Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Asymptotics of the eigenvalues of the Dirichlet-Laplace problem in a domain with thin tube excluded


Author: X. Claeys
Journal: Quart. Appl. Math. 74 (2016), 595-605
MSC (2010): Primary 35B25
DOI: https://doi.org/10.1090/qam/1436
Published electronically: June 17, 2016
MathSciNet review: 3539023
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Abstract: We consider a Laplace problem with Dirichlet boundary condition in a three dimensional domain containing an inclusion taking the form of a thin tube with small thickness $ \delta $. We prove convergence in operator norm of the resolvent of this problem as $ \delta \to 0$, establishing that the perturbation induced by the inclusion on the resolvent is not greater than $ O(\vert \ln \delta \vert ^{-\gamma })$ for some $ \gamma >0$. We deduce convergence of the eigenvalues of the perturbed operator toward the limit operator.


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Additional Information

X. Claeys
Affiliation: Laboratoire Jacques-Louis Lions, UPMC University of Paris 6 and CNRS UMR 7598, 75005, Paris, France;; INRIA-Paris-Rocquencourt, EPC Alpines, Le Chesnay Cedex, France
Email: claeys@ann.jussieu.fr

DOI: https://doi.org/10.1090/qam/1436
Received by editor(s): February 25, 2015
Published electronically: June 17, 2016
Article copyright: © Copyright 2016 Brown University

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