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Journal Articles Inventiones Mathematicae Year : 2016

Convergence of the solutions of the discounted Hamilton–Jacobi equation

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Albert Fathi
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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.

Dates and versions

ensl-01412048 , version 1 (07-12-2016)

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Albert Fathi. Convergence of the solutions of the discounted Hamilton–Jacobi equation. Inventiones Mathematicae, 2016, ⟨10.1007/s00222-016-0648-6⟩. ⟨ensl-01412048⟩
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