Regularity of \(C^ 1\) solutions of the Hamilton-Jacobi equation.
type: Item
instance of: scholarly article
title: Regularity of \(C^ 1\) solutions of the Hamilton-Jacobi equation.
zbMATH Open document ID: 1059.37047
author: Albert Fathi
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2005-01-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73614
review text: Under some regularity conditions for a Hamiltonian, it is shown that \(C^1\) solutions of the Hamilton-Jacobi equations are necessarily \(C^{1,1}\). Similar results were obtained before by \textit{P.L. Lions} [Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, 69, Boston-London-Melbourne: Pitman Advanced Publishing Program (1982; Zbl 0497.35001)], for the particular case when the Hamiltonian depends on the fibre coordinates only. The results obtained by the author are stronger, in Theorem 3.2, it is shown that the Lipschitz constant of the derivative of the solution of the Hamilton-Jacobi equation can be locally controlled. This implies some results regarding the compactness of sets of \(C^1\) solutions of Hamilton-Jacobi equation.
Mathematics Subject Classification ID: 37J50
Mathematics Subject Classification ID: 35F20
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 70H20
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 2124650
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: Euler-Lagrange equation
zbMATH Keywords: Legendre transformation
reviewed by: Ioan Bucataru
MaRDI profile type: Publication
OpenAlex ID: W2321838087
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DOI: 10.5802/AFST.1059
description: scientific article; zbMATH DE number 2124650