Coercivity of integrals versus coercivity of integrands
type: Item
instance of: scholarly article
title: Coercivity of integrals versus coercivity of integrands
zbMATH Open document ID: 0866.49002
author: Bernard Dacorogna
author: Lucio Boccardo
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-07-25T00:00:00Z
review text: Considering the integral functional of the calculus of variations \[ I(u)=\int_{\Omega} f(x,u(x),\nabla u(x)) dx \] where \(\Omega\subset \mathbb{R}^n, u:\Omega \to \mathbb{R},\) the authors discuss the possibility that a coercivity assumption on \(I\) implies an analogous property on the integrand. More precisely, they consider the conditions \[ \begin{alignedat}{2} I(u)&\geq \alpha |\nabla u|^p_{L^p}+\beta, \qquad\qquad &&u\in W^{1,\infty}_0(\Omega) \quad [ u\in W^{1,\infty}(\Omega) ] \tag{1}\\ f(x,u,\xi) &\geq a|\xi|^p+b, &&(x,u,\xi)\in \Omega\times\mathbb{R}\times \mathbb{R}^n.\tag{2}\end{alignedat} \] They prove that (1) and (2) are equivalent if \(f=f(\xi)\) but the implication \((1)\Longrightarrow (2)\) is false for integrands of the type \(f(x,\xi)\) or \(f(u,\xi).\) They also extend these results to the vectorial case, i.e., for \(u:\Omega\to \mathbb{R}^m\), \(m>1\).
Mathematics Subject Classification ID: 49J10
zbMATH DE Number: 765087
zbMATH Keywords: integral functional
zbMATH Keywords: coercivity
reviewed by: Anna Salvadori
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1039
OpenAlex ID: W2031860970
DOI: 10.1006/JMAA.1995.1039
description: scientific article; zbMATH DE number 765087