HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation

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.
Document type :
Complete list of metadata

Contributor : Rapport de Recherche Inria Connect in order to contact the contributor
Submitted on : Tuesday, May 23, 2006 - 7:20:58 PM
Last modification on : Wednesday, May 11, 2022 - 12:06:04 PM
Long-term archiving on: : Sunday, April 4, 2010 - 10:45:57 PM


  • HAL Id : inria-00071952, version 1



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



Record views


Files downloads