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1995 Eikonal equations with discontinuities
Richard T. Newcomb II, Jianzhong Su
Differential Integral Equations 8(8): 1947-1960 (1995).


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.


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Richard T. Newcomb II. Jianzhong Su. "Eikonal equations with discontinuities." Differential Integral Equations 8 (8) 1947 - 1960, 1995.


Published: 1995
First available in Project Euclid: 20 May 2013

zbMATH: 0854.35022
MathSciNet: MR1348959

Primary: 35F20
Secondary: 35B05 , 35D05

Rights: Copyright © 1995 Khayyam Publishing, Inc.


Vol.8 • No. 8 • 1995
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