Mixed Boundary Value Problems in Singularly Perturbed Two-Dimensional Domains with the Steklov Spectral Condition
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Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
MIXED BOUNDARY VALUE PROBLEMS IN SINGULARLY PERTURBED TWO-DIMENSIONAL DOMAINS WITH THE STEKLOV SPECTRAL CONDITION V. Chiad` o Piat Politecnico di Torino, DISMA 24, Corso Duca degli Abruzzi, Torino 10129, Italy [email protected]
S. A. Nazarov ∗ St.-Petersburg State University 7-9, Universitetskaya nab., St. Petersburg 199034, Russia Institute of Problems of Mechanical Engineering RAS, 61, V.O., Bolshoj pr., St. Petersburg 199178, Russia [email protected]
UDC 517.958:539.3(3):621.372.8
We study the asymptotic behavior of the spectrum of the Laplace equation with the Steklov, Dirichlet, Neumann boundary conditions or their combination in a twodimensional domain with small holes of diameter O(ε) as ε → +0. We derive and justify asymptotic expansions of eigenvalues and eigenfunctions of two types: series in z = | ln ε|−1 and power series with rational and holomorphic terms in z respectively. For the overall Steklov problem we obtain asymptotic expansions in the low and middle frequency ranges of the spectrum. Bibliography: 18 titles.
1
Introduction
1.1. Formulation of the problems. We consider the Laplace equation −Δx uε (x) = 0,
x ∈ Ωε ,
(1.1)
ω εj ,
(1.2)
in the planar domain Ωε = Ω \
J j=1
where Ω and ωj , j = 1, . . . , J, are domains in the plane R2 enveloped by simple closed smooth ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 91-124. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0655
655
(for the sake of simplicity, of class C ∞ ) contours Γ = ∂Ω and γj = ∂ωj respectively. Furthermore, ωjε = {x : ξ j = ε−1 (x − xj ) ∈ ωj },
(1.3)
where x1 , . . . , xJ are fixed points in Ω, xj = xk for j = k, ε ∈ (0, ε0 ], ε0 > 0, is a small parameter, and ω εj ⊂ Ω, ω εj ∩ ω εk = ∅, j, k = 1, . . . , J, j = k. If necessary, we diminish ε0 , but keep the notation. We assume that ωj contains the origin. We consider Equation (1.1) with various boundary conditions on the exterior Γ and interior γ1ε , . . . γJε parts of the boundary Γε = ∂Ωε including at least one of the Steklov spectral conditions ∂ν uε (x) = λε uε (x),
x ∈ Γ,
∂ν uε (x) = λε uε (x),
x ∈ γjε ,
(1.4) j = 1, . . . , J,
(1.5)
where ∂ν is the outward normal derivative and λε is the spectral parameter. In addition to the overall Steklov problem (1.1), (1.4), (1.5), we consider all possible variants of boundary value problems obtained by replacing one of the Steklov conditions (1.4) or (1.5) with either the Neumann conditions ∂ν uε (x) = 0,
x ∈ Γ,
∂ν uε (x) = 0,
x ∈ γjε ,
(1.6) j = 1, . . . , J,
(1.7)
or the Dirichlet conditions uε (x) = 0,
x ∈ Γ,
uε (x) = 0,
x ∈ γjε ,
(1.8) j = 1, . . . , J.
(1.9)
The variational formulation of the problem is to find an eigenpair {λε , uε } ∈ R×H ε satisfying the integral identity (∇x uε , ∇x v ε )Ωε = λε (ρε uε , v ε )Γε ∀v ε ∈ H ε , (1.10) where ∇x = grad, ρε = 0 on Γ or γ ε = γ1ε ∪ . . . ∪ γJε if the conditions (1.4) or (1.5) are
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