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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Layer potential techniques in spectral analysis. Part I: Complete asymptotic expansions for eigenvalues of the Laplacian in domains with small inclusions


Authors: Habib Ammari, Hyeonbae Kang, Mikyoung Lim and Habib Zribi
Journal: Trans. Amer. Math. Soc. 362 (2010), 2901-2922
MSC (2000): Primary 35B30
Published electronically: January 5, 2010
MathSciNet review: 2592941
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Abstract: We provide a rigorous derivation of new complete asymptotic expansions for eigenvalues of the Laplacian in domains with small inclusions. The inclusions, somewhat apart from or nearly touching the boundary, are of arbitrary shape and arbitrary conductivity contrast vis-à-vis the background domain, with the limiting perfectly conducting inclusion. By integral equations, we reduce this problem to the study of the characteristic values of integral operators in the complex plane. Powerful techniques from the theory of meromorphic operator-valued functions and careful asymptotic analysis of integral kernels are combined for deriving complete asymptotic expansions for eigenvalues. Our asymptotic formulae in this paper may be expected to lead to efficient algorithms not only for solving shape optimization problems for Laplacian eigenvalues but also for determining specific internal features of an object based on scattering data measurements.


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Additional Information

Habib Ammari
Affiliation: Centre de Mathématiques Appliquées, CNRS UMR 7641, École Polytechnique, 91128 Palaiseau Cedex, France
Email: ammari@cmapx.polytechnique.fr

Hyeonbae Kang
Affiliation: Department of Mathematical Sciences and RIM, Seoul National University, Seoul 151-747, Korea
Address at time of publication: Department of Mathematics, Inha University, Incheon, 402-751, Korea
Email: hkang@math.snu.ac.kr, hbkang@inha.ac.kr

Mikyoung Lim
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
Address at time of publication: Department of Mathematics, Korean Advanced Institute of Science and Technology, Daejeon 305-701, Korea
Email: lim@math.colostate.edu, mklim@kaist.ac.kr

Habib Zribi
Affiliation: Centre de Mathématiques Appliquées, CNRS UMR 7641, École Polytechnique, 91128 Palaiseau Cedex, France
Address at time of publication: Department of Mathematics, Korean Advanced Institute of Science and Technology, Daejeon 305-701, Korea
Email: zribi@cmapx.polytechnique.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04695-7
Keywords: Eigenvalue problem, Laplacian, small inclusion, full-asymptotic expansions, boundary integral equation
Received by editor(s): January 27, 2006
Received by editor(s) in revised form: December 9, 2007
Published electronically: January 5, 2010
Article copyright: © Copyright 2010 American Mathematical Society