Necessary conditions for an extremum for general functionals of calculus of variations with a nonsmooth integrand
type: Item
instance of: scholarly article
title: Necessary conditions for an extremum for general functionals of calculus of variations with a nonsmooth integrand
zbMATH Open document ID: 0713.49027
DOI: 10.1007/BF00970281
author: Yu G. Reshetnyak
author: N. A. Kudryavtseva
published in: Siberian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: The variational problem in a Sobolev space is considered, i.e. \[ J_ F(u)=\int_{G}F[x,u(x),u'(x)]dx\to \inf, \] where the function F(x,s,\(\cdot)\) is convex for all x,s. The authors show that if u is a minimizer then there is a function h: \(G\times {\mathbb{R}}\to {\mathbb{R}}^ n\) such that \[ \int_{G}\{\frac{\partial F}{\partial S}[x,u(x),u'(x)]\phi (x)+<h(x,u(x)),\phi '(x)>\}dx=0 \] for all \(\phi \in C_ 0^{\infty}(G)\), where h(x,u(x)) belongs to the subdifferential of the function F(x,u(x),\(\cdot)\) at \(u'(x)\) for almost all \(x\in G\).
Mathematics Subject Classification ID: 49K10
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 4173766
zbMATH Keywords: subdifferential
MaRDI profile type: Publication
cites work: Q3947877
cites work: Q3218622
cites work: Q3787019
full work available at URL: https://doi.org/10.1007/bf00970281
OpenAlex ID: W1988817120
description: scientific article; zbMATH DE number 4173766