Eikonal equations with discontinuities

Abstract

This paper is concerned with the Hamilton-Jacobi equation of eikonal type $$ H(Du)=n(x) \qquad x\in \Omega \subset {\Bbb R}^N ,\tag E $$ where $H$ is convex, $Du$ represents the gradient of $u$ with respect to $x$, and $n(x)$ is lower semi-continuous. In this work, a new notion of generalized solution for (E) is developed which is appropriate for this class of discontinuous right-hand sides $n(x)$. Such solutions we term Monge solutions. The Monge notion arises in a natural way from the variational formulation of (E) and is consistent with the well-known viscosity notion when $n(x)$ is continuous. In the class of lower semi-continuous $n(x)$, we establish the comparison principle for Monge subsolutions and supersolutions, existence and uniqueness results for (E) with Dirichlet boundary conditions, and a stability result. Moreover, we show that the Monge solution can be smaller than the maximal Lipschitz subsolution.