Averaging of Hamilton-Jacobi equations over Hamiltonian flows

TL;DR

This paper investigates the long-term behavior of solutions to Hamilton-Jacobi equations with large drift terms, focusing on the averaging effect in the viscosity sense for general Hamiltonians.

Contribution

It extends averaging results for Hamilton-Jacobi equations to more general Hamiltonians under viscosity boundary conditions.

Findings

Established averaging results for Hamilton-Jacobi equations with general Hamiltonians.

Extended previous convex Hamiltonian results to non-convex cases.

Analyzed the asymptotic behavior of solutions with large drift terms.

Abstract

We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by the Hamiltonian vector fields of Hamiltonian $H$. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians ($G$ below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense, and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian $G$.