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