Weak KAM Research Articles

Weak KAM Theory topics in the stationary ergodic setting

We perform a qualitative analysis of the critical equation associated with a stationary ergodic Hamiltonian through a stochastic version of the metric method, where the notion of closed random stationary set, issued from stochastic geometry, plays a major role. Our purpose is to give an appropriate notion of random Aubry set, to single out characterizing conditions for the existence of exact or approximate correctors, and write down representation formulae for them. For the last task, we make use of a Lax-type formula, adapted to the stochastic environment. This material can be regarded as a first step of a long-term project to develop a random analog of Weak KAM Theory, generalizing what done in the periodic case or, more generally, when the underlying space is a compact manifold.

On weak KAM theory for N-body problems

AbstractWe consider N-body problems with potential 1/r2κ, where κ∈(0,1), including the Newtonian case (κ=1/2). Given R>0 and T>0, we find a uniform upper bound for the minimal action of paths binding, in time T, any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates the Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding two configurations in free time). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax–Oleinik semigroup, and we show that they are global viscosity solutions of the corresponding Hamilton–Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

Hölder regularity of weak KAM solutions in a priori unstable systems

Hölder regularity of weak KAM solutions in a priori unstable systems

Weak KAM aspects of convex Hamilton–Jacobi equations with Neumann type boundary conditions

We study convex Hamilton–Jacobi equations H ( x , Du ) = 0 and u t + H ( x , Du ) = 0 in a bounded domain Ω of R n with the Neumann type boundary condition D γ u = g in the viewpoint of weak KAM theory, where γ is a vector field on the boundary ∂ Ω pointing a direction oblique to ∂ Ω. We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex Hamilton–Jacobi equations, some comparison and existence results for convex and coercive Hamilton–Jacobi equations with the Neumann type boundary condition as well as existence results for the Skorokhod problem. We define the Aubry set associated with the Neumann type boundary problem and establish some properties of the Aubry set including the existence results for the “calibrated” extremals for the corresponding action functional (or variational problem).

Weak KAM methods and ergodic optimal problems for countable Markov shifts

Let σ: Σ → Σ be the left shift acting on Σ, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of σ-invariant Borel probabilities that maximize the integral of a given locally Holder continuous potential A: Σ → ℝ. Under certain conditions, we are able to show not only that A-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions).

Weak KAM for commuting Hamiltonians

For two commuting Tonelli Hamiltonians, we recover the commutation of the Lax–Oleinik semi-groups, a result of Barles and Tourin (2001 Indiana Univ. Math. J. 50 1523–44), using a direct geometrical method (Stoke's theorem). We also obtain a ‘generalization’ of a theorem of Maderna (2002 Bull. Soc. Math. France 130 493–506). More precisely, we prove that if the phase space is the cotangent of a compact manifold then the weak KAM solutions (or viscosity solutions of the critical stationary Hamilton–Jacobi equation) for G and for H are the same. As a corollary we obtain the equality of the Aubry sets and of the Peierls barrier. This is also related to works of Sorrentino (2009 On the Integrability of Tonelli Hamiltonians Preprint) and Bernard (2007 Duke Math. J. 136 401–20).

Continuity of optimal control costs and its application to weak KAM theory

We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the Aubry–Mather problems corresponding to these systems.

Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory

In this article, following [29], we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function $c: M \times M\to \R$ defined on a smooth connected manifold is locally semiconcave and satisfies twist conditions, then there exists a $C^{1,1}$ critical subsolution strict on a maximal set (namely, outside of the Aubry set). We also explain how this applies to costs coming from Tonelli Lagrangians. Finally, following ideas introduced in [18] and [26], we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analog of Mather's $\alpha$ function on the cohomology.

Homogenization of generalized characteristics associated to solutions of Hamilton–Jacobi equations

It was proved in Arch. Ration. Mech. Anal. 162 (2002), 1-23 that singularities of solutions for Hamilton-Jacobi equa- tions will propagate along generalized characteristics. In this paper, we consider the homogenization of generalized character- istics. It is natural to think that (∗) generalized characteristics associated to viscosity solutions of oscillatory Hamilton-Jacobi equations will converge to generalized characteristics associated to solutions of the effective equation. We show that this is indeed correct if the spatial dimension is 1. For high dimensions, we need to add some extra assumptions of singularities near generalized characteristics. We provide a counterexample that (∗) fails without those assumptions. Some issues related to the weak KAM theory are also discussed.

Lagrangian dynamics on an infinite-dimensional torus; a Weak KAM theorem

The space L 2 ( 0 , 1 ) has a natural Riemannian structure on the basis of which we introduce an L 2 ( 0 , 1 ) -infinite-dimensional torus T . For a class of Hamiltonians defined on its cotangent bundle we establish existence of a viscosity solution for the cell problem on T or, equivalently, we prove a Weak KAM theorem. As an application, we obtain existence of absolute action-minimizing solutions of prescribed rotation number for the one-dimensional nonlinear Vlasov system with periodic potential.

Singular Dynamics for Semiconcave Functions

Semiconcave functions are a well-known class of nonsmooth functions that possess deep connections with optimization theory and nonlinear pde’s. Their singular sets exhibit interesting structures that we investigate in this paper. First, by an energy method, we analyze the curves along which the singularities of semiconcave solutions to Hamilton–Jacobi equations propagate—the socalled generalized characteristics. This part of the paper improves the main result in [P. Albano, P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations , Arch. Ration. Mech. Anal. 162 (2002), 1–23] and simplifies the construction therein. As applications, we recover some known results for gradient flows and conservation laws. Then we derive a simple dynamics for the propagation of singularities of general semiconcave functions. This part of the work is also used to study the singularities of generalized solutions to Monge–Ampère equations. We conclude with a global propagation result for the singularities of solutions e in weak KAM theory.

Fast Transport Optimization for Monge Costs on the Circle

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circle, both of which have the same total mass, find an optimal transport plan with respect to a given cost function satisfying the Monge condition. In the circular case the sorting strategy fails to provide a unique candidate solution and a naive approach requires a quadratic number of operations. For the case of $N$ real-valued point masses we present an O(N |log epsilon|) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.

Birkhoff’s theorem and viscosity solutions of Hamilton-Jacobi equations

We obtain a partial generalization of Birkhoff’s theorem of invariant curve to higher dimesional case in the context of viscosity solutions of Hamilton-Jacobi equations, or weak KAM theory. This is a new approach after Herman’s proof.

Singular perturbations of weakly coupled systems of Hamilton–Jacobi equations

In this paper we study singular perturbations of weakly coupled systems of elliptic equations. A model problem is given by small random perturbations of random evolution processes. In this setting we give a PDE proof of large deviation results, analogous to those studied in Ann. Probab. 15 (1987), 646-658, Stochastics Stochastics Rep. 33 (1990), 111-148, Probab. Theory Related Fields 94 (1993), 335-374. An essential tool in our approach is the weak KAM theory introduced in Calc. Var. Partial Differential Equations 22 (2005), 185-228.

Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem

Aim of this paper is to show that some of the results in the weak KAM theoryfor $1^{s t}$ order convex Hamilton-Jacobiequations (see [11], [13])can be extended to systems of convex Hamilton-Jacobi equations with implicit obstacles and to the obstacle problem.We obtain two results:a comparison theorem for systems lacking strict monotonicity; a representation formulafor the obstacle problem involving the distance functionassociated to the Hamiltonian of the equation.

Further PDE methods for weak KAM theory

We introduce and make estimates for several new approximations that in appropriate asymptotic limits yield the key PDE for weak KAM theory, namely a Hamilton–Jacobi type equation for a potential u and a coupled transport equation for a measure σ. We revisit as well a singular variational approximation introduced in Evans (Calc Vari Partial Differ Equ 17:159–177, 2003) and demonstrate “approximate integrability” of certain phase space dynamics related to the Hamiltonian flow. Other examples include a pair of strongly coupled PDE suggested by the Lions–Lasry theory (Lasry and Lions in Japan J Math 2:229–260, 2007) of mean field games and a new and extremely singular elliptic equation suggested by sup-norm variational theory.

Long-time Behavior of Solutions of Hamilton–Jacobi Equations with Convex and Coercive Hamiltonians

We investigate the long-time behavior of viscosity solutions of Hamilton–Jacobi equations in \({\mathbb{R}^n}\) with convex and coercive Hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity. We apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples. We take a dynamical approach, based on tools from weak KAM theory such as extremal curves, Aubry sets and representation formulas for solutions, for these investigations.

The dynamics of pseudographs in convex Hamiltonian systems

We study the evolution, under convex Hamiltonian flows on cotangent bundles of compact manifolds, of certain distinguished subsets of the phase space. These subsets are generalizations of Lagrangian graphs, which we call pseudographs. They emerge in a natural way from Fathi’s weak KAM theory. By this method, we find various orbits which connect prescribed regions of the phase space. Our study was inspired by works of John Mather. As an application, we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem. We hope that our method will have applications to more examples. Résumé. Nous étudions l’évolution, par le flot d’un Hamiltonien convexe sur une variété compacte, de certains ensembles de l’espace des phases. Nous appelons pseudographes ces ensembles, qui sont des généralisations de graphes Lagrangiens apparaissant de manière naturelle dans la théorie KAM faible de Fathi. Par cette méthode, nous trouvons diverses orbites qui joignent des domaines donnés de l’espace des phases. Notre étude s’inspire de travaux de John Mather. Nous obtenons l’existence de diffusion dans une large classe de systèmes à priori instables comme application de cette méthode, qui permet de résoudre le probleme de l’écart entre les tores invariants. Nous espérons que la méthode s’appliquera à d’autres exemples.

Minimizers of Dirichlet functionals on the n-torus and the weak KAM theory

Given a probability measure μ on the n-torus Tn and a rotation vector P∈Rn, we ask whether there exists a minimizer to the integral ∫Tn|∇ϕ+P|2dμ. This problem leads, naturally, to a class of elliptic PDE and to an optimal transportation (Monge–Kantorovich) class of problems on the torus. It is also related to higher dimensional Aubry–Mather theory, dealing with invariant sets of periodic Lagrangians, and is known as the “weak-KAM theory”.

Weak kam theorem on non compact manifolds

In this paper, we consider a time independent C2 Hamiltonian, satisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide.