LAYER POTENTIAL TECHNIQUES IN SPECTRAL ANALYSIS. PART I: COMPLETE ASYMPTOTIC EXPANSIONS FOR EIGENVALUES OF THE LAPLACIAN IN DOMAINS WITH SMALL INCLUSIONS

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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-a-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.
Publisher
AMER MATHEMATICAL SOC
Issue Date
2010-06
Language
English
Article Type
Article
Citation

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v.362, no.6, pp.2901 - 2922

ISSN
0002-9947
DOI
10.1090/S0002-9947-10-04695-7
URI
http://hdl.handle.net/10203/96749
Appears in Collection
MA-Journal Papers(저널논문)
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