Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations
type: Item
instance of: scholarly article
title: Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations
zbMATH Open document ID: 1053.35044
author: Martino Bardi
author: Yoshikazu Giga
published in: Communications on Pure and Applied Analysis
publication date: 2004-03-07T00:00:00Z
review text: The authors study the Cauchy problem for Hamilton-Jacobi equations \[ u_{t}+H(x,Du)=0, \text{ in } \mathbb{R}^{N}\times(0,T), u(x,0)=g(x), \text{ for } x\in\mathbb{R}^{N}, \] where the initial data \(g\) is an upper semicontinuous map from \(\mathbb{R}^{N}\) into the real line (that is, not necessarily continuous). Moreover, they impose the following hypothesis on the Hamiltonian \[ | H(x,p)-H(x,q)| \leq L| p-q| , \text{ for all } x,p,q\in \mathbb{R}^{N}, \] and \[ | H(x,p)-H(y,p)| \leq C_{R}(1+| p| )| x-y| , \text{ for all } | x| ,| y| \leq R,p\in \mathbb{R}^{N}. \] First, the authors establish that the generalized solution \(u(x,t)\) can be represented as the value function \(U(x,t)\) of a differential game. The formula is too much involved to be described here. Second, using this representation formula the authors obtain necessary and sufficient conditions in order for the solution to attain the initial data g through the so-called right accessibility, that is, along sequences \(\{t_{n}\}\), \(\{x_{n}\}\), with \(t_{n} >0\), \(x_{n}\in \mathbb{R}^{N}\), such that \(t_{n}\rightarrow 0\), \(x_{n}\rightarrow x_{0}\) and \(u(x_{n},t_{n})\rightarrow g(x_{0})\). It is worthly pointing out that no convexity assumptions on \(H\) are imposed.
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 2052054
zbMATH Keywords: Hamilton-Jacobi equations
zbMATH Keywords: Cauchy problem
zbMATH Keywords: semicontinuous viscosity solutions
zbMATH Keywords: right accessibility
zbMATH Keywords: differential games
reviewed by: Antonio Linero Bas
MaRDI profile type: Publication
full work available at URL: cpaa.2003.2.447
OpenAlex ID: W2141978100
DOI: 10.3934/CPAA.2003.2.447
description: scientific article; zbMATH DE number 2052054