Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function

TL;DR

This paper proves the uniform convergence of solutions to a class of Hamilton-Jacobi equations depending nonlinearly on the unknown function as a parameter tends to zero, characterizing the limit via variational measures.

Contribution

It establishes the convergence of solutions for nonlinear Hamilton-Jacobi equations depending on the unknown, extending previous results to more general Hamiltonians.

Findings

Solutions $u^\lambda$ converge uniformly as $\lambda o 0$

Limit solution characterized by Peierls barrier and Mather measures

Results apply under broad conditions on the Hamiltonian

Abstract

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function. Let $H(x,p,u)$ be a continuous Hamiltonian which is strictly increasing in $u$, and is convex and coercive in $p$. For each parameter $\lambda>0$, we denote by $u^\lambda$ the unique viscosity solution of the H-J equation \[H( x,Du(x),\lambda u(x) )=c.\] Under quite general assumptions, we prove that $u^\lambda$ converges uniformly, as $\lambda$ tends to zero, to a specific solution of the critical H-J equation $ H(x,Du(x),0)=c.$ We also characterize the limit solution in terms of Peierls barrier and Mather measures.