Identification problems in linear elasticity
type: Item
instance of: scholarly article
title: Identification problems in linear elasticity
zbMATH Open document ID: 0892.35154
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-07-14T00:00:00Z
review text: Assume that \({\mathcal E}\) is an elastic isotropic body consisting of two homogeneous parts \({\mathcal E}_1\) and \({\mathcal E}_2\) with Lamé coefficients \(({\widetilde \mu},{\widetilde \lambda})\) and \((\mu,\lambda)\), respectively, such that \(({\widetilde \mu},{\widetilde \lambda})\neq (\mu,\lambda)\). We denote by \(\Omega\) and \(D\) the reference open sets related to \({\mathcal E}\) and \({\mathcal E}_1\) and assume that \(\Omega\) and \(D\) have boundaries which are, respectively, of class \(C^2\) and Lipschitz continuous. As a consequence, if we assume to know the boundary displacement \({\varphi}\) of \({\mathcal E}\), the displacement vector \({\mathbf u}\) solves the Dirichlet problem \[ \sum_{j=1}^n D_i\big[\mu(x)(D_ju_i+D_iu_j) + \lambda(x)D_ju_j\big] = 0 \qquad \text{in } \Omega,\quad i=1,\ldots,n \tag{1} \] \[ u_i=\varphi_i \qquad \text{on } \partial \Omega,\quad i=1,\ldots,n \tag{2} \] where \(\mu(x)={\widetilde \mu}\chi_D(x) + \mu \chi_{{\mathbb{R}}^n\setminus D}(x)\) and \(\lambda(x)={\widetilde \lambda}\chi_D(x)+\lambda \chi_{{\mathbb{R}}^n\setminus D}(x)\), \(\chi_F\) denoting the characteristic function of a set \(F\). Denote by \(\Lambda_D\) the Dirichlet to Neumann map defined on \(\partial \Omega\) by \(\Lambda_D\phi=\mu [\nabla {\mathbf u}+(\nabla {\mathbf u})^t] {\nu} + \lambda (\text{div}){\nu}\), \({\mathbf u}\) and \({\nu}\) being, respectively, the solution to problem (1)--(2) and the outward unit vector normal to \(\partial \Omega\). Concerning the unique determination of \(D\) the authors prove the following result. Theorem. Let \((\mu-{\widetilde \mu})(\lambda-{\widetilde \lambda})\geq 0\) and let \(D_i\) \((i=1,2)\) be two Lipschitz domains such that \({\overline D}_i \subset \Omega\) and \(\Omega \setminus {\overline D}_i\) is connected \((i=1,2)\). Then \(\Lambda_{D_1}=\Lambda_{D_2}\) implies \(D_1=D_2\). In the case when it is a priori known that the one of the two domains is contained in the latter, the authors show that …
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 74G30
Mathematics Subject Classification ID: 74H25
zbMATH DE Number: 1115430
zbMATH Keywords: Dirichlet to Neumann map
zbMATH Keywords: piecewise nonhomogeneous elastic bodies
zbMATH Keywords: determination of the separating boundaries
zbMATH Keywords: uniqueness
author: Hyunseok Kim
author: Jin Keun Seo
reviewed by: Alfredo Lorenzi
MaRDI profile type: Publication
OpenAlex ID: W2081788649
cites work: L'intégrale de Cauchy définit un opératuer borne sur \(L^ 2 \)pour les courbes lipschitziennes
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cites work: Q3894996
DOI: 10.1006/JMAA.1997.5656
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description: scientific article; zbMATH DE number 1115430