An abstract transmission problem in a thin layer, I: Sharp estimates
type: Item
instance of: scholarly article
title: An abstract transmission problem in a thin layer, I: Sharp estimates
zbMATH Open document ID: 1231.34103
author: Giovanni Dore
author: Angelo Favini
author: Rabah Labbas
author: Keddour Lemrabet
published in: Journal of Functional Analysis
publication date: 2011-09-22T00:00:00Z
review text: The authors study the following abstract elliptic transmission problem \[ \begin{cases} u''_{-}(x)+Au_{-}(x)=-g_{-}(x),\quad x\in (-1,0),\\ u''_{+}(x)+Au_{+}(x)=-g_{+}(x),\quad x\in (0,\delta),\\ u_{-}(-1)=f_{-},\\ u'_{+}(\delta)=f_{+},\\ u_{-}(0)=u_{+}(0),\\ p_{-}u'_{-}(0)=p_{+}u'_{+}(0)+\chi, \end{cases} \] where \(A\) is a linear closed operator in a Banach space \(X\), \(p_{-},p_{+}\in\mathbb R^+\), \(\delta\) is a small parameter, \(g_{-}\in L^p((-1,0);X)\), \(g_{+}\in L^p((0,\delta);X)\), \(f_{+},f_{-},\chi\in X\). Existence, uniqueness, and maximal regularity of the solution \((u_{-},u_{+})\) of the above problem in the sense of \(L^p\)-spaces and its behavior for small values of the parameter \(\delta\) are established in the paper. The main tools of the authors' approach are impedance and admittance operators (i.e., Dirichlet-to-Neumann and Neumann-to-Dirichlet operators) and \(H^{\infty}\) functional calculus for sectorial operators in Banach spaces. First, a few physical models are presented in the paper as a motivation of the abstract study. At the end of the paper, the authors solve these model problems using the obtained abstract results. The authors also announced in the paper, that in a forthcoming paper they will study the limit problem for \(\delta\to 0\).
reviewed by: Yakov Yakubov
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 5949072
zbMATH Keywords: abstract differential equation
zbMATH Keywords: elliptic problem
zbMATH Keywords: transmission problem
zbMATH Keywords: \(H^{\infty }\) functional calculus
zbMATH Keywords: impedance operator
zbMATH Keywords: admittance operator
zbMATH Keywords: UMD Banach space
zbMATH Keywords: sectorial operator
MaRDI profile type: Publication
full work available at URL: j.jfa.2011.05.021
OpenAlex ID: W1992601831
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DOI: 10.1016/J.JFA.2011.05.021
description: scientific article; zbMATH DE number 5949072