Weak KAM theory for general Hamilton-Jacobi equations I: the solution semigroup under proper conditions

TL;DR

This paper extends weak KAM theory to more general Hamilton-Jacobi equations where the Hamiltonian explicitly depends on the unknown function, by developing a variational principle and solution semigroup under certain conditions.

Contribution

It introduces an implicit solution semigroup and extends weak KAM theory to Hamilton-Jacobi equations with explicit dependence on the solution.

Findings

Viscosity solutions tend asymptotically to weak KAM solutions.

Established a variational principle for generalized Hamilton-Jacobi equations.

Extended Fathi's weak KAM theory to broader classes of Hamiltonians.

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*} Under some assumptions on $H(x,u,p)$ with respect to $p$ and $u$, we provide a variational principle on the evolutionary Hamilton-Jacobi equation. By introducing an implicitly defined solution semigroup, we extend Fathi's weak KAM theory to certain more general cases, in which $H$ explicitly depends on the unknown function $u$. As an application, we show the viscosity solution of the evolutionary Hamilton-Jacobi equation with initial condition tends asymptotically to the weak KAM solution of the following stationary Hamilton-Jacobi equation: \begin{equation*} H(x,u(x),\partial_xu(x))=0. \end{equation*}.