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Ufimsk. Mat. Zh., 2017, Volume 9, Issue 2, Pages 3–16 (Mi ufa371)  

On spectral properties of one boundary value problem with a surface energy dissipation

O. A. Andronovaa, V. I. Voytitskiyb

a Academy of Construction and Architecture of the Federal State Autonomous Educational Institution of Higher Education V.I.Vernadsky rimean Federal University
b Crimea Federal University, Simferopol

Abstract: We study a spectral problem in a bounded domain ${\Omega \subset \mathbb{R}^{m}}$, depending on a bounded operator coefficient $Q>0$ and a dissipation parameter $\alpha>0$. In the general case we establish sufficient conditions ensuring that the problem has a discrete spectrum consisting of countably many isolated eigenvalues of finite multiplicity accumulating at infinity. We also establish the conditions, under which the system of root elements contains an Abel-Lidskii basis in the space $ L_2(\Omega)$. In model one- and two-dimensional problems we establish the localization of the eigenvalues and find critical values of $\alpha$.

Keywords: spectral parameter, quadratic operator pencil, localization of eigenvalues, compact operator, Schatten-von-Neumann classes $S_p$, Abel-Lidskii basis property.

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English version:
Ufa Mathematical Journal, 2017, 9:2, 3–16 (PDF, 450 kB); https://doi.org/10.13108/2017-9-2-3

Bibliographic databases:

UDC: 517.98+517.9:532
MSC: 35P05, 35P10
Received: 01.02.2016

Citation: O. A. Andronova, V. I. Voytitskiy, “On spectral properties of one boundary value problem with a surface energy dissipation”, Ufimsk. Mat. Zh., 9:2 (2017), 3–16; Ufa Math. J., 9:2 (2017), 3–16

Citation in format AMSBIB
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