Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains

TL;DR

This paper investigates the asymptotic behavior of solutions to fully nonlinear Hamilton-Jacobi equations on noncompact domains as a parameter approaches zero, introducing a variational approach and a selection principle.

Contribution

It extends localization techniques and develops a modified variational formula for nonlinear Hamilton-Jacobi equations, providing new insights into their vanishing discount limits.

Findings

Derived limiting Mather-type measures.

Formulated a new selection principle.

Extended localization techniques to nonlinear equations.

Abstract

We study the asymptotic behavior of solutions to the fully nonlinear Hamilton-Jacobi equation $H(x, Du, \lambda u) = 0$ in $\mathbb{R}^n$ as $\lambda \to 0^+$. Under the assumption that the Aubry set is localized, we employ a variational approach to derive limiting Mather-type measures and formulate a selection principle. Central to our analysis is a modified variational formula that bridges global and local state-constraint solutions, thereby extending localization techniques to the nonlinear framework.