On global discontinuous solutions of Hamilton-Jacobi equations
type: Item
instance of: scholarly article
title: On global discontinuous solutions of Hamilton-Jacobi equations
zbMATH Open document ID: 1002.35029
DOI: 10.1016/S1631-073X(02)02228-8
author: Bo Su
author: Gui-Qiang Chen
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-01-13T00:00:00Z
review text: The aim of this paper is to study global discontinuous solutions of the Cauchy problem for the Hamilton-Jacobi equations: \[ \begin{cases} u_t+H(t,x,u, Du)=0,\;x\in\mathbb{R}^d,\;t>0,\\ u(0,x)=\varphi(x). \end{cases}\tag{1} \] Under some natural conditions on \(H\) the authors prove that the classical semicontinuous solution of (1) is unique, providing that \(\varphi(x)\) is almost everywhere continuous (a.e.-continuous, for short). Moreover they show that, in the space of a.e.-continuous functions, (1) is well posed, and they establish the Lipschitz regularity of the discontinuous solutions after finite time. Note that this regularization effect is new even for conventional continuous viscosity solutions.
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 1744363
zbMATH Keywords: Lipschitz regularity
zbMATH Keywords: viscosity solutions
Wikidata QID: Q57386535
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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full work available at URL: s1631-073x(02)02228-8
OpenAlex ID: W2036153248
description: scientific article; zbMATH DE number 1744363