Weak KAM theory for general Hamilton-Jacobi equations II: the fundamental solution under Lipschitz conditions

TL;DR

This paper develops a variational framework for evolutionary Hamilton-Jacobi equations with Lipschitz continuous Hamiltonians, introducing a fundamental solution and analyzing long-term behavior of viscosity solutions.

Contribution

It establishes a variational principle and a fundamental solution for Hamilton-Jacobi equations under Lipschitz conditions, linking viscosity solutions to minimal characteristics.

Findings

Introduces an implicit fundamental solution for the equations.

Provides a variational representation formula for viscosity solutions.

Analyzes the large time behavior of solutions.

Abstract

We consider the following evolutionary Hamilton-Jacobi equation with initial condition: \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x), \end{cases} \end{equation*} where $\phi(x)\in C(M,\mathbb{R})$. Under some assumptions on the convexity of $H(x,u,p)$ with respect to $p$ and the uniform Lipschitz of $H(x,u,p)$ with respect to $u$, we establish a variational principle and provide an intrinsic relation between viscosity solutions and certain minimal characteristics. By introducing an implicitly defined {\it fundamental solution}, we obtain a variational representation formula of the viscosity solution of the evolutionary Hamilton-Jacobi equation. Moreover, we discuss the large time behavior of the viscosity solution of the evolutionary Hamilton-Jacobi equation and provide a dynamical representation formula of the viscosity solution of the stationary Hamilton-Jacobi equation with strictly increasing $H(x,u,p)$ with respect to $u$.