On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case

Abstract We study the asymptotic behavior of the viscosity solutions of the Hamilton-Jacobi (HJ) equation as the positive discount factor λ tends to 0, where is the perturbation of a Hamiltonian –periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential The constant c(G) appearing above is defined as the infimum of values for which the HJ equation in admits bounded viscosity subsolutions. We prove that the functions locally uniformly converge, for to a specific solution of the critical equation We identify in terms of projected Mather measures for G and of the limit to the unperturbed periodic problem. Our work also includes a qualitative analysis of the critical equation with a weak KAM theoretic flavor.