABLE]

i.e. with respect to deformations of $M$, generated by compactly supported vector fields. The norm of the gradient $\nabla_{g}u$ is measured with respect to the metric $g$ and we note that

[TABLE]

where $\bar{g}_{p}$ denotes the induced inner product on covectors $\xi\in T^{*}_{p}(M)$. Take $X\in\Gamma(TM)$ a smooth, compactly supported vector field on $M$ and let $\phi:\mathbb{R}\times M\to M$ be its associated flow:

[TABLE]

For all $t\in\mathbb{R}$, $\phi_{t}:M\to M$ defines a diffeomorphism of $M$ onto itself with inverse $(\phi_{t})^{-1}=\phi_{-t}$, generated by $(-X)$. Fix a function $u\in H^{1}_{\text{loc}}(M)$ and set

[TABLE]

where $(\phi_{-t})^{*}$ denotes the pullback by $\phi_{-t}$. Then $u_{t}\in H^{1}_{\text{loc}}(M)$ and we are interested in computing

[TABLE]

where $D\subset M$ is a compact subset of $M$, containing the support of $X$.

Proposition A.1**.**

Assume the above setup. Then the first and second inner variations of $I_{\varepsilon}(\cdot,M)$ at $u$ along the vector field $X$ are given by

[TABLE]

where

[TABLE]

and $L_{X}$ denotes the $Lie$ derivative along $X$.

We refer the reader to the book of Lee [Lee13, Chapter 12] for a discussion of the many nice properties that the Lie derivative enjoys. We recall that in local coordinates $\{x^{i}\}$ of $M$, the Lie derivative of a $(2,0)$ tensor field $S=S^{ij}\partial_{x^{i}}\otimes\partial_{x^{j}}$ takes the form

[TABLE]

where we have adopted the standard summation convention over repeated indices. For a domain $M=\Omega\subseteq\mathbb{R}^{n}$ of Euclidean space, equipped with the Euclidean metric $g=\delta$, the expressions for $L_{X}\bar{\delta}$ and $L_{X}^{2}\bar{\delta}$ in the standard coordinates then take the form

[TABLE]

Proof of Proposition A.1.

After changing variables, $y=\phi_{t}(x)$, we get

[TABLE]

Since the differential $d$ commutes with pullbacks, we can rewrite the expression for $h_{t}$ as:

[TABLE]

We can view $\bar{g}\in\Gamma(TM\otimes TM)$ as a contravariant $(2,0)$ tensor field and $\bar{g}(\omega_{1},\omega_{2})$, where $\omega_{1},\omega_{2}$ are 1-forms, as the corresponding contraction of the (2,2) tensor field $\bar{g}\otimes\omega_{1}\otimes\omega_{2}$. Using the fact that pullbacks and contractions commute, and that pullbacks distribute over tensor products, we can further simplify

[TABLE]

since $\phi_{t}^{*}\phi_{-t}^{*}=(\phi_{-t}\circ\phi_{t})^{*}=\text{id}^{*}=\text{id}$. In (A.5), $\mu_{t}:=\phi_{t}^{*}\bar{g}$ denotes the pullback of the tensor field $\bar{g}$ by $\phi_{t}$. The $t$-derivatives of the tensor fields $\mu_{t}$ and $\nu_{t}$ can now be computed using the celebrated formula [Lee13, Proposition 12.36]

[TABLE]

We obtain

[TABLE]

Similarly,

[TABLE]

It is well known ([Lee13, pp. 425]) that the Lie derivative of $\mathrm{vol}_{g}$ computes to

[TABLE]

and by using the property that $L_{X}$ is a derivation, we can further calculate

[TABLE]

Based on the preceding observations, we see that $t\to I_{\varepsilon}(u_{t},D)$ is a smooth function, whose first derivative at $t=0$ is given by

[TABLE]

and whose second derivative at $t=0$ is

[TABLE]

according to the computations in (A.5)–(A.9). ∎

We end this section by fleshing out the divergence structure in the integrands $V_{1}$ and $V_{2}$ of (A.1) and (A.2). For ease of notation, we will drop subscripts $g$ and denote

[TABLE]

Lemma A.2**.**

Assume that $u\in C^{2}(W)$ and $\mathcal{F}_{\varepsilon}(u)\in C^{1}(W)$ in an open subset $W\subseteq M$, $\varepsilon\geq 0$. Then

[TABLE]

Proof.

We compute in $W$:

[TABLE]

where we used the fact that $L_{X}$ commutes with the differential $d$. Adding the two equalities above, we obtain (A.10). ∎

As an easy corollary, we get the following well known result.

Proposition A.3**.**

Let $\varepsilon>0$. If $u\in C^{2}(M)$ and $\mathcal{F}_{\varepsilon}(u)\in C^{1}(M)$, then

[TABLE]

for all compactly supported, smooth vector fields $X\in\Gamma(TM)$. In particular, if $u\in C^{2}(M)$ is a positive critical point of $I_{\varepsilon}$, then the first inner variation $\delta I_{\varepsilon}(u,M)=0$.

In the next lemma we provide the divergence structure within $V_{2}$.

Lemma A.4**.**

Assume that $u\in C^{3}(W)$ and $\mathcal{F}_{\varepsilon}(u)\in C^{1}(W)$ in an open subset $W\subseteq M$, $\varepsilon\geq 0$. Then we have in $W$:

[TABLE]

Proof.

We manipulate the terms comprising $V_{2}$ as follows:

[TABLE]

Hence, after adding the three equalities, we obtain

[TABLE]

where

[TABLE]

∎

Proposition A.5**.**

Let $u\in C^{3}(M)$ be a critical point of $I_{\varepsilon}$ such that $f_{\varepsilon}(u)=\frac{1}{2}\mathcal{F}^{\prime}_{\varepsilon}(u)\in C^{1}(M)$, $\varepsilon>0$. Then

[TABLE]

Proof.

Since $u\in C^{3}(M)$ is a critical point of $I_{\varepsilon}$, we have $2\Delta u=\mathcal{F}^{\prime}_{\varepsilon}(u)$. After integration, the divergence terms in (A.11) vanish and we are left with

[TABLE]

after another application of the Divergence theorem. ∎

Appendix B First and second inner variations for regular free boundaries

We will apply the formulas in Lemma A.2 and A.4 to simplify the expressions for the first and second inner variations of the Alt-Caffarelli energy $I_{0}$ in the Euclidean setting.

Definition B.1**.**

Let $W\subset\mathbb{R}^{n}$ be a open set. We say that a point $p\in\partial W$ is $C^{1}$-regular if there exists $r>0$ and a $C^{1}$ function $g:\mathbb{R}^{n-1}\to\mathbb{R}$ such that in a suitable Euclidean coordinate system

[TABLE]

Otherwise, we call p singular. We will denote by $\text{Reg}(\partial W)$ the (relatively open) subset of $C^{1}$-regular points of $\partial W$.

Proposition B.2**.**

Let $\Omega\subseteq\mathbb{R}^{n}$ be a Euclidean domain and assume that $u\in H^{1}_{\text{loc}}(\Omega)\cap C(\Omega)$ is a nonnegative inner-stationary solution of (1.7) in $\Omega$ that satisfies

• •

$u$* is harmonic in $\Omega^{+}_{0}(u)=\{x\in\Omega:u(x)>0\}$;*

• •

$u$* is $C^{1}$ up to $\text{Reg}(F_{0}(u))$.*

Then $|\nabla u|(p)=1$ at every $C^{1}$-regular point $p\in F_{0}(u)$.

Proof.

Pick a regular point $p\in F_{0}(u)$ and let $B$ be a small enough ball centered at $p$ such that $W:=\Omega^{+}_{0}(u)\cap B$ is the supergraph of a $C^{1}$ function. Let $X\in C^{\infty}_{c}(B;\mathbb{R}^{n})$. Since $u\in C^{\infty}(W)\cap C^{1}\left(W\cup\text{Reg}(F_{0}(u))\right)$ and $\mathcal{F}_{0}(u)=1$ in $W$, (A.10) tells us that

[TABLE]

as $\Delta u=0$ in $W$. Now, since $\nabla u=0$ a.e. in $\{u=0\}$, we see that

[TABLE]

where the last equality is a consequence of the Divergence Theorem and $\nu$ denotes the outer unit normal to $\partial D$. As $\langle 2L_{X}u\nabla u,\nu\rangle=2|\nabla u|^{2}\langle X,\nu\rangle$, we deduce

[TABLE]

Since $X\in C^{\infty}_{c}(B;\mathbb{R}^{n})$ can be taken arbitrary, we conclude that $|\nabla u(p)|=1$. ∎

Proposition B.3**.**

Let $\Omega\subseteq\mathbb{R}^{n}$ be a Euclidean domain and suppose that $u\in H^{1}_{\text{loc}}(\Omega)\cap C(\Omega)$ satisfies

• •

$u$* is an inner-stationary solution of (1.7): $\delta I_{0}(u,\Omega)=0$;*

• •

$u$* is harmonic in $\Omega^{+}_{0}(u)=\{x\in\Omega:u(x)>0\}$;*

• •

$u$* is $C^{2}$ up to the $\text{Reg}(F_{0}(u))$.*

Then for every vector field $X\in C^{\infty}_{c}(\Omega,\mathbb{R}^{n})$ supported away from the singular part of $F_{0}(u)$, the second inner variation of $I_{0}$ at $u$, along $X$, equals

[TABLE]

where $H$ denotes the mean curvature of the regular free boundary $\text{Reg}(F_{0}(u))$ with respect to the outer unit normal $\nu=-\nabla u$.

Proof.

Since $\nabla u=0$ a.e. in $\{u=0\}$, the integration in the formula for $\delta^{2}I_{0}(u,\Omega)[X]$ can be taken only over the positive phase $W:=\Omega^{+}(u)$. In $W$ $u$ is smooth and $\mathcal{F}_{0}^{\prime}(u)=0$, so that we have the validity of formulas (A.11)-(A), indicating

[TABLE]

on account of the fact that $\Delta u=0$ in $D$, where $Y$ is given by (A). Denote $\Sigma:=\text{Reg}(F_{0}(u))$. Since $V_{2}$ is supported away from the singular part of $F_{0}(u)$, we may apply the Divergence Theorem to obtain

[TABLE]

where $Y_{0}$ is the continuous vector field on $\Sigma$, defined by

[TABLE]

with $Y(x)$ given by (A). Note that in (B.2) we have used Proposition B.2 that the outer unit normal to $\partial\Sigma$, $\nu=-\nabla u$.

We claim that

[TABLE]

where $X^{T}$ denotes the component of $X$ tangential to $\Sigma$, and $\text{div}_{\Sigma}\,Z$ denotes the surface divergence of a vector field $Z$ on $\Sigma$:

[TABLE]

Once we establish (B.3), the formula (B.1) will be a consequence of (B.2) and the Divergence Theorem, applied in $\Sigma$.

Pick any point $p\in\Sigma$. It will be convenient to work in a Euclidean coordinate system $(x_{1},\ldots x_{n})$ centered at $p$, such that the unit vector along $x_{n}$, $e_{n}=\nabla u(p)$. With this choice, $u_{i}(p)=0$ for $i\in S:=\{1,2,\ldots,n-1\}$, $u_{n}(p)=1$ and

[TABLE]

Since $|\nabla u|=1$ on $\Sigma$, we have $u_{ni}(p)=0$ for $i\in S$. Furthermore, because of harmonicity and the fact that $|\nabla u|(p)=1$, the mean curvature of $\Sigma$ with respect to the outer unit normal $\nu=-\nabla u$,

[TABLE]

With all this in mind, let us calculate the left-hand side of (B.3), using the coordinates above. Since

[TABLE]

we have at $p$,

[TABLE]

On the other hand, as $X^{T}=X-\langle X,\nabla u\rangle\nabla u$ on $\Sigma$, the right-hand side of (B.3) equals

[TABLE]

where we used the harmonicity of $u$ and the formula (B.4) for the mean curvature of $\Sigma$ to obtain the last line. Now, (B.5) and (B.6) give (B.3), thereby completing the proof of the proposition.

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