Convex approximation of an inhomogeneous anisotropic functional
type: Item
instance of: scholarly article
title: Convex approximation of an inhomogeneous anisotropic functional
zbMATH Open document ID: 0809.65069
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1994-10-10T00:00:00Z
full work available at URL: https://eudml.org/doc/244308
review text: This paper is concerned with numerical minimization of a model functional which is of a form encountered in applications such as phase transitions and crystal growth: the functional is \[ {\mathcal F}(u)= \int_ \Omega \phi(x,\nu_ u)| Du|+ \int_{\partial\Omega} \mu u d{\mathcal H}^{n-1}- \int_ \Omega \kappa u dx \] in which \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^ n\), \(\phi\) a convex, positively homogeneous and continuous function with linear growth, and where \({\mathcal H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure in \(\mathbb{R}^ n\). The minimum is sought on the set \(BV(\Omega;[-1,1])\). The functional \(\mathcal F\) is not strictly convex, and is therefore regularized by a family \({\mathcal F}_ \varepsilon\) of strictly convex functionals. Then in addition, a regular family of partitions of \(\Omega\) parametrized by a mesh parameter \(h\) is introduced, as is the piecewise linear finite element space, and corresponding discrete functionals \({\mathcal F}_ h\) and \({\mathcal F}_{\varepsilon,h}\) are defined on the finite-dimensional spaces. Results on the uniform convergence of the discrete functionals to the continuous functionals are proved, and the main result of the paper is the \(\Gamma\)-convergence of \({\mathcal F}_{\varepsilon,h}\) to \(\mathcal F\) when \(\varepsilon\) and \(h\) go to zero independently. A convergence result for the minimizers \(u_{\varepsilon,h}\) of the discrete regularized functional to the minimizer \(u\) of \(\mathcal F\) is also proved: \(\{u_{\varepsilon,h}\}\) contains a subsequence which converges to \(u\) and, furthermore, \({\mathcal F}_{\varepsilon,h}(u_{\varepsilon,h})\) converges to \({\mathcal F}(u)\).
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 49M15
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 35J85
zbMATH DE Number: 650745
zbMATH Keywords: convex approximation
zbMATH Keywords: anisotropic functional
zbMATH Keywords: phase transitions
zbMATH Keywords: crystal growth
zbMATH Keywords: finite element
zbMATH Keywords: uniform convergence
author: Giovanni Bellettini
author: Maurizio Paolini
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 650745