Asymptotic analysis of periodically-perforated nonlinear media.
type: Item
instance of: scholarly article
title: Asymptotic analysis of periodically-perforated nonlinear media.
zbMATH Open document ID: 1036.35021
DOI: 10.1016/S0021-7824(01)01226-0
author: Nadia Ansini
author: Andrea Braides
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-09-07T00:00:00Z
review text: Given \(\Omega\) a bounded open set in \({\mathbb R}^n\) (\(n \geqslant 3\)), \(p \in (1,n)\), \(B_i^{\delta}\) the ball of centre \(i\delta\) (\(i \in {\mathbb Z}^n\) and \(\delta > 0\)) and radius \(\delta^{n/(n-p)}\), \(\Omega_{\delta} = \Omega \setminus \bigcup_{i \in {\mathbb Z}^n} \overline{B_i^{\delta}}\), consider a sequence of positive numbers \(\{\delta_j\}_j\) converging to zero and the nonlinear sequence of the minimum problem \[ \inf \Big\{ F_j u = \int_{\Omega_{\delta_j}} f(Du) \text{d}x + \langle \phi, u\rangle \;\Big| \;u \in W^{1,p}_0(\Omega_{\delta_j}; {\mathbb R}^m) \Big\} = m_j \] where \(\phi \in W^{-1,p'}_0(\Omega; {\mathbb R}^m)\) and \(f:M^{m\times n} \to [0,+\infty)\) (\(M^{m\times n}\) the space of \(m \times n\) matrices) is a Borel function satisfying \[ f(0) = 0, \quad c_1 (| A| ^p - 1) \leqslant f(A) \leqslant c_2 (| A| ^p + 1) \quad \text{ for every } A \in M^{m\times n} \] with \(c_1, c_2\) positive constants. Denote by \(Qh\), \(h:M^{m\times n} \to [0,+\infty)\) a Borel function, the \(W^{1,p}\)-quasiconvexification of \(h\) \[ Q h (A) = \inf \Big\{ \int_{(0,1)^n} h(A+Du) \text{d}x \;\Big| \;u \in W^{1,p}_0((0,1)^n; {\mathbb R}^m) \Big\}. \] Under the assumption that there exists the limit \[ g(z) = \lim_j \delta_j^{{np}\over{n-p}} Qf (\delta_j^{-{{np}\over{n-p}}} z) \quad \text{ for every } z \in {\mathbb R}^m \] and define \[ F u = \int_{\Omega} (Qf(Du) + \varphi (u)) \text{d}x + \langle \phi, u\rangle, \quad u \in W^{1,p}_0(\Omega; {\mathbb R}^m) \] where \[ \varphi (z) = \inf \Big\{ \int_{{\mathbb R}^n} g(D \zeta ) \text{d}x \;\Big| \;\zeta - z \in W^{1,p}({\mathbb R}^n; {\mathbb R}^m) \, , \zeta =0 \text{ on } B_1(0) \Big\} \] the authors prove that \[ F_j\;\Gamma \text{-converge to } F \text{ with respect to } L^p(\Omega ; {\mathbb R}^m). \] In particular \(m_j \to m = \min Fu\) and, if \(u_j\) is such that \(F_j u_j = m_j + o(1)\) then, up to a subsequence, \(u_j \to u\) in \(W^{1,p}_0(\Omega; {\mathbb R}^m)\)-weak where \(u\)…
reviewed by: Fabio Paronetto
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 74Q05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1975940
zbMATH Keywords: perforated domains
zbMATH Keywords: relaxed Dirichlet problems
zbMATH Keywords: nonlinear capacity
zbMATH Keywords: Gamma-limit
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1975940