Convergence of the solutions of the discounted Hamilton–Jacobi equation
Abstract
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.