Variational and viscosity operators for the evolutive Hamilton-Jacobi equation

TL;DR

This paper develops a variational operator for the evolutive Hamilton-Jacobi equation with non-convex Hamiltonians, extending to viscosity solutions and providing Lipschitz estimates, thus broadening the solution framework.

Contribution

It introduces a new variational operator for non-convex Hamiltonians and connects it with viscosity solutions, extending classical methods.

Findings

Constructed a variational operator for non-convex Hamiltonians.

Extended the operator to viscosity solutions with Lipschitz estimates.

Reproduced the Lax-Oleinik semigroup in convex/concave cases.

Abstract

We study the Cauchy problem for the first order evolutive Hamilton-Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax-Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable.