Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube

Abstract

We construct and justify asymptotic representations for the eigenvalues and eigenfunctions of boundary value problems for the Laplace operator in a three-dimensional domain Ω(ε) = Ω \ Γ̄ε with a thin singular set Γε lying in the cεneighborhood of a simple smooth closed contour Γ. We consider the Dirichlet problem, a mixed boundary value problem with the Neumann conditions on ∂Γε, and also a spectral problem with lumped masses on Γε. The asymptotic representations are of diverse character: we find an asymptotic series in powers of the parameter |ln ε|−1 or ε. The most comprehensive and complicated analysis is presented for the lumped mass problem; namely, we sum the series in powers of |ln ε|−1 and obtain an asymptotic expansion with the leading term holomorphically depending on |ln ε|−1 and with the remainder O(εδ), δ ∈ (0, 1). The main role in asymptotic formulas is played by solutions of the Dirichlet problem in Ω \ Γ with logarithmic singularities distributed along the contour Γ. 1. Statement of the problems. Description of the methods and results 1.1. Domain and boundary value problems. Let Γ be a simple smooth (C∞) closed contour on the plane R. In a neighborhood V of Γ, we introduce the natural curvilinear coordinates (n, s), where s is the arc length parameter and n is the signed distance from Γ positive outside the domain surrounded by Γ (Figure 1). In what follows, we slightly abuse the notation by writing s ∈ Γ to mean the point of Γ with coordinate s and by denoting the set {x ∈ R : (x1, x2) ∈ Γ, x3 = 0} in the space R again by Γ. Let ω be a bounded domain on the plane (Figure 2(a)), let U be a neighborhood of Γ in R where the coordinate system (n, s, x3) is defined, and let (1.1) Γε = {x ∈ U : s ∈ Γ, η = (ε−1n, εx3) ∈ ω}. Here ε > 0 is a small parameter; i.e., Γε is a thin toroidal set (Figure 2(b)). Finally, let Ω be a domain in R containing Γ (and hence containing the set (1.1) for small ε ∈ (0, ε0], ε0 > 0). For simplicity, we assume that the boundaries ∂Ω and ∂ω are smooth and place the origin η = 0 in the interior of the set ω ⊂ R. The aim of this paper is to study asymptotic properties of the spectra of several boundary value problems. First, this is the Dirichlet problem in the singularly perturbed 2010 Mathematics Subject Classification. Primary 35J25; Secondary 35B25, 35B40, 35B45, 35P20, 35S05.