Abnormal Behavior of Eigenvalues of Mixed Boundary Value Problems for the Laplace Operator in Truncated, but Long Cylind
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
ABNORMAL BEHAVIOR OF EIGENVALUES OF MIXED BOUNDARY VALUE PROBLEMS FOR THE LAPLACE OPERATOR IN TRUNCATED, BUT LONG CYLINDERS S. A. Nazarov St.-Petersburg State University 7-9, Universitetskaya nab., St. Petersburg 199034, Russia [email protected]
UDC 517.956.8:517.956.227
In the spectrum of the mixed boundary value problem for the Laplace operator in a finite, but long cylinder Ω with curved ends, we detect a series of eigenvalues of the strange behavior as the length 2 unlimitedly increases. In addition to usual hard-movable eigenvalues not leaving small neighborhoods of fixed points, we detect eigenvalues gliding down with a large speed along the real axis, but smoothly landing on the lower bound λ† = Λ1 of the continuous spectrum of the limit problem in the half-cylinder Π± obtained by elongating Ω in one of two directions. We give a complete description of the low-frequency range of the spectrum and show that each point in (Λ1 , Λ2 ) is a blinking eigenvalue. Above the second threshold Λ2 of the continuous spectrum of the problem in Π± , the behavior of eigenvalues of the problem in the waveguide Ω as → +∞ is chaotic and their asymptotics can be constructed only in particular cases. Bibliography: 40 titles. Illustrations: 2 figures.
1
Statement of the Mixed Boundary Value Problem
Let Ω = ω × R be a cylinder with cross-section ω ⊂ Rd−1 , d 2, smooth (of class C ∞ for the sake of simplicity) boundary ∂ω, and compact closure ω = ω ∪ ∂ω. Assume that is a large positive number and H± are smooth profile functions on ω. Without loss of generality we can assume that (1.1) H± (y) H0 > 0, y ∈ ω. In the long, but finite waveguide (cf. Figure 1) Ω = x = (y, z) : y = (x1 , . . . , xd−1 ) ∈ ω, −H− (y) − < xd < + H+ (y)
(1.2)
we consider the spectral mixed boundary value problem − Δx u (x) = λ u (x), u (x) = 0,
x ∈ Ω ,
x ∈ Γ ,
(1.3) (1.4)
Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 147-173. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0351
351
∂n u (x) = 0,
x ∈ ± ,
(1.5)
where Δx is the Laplace operator written in the Cartesian coordinates x, ∂n is the directional derivative along the outward normal to the ends = x : y ∈ ω, z := xd = ± ± H± (y) , ±
(1.6)
∪ ) is the lateral surface of the waveguide (1.2). The spectrum of the and Γ = ∂Ω \ (+ − problem (1.3)–(1.5) is a monotone positive unbounded sequence of eigenvalues
λ1 < λ2 λ3 . . . λm . . . .
(1.7)
The main goal of this paper is to study the asymptotic behavior of terms of the sequence (1.7) as → +∞ and extract groups of eigenvalues with different behavior under unlimited elongation of the cylinder (1.2). Namely, under certain assumptions, we find stereotype hardmovable eigenvalues that do not leave a small neighborhood of some point λ• ∈ R+ = (0, +∞) for sufficiently large > • , as well as gliding eigenvalues that glide down with a large speed along the real axis as → +∞, but smoothly l
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