Partial regularity for optimal design problems involving both bulk and surface energies
type: Item
instance of: scholarly article
title: Partial regularity for optimal design problems involving both bulk and surface energies
zbMATH Open document ID: 0946.49034
DOI: 10.1142/S0252959999000175
author: Robert V. Kohn
author: Fanghua Lin
published in: Chinese Annals of Mathematics. Series B
publication date: 2000-10-18T00:00:00Z
review text: The paper, which is a refinement and an improvement of a previous work of the first author [Calc. Var. Partial Differ. Equ. 1, No. 2, 149-168 (1993; Zbl 0794.49038)], deals with the regularity of variational problems involving both a volume and a surface energy. Specifically, one considers pairs \((u,A)\) with \(u\in H^1(\Omega)\) and \(A\subset\Omega\) with finite perimeter in \(\Omega\) minimizing the energy (possibly with volume constraints) \[ (u,A)\mapsto \int_\Omega [F(x,u,\nabla u)+\chi_AG(x,u,\nabla u)] dx+ \int_{\partial^* A}\psi(x,\nu_A) d{\mathcal H}^{n-1} \] where \(\chi_A:\Omega\to\{0,1\}\) is the characteristic function of \(A\), \(\partial^* A\) is the essential boundary of \(A\) and \(\nu_A\) is its inner normal. Under suitable smoothness and ellipticity assumptions on the data and on the energy densities the authors prove that \(\partial^* A\) is closed and \(C^{1,\alpha}\) out of a \({\mathcal H}^{n-1}\)-negligible closed set \(\Sigma\). Related results have been obtained by \textit{L. Ambrosio, N. Fusco} and \textit{D. Pallara} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 1, 39-62 (1997; Zbl 0896.49024)] in the more general framework of free discontinuity problems, in which the surface energy density is not necessarily supported on a boundary.
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 1340795
zbMATH Keywords: free boundary problems
zbMATH Keywords: partial regularity
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
full work available at URL: s0252959999000175
OpenAlex ID: W1978717788
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description: scientific article; zbMATH DE number 1340795