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On Some Class of Self-adjoint Boundary Value Problems with the Spectral Parameter in the Equations and the Boundary Conditions

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Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 221))

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Abstract

The aim of this work is to study the spectral properties of some class of self-adjoint linear boundary value and transmission problems (in domains with Lipschitz boundaries) where equations and boundary conditions depend linearly on the eigenparameter λ. We consider some abstract general problem that can be formulated on the basis of the abstract Green’s formula for a triple of Hilbert spaces and a trace operator.W e prove that the spectrum of the abstract general problem consists of real normal eigenvalues with unique limit point ∞, and the system of corresponding eigenelements forms an orthonormal basis in some Hilbert space.W e find also some asymptotic formulas for positive or positive and negative branches of eigenvalues.A s examples, we consider three multicomponent transmission problems arising in mathematical physics and some abstract general transmission problem.

Mathematics Subject Classification (2000). Primary 35P05; Secondary 35P10.

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Authors and Affiliations

  1. Taurida National University Prosp.of Acad. V.I. Vernadsky, 4, Simferopol, 95007, Ukraine

    Victor Voytitsky

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  1. Victor Voytitsky
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Correspondence to Victor Voytitsky .

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Editors and Affiliations

  1. , Abt. Mathematik V, Universität Ulm, Ulm, 89069, Germany

    Wolfgang Arendt

  2. Department of Mathematics, Virginia Polytechnic Institute, McBryde Hall 524, Blacksburg, 24061, Virginia, USA

    Joseph A. Ball

  3. Institut für Mathematik, AG Modellierung, Numerik und, TU Berlin, Straße des 17 Juni 136, Berlin, 10623, Berlin, Germany

    Jussi Behrndt

  4. Fak. II Mathematik & Naturwissenschaften, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Karl-Heinz Förster

  5. Fak. II Mathematik & Naturwissenschaften, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Volker Mehrmann

  6. , Institut für Mathematik, TU Ilmenau, Weimarer Str. 25, Ilmenau, 98693, Thüringen, Germany

    Carsten Trunk

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Voytitsky, V. (2012). On Some Class of Self-adjoint Boundary Value Problems with the Spectral Parameter in the Equations and the Boundary Conditions. In: Arendt, W., Ball, J., Behrndt, J., Förster, KH., Mehrmann, V., Trunk, C. (eds) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Operator Theory: Advances and Applications, vol 221. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0297-0_38

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