Convergence of the solutions of the nonlinear discounted Hamilton–Jacobi equation: The central role of Mather measures

Abstract

Given a continuous Hamiltonian H:(x,p,u)↦H(x,p,u) defined on T⁎M×R, where M is a closed connected manifold, we study viscosity solutions, uλ:M→R, of discounted equations:H(x,dxuλ,λuλ(x))=cin M where λ>0 is called a discount factor and c is the critical value of H(⋅,⋅,0).When the Hamiltonian H is convex and superlinear in p and non–decreasing in u, under an additional non–degeneracy condition, we obtain existence and uniqueness (with comparison principles) results of solutions and we prove that the family of solutions (uλ)λ>0 converges to a specific solution u0 ofH(x,dxu0,0)=cin M. Our non–degeneracy condition requires H to be increasing (in u) on localized regions linked to the support of Mather measures, whereas usual similar results are obtained for Hamiltonians that are everywhere increasing in u.