Convex approximations of functionals with curvature
type: Item
instance of: scholarly article
title: Convex approximations of functionals with curvature
zbMATH Open document ID: 0754.65066
author: Giovanni Bellettini
author: Maurizio Paolini
author: Claudio Verdi
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/244157
review text: For a set \(\Omega\) in \(\mathbb{R}^ n\) with Lipschitz boundary \(\partial\Omega\), \(dH\) the \((n-1)\) dimensional Hausdorff measure in \(\mathbb{R}^ n\) and bounded functions \(\mu\) on \(\partial \Omega\), \(\kappa\) on \(\Omega\), define \(BV(\Omega)\) as space of functions of bounded variation on \(\Omega\), with \(\int| Dv|\) the total variation, \(\int(| 1+| Dv|^ 2)^{1/2}\) the area of \(v\). The authors study the numerical minimization of the functional \(\int_ \Omega| Dv|+\int_ \Omega\mu vdH-\int_ \Omega \kappa vdx\). They first turn the functional into a strictly convex one by replacing \(| Dv |\) by (\(\varepsilon^ 2+| Dv|^ 2)^{1/2}\) and then minimize the new functional by finite elements. They prove that the solution of the modified functional tends to the solution of the original one for \(\varepsilon\to 0\) and that the computed solutions tend to the true solution for the meshsize \(\to 0\) and \(\varepsilon\to 0\).
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49M15
Mathematics Subject Classification ID: 49Q10
zbMATH DE Number: 61697
zbMATH Keywords: calculus of variations
zbMATH Keywords: surfaces with prescribed mean curvature
zbMATH Keywords: convergence
zbMATH Keywords: computer vision
zbMATH Keywords: convex approximations
zbMATH Keywords: finite elements
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 61697