Weak KAM theory for general Hamilton-Jacobi equations III: the variational principle under Osgood conditions

TL;DR

This paper develops a variational principle for a class of Hamilton-Jacobi equations with Osgood growth conditions, linking viscosity solutions to minimal characteristics and providing a new representation formula.

Contribution

It introduces an implicit variational principle for Hamilton-Jacobi equations under Osgood conditions, extending the weak KAM theory to more general settings.

Findings

Established an implicit variational principle under Osgood conditions.

Linked viscosity solutions with minimal characteristics.

Derived a new representation formula for 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 Osgood growth of $H(x,u,p)$ with respect to $u$, we establish an implicitly variational principle and provide an intrinsic relation between viscosity solutions and certain minimal characteristics. Moreover, we obtain a representation formula of the viscosity solution of the evolutionary Hamilton-Jacobi equation.