Almost continuous solutions of geometric Hamilton--Jacobi equations
type: Item
instance of: scholarly article
title: Almost continuous solutions of geometric Hamilton--Jacobi equations
zbMATH Open document ID: 1029.35067
DOI: 10.1016/S0294-1449(02)00010-0
author: Antonio Siconolfi
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2003-05-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78578
review text: The Hamilton-Jacobi time dependent equation \(u_t+H(x,Du)=0\) in \(\mathbb{R}^N \times (0,+\infty)\), where \(H\) has constant sign and is homogeneous in the second argument, but does not verify any convexity or uniform continuity assumptions with respect to the state variable, is studied. The author is interested in discontinuous viscosity solutions verifying certain initial conditions with discontinuous data. More precisely, the locally bounded solutions \(u\) verifying \[ \limsup_{(x,t) \to(x_0,0)}u^* (x,t)\leq u_0^* (x_0),\;\liminf_{(x,t) \to(x_0,0)} u_*(x,t)\geq u_{0_*}(x_0) \] for any \(x_0\in \mathbb{R}^N\), where \(u_0\) is a locally bounded initial datum and \(u^*(u_*)\) denotes the upper (lower) semicontinuous envelope of \(u\), are seeked. The aim of this paper is to give representation formulae as well as uniqueness and stability results.
Mathematics Subject Classification ID: 35F20
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 35C99
zbMATH DE Number: 1912450
zbMATH Keywords: Hamilton-Jacobi equations
zbMATH Keywords: Discontinuous viscosity solutions
zbMATH Keywords: Representation formulae
zbMATH Keywords: uniqueness and stability results
reviewed by: Vasile Iftode
MaRDI profile type: Publication
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OpenAlex ID: W2030084745
description: scientific article; zbMATH DE number 1912450