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.
Convergence of the solutions of the discounted Hamilton–Jacobi equation: Convergence of the discounted solutions / Davini, Andrea; Fathi, Albert; Iturriaga, Renato; Zavidovique, Maxime. - In: INVENTIONES MATHEMATICAE. - ISSN 0020-9910. - STAMPA. - 206:1(2016), pp. 29-55. [10.1007/s00222-016-0648-6]
Convergence of the solutions of the discounted Hamilton–Jacobi equation: Convergence of the discounted solutions
DAVINI, ANDREA;
2016
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.File | Dimensione | Formato | |
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