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

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.

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 : X×X → 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 subsolutions are not necessarily continuous. We will show how to define a Mañe 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 Mañe potential, and also from critical sub-solutions.

Following the random approach of [27], we define a Lax-Oleinik formula adapted to evolutive weakly coupled systems of Hamilton-Jacobi equations. It is reminiscent of the corresponding scalar formula, with the relevant difference that it has a stochastic character since it involves, loosely speaking, random switchings between the various associated Lagrangians. We prove that the related value functions are viscosity solutions to the system, and establish existence of minimal random curves under fairly general hypotheses. Adding Tonelli like assumptions on the Hamiltonians, we show differentiability properties of such minimizers, and existence of adjoint random curves. Minimizers and adjoint curves are trajectories of a twisted generalized Hamiltonian dynamics.