The nonlinear membrane energy: variational derivation under the constraint ``\(\det \nabla u\neq 0\)''
type: Item
instance of: scholarly article
title: The nonlinear membrane energy: variational derivation under the constraint ``\(\det \nabla u\neq 0\)''
zbMATH Open document ID: 1114.35003
author: Omar Anza Hafsa
author: Jean-Philippe Mandallena
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2007-02-26T00:00:00Z
review text: Let \( \Sigma_{\varepsilon} \subset \mathbb R^3\) be of the form \(\Sigma_{\varepsilon}=\Sigma \times (-\frac{1}{\varepsilon}, \frac{1}{\varepsilon})\) where \(\Sigma \subset \mathbb R^2\) is Lipschitz open and bounded, and define the integral functionals of the form \[ E_{\varepsilon}(u)= \frac{1}{\varepsilon} \int_{\Sigma_{\varepsilon}} W(Du)\,dx \] where \(W(z)\) be the continuous stored-energy function such that \(W(z)=+\infty\) if and only if \(\det z=0\) and is coercive when \(\det z \geq \delta\). The authors prove that as \(\varepsilon \rightarrow 0\) the three-dimensional free energy \(E_{\varepsilon}\) converges in a variational sense, which is a variant of De Giorgi \(\Gamma\)-convergence, to the two-dimensional free energy \[ E_{\text{mem}}(u)= \int_{\Sigma} W_{\text{mem}}(Du)\,dx \] and give a representation formula for \(W_{0}\). The result permits to obtain the nonlinear membrane energy \(E_{\text{mem}}\) under the constraint \(\det Du \neq 0\).
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 74K15
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 74G65
zbMATH DE Number: 5129882
zbMATH Keywords: dimensional reduction
zbMATH Keywords: \(\Gamma\)-convergence
zbMATH Keywords: determinant constraint
reviewed by: Elvira Mascolo
MaRDI profile type: Publication
full work available at URL: j.matpur.2006.01.004
OpenAlex ID: W2041217130
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DOI: 10.1016/J.MATPUR.2006.01.004
description: scientific article; zbMATH DE number 5129882