Convergence of the solutions of the discounted Hamilton–Jacobi equation: Convergence of the discounted solutions

We 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 equation λuλ(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 equation H(x,dxu) = c(H). We characterize u0 in 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.