Mather measures for space-time periodical nonconvex hamiltonians

TL;DR

This paper develops a method to construct Mather measures for space-time periodic nonconvex Hamiltonians, expanding the understanding of variational principles in complex dynamical systems.

Contribution

It introduces a novel approach to construct Mather measures specifically for nonconvex Hamiltonians with space-time periodicity.

Findings

Successfully constructs Mather measures for the specified class of Hamiltonians.

Provides new insights into the variational structure of nonconvex Hamiltonian systems.

Extends classical Mather theory to more general, nonconvex settings.

Abstract

The main goal of this paper is to construct Mather measures for space-time periodical nonconvex Hamiltonians.

Equations

$$L(x,v,t)=\underset{p}{\max }pv-H(x,p,t),$$

$$\dot{x}=D_{p}H(x,p,t),$$

$$\dot{p}=-D_{x}H(x,p,t).$$

$$\dot{X}=D\bar{H}(P),$$

$$\dot{P}=0.$$

$$(p,x)\to (P,X)$$

$$u_t+H(x,D_{x}u,t)=\bar{H}(P).$$

$$v_t+H(x,D_{x}v(x,t),t)\ge \bar{H}(P),$$

$$v_t+H(x,D_{x}v(x,t),t)\le \bar{H}(P).$$

$$\{F,G\}:=D_{p}F\cdot D_{x}G-D_{x}F\cdot D_{p}G.$$

$${\int }_{{\mathbb{T}}^{d+1}\times {\mathbb{R}}^d}{\varphi }_s+\{H,\varphi \}d\mu =0$$

$${\int }_{{\mathbb{T}}^{d+1}\times {\mathbb{R}}^d}\varphi (x,p,t)d\mu (x,p,t)={\int }_{{\mathbb{T}}^{d+1}\times {\mathbb{R}}^d}\varphi (x,D_{v}L(x,v,t),t)d\nu (x,v,t)$$

$$\mathcal{D}=\Big{\{}\nu\in\mathcal{P}(\Omega):\int_{\Omega}\phi_{t}+\{H,\phi\}d\Phi_{\#}\nu(x,p,t)=0\text{ for every }\phi\in C_{c}^{1}(\Omega)\Big{\}},$$

$$\mathcal{F}=\{\nu \in \mathcal{P}(\Omega ):{\int }_{\Omega }{\psi }_t+v\cdot D\psi (x,t)d\nu (x,v,t)=0,\text{ for every }\psi \in C^1({\mathbb{T}}^{d+1})\}.$$

$$\underset{v\in \mathcal{F}}{\min }{\int }_{\Omega }L(x,v,t)d\nu (x,v,t),$$

$$\underset{\nu \in \mathcal{F}}{\min }{\int }_{\Omega }(L(x,v,t)-P\cdot v)d\nu .$$

$${\int }_{\Omega }{\varphi }_t+\{H,\varphi \}d\mu (x,p,t)=0,\text{ for every }\varphi \in C_c^1(\Omega )$$

$$\Sigma :=\{(x,p,t)\in {\mathbb{T}}^{d+1}\times {\mathbb{R}}^d:p=P+D_{x}u(x,P,t)\}$$

$$-\underset{\nu \in \mathcal{F}}{\min }{\int }_{\Omega }(L(x,v,t)-P\cdot v)d\nu (x,v,t),$$

$${\int }_{\Omega }{\varphi }_t+H(x,p,t)d\mu =\bar{H}(0),$$

$$H(x,p,t)=p\cdot D_{p}H(x,p,t)-L(x,D_{p}H(x,p,t),t),$$

$${\int }_{\Omega }{\varphi }_t+p\cdot D_{p}H(x,p,t)d\mu =0.$$

$${\int }_{\Omega }{\psi }_t+v\cdot D\psi (x,t)d\nu ={\int }_{\Omega }{\psi }_t+D_{p}H(x,p,t)\cdot D\psi (x,t)=0,$$

$$\bar{H}(0)={\int }_{\Omega }{\varphi }_t+H(x,p,t)d\mu ={\int }_{\Omega }{\varphi }_t+p\cdot D_{p}H-Ld\mu ={\int }_{\Omega }-Ld\mu .$$

$${\int }_0^{\infty }\chi (u{)}^{-1}du=\infty \text{ and }|H(x,p,t)|\le \chi (|p|).$$

$$H(x,p,t)=|p_1|-|p_2|+V(x,t)$$

$${\int }_0^{\infty }\chi (u{)}^{-1}du=\infty$$

$${\varphi }_t^{\epsilon }+\epsilon \Delta {\varphi }^{\epsilon }+H(x,P+D{\varphi }^{\epsilon }(x,t),t)=\bar{H}(P).$$

$${\varphi }_t^{\epsilon }+\Delta {\varphi }^{\epsilon }+H(x,D{\varphi }^{\epsilon },t)-\epsilon {\varphi }^{\epsilon }=0$$

$$\sup \Vert D{\varphi }^{\epsilon }(\cdot ,t){\Vert }_{\infty }\le K$$

$$L_{\epsilon ,P}\psi ={\psi }_t+\epsilon \Delta \psi +D_{p}H(x,P+D{\varphi }^{\epsilon }(x,t),t)D\psi ,$$

Full text

1. Introduction.

1.1. The Convex Case.

Let $\mathbb{T}^{d+1}$ be the $d+1$ - torus and consider a smooth periodic Tonelli Hamiltonian $H:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{R}$.

Let $L:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{R}$ be the Lagrangian associated to the Hamiltonian:

[TABLE]

for every $(x,v,t)\in\mathbb{T}^{d+1}\times\mathbb{R}^{d}$.

Now we consider the corresponding flow of the time dependent Hamiltonian:

[TABLE]

Now the dynamics transforms to

[TABLE]

under the change of variables

[TABLE]

where $p=P+D_{x}u(x,P,t)$, $X=x+D_{P}u(x,P,t)$ if we suppose that both $u(x,P,t)$ and $\bar{H}(P)$ are smooth functions and $u(x,P,t)$ satisfies the time dependent Hamilton-Jacobi equation

[TABLE]

Definition 1**.**

A continuous function $u:\mathbb{T}^{d+1}\to\mathbb{R}$ is called a forward viscosity solution of (4) if it satisfies the two properties.

1. (1)

If $v$ is a $C^{1}$ function and $u-v$ has a local maximum at $(x,t)$, then

[TABLE] 2. (2)

If $v$ is a $C^{1}$ function and $u-v$ has a local minimum at $(x,t)$, then

[TABLE]

Backward viscosity solutions are defined by reversing both inequalities.

It is known ([CIS], [EG]) that there is only one value $\bar{H}(P)$, such that (4) has a time periodic viscosity solution.

As a consequence of the semilinearity and convexity there is a consequence map $\Phi:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{T}^{d+1}\times\mathbb{R}^{d},$ given by $\Phi(x,v,t)=(x,D_{v}L(x,v,t),t),$ wich is well defined and one-to-one.

Recall the Poisson bracket,

[TABLE]

In Hamiltonian coordinates, the property of invariance for a probability measure $\nu$ can be written as

[TABLE]

for every $\phi\in C_{c}^{1}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$, where $\mu=\Phi_{\#}\nu$ is the push-forward of the measure $\nu$ with respect to the map $\Phi$, i.e. the measure $\mu$ such that

[TABLE]

for every $\phi\in C_{c}^{1}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$.

Denoting by $\mathcal{P}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$ the class of probability measures on $\mathbb{T}^{d+1}\times\mathbb{R}^{d}$, and taking $\Omega=\mathbb{T}^{d+1}\times\mathbb{R}^{d}$, where $(x,v,t)=z$ represents a generic point $z\in\Omega$ with $(x,t)\in\mathbb{T}^{d+1}$ and $v\in\mathbb{R}^{d}$.

Now let $\mathcal{D}$ be the class of probability measures in $\Omega$ that are invariant under the Euler-Lagrange flow, so we have

[TABLE]

and the set of holonomic measures

[TABLE]

We recall the Mather problem

[TABLE]

a more general version of (5) consists in studying for each $P\in\mathbb{R}^{d}$ fixed

[TABLE]

Any minimizer of (6) is a Mather measure, now the following proposition will be helpful to prove an important result.

Proposition 2**.**

Let $H:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{R}$ be a smooth function that satisfies the classical hypotheses. Let $P\in\mathbb{R}^{d}$, $\nu\in\mathcal{P}(\Omega)$ be a minimizer of (6) and set $\mu=\Phi_{\#}\nu$. Then

1. (i)

$\mu$* is invariant under the Hamiltonian dynamics, i.e.,*

[TABLE] 2. (ii)

$\mu$* is supported on the graph*

[TABLE]

where $u$ is any viscosity solution of (4).

The proof of the proposition is a consequence of results in [B] and [CIS].

As in [CGT] the following theorem gives a characterization of Mather measures in the time dependent convex case.

Theorem 3**.**

Assume $H:\mathbb{T}^{d+1}\times\mathbb{R}^{d}\to\mathbb{R}$ is a smooth function that satisfies the classical hypotheses of convexity, superlinearity, and periodicity and let $P\in\mathbb{R}$. Then $\nu\in\mathcal{P}(\Omega)$ is a solution of

[TABLE]

if and only if

1. (a)

$\int_{\Omega}\,\phi_{t}+H(x,p,t)d\mu=\bar{H}(P)=H(x,p,t)\,$* $\,\mu$ a.e.,* 2. (b)

$\int_{\Omega}\phi_{t}+(p+P)\cdot\,D_{p}H(x,p,t)\,d\mu(x,p,t)=0$, 3. (c)

$\int_{\Omega}\phi_{t}+D_{p}H(x,p,t)\cdot D\phi(x,p,t)=0$, for every $\phi\in C^{1}(\mathbb{T}^{d+1})$.

where $\mu=\Phi_{\#}\nu$ and $\bar{H}(P)$ is the unique value such that (4) has a time periodic viscosity solution.

Proof.

To simplify, we will assume $P=0$. Let us prove that $\mu=\Phi_{\#}\nu$ satisfies (a)-(c). From (ii) of the last proposition, and (1), we have that

[TABLE]

so (a) holds.

Now, we know that

[TABLE]

and from (a) it follows that

[TABLE]

Finally (c) follows from that $\nu\in\mathcal{F}$.

Reciprocally let $\mu\in\mathcal{P}({\Omega})$ such that (a), (b) and (c) holds, and we will show that $\nu=\Phi_{\#}\mu$ is a minimizer of (6).

Now observe that $\nu\in\mathcal{F}$, then

[TABLE]

for every $\psi\in C^{1}(\mathbb{T}^{d+1})$.

The fact that $\nu$ is a minimizer is obtained by using (a) and (b)

[TABLE]

∎

The previous characterization will help us to define Mather measures in the nonconvex case.

1.2. The Nonconvex Case

Throughout the paper, we will assume that

1. i.

$H$ is smooth, 2. ii.

$H(\cdot,p,t)$ is $\mathbb{Z}^{d+1}$-periodic for $(p,t)\in\mathbb{R}^{d+1}$, 3. iii.

There exists a continuous function $\chi:[0,+\infty)\to\mathbb{R}$ such that

[TABLE]

Example 1**.**

Consider

[TABLE]

If we take $\chi(u)=2u+C$,

[TABLE]

and $|H(x,p,t)|=||p_{1}|-|p_{2}|+V(x,t)|\leq\chi(|p|)$.

We extend the definition of Mather measure in the nonconvex and time dependent setting:

Definition 4**.**

We say that a measure $\mu\in\mathcal{P}(\Omega)$ is a Mather measure if there exists $P\in\mathbb{R}^{d}$ such that properties (a)-(c) in Theorem 3 are satisfied.

Our main result is:

Theorem 5**.**

Assume that the Hamiltonian is a smooth function that satisfies the conditions (i.)-(iii.) and let $\{\mu^{\varepsilon}\}_{\varepsilon>0}$ be the family of measures defined in (16). Then there exist a Mather measure $\mu$ and a nonnegative and symmetric $d\times d$ matrix $(m_{kj})_{k,j=1,\dots d}$ of Borel measures called the dissipation measure, such that:

1. (1)

$\mu^{\varepsilon}\overset{*}{\rightharpoonup}\mu$* in the sense of measures up to subsequences,* 2. (2)

$\int_{\Omega}\varphi_{t}+\{H,\varphi\}d\mu+\int_{\Omega}\varphi_{p_{k}p_{j}}dm_{kj}=0\text{ for all }\varphi\in C_{c}(\Omega),$** 3. (3)

$\operatorname{supp}\mu$* and $\operatorname{supp}m$ are compact.*

2. Uniform Derivate Bounds

Let us consider the equation:

[TABLE]

Lemma 6**.**

The periodic solutions of (8) have first derivatives, uniformly bounded in $\varepsilon$.

Sketch of the proof..

For every $\varepsilon>0$ let us consider the following problem

[TABLE]

The above equation has a unique smooth solution $\phi^{\varepsilon}$ in $\mathbb{R}^{d+1}$ which is $\mathbb{Z}^{d+1}$ periodic [GV]. First, we proved that $D\phi^{\varepsilon}$ is uniformly bounded, by following [BS] we proved that there exists $K>0$ depending only on $H$ such that

[TABLE]

Finally if we take $g=d_{t}^{2}phi^{\varepsilon}+|D\phi^{\varepsilon}|^{2}$and using the Bernstein’s method we prove that $d_{t}\phi^{\varepsilon}$ is uniformly bounded.

∎

Theorem 7**.**

For every $\varepsilon>0$ and every $P\in\mathbb{R}^{d}$, there exists a unique number $H^{\varepsilon}(P)\in\mathbb{R}$ such that the equation (8) admits a unique (up to constants) $\mathbb{Z}^{d+1}$ periodic viscosity solution. Moreover, for every $P\in\mathbb{R}^{d}$ $\underset{\varepsilon\to 0^{+}}{\lim}\bar{H}^{\varepsilon}(P)\to\bar{H}(P)$ and $\phi^{\varepsilon}\to\phi_{0}$ uniformly (up to subsequences), where $\phi_{0}:\mathbb{T}^{d+1}\to\mathbb{R}$ is that (4) is satisfied in the viscosity sense.

Proof.

The theorem follows by Lemma 6, the stability theorem for viscosity solutions and the Arzela-Ascoli Theorem. ∎

3. Stochastic Measures

Definition 8**.**

Let $\varepsilon>0$ and $P\in\mathbb{R}^{d}$. The linearized operator associated to (8) is defined as $L_{\varepsilon,P}:C^{2}(\mathbb{T}^{d+1})\to C(\mathbb{T}^{d+1}):$

[TABLE]

for every $\psi\in C^{2}(\mathbb{T}^{d+1}).$

As in [CGT], we denote by $\beta$ either a direction in $\mathbb{R}^{d}$ (i.e., $\beta\in\mathbb{R}^{d}$ with $|\beta|=1$) or a parameter (for example $\beta=P_{i}$ for some $i\in\{1,\dots,d\}$). When $\beta=P_{i}$ for some $i\in\{1,\dots,d\}$ the symbols $H_{\beta}$ and $H_{\beta\beta}$ have to be understood as $H_{p_{i}}$ and $H_{p_{i}p_{i}}$ respectively. If we derive (8) with respect to $\beta$ and recalling (11) we get

[TABLE]

so

[TABLE]

As before, let $\Omega=\mathbb{T}^{d+1}\times\mathbb{R}^{d}$, where $(x,v,t)$ represents a generic point with $(x,t)\in\mathbb{T}^{d+1}$ and $v\in\mathbb{R}^{d}$. We need to introduce a probability space $(\Omega,{\mathcal{B}},\mathbb{P})$ endowed with a Brownian motion $W(t):\Omega\to\mathbb{T}^{d}$ on the flat $d$-torus. Let $\varepsilon>0$, to simplify we set $P=0$ and we introduce the time dependent vector field [Fl], $U_{\varepsilon}(x,t)=D_{p}H(x,D\phi_{\varepsilon}(x,t),t)$ and consider the solution $X_{\varepsilon}(s)$ of the stochastic differential equation

[TABLE]

And the momentum variable is defined as

[TABLE]

Now suppose $\mathbf{z}:[0,+\infty)\to\mathbb{R}^{d}$ is a solution to the stochastic differential equation

[TABLE]

with $a_{i}$ and $b_{ij}$ bounded and progressively measurable processes. Let $\varphi:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be a smooth function where $\varphi(\mathbf{z},t)$ satisfies the Itô formula:

[TABLE]

From hereafter, we will use Einstein’s convention for repeated indices in a sum. Here, we have $a_{i}=D_{p}H(x,D\phi^{\varepsilon}(x,t),t)$ and $b_{ij}=\sqrt{2\varepsilon}\delta_{ij}$.

Therefore, from (13), (14) and (12),

[TABLE]

Thus $(X_{\varepsilon},p_{\varepsilon})$ satisfies the following stochastic version of the Hamiltonian dynamics

[TABLE]

Now we are going to study the solution $\phi^{\varepsilon}$ of (8) along the trajectory $X_{\varepsilon}(s)$. Due to the Itô formula, and the equations (8) and (13).

[TABLE]

And using the Dynkin formula, we obtain

[TABLE]

Now we will associate to each trajectory $(X_{\varepsilon},p_{\varepsilon},t)$ of (15) a probability measure $\mu_{\varepsilon}\in\mathcal{P}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$ defined by

[TABLE]

for every $\phi\in C_{c}(\mathbb{T}^{d+1}\times\mathbb{R}^{d}).$ Here, the definition makes sense provided the limit is taken over an appropiate subsequence. Then using Dynkin’s formula, we have that for every $\varphi\in C_{c}(\mathbb{T}^{d+1}\times\mathbb{R}^{d}),$

[TABLE]

Dividing the equation (17) by $T$ and taking the limit when $T\to\infty$ along a suitable subsequence we obtain:

[TABLE]

Let us define the projected measure $\theta_{\mu^{\varepsilon}}\in\mathcal{P}(\mathbb{T}^{d+1})$ as follows

[TABLE]

for all $\varphi\in C(\mathbb{T}^{d+1})$. And using test functions that do not depend on $p$ in the last definition:

[TABLE]

for all $\varphi\in C^{2}(\mathbb{T}^{d+1})$.

Given $\phi^{\varepsilon}$, let us consider the partial differential equation

[TABLE]

From lemma 32 and lemma 33 in [GV], we have that 0 is the principal value of Fokker-Planck operator

[TABLE]

and so $\mu^{\varepsilon}$ can be defined as a unique measure such that

[TABLE]

for every $\psi\in C_{c}(\mathbb{T}^{d+1}\times\mathbb{R}^{d})$.

3.1. Uniform Estimates.

Lemma 9**.**

We have the following estimates:

[TABLE]

[TABLE]

[TABLE]

Proof.

Recalling (11), we obtain

[TABLE]

Thus

[TABLE]

Integrating with respect to $\theta_{\mu^{\varepsilon}}$ and using (19), we get (20).

To obtain (21) we differentiate (12) with respect to $\beta$, we have:

[TABLE]

Integrating again with respect to $\theta_{\mu^{\varepsilon}}$ and using (19) we get (21).

On the other hand

[TABLE]

using (23) we obtain

[TABLE]

once again, integrating with respect to $\theta_{\mu^{\varepsilon}}$ and by (19) we get (22). ∎

Following the techniques of [CGT], [E1] and [T], we will obtain several estimates that will be useful in the future.

Proposition 10**.**

We have the following

[TABLE]

[TABLE]

[TABLE]

Proof.

Taking $\beta=x_{1},x_{2},\dots,x_{d+1}$ respectively in (20), we have

[TABLE]

and adding these $d+1$ identities we obtain

[TABLE]

now, due to Lemma. 6, $|D_{x}\phi^{\varepsilon}|$, $|D_{x}H|$ are uniformly bounded, thus we get (24).

Now, the relation (25) follows by taking $\beta=P_{1},P_{2},\dots,P_{d+1}$ in (20), adding the $d+1$ identities

[TABLE]

And using the Young’s inequality.

To obtain (26), taking $\beta=x_{i}$ in (22)

[TABLE]

Due to Lemma 6, we have that $|H_{x_{i},x_{i}}|,\,|D_{p}H_{x_{i}}|,\,|D^{2}_{pp}H|\leq C$ on the support of $d\theta_{\mu^{\varepsilon}}$, so

[TABLE]

∎

4. Existence of Mather measures.

Now we are able to prove the existence of Mather measures.

Proof of Theorem 5..

The proof straightforward noticing that $\{\phi^{\varepsilon}\}$ have a uniform Lipschitz estimate, therefore there exists a compact set $K\subset\Omega$ such that $\operatorname{supp}\mu^{\varepsilon}\subset K\,\forall\varepsilon>0$. Moreover, up to subsequences, we have $\mu^{\varepsilon}\overset{*}{\rightharpoonup}\mu$, that is

[TABLE]

for every function $\phi\in C_{c}(\Omega)$, for some probability measure $\mu\in\mathcal{P}(\Omega)$, and also it follows that $\operatorname{supp}\mu\subset K$.

To obtain (2), let us remember (18) particularly the second term

[TABLE]

But

[TABLE]

by using the estimates in Proposition 10, so the

[TABLE]

Note that $\varepsilon\int_{\Omega}(\phi^{\varepsilon}_{x_{i}x_{k}}\phi^{\varepsilon}_{x_{i}x_{j}}\varphi_{p_{k}p_{j}})\,d\mu^{\varepsilon}$ does not vanish in the limit, through a subsequence for every $k,j=1,\dots,n$ we have

[TABLE]

for some nonnegative, symmetric $d\times d$ matrix $(m_{kj})\,\,k,j=1,\dots,n$ of Borel measures, so condition 2, follows. To obtain (3), recall that $\sup|p^{\varepsilon}(t)|<\infty$ and the periodicity in time.

Now we will prove that $\mu$ satisfies the conditions (a)-(c) in Definition 4 with $P=0$. Following [CGT], [E1] and [T] consider

[TABLE]

when $\varepsilon\to 0$ due to (8) and (24), thus (a) occurs.

Recalling the equation (18) and choosing as a test function $\varphi=\psi(\phi^{\varepsilon}(x,t))$

[TABLE]

if $\varepsilon$ goes to zero, we obtain $\int_{\Omega}\psi^{\prime}(\phi)[\phi_{t}+p\cdot D_{p}H]d\mu=0$, choosing $\psi(\phi)=\phi$, we obtain b). Now part c) follows by choosing in (2) test functions $\varphi$ that do not depend on the variable p. ∎