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

Abstract

Abstract. We study a spectral problem in a bounded domain Ω ⊂ R depending on a bounded operator coefficient > 0 and a dissipation parameter > 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 2 (Ω). In model one-and two-dimensional problems we establish the localization of the eigenvalues and find critical values of .Keywords: spectral parameter, quadratic operator pencil, localization of eigenvalues, compact operator, Schatten-von-Neumann classes , Abel-Lidskii basis property. Mathematics Subject Classification: 35P05, 35P10To the memory of Tomas Yakovlevich Azizov, whose results and lectures helped the authors in writing this work. IntroductionIn this work we study the spectral properties of one linear problem in mathematical physics depending on the dimension of a domain Ω ⊂ R (with a piece-wise smooth boundary Ω), a bounded in 2 (Ω) operator coefficient > 0 and a parameter > 0 modeling the intensity of the energy dissipation on a part of the boundary Γ. Namely, we study the problem= 0 (on := Ω ∖ Γ),for an unknown field = ( ) ( ∈ Ω) and a spectral parameter ∈ C.This spectral problem is generated by the initial boundary value problem arising as a lin-