A monotone boundary condition for a domain with many tiny spherical holds
type: Item
instance of: scholarly article
title: A monotone boundary condition for a domain with many tiny spherical holds
zbMATH Open document ID: 0577.35042
author: Satoshi Kaizu
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: This work considers the behavior, as \(\epsilon \to 0^+\), of the solutions \(u_{\epsilon}\) of the problems \[ -\Delta u_{\epsilon}=f\quad a.e.\quad in\quad \Omega_{\epsilon},\quad \partial u_{\epsilon}/\partial \nu +\beta_{\epsilon}(u_{\epsilon})=0\quad a.e.\quad on\quad \partial \Omega_{\epsilon} \] where \(\nu\) is the outer normal to \(\partial \Omega_{\epsilon}\), \(\beta_{\epsilon}\) is a suitable monotone function; \(\Omega_{\epsilon}=\Omega \setminus F_{\epsilon}\), \(\Omega\) is a bounded domain in \(R^ N\) with smooth boundary \(\partial \Omega\), \(F_{\epsilon}\) is the union of all balls \(B^ i(r_{\epsilon})\) (contained in \(\Omega)\) such that \(dist(B^ i(r_{\epsilon}),\partial \Omega)\leq \epsilon\), \(B^ i(r_{\epsilon})\) has radius \(r_{\epsilon}<\epsilon\) and the same center as \(C^ i_{\epsilon}\), and finally \(C^ i_{\epsilon}\) are cubes, with volume \((2\epsilon)^ N\), which divide \(R^ N\).
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 3923237
zbMATH Keywords: monotone boundary conditions
zbMATH Keywords: domain with many holes
zbMATH Keywords: limit
zbMATH Keywords: behavior
reviewed by: Maurizio Chicco
MaRDI profile type: Publication
full work available at URL: pjaa.61.140
OpenAlex ID: W1997375159
cites work: Q3961204
cites work: Q4120903
cites work: Probabilistic methods in some problems of scattering theory
cites work: The Robin problems on domains with many tiny holes
cites work: On an elaboration of M. Kac's theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles
cites work: Potential and scattering theory on wildly perturbed domains
DOI: 10.3792/PJAA.61.140
description: scientific article; zbMATH DE number 3923237