International audienceWe consider a continuous coercive Hamiltonian H on the cotangent bundle of the compact connected manifold M which is convex in the momentum. If uλ:M→ℝ is the viscosity solution of the discounted equationλuλ(x)+H(x,dxuλ)=c(H),where c(H) is the critical value, we prove that uλ converges uniformly, as λ→0, to a specific solution u0:M→ℝ of the critical equationH(x,dxu)=c(H).We characterize u0in terms of Peierls barrier and projected Mather measures. As a corollary, we infer that the ergodic approximation, as introduced by Lions, Papanicolaou and Varadhan in 1987 in their seminal paper on homogenization of Hamilton–Jacobi equations, selects a specific corrector in the limit

We consider the Hamilton-Jacobi equation ∂tu + H(x, Du) = 0 in (0, +∞) × T N where T N is the flat N-dimensional torus, and the Hamiltonian H(x, p) is assumed continuous in x and strictly convex and coercive in p. We study the large time behavior of solutions, and we identify the limit through a Lax-type formula. Some convergence results are also given for H solely convex. Our qualitative method is based on the analysis of the dynamical properties of the so called Aubry set, performed in the spirit of [13]. This can be viewed as a generalization of the techniques used in [11] and [18]. Analogous results have been obtained in [4] using PDE methods.

We perform a qualitative investigation of critical Hamilton-Jacobi equations, with stationary ergodic Hamiltonian, in dimension 1. We show the existence of approximate correctors, give characterizing conditions for the existence of correctors, provide Lax-type representation formulae and establish comparison principles. The results are applied to look into the corresponding effective Hamiltonian and to study a homogenization problem. In the analysis a crucial role is played by tools from stochastic geometry such as, for instance, closed random stationary sets

International audienceWe derive a discrete version of the results of Davini et al. (Convergence of the solutions of the discounted Hamilton–Jacobi equation. Invent Math, 2016). If M is a compact metric space, c:M×M→ℝ a continuous cost function and λ∈(0,1), the unique solution to the discrete λ-discounted equation is the only function uλ:M→ℝ such that∀x∈M,uλ(x)=miny∈Mλuλ(y)+c(y,x).We prove that there exists a unique constant α∈ℝ such that the family of uλ+α/(1−λ) is bounded as λ→1 and that for this α, the family uniformly converges to a function u0:M→ℝ which then verifies∀x∈X,u0(x)=miny∈Xu0(y)+c(y,x)+α.The proofs make use of Discrete Weak KAM theory. We also characterize u0 in terms of Peierls barrier and projected Mather measures

Abstract. We introduce a notion of Aubry set for weakly coupled systems of Hamilton-Jacobi equations on the torus and characterize it as the region where the obstruction to the existence of globally strict critical subsolutions concentrates. As in the case of a single equation, we prove the existence of critical subsolutions which are strict and smooth outside the Aubry set. This allows us to derive in a simple way a comparison result among critical sub and supersolutions with respect to their boundary data on the Aubry set, showing in particular that the latter is a uniqueness set for the critical system. We also highlight some rigidity phenomena taking place on the Aubry set.