Lipschitz stability in the lateral Cauchy problem for elasticity system
type: Item
instance of: scholarly article
title: Lipschitz stability in the lateral Cauchy problem for elasticity system
zbMATH Open document ID: 1067.35142
author: Yanyan Li
published in: Journal of Mathematics of Kyoto University
publication date: 2004-12-29T00:00:00Z
review text: The authors deal with the problem of determining the displacement vector \({\mathbf u}\) in the isotropic elasticity system \[ \rho D_t^2{\mathbf u}-\mu[\Delta {\mathbf u}+\nabla(\nabla^T {\mathbf u})]-\nabla(\lambda \nabla^T{\mathbf u}) -\sum_{j=1}^3\,\nabla \mu\cdot (\nabla u_j+D_{x_j}{\mathbf u}){\mathbf e}_j,\quad \text{ in\;} \Omega\times (0,T)=: Q, \tag{1} \] subject to the lateral Cauchy conditions \[ {\mathbf u}={\mathbf g} \quad \text{ on\;} \partial \Omega\times (0,T),\qquad D_\nu{\mathbf u}={\mathbf h} \quad \text{ on\;} \Gamma\times (0,T). \tag{2} \] They assume that \(\Omega\) is a bounded domain with a \(C^2\)-boundary and \(\Gamma\) is a (smooth) subset of \(\partial \Omega\). Further, they assume that \(\rho\in C^2({\overline Q})\), \(\lambda,\mu\in C^3({\overline Q})\) and \(\rho(x)>0\), \(3\lambda(x)+\mu(x)>0\) for all \(x\in {\overline Q}\). Using a procedure based on Carleman estimates and diagonalization of the principal part of system (1), which makes use of the auxiliary function \(v=\nabla^T{\mathbf u}\), under suitable assumptions on \(\Omega\), \(\Gamma\) and functions \(a=\rho/\mu\) and \(b=\rho/(\lambda+2\mu)\), the authors prove the following estimate \[ \begin{aligned} \| D_t{\mathbf u}(\cdot,t)\| _{H^2(\Omega)} &+ \| {\mathbf u}(\cdot,t)\| _{H^1(\Omega)} + \| D_t\nabla^T{\mathbf u}(\cdot,t)\| _{H^2(\Omega)} + \| \nabla^T{\mathbf u}(\cdot,t)\| _{H^1(\Omega)}\\ \leq C\{ \| {\mathbf h}\| _{H^2(\Gamma \times (0,T))} &+ \| {\mathbf g}\| _{H^1(\partial \Omega\times (0,T))} + \| \nabla^T{\mathbf u}\| _{H^1(\partial \Omega)\times (0,T))} + \| D_\nu\nabla^T{\mathbf u}\| _{H^1(\Gamma \times (0,T))} \}. \end{aligned} \tag{3} \] Moreover, they get rid of the term \(\nabla^T{\mathbf u}\) occurring in the right-hand side in (3), when the lateral conditions (2) are replaced with \[ {\mathbf u}={\mathbf 0} \quad \text{ on\;} \partial \Omega\times (0,T),\qquad \sigma({\mathbf u})\nu={\mathbf h} \quad \text{ on\;} \Gamma\times (0,T), \] where \(\sigma({…
reviewed by: Alfredo Lorenzi
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 74B05
Mathematics Subject Classification ID: 74H55
zbMATH DE Number: 2124048
zbMATH Keywords: Isotropic elasticity system
zbMATH Keywords: Cauchy lateral problem
zbMATH Keywords: continuous dependence results
zbMATH Keywords: Carleman estimates
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283691
OpenAlex ID: W2122432836
DOI: 10.1215/KJM/1250283691
description: scientific article; zbMATH DE number 2124048