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

Abstract

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.

Asymptotic Analysis of High-Contrast Phononic Crystals and a Criterion for the Band-Gap Opening

“…6) and the mapping u −→ (φ, ψ) from solutions of (3.1) in H 1 (Y ) 2 to solutions to the system of integral equations (3.6) …”

“…Generalized Rouché's theorem and other techniques from the theory of meromorphic operator-valued functions are combined with careful asymptotic expansions of integral kernels to obtain full asymptotic expansions for eigenvalues. This method was first used in [8], and then successfully applied to obtain an asymptotic formula for the eigenvalues of Laplacian under singular perturbations [6] and high contrast asymptotics for the photonic crystals [7]. See also [4].…”

Artificial conditions for the linear elasticity equations

“…This problem enters the general framework of local perturbations for elliptic problems, which have been deeply studied. Among others, let us mention the following works using potential theory [38,5,3,4] and the reference monographs [30,31] for multiscale expansions. Following [10,36], the solution of (1.1) is given to first order by…”

“…We choose here the boundary to be the circle ∂B R , where R is assumed to be large. We show in the next section that the problem is reduced to seeking solutions of the following boundary value problem: 4) with G ∈ H 1/2 (∂ω), ω is a C ∞ perturbation of the unit ball and E, ν the Young's modulus and Poisson's ratio are linked to the Lamé coefficient by relation (2.9) while ∆ τ denotes the Laplace-Beltrami operator on ∂Ω. Let us recall the definition of the tangential differential operators: the tangential gradient on ∂Ω of u ∈ H 2 (Ω) is ∇ τ u = ∇u−∂ n u n, the tangential divergence div τ is its L 2 (∂Ω) adjoint and the Laplace-Beltrami operator is div τ (∇ τ ·).…”

Multiscale expansion and numerical approximation for surface defects

“…The multiscale superposition method is used in the preliminary step of crack detection, a continuum-discrete damage model is involved for its propagation. The question of domains with small inclusions or inhomogeneities has been widely studied, especially in the case of electromagnetics and inverse problem, see for example [1,2,4,11,15,20,24,25].…”