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The weak eigenfunctions of boundary-value problem with symmetric discontinuities

Olğar Hayati, Mukhtarov Oktay S., Muhtarov Fahreddin S., Aydemir Kadriye

Journal of Applied Analysis

2022

Vol. 28, nr 2

275--283

The main goal of this study is the investigation of discontinuous boundary-value problems for second-order differential operators with symmetric transmission conditions.We introduce the new notion of weak functions for such type of discontinuous boundary-value problems and develop an operator-theoretic method for the investigation of the spectrum and completeness property of the weak eigenfunction systems. In particular, we define some self-adjoint compact operators in suitable Sobolev spaces such that the considered problem can be reduced to an operator-pencil equation. The main result of this paper is that the spectrum is discrete and the set of eigenfunctions forms a Riesz basis of the suitable Hilbert space.

Imperfect transmission conditions for a thin weakly compressible interface. 2D problems

Mishuris G.

Archives of Mechanics

2004

Vol. 56, nr 2

103-115

Imperfect transmission conditions are evaluated in the case of weakly compressible elastic interphase between different bonded elastic materials. It is shown that the corresponding conditions can be different in comparison with those for the elastic compressible interface.