Free boundary problems and density perimeter
type: Item
instance of: scholarly article
title: Free boundary problems and density perimeter
zbMATH Open document ID: 0856.35137
author: Dorin Bucur
author: J. P. Zolesio
published in: Journal of Differential Equations
publication date: 1996-06-30T00:00:00Z
review text: This paper is devoted to study the existence of optimal domains which minimize the domain energy functional \(\Omega\to E(\Omega)\). Special cases are studied in which a family of open subsets of a fixed bounded set is compact in the char-topology defined on the family of measurable subsets of \(\mathbb{R}^n\) by the \(L^2\)-matrix, and in which the \(H^c\)-convergence implies the char-convergence at the same limit. The key technique is the introduction of the identity perimeter, as the authors call it, in free boundary problems in order to get zero measure boundaries. Putting a boundedness constraint on the density perimeter, the authors are able to prove that the \(H^c\)-convergence implies the char-convergence while each limiting term has zero boundary measure. Finally, some existence results for Bernoulli like free boundary problems are proved by aid of a penalty method with density perimeter. For a variational problem arisen from the computer version, the authors prove the existence of the minimizing terms under the same penalty conditions and make some considerations involving a new concept: the maximal density curvature, precisely, they deduce the \(\Gamma\)-convergence of the density perimeter for \(\gamma\to 0\) to the Minkowski content in the family of closed sets with bounded maximal density curvature.
reviewed by: Yang Yingchen
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49K20
zbMATH DE Number: 880556
zbMATH Keywords: domain energy functional
zbMATH Keywords: \(H^ c\)-convergence
zbMATH Keywords: char-convergence
zbMATH Keywords: identity perimeter
zbMATH Keywords: existence results for Bernoulli like free boundary problem
zbMATH Keywords: maximal density curvature
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0050
OpenAlex ID: W2047704663
DOI: 10.1006/JDEQ.1996.0050
Recommended article: Q4844607
Recommended article: Q4240544
Recommended article: Q3833117
Recommended article: Q3766152
Recommended article: Q3338613
Recommended article: Q4830049
Recommended article: Q3482274
Recommended article: On a Free Boundary Problem
Recommended article: Q4850808
Recommended article: Q4289191
description: scientific article; zbMATH DE number 880556