Homogenization of Poisson's kernel and applications to boundary control
type: Item
instance of: scholarly article
title: Homogenization of Poisson's kernel and applications to boundary control
zbMATH Open document ID: 0617.35014
author: Marco Avellaneda
author: Fanghua Lin
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1989-01-01T00:00:00Z
review text: We prove an asymptotic expansion for the Poisson kernels \(P_{\epsilon}(x,y)\) of a family of elliptic operators of the form \[ (1)\quad L_{\epsilon}=-(\partial /\partial x^ i)[a^{ij}(x/\epsilon)\partial /\partial x_ j] \] with periodic \(a^{ij}\)'s, in a \(C^{1,\alpha}\) domain. Accordingly, \[ (2)\quad P_{\epsilon}(x,y)=<\partial \Phi ^ *_{\epsilon}/\partial v_{a,\epsilon},\quad n(y)>\partial G_ 0(x,y)/\partial n(y)+R_{\epsilon}(x,y),\quad x\in D,\quad y\in \partial, \] here \(\Phi ^ *_{\epsilon}\) is the solution of \[ (3)\quad L^ *_{\epsilon} \Phi ^ *_{\epsilon} =0\quad in\quad D;\quad \Phi ^ *_{\epsilon}=x\quad on\quad \partial D, \] \(\partial /\partial v_{a,\epsilon}\) is the conormal derivative, \(G_ 0(x,y)\) is the Green's function of the homogenized problem and \(R_{\epsilon}(x,y)\) is a function satisfying \[ (4)\quad | R_{\epsilon}(x,y)| \leq C dist(x,\partial D)/| x-y| ^ n,\quad x\in D,\quad y\in \partial D, \] and \[ (5)\quad \lim _{\epsilon \downarrow 0} \sup _{dist(x,\partial D)>\delta} \sup _{y\in \partial D}| R_{\epsilon}(x,y)| =0, \] for all \(\delta >0\). The expansion (2) is used to identify the limit as \(\epsilon\) \(\to 0\) of solutions to a problem of exact boundary controllability for the hyperbolic operator \(\partial ^ 2_{tt}+L_{\epsilon}\).
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 93D99
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 93B05
zbMATH DE Number: 4000482
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: Poisson kernels
zbMATH Keywords: Green's function
zbMATH Keywords: homogenized problem
zbMATH Keywords: exact boundary controllability
MaRDI profile type: Publication
full work available at URL: j.matpur.2005.12.001
OpenAlex ID: W1981973570
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Homogenization of Poisson's kernel and applications to boundary control (DOI)
DOI: 10.1016/J.MATPUR.2005.12.001