A min-max theorem for multiple integrals of the calculus of variations and applications
type: Item
instance of: scholarly article
title: A min-max theorem for multiple integrals of the calculus of variations and applications
zbMATH Open document ID: 0831.49012
author: David Arcoya
author: Lucio Boccardo
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1996-02-13T00:00:00Z
full work available at URL: https://eudml.org/doc/244334
review text: Summary: In this paper we deal with the existence of critical points for functionals defined on the Sobolev space \(W^{1,2}_0(\Omega)\) by \(f(v)= \int_\Omega J(x, v, Dv) dx\), \(v\in W^{1,2}_0(\Omega)\), where \(\Omega\) is a bounded, open subset of \(\mathbb{R}^N\). Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of calculus of variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions. Then we present some applications of this theorem to the study of the existence and multiplicity of nonnegative critical points for multiple integrals of the calculus of variations.
Mathematics Subject Classification ID: 49J35
zbMATH DE Number: 817260
zbMATH Keywords: Ambrosetti-Rabinowitz mountain pass theorem
zbMATH Keywords: existence
zbMATH Keywords: critical points
zbMATH Keywords: functionals
zbMATH Keywords: Sobolev space
zbMATH Keywords: multiple integrals
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 817260