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

We study the asymptotic behavior of the viscosity solutions u(G)(lambda) of the Hamilton-Jacobi (HJ) equationlambda u(x) + G(x, u') = c(G) in Ras the positive discount factor lambda tends to 0, where G(x, p) := H(x, p) - V(x) is the perturbation of a Hamiltonian H is an element of C(R x R), Z -periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential V is an element of C-c(R). The constant c(G) appearing above is defined as the infimum of values a is an element of R for which the HJ equation G(x, u') = a in R admits bounded viscosity subsolutions. We prove that the functions u(G)(lambda) locally uniformly converge, for lambda -> 0(+), to a specific solution u(G)(0) of the critical equationG(x, u') = c(G) in R.We identify u(G)(0) in terms of projected Mather measures for G and of the limit u(H)(0) to the unperturbed periodic problem. Our work also includes a qualitative analysis of the critical equation with a weak KAM theoretic flavor.