Shape derivative in the wave equation with Dirichlet boundary conditions
type: Item
instance of: scholarly article
title: Shape derivative in the wave equation with Dirichlet boundary conditions
zbMATH Open document ID: 0959.35017
author: John Cagnol
author: J. P. Zolesio
published in: Journal of Differential Equations
publication date: 2001-04-26T00:00:00Z
review text: The authors study the second-order Dirichlet hyperbolic problem \[ \begin{aligned} & \partial^2_ty-\text{div}(K\nabla y)=f\quad \text{on } (0,T)\times \Omega,\\ & y=g\quad \text{on }(0,T)\times \partial\Omega,\\ & y(0)=\varphi,\quad \partial_ty(0)=\psi\quad\text{on }\Omega,\end{aligned} \] where \(K\) be a coercive and symmetric \(N\times N\)-matrix of functions, and \(\Omega\subset\mathbb{R}^n\) is a a domain with the boundary \(\partial\Omega\in C^k\) for \(k \geq 2\). First of all the authors prove that the solution of the given problem is shape and material differentiable under a strong regularity of the data. Then using the hidden regularity they prove the existence of a shape derivative under weak regularity conditions of the data. The implicit function theorem used in the smooth cases does not work to solve the problem of a weak regularity of the data. Nevertheless by a more technical approach the authors prove analogical results as for the regular data.
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 1384452
zbMATH Keywords: shape boundary derivative
zbMATH Keywords: material derivative
zbMATH Keywords: shape and material differentiability
zbMATH Keywords: weak regularity of the data
reviewed by: Vladimír Ďurikovič
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3643
OpenAlex ID: W1989499876
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DOI: 10.1006/JDEQ.1999.3643
description: scientific article; zbMATH DE number 1384452