Existence theory for a new class of variational problems
type: Item
instance of: scholarly article
title: Existence theory for a new class of variational problems
zbMATH Open document ID: 0711.49064
DOI: 10.1007/BF00376024
published in: Archive for Rational Mechanics and Analysis
publication date: 1990-01-01T00:00:00Z
review text: The paper deals with the existence of minimizers of functionals of the form \[ \int_{\Omega \setminus \Gamma}f(x,u,\nabla u)dx+\int_{\Gamma}\phi (x,u^+,u^-,\nu)d{\mathcal H}^{n-1} \] whese \(\Omega \subset {\mathbb{R}}^ n\) and \({\mathcal H}^{n-1}\) is the Hausdorff (n- 1)-dimensional measure. The function u: \(\Omega\to {\mathbb{R}}^ k\) need not be continuous: the surface energy is obtained by integrating on the discontinuity set \(\Gamma\) of u an energy density \(\phi\) depending on x, the normal \(\nu\) to \(\Gamma\) and the asymptotic value \(u^+,u^-\) of u near x. The appropriate function space of the functional is a suitable subset \(SBV(\Omega,{\mathbb{R}}^ k)\) of vector valued functions of bounded variation. General compactness and lower semicontinuity results are established.
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 26B30
zbMATH DE Number: 4169545
zbMATH Keywords: existence of minimizers
zbMATH Keywords: compactness
zbMATH Keywords: lower semicontinuity
author: Luigi Ambrosio
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00376024
OpenAlex ID: W2083623174
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description: scientific article; zbMATH DE number 4169545