Almost classical solutions of Hamilton-Jacobi equations
type: Item
instance of: scholarly article
title: Almost classical solutions of Hamilton-Jacobi equations
zbMATH Open document ID: 1161.26005
DOI: 10.4171/RMI/564
author: Robert Deville
author: Jesús A. Jaramillo
published in: Revista Matemática Iberoamericana
publication date: 2009-02-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1228834302
full work available at URL: https://eudml.org/doc/42404
review text: The authors study the existence of almost classical solutions of fairly general Hamilton-Jacobi equations on open subsets of \(\mathbb R^d\) and on \(d\)-dimensional smooth manifolds for \(d \geq 2\). Let \(d \geq 2\) and let \(\Omega\) be an open subset of \(\mathbb R^d\). Let \(F : \mathbb R \times \Omega \times \mathbb R^{d} \rightarrow \mathbb R\) and \(u_{0} : \partial \Omega \rightarrow \mathbb R\) be continuous functions. We say that \(u : \overline \Omega \rightarrow \mathbb R\) is an almost classical solution of \(F(u(x),x,\nabla u(x))=0\) with Dirichlet condition \(u|_{ \partial \Omega} = u_{0}\) if {\parindent=4mm \begin{itemize}\item[{\(\bullet\)}] \(u(x) = u_{0}(x)\), \(x \in \partial \Omega\); \item[{\(\bullet\)}] for every \(x \in \Omega\), \(u\) is differentiable at \(x\) and \(F(u(x),x,\nabla u(x)) \leq 0\); \item[{\(\bullet\)}] \(u\) satisfies \(F(u(x),x,\nabla u(x))=0\) for almost every \(x \in \Omega\) in the sense of Lebesgue measure on \(\mathbb R^d\). \end{itemize}} The authors obtain the following. Theorem. Let \(d \geq 2\), let \(\Omega\) be an open subset of \(\mathbb R^d\) and let \(F : \mathbb R \times \Omega \times \mathbb R^{d} \rightarrow \mathbb R\) be a continuous function. Suppose that {\parindent=7mm \begin{itemize}\item[(A)] there exists a continuous function \(u_{0} : \overline \Omega \rightarrow \mathbb R\) such that \(u_{0}\) is \(C^{1}\)-smooth on \(\Omega\) and \(F(u_{0}(x),x,\nabla u_{0}(x)) \leq 0\) \((x \in \Omega)\); \item[(B)] for each compact set \(K \subseteq \Omega\) there exist an \(M_{K}>0\) and an \(\alpha_{K} > 0\) such that for every \(x \in K\), \(u \in [0,\alpha_{K}]\) and \(p \in \mathbb R ^{d}\) satisfying \(\|p\| \geq M_{K}\) we have \(F(u_{0}(x)+u,x,p) > 0\). \end{itemize}} Then there exists an almost classical solution \(u\) of \(F(u(x),x,\nabla u(x))=0\) with Dirichlet condition \(u|_{ \partial \Omega} = u_{0}\). Moreover, if \(u_{0}\) is \(C^{1}\)-smooth on \(\mathbb R^d\) then \(u\) can be extended to a…
reviewed by: Tamás Mátrai
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 58J32
zbMATH DE Number: 5509271
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: eikonal equation on manifolds
zbMATH Keywords: almost everywhere solution
MaRDI profile type: Publication
cites work: Nonsmooth analysis and Hamilton--Jacobi equations on Riemannian manifolds
cites work: Viscosity solutions of Hamilton-Jacobi equations
cites work: Triangulations and the stability theorem for foliations
cites work: Solution to the gradient problem of C. E. Weil
cites work: User’s guide to viscosity solutions of second order partial differential equations
cites work: Infinite games, Banach space geometry and the Eikonal equation
cites work: Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation
cites work: A note on Buczolich's solution of the Weil gradient problem: a construction based on an infinite game
cites work: Hamilton-Jacobi equations and distance functions on Riemannian manifolds
cites work: On Properties of Derivatives
cites work: Q3253193
OpenAlex ID: W2104883868
description: scientific article; zbMATH DE number 5509271