Convergence of the solutions of the nonlinear discounted Hamilton-Jacobi equation: The central role of Mather measures
Résumé
Given a continuous Hamiltonian $H : (x,p,u) \mapsto H(x,p,u)$ defined on $ T^*M \times \mathbb R $, where $M$ is a closed connected manifold, we study viscosity solutions, $u_λ: M\to \mathbb R$, of discounted equations: $ H(x, d_x u_λ, λu_λ(x))=c$ in $M$, where $λ>0$ is called a discount factor and $c$ is the critical value of $H(\cdot, \cdot , 0)$. When $H$ is convex and superlinear in $p$ and non--decreasing in $u$, under an additional non--degeneracy condition, we obtain existence and uniqueness (with comparison principles) results of solutions and we prove that the family of solutions $(u_λ)_{λ>0}$ converges to a specific solution $u_0$ of $ H(x, d_x u_0, 0)=c$ in $M$. Our degeneracy condition requires $H$ to be increasing (in $u$) on localized regions linked to the support of Mather measures, whereas usual similar results are obtained for Hamiltonians that are everywhere increasing in $u$.
Mots clés
Analysis of PDEs (math.AP)
FOS: Mathematics
Discounted Hamilton-Jacobi equations viscosity solutions comparison principle weak KAM Theory Mather measures 2010 Mathematics Subject Classification. 35B40 35F21 37J50 49L25
Discounted Hamilton-Jacobi equations
viscosity solutions
comparison principle
weak KAM Theory
Mather measures 2010 Mathematics Subject Classification. 35B40
35F21
37J50
49L25
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)