On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

TL;DR

This paper investigates the long-term behavior of solutions to Hamilton-Jacobi equations with nonlinear boundary conditions, providing convergence results through PDE methods and an optimal control approach, applicable under various convexity assumptions.

Contribution

It introduces new convergence results for Hamilton-Jacobi equations with nonlinear boundary conditions using PDE and control methods, including formulas for asymptotic solutions.

Findings

Established convergence of viscosity solutions for nonlinear boundary conditions.

Developed PDE-based methods applicable without convexity assumptions.

Provided formulas for asymptotic solutions using the weak KAM approach.

Abstract

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods : the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the "weak KAM approach" which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets.