\(\Gamma \)-convergence of some super quadratic functionals with singular weights
type: Item
instance of: scholarly article
title: \(\Gamma \)-convergence of some super quadratic functionals with singular weights
zbMATH Open document ID: 1205.82069
author: Giampiero Palatucci
author: Yannick Sire
published in: Mathematische Zeitschrift
publication date: 2010-10-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/0903.0984
review text: Fix an open bounded set \(\Omega\) of \(\mathbb R^3\) and fix \(\alpha= 2-p\) with \(p>2\). Let \(u\) be a scalar density function and let \(Tu\) denote its trace on \(\partial\Omega\). The distance function to the boundary is denoted by \(d(x,\partial\Omega)\). This paper studies the \(\Gamma\)-convergence of the following functional \(F_\varepsilon(u)\): \[ \varepsilon^{p-2} \int_\Omega |Du|^p \,d(x,\partial\Omega)^\alpha\,dx+ \varepsilon^{-\frac{p-2}{p-1}} \int_\Omega W(u)\,d(x,\partial\Omega)^{-\frac{\alpha}{p-1}}\,dx+ \varepsilon^{-1/2} \int_{\partial\Omega} V(Tu) \,d{\mathcal H}^2, \] where \({\mathcal H}^n\) is the \(n\)-dimensional Hausdorff measure, \(W\) and \(V\) are two nonnegative continuous functions vanishing at \(\alpha\), \(\beta\) and \(\alpha'\), \(\beta'\), respectively. It is shown that the singular limit of the energies \(F_\varepsilon\) leads to a coupled problem of bulk and surface phase transitions.
reviewed by: Mu-Fa Chen
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 5797334
zbMATH Keywords: phase transitions
zbMATH Keywords: line tension
zbMATH Keywords: weighted Sobolev spaces
zbMATH Keywords: nonlocal variational problems
zbMATH Keywords: \(\Gamma\)-convergence
zbMATH Keywords: functions of bounded variation
MaRDI profile type: Publication
OpenAlex ID: W2034615842
arXiv ID: 0903.0984
cites work: Q5318420
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cites work: Q3972449
DOI: 10.1007/S00209-009-0584-X
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description: scientific article