Convergence of the solutions of the discounted Hamilton–Jacobi equation

Abstract

International audienceWe 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 equationλuλ(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 equationH(x,dxu)=c(H).We characterize u0in 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

The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space

“…this maximum does exist in force of (A3). The function u ≡ − b λ is subsolution to (DP) for any λ > 0, so that λ u λ ≥ −b, and accordingly (7) 0…”

“…Further, due to H ≥ H, any subsolution of λ u + H[u] is also subsolution to (DP), which implies that the maximal subsolution to λ u + H[u] = 0 is less than or equal to u λ . On the other side, since by (7) H(x, Du λ (x)) = H(x, Du λ (x)) for a.e. x the function u λ itself is subsolution to λ u + H[u] = 0.…”

“…As in [7], we derive the asymptotic behavior of solutions from weak convergence of suitable associated measures. Our method however is rather different.…”

The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus

“…Hamilton-Jacobi equations in [8] (first-order case), [1] (first-order case with Neumanntype boundary condition), [20] (degenerate viscous case). In the case of first-order equations [1,8], optimal control formulas are used and the Mather measures, introduced and developed in [18,19], play a central role.…”

“…In the case of first-order equations [1,8], optimal control formulas are used and the Mather measures, introduced and developed in [18,19], play a central role. The proof in [20] for a degenerate viscous case is based on the nonlinear adjoint method, stochastic Mather measures and a regularization procedure by mollifications to handle the delicate viscous term using a commutation lemma.…”