Homogenization of variational inequality for the Laplace operator with nonlinear constraint on the flow in a domain perforated by arbitrary shaped sets. Critical case
type: Item
instance of: scholarly article
title: Homogenization of variational inequality for the Laplace operator with nonlinear constraint on the flow in a domain perforated by arbitrary shaped sets. Critical case
zbMATH Open document ID: 1402.35035
author: M. N. Zubova
author: T. A. Shaposhnikova
published in: Journal of Mathematical Sciences (New York)
publication date: 2018-08-16T00:00:00Z
review text: In the paper under review, the authors study the homogenization of a variational inequality for the Laplace equation with nonlinear constraint in a perforated domain by small holes of critical size. More precisely, they consider the asymptotic behavior of the solutions \(u_\epsilon\) of a variational inequality for the Laplacian in a set \(\Omega_\epsilon \subseteq \mathbb{R}^n\) (\(n \geq 3\)), perforated by arbitrary shaped sets \(G_\epsilon^j\) of radius equal to \(C_0 \epsilon^{\frac{n}{n-2}}\) with \(C_0>0\). On the boundary of the sets \(G_\epsilon^j\) they impose constraints of the type \[ u_\epsilon \geq 0\, , \quad \partial_\nu u_\epsilon\geq -\epsilon^{\frac{n}{2-n}}\sigma(u_\epsilon)\, , \quad u_\epsilon \big(\partial_\nu u_\epsilon+\epsilon^{\frac{n}{2-n}}\sigma(u_\epsilon)\big)=0\, , \] whereas they impose \( u_\epsilon=0 \) on the remaining part of the boundary of \(\Omega_\epsilon\). Then they construct special test functions and establish the asymptotics of the solutions to the variational inequality.
reviewed by: Paolo Musolino
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J87
zbMATH DE Number: 6919794
zbMATH Keywords: critical size
zbMATH Keywords: Laplace operator, nonlinear constraint
MaRDI profile type: Publication
full work available at URL: s10958-018-3888-8
OpenAlex ID: W2807034846
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DOI: 10.1007/S10958-018-3888-8
description: scientific article; zbMATH DE number 6919794