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Convergence of the solutions of the discounted Hamilton-Jacobi equation. Convergence of the discounted solutions. (English) Zbl 1362.35094

Summary: We consider a continuous coercive Hamiltonian \(H\) on the cotangent bundle of the compact connected manifold \(M\) which is convex in the momentum. If \(u_\lambda :M\rightarrow \mathbb {R}\) is the viscosity solution of the discounted equation \[ \lambda u_\lambda (x)+H(x,\mathrm{d}_x u_\lambda)=c(H), \] where \(c(H)\) is the critical value, we prove that \(u_\lambda \) converges uniformly, as \(\lambda \rightarrow 0\), to a specific solution \(u_0:M\rightarrow \mathbb {R}\) of the critical equation \[ H(x,\mathrm{d}_x u)=c(H). \] We characterize \(u_0\) 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.

MSC:

35F21 Hamilton-Jacobi equations
35B40 Asymptotic behavior of solutions to PDEs
37J50 Action-minimizing orbits and measures (MSC2010)
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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