Summary: LAYER POTENTIAL TECHNIQUES IN SPECTRAL ANALYSIS.
PART I: COMPLETE ASYMPTOTIC EXPANSIONS FOR
EIGENVALUES OF THE LAPLACIAN IN DOMAINS WITH
HABIB AMMARI, HYEONBAE KANG, MIKYOUNG LIM, AND HABIB ZRIBI
Abstract. We provide a rigorous derivation of new complete asymptotic ex-
pansions 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 equa-
tions, we reduce this problem to the study of the characteristic values of in-
tegral 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.
Let be a bounded domain in Rd