Differential and Integral Equations

Eikonal equations with discontinuities

Richard T. Newcomb and Jianzhong Su

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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

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

First available in Project Euclid: 20 May 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35F20: Nonlinear first-order equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35D05


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

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