Viscosity solutions of discontinuous Hamilton–Jacobi equations

Abstract

We define viscosity solutions for the Hamilton-Jacobi equation ${\varphi }_t=v(x,t)H(\nabla \varphi )$ in ${\mathbb{R}}^N\times (0,\infty )$ where $v$ is positive and bounded measurable and $H$ is non-negative and Lipschitz continuous. Under certain assumptions, we establish the existence and uniqueness of Lipschitz continuous viscosity solutions. The uniqueness result holds in particular for those $v$ which are independent of $t$ and piecewise continuous with discontinuous sets consisting of finitely many smooth lower dimensional surfaces not tangent to each other at any point of their intersection.ets are determined by the solution of a disc packing problem.