Vol. 15, No. 5, 2022

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Convergence of solutions for some degenerate discounted Hamilton–Jacobi equations

Maxime Zavidovique

Vol. 15 (2022), No. 5, 1287–1311
Abstract

We study solutions of Hamilton–Jacobi equations of the form

λα(x)uλ(x) + H(x,Dxuλ) = c,

where α is a nonnegative function, λ a positive constant, c a constant and H a convex coercive Hamiltonian. Under suitable conditions on α we prove that the functions uλ converge as λ 0 to a function u0 that is a solution of the critical equation H(x,Dxu0) = c.

Keywords
discounted Hamilton–Jacobi equations, viscosity solutions, weak KAM theory, Mather measures
Mathematical Subject Classification
Primary: 35D40, 35F21, 49L25
Milestones
Received: 10 June 2020
Accepted: 18 January 2021
Published: 29 September 2022
Authors
Maxime Zavidovique
Sorbonne Université and Université de Paris, CNRS, IMJ-PRG
Paris
France