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Asymptotic analysis of shape functionals

Serguei Nazarov 1 Jan Sokolowski
1 ONDES - Modeling, analysis and simulation of wave propagation phenomena
Inria Paris-Rocquencourt, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR2706
Abstract : A family of boundary value problems is considered in domains $\Omega(\varepsilon)=\Omega\setminus\overline\omega_\varepsilon \subset\mathbb{R}^n$, $n\ge 3$, with cavities $\omega_\varepsilon$ depending on a small parameter $\varepsilon\in (0,\varepsilon_0]$. An approximation ${\mathcal U}(\varepsilon, x)$, $x\in\Omega(\varepsilon)$, of the solution $u(\varepsilon, x)$, $x\in\Omega(\varepsilon)$, to the boundary value problem is obtained by an application of the methods of matched and compound asymptotic expansions. The asymptotic expansion is constructed with precise a priori estimates for solutions and remainders in Hölder spaces, i.e., pointwise estimates are established as well. The asymptotic solution ${\mathcal U}(\varepsilon, x)$ is used in order to derive the first term of the asymptotic expansion with respect to $\varepsilon$ for the shape functional ${\mathcal J}(\Xi(\varepsilon))={\mathbb{J}}_\varepsilon(u) \cong{\mathbb{J}}_\varepsilon({\mathcal U})$. In particular, we obtain the {\it topological derivative} $\cT(x)$ of the shape functional $\cJ(\Xi)$ at a point $x\in\O$. Volume and surface functionals are considered in the paper.
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Submitted on : Tuesday, May 23, 2006 - 7:20:58 PM
Last modification on : Wednesday, July 3, 2019 - 10:48:03 AM
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  • HAL Id : inria-00071952, version 1



Serguei Nazarov, Jan Sokolowski. Asymptotic analysis of shape functionals. RR-4633, INRIA. 2002. ⟨inria-00071952⟩



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