Discontinuous Solutions of a Hamilton--Jacobi Equation with Infinite Speed of Propagation

Abstract

We consider the Cauchy problem \begin{eqnarray*} u\Rn\times (0,+\infty),$} u\Rn\times\{t=0\},$} \end{eqnarray*} where A(x) is a positive locally Lipschitz map from ${\Bbb R}n$ to the space of symmetric matrices. Since no growth condition on A is assumed, pathological phenomena can occur. We study the problem in the framework of discontinuous viscosity solutions and we get some comparison results and a representation formula for the minimal solution of the problem.