Spectral properties of elliptic operator with double-contrast coefficients near a hyperplane

2014/04/09 by Khrabustovskyi, Andrii, Plum, Michael
#35B27 #35B40 #35J25 #35P20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1404.2555

Abstract

In this paper we study the asymptotic behaviour as ε→ 0 of the spectrum of the elliptic operator A^ε=-1\over b^εdiv(a^ε∇) posed in a bounded domain Ω⊂ℝⁿ (n ≥ 2) subject to Dirichlet boundary conditions on ∂Ω. When ε→ 0 both coefficients a^ε and b^ε become high contrast in a small neighborhood of a hyperplane Γ intersecting Ω. We prove that the spectrum of A^ε converges to the spectrum of an operator acting in L²(Ω)⊕ L²(Γ) and generated by the operation -Δ in Ω∖Γ, the Dirichlet boundary conditions on ∂Ω and certain interface conditions on Γ containing the spectral parameter in a nonlinear manner. The eigenvalues of this operator may accumulate at a finite point. Then we study the same problem, when Ω is an infinite straight strip ("waveguide") and Γ is parallel to its boundary. We show that A^ε has at least one gap in the spectrum when ε is small enough and describe the asymptotic behaviour of this gap as ε→ 0. The proofs are based on methods of homogenization theory.

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