Homogenization for the \(p\)-Laplace operator and nonlinear Robin boundary conditions in perforated media along \((n - 1)\)-dimensional manifolds
type: Item
instance of: scholarly article
title: Homogenization for the \(p\)-Laplace operator and nonlinear Robin boundary conditions in perforated media along \((n - 1)\)-dimensional manifolds
zbMATH Open document ID: 1305.35136
DOI: 10.1134/S1064562414010049
published in: Doklady Mathematics
publication date: 2014-08-29T00:00:00Z
review text: From the authors' introduction: We study a homogenization problem for the \(p\)-Laplace operator posed in a domain \(\Omega_\varepsilon\) which is obtained by removing balls \(G_\varepsilon\), the cavities, of diameter \(O(\varepsilon^\alpha)\) from a fixed domain \(\Omega\) of \(\mathbb{R}^n\). The balls are \(\varepsilon\)-periodically distributed along a \((n-1)\)-dimensional manifold and nonlinear Robin type conditions are imposed on the boundary of the cavities with a large adsorption constant \(\varepsilon^{-k}\). Here \(\varepsilon\) denotes a small parameter and we shall let \(\varepsilon\to 0\), \(\alpha> 1\) and \(k>0\). For the critical case \(\alpha={n-1\over n-p}\) and \(k={(p-1)(n-1)\over n-p}\) with \(p\) and \(n\) fixed, \(p\in[2, n)\), the homogenized problem is obtained. This problem has a nonlinear transmission condition which depends on the nonlinearity arising both in the equation and on the boundary conditions. It should be noted that the results in this paper extend those by \textit{M. Lobo} et al. [ibid. 83, No. 1, 34--38 (2011); translation from Dokl. Akad. Nauk 436, No. 2, 163--167 (2011; Zbl 1233.35075)] adding extra nonlinearity from the operator.
reviewed by: Anthony D. Osborne
Mathematics Subject Classification ID: 35Q74
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 74Q05
zbMATH DE Number: 6336259
MaRDI profile type: Publication
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full work available at URL: s1064562414010049
OpenAlex ID: W2048227990
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author: D. Gómez
author: Maria Lobo
author: E. Pérez
author: A. V. Podolskiy
author: T. A. Shaposhnikova
description: scientific article; zbMATH DE number 6336259