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

Given a continuous Hamiltonian $H:(x,p,u)\mapsto H(x,p,u)$ defined on $T^{\ast }M\times \mathbb{R}$, where $M$ is a closed connected manifold, we study viscosity solutions, $u_{\lambda }:M\to \mathbb{R}$, of discounted equations: $H(x,d_x{u}_{\lambda },\lambda u_{\lambda }(x))=c$ in $M$, where $\lambda >0$ is called a discount factor and $c$ is the critical value of $H(\cdot ,\cdot ,0)$. When $H$ 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 $(u_{\lambda }{)}_{\lambda >0}$ converges to a specific solution $u_0$ of $H(x,d_x{u}_0,0)=c$ in $M$. Our 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 $u$.