Weak KAM Research Articles

Long Time Average of First Order Mean Field Games and Weak KAM Theory

We show that the long time average of solutions of first order mean field game systems in finite horizon is governed by an ergodic system of mean field game type. The well-posedness of this later system and the uniqueness of the ergodic constant rely on weak KAM theory.

Regularization of Subsolutions in Discrete Weak KAM Theory

AbstractWe expose different methods of regularizations of subsolutions in the context of discrete weak KAM theory that allow us to prove the existence and the density of C1,1 subsolutions. Moreover, these subsolutions can be made strict and smooth outside of the Aubry set.

Ground States for Mean Field Models with a Transverse Component

We investigate global logarithmic asymptotics of ground states for a family of quantum mean field models. Our approach is based on a stochastic representation and a combination of large deviation and weak KAM techniques. The spin- 1/2 case is worked out in more detail.

Translation invariance of weak KAM solutions of the Newtonian $N$-body problem

We consider in this note the Hamilton-Jacobi equation H(x, dx u) = c, where c \geq 0, of the classical N-body problem in an Euclidean space E of dimension k \geq 2. The fixed points of the Lax-Oleinik semigroup are global viscosity solutions for the critical value of the constant (c = 0) also called weak KAM solutions. We show that all these solutions are invariant under the action of E by translations on the space of configurations. We deduce the existence of non-invariant solutions for the super-critical equations (c > 0).

NEW ESTIMATES FOR EVANS' VARIATIONAL APPROACH TO WEAK KAM THEORY

We consider a recent approximate variational principle for weak KAM theory proposed by Evans. As in the case of classical integrability, for one-dimensional mechanical Hamiltonian systems all the computations can be carried out explicitly. In this setting, we illustrate the geometric content of the theory and prove new lower bounds for the estimates related to its dynamic interpretation. These estimates also extend to the case of n degrees of freedom.

Weak KAM Solutions of a Discrete-Time Hamilton-Jacobi Equation in a Minimax Framework

The purpose of this paper is to study the existence of solutions of a Hamilton-Jacobi equation in a minimax discrete-time case and to show different characterizations for a real number called the critical value, which plays a central role in this work. We study the behavior of solutions of this problem using tools of game theory to obtain a “fixed point” of the Lax operator associated, considering some facts of weak KAM theory to interpret these solutions as discrete viscosity solutions. These solutions represent the optimal payoff of a zero-sum game of two players, with increasingly long time payoffs. The developed techniques allow us to study the behavior of an infinite time game without using discount factors or average actions.

Systems of Weakly Coupled Hamilton-Jacobi Equations with Implicit Obstacles

Abstract In this paper we study systems of weakly coupled Hamilton-Jacobi equations with implicit obstacles that arise in optimal switching problems. Inspired by methods from the theory of viscosity solutions and weak KAM theory, we extend the notion of Aubry set for these systems. This enables us to prove a new result on existence and uniqueness of solutions for the Dirichlet problem, answering a question of F. Camilli, P. Loreti, and N. Yamada.

The Lax–Oleinik semi-group: a Hamiltonian point of view

The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on ℝd and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications.

Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set

We introduce the notion of a viscosity solution for the first-order Hamilton–Jacobi equation, in the more general setting of manifolds, to obtain a weak KAM theory using only tools from partial differential equations. This work should be accessible to people with no prior knowledge of the subject.

Weak KAM theory for nonregular commuting Hamiltonians

In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax--Oleinik semigroups. This is equivalent to the solvability of an associated multi--time Hamilton--Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a result recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical subsolution, strict outside their Aubry set. This subsolution can be taken of class $C^{1,1}$ in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable differentiability properties of the critical subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest.

Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures

In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as:•the Lyapunov exponents of minimizing measures;•the weak KAM solutions. In particular, we deduce that the support of every minimizing measure μ, all of whose Lyapunov exponents are zero, is C1-regular μ-almost everywhere.

A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics

We extend the metric proof of the converse Lyapunov Theorem,given in [13] for continuous multivalued dynamics, bymeans of tools issued from weak KAM theory, to the case where theset-valued vector field is just upper semicontinuous. Thisgenerality is justified especially in view of application todiscontinuous ordinary differential equations. The more relevantnew point is that we introduce, to compensate the lack ofcontinuity, a family of perturbed dynamics, obtained throughinternal approximation of the original one, and perform somestability analysis of it.

Uniqueness sets for minimization formulas

In this paper, we consider minimization formulas which arise typically in optimal control and weak KAM theory for Hamilton-Jacobi equations. Given a minimization formula, we define a uniqueness set for the formula, which replaces the original region of minimization without changing its values. Our goal is to provide a necessary and sufficient condition that a given set be a uniqueness set. We also provide a characterization of the existence of a minimal uniqueness set with respect to set inclusion.

Mather Measures Associated with a Class of Bloch Wave Functions

In this paper we study the Wigner transform for a class of smooth Bloch wave functions on the flat torus $${\mathbb{T}^n = \mathbb{R}^n /2\pi \mathbb{Z}^n}$$ : $$\psi_{\hbar,P} (x) = a (\hbar,P,x) {\rm e}^{ \frac{i}{\hbar} ( P\cdot x + \hat{v}(\hbar,P,x) )}.$$ On requiring that $${P \in \mathbb{Z}^n}$$ and $${\hbar = 1/N}$$ with $${N \in \mathbb{N}}$$ , we select amplitudes and phase functions through a variational approach in the quantum states space based on a semiclassical version of the classical effective Hamiltonian $${{\bar H}(P)}$$ which is the central object of the weak KAM theory. Our main result is that the semiclassical limit of the Wigner transform of $${\psi_{\hbar,P}}$$ admits subsequences converging in the weak* sense to Mather probability measures on the phase space. These measures are invariant for the classical dynamics and Action minimizing.

Generalized Maupertuis’ principle with applications

We give a rigorous proof of the equivalence of Mane’s supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians, and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather’s α-function is discussed.

On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton–Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy–Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the “weak KAM approach”, which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry–Mather sets.

Strict sub-solutions and Mañé potential in discrete weak KAM theory

In this paper, we explain some facts on the discrete case of weak KAM theory. In that setting, the Lagrangian is replaced by a cost c\colon X\times X \rightarrow \mathbb{R} , on a “reasonable” space X . This covers for example the case of periodic time-dependent Lagrangians. As is well known, it is possible in that case to adapt most of weak KAM theory. A major difference is that critical sub-solutions are not necessarily continuous. We will show how to define a Mañé potential. In contrast to the Lagrangian case, this potential is not continuous. We will recover the Aubry set from the set of continuity points of the Mañé potential, and also from critical sub-solutions.

Connecting orbits for families of Tonelli Hamiltonians

We investigate the existence of Arnold diffusion-type orbits forsystems obtained by iterating in any order the time-one maps of afamily of Tonelli Hamiltonians. Such systems are known as'polysystems' or 'iterated function systems'. When specialized tofamilies of twist maps on the cylinder, our results are similar tothose obtained by Moeckel [20] and Le Calvez [15]. Ourapproach is based on weak KAM theory and is close to the one used byBernard in [3] to study the case of a single TonelliHamiltonian.

A New Kind of Lax-Oleinik Type Operator with Parameters for Time-Periodic Positive Definite Lagrangian Systems

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.

On smooth time functions

AbstractWe are concerned with the existence of smooth time functions on connected time-oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold M where is just defined a field of tangent convex cones (Cx)x ∈ M enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function.Our approach is based on the definition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory of Cx. We find connections with topics pertaining to Hamilton–Jacobi equations, and make use of tools and results issued from weak KAM theory.