Existence of minimizers and lower semicontinuity of integral functionals in the vectorial case
type: Item
instance of: scholarly article
title: Existence of minimizers and lower semicontinuity of integral functionals in the vectorial case
zbMATH Open document ID: 0945.49008
author: Silvia Bertirotti
author: Roberto van der Putten
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2000-09-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108497
review text: The main existence result of the paper concerns the Monge-Ampère boundary problem \[ \begin{cases}\text{det }D^2u(x)=f(x),\qquad\text{a.e. in }\Omega,\\ Du(x)=x,\qquad\text{on }\partial\Omega,\end{cases} \] where \(\Omega\) is a bounded open set of \(\mathbb R^n\) with \(|\partial\Omega|=0\), and \(f\) is a positive continuous function on \(\overline\Omega\) such that \(\int_\Omega f=|\Omega|\). The authors prove the existence of a solution \(u\in W^{2,p}(\Omega)\cap C^1(\Omega)\) for every \(1\leq p<\infty\) which minimizes the underlying functional \(J\) in a suitable set \(W\), extending a previous result due to \textit{B. Dacorogna} and \textit{J. Moser} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 1, 1-26 (1990; Zbl 0707.35041)]. With the same technique other boundary problems are solved, and also criteria for lower semicontinuity of the functionals \(\int_\Omega F(u(x),\text{det }Du(x))\) are given under assumptions usually verified in nonlinear elasticity.
reviewed by: Patrizia Pucci
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 1395894
zbMATH Keywords: integral functionals
zbMATH Keywords: Monge-Ampère boundary problem
zbMATH Keywords: lower semicontinuity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1395894