Article
Ufa Mathematical Journal
Volume 9, Number 2, pp. 3-16
On spectral properties of one boundary value problem with a surface energy dissipation
Andronova O.A., Voytitskiy V.I.
DOI:10.13108/2017-9-2-3
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We study a spectral problem in a bounded domain $\Omega \subset \mathbb{R}^{m}$ depending on a bounded operator coefficient $S>0$ 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 $\alpha$.