## Differential and Integral Equations

- Differential Integral Equations
- Volume 8, Number 8 (1995), 1947-1960.

### Eikonal equations with discontinuities

Richard T. Newcomb, II and Jianzhong Su

#### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 8, Number 8 (1995), 1947-1960.

**Dates**

First available in Project Euclid: 20 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1369056134

**Mathematical Reviews number (MathSciNet)**

MR1348959

**Zentralblatt MATH identifier**

0854.35022

**Subjects**

Primary: 35F20: Nonlinear first-order equations

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35D05

#### Citation

Newcomb, Richard T.; Su, Jianzhong. Eikonal equations with discontinuities. Differential Integral Equations 8 (1995), no. 8, 1947--1960. https://projecteuclid.org/euclid.die/1369056134