Discontinuous solutions in \(L^\infty\) for Hamilton-Jacobi equations
type: Item
instance of: scholarly article
title: Discontinuous solutions in \(L^\infty\) for Hamilton-Jacobi equations
zbMATH Open document ID: 0953.35032
DOI: 10.1142/S0252959900000200
author: Bo Su
author: Gui-Qiang Chen
published in: Chinese Annals of Mathematics. Series B
publication date: 2000-06-21T00:00:00Z
review text: This paper is concerned with the global discontinuous solutions in \(L^\infty\) of the Cauchy problem for Hamilton-Jacobi equations \[ \begin{cases} u_t+ H(t,x,u,Du)=0,\;x\in\mathbb{R}^n,\;0\leq t\leq T,\\ u(0,x)= \varphi(x). \end{cases} \] The approach introduced to construct global discontinuous solutions allows the initial data only in \(L^\infty\) and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in \(L^\infty\). The existence of global discontinuous solutions in \(L^\infty\) is established by taking the infimum of the supsolution set or the supremum of the subsolution set. These solutions in \(L^\infty\) coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is studied to analyze the stability of discontinuous solutions in \(L^\infty\). The analysis also shows that the global discontinuous solutions are determined by the topology in which the initial data are approximated.
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 1462259
zbMATH Keywords: minimax solutions
zbMATH Keywords: nonconvex Hamiltonians
reviewed by: Vasile Iftode
Wikidata QID: Q57386549
MaRDI profile type: Publication
full work available at URL: s0252959900000200
OpenAlex ID: W2320295996
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description: scientific article; zbMATH DE number 1462259