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Second Order Elliptic Differential-Operator Equations with Unbounded Operator Boundary Conditions in UMD Banach Spaces

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Abstract

In a UMD Banach space E, we consider a boundary value problem for a second order elliptic differential-operator equation with a spectral parameter when one of the boundary conditions, in the principal part, contains a linear unbounded operator in E. A theorem on an isomorphism is proved and an appropriate estimate of the solution with respect to the space variable and the spectral parameter is obtained. In this way, Fredholm property of the problem is shown. Moreover, discreteness of the spectrum and completeness of a system of root functions corresponding to the homogeneous problem are established. Finally, applications of obtained abstract results to nonlocal boundary value problems for elliptic differential equations with a parameter in non-smooth domains are given.

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Authors and Affiliations

  1. National Academy of Sciences of Azerbaijan, Institute of Mathematics and Mechanics, 9, F. Agayev str., Baku, 1141, Azerbaijan Republic

    B. A. Aliev

  2. Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, 69978, Israel

    Ya. Yakubov

Authors
  1. B. A. Aliev
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  2. Ya. Yakubov
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Correspondence to Ya. Yakubov.

Additional information

Ya. Yakubov was supported by the Israel Ministry of Absorption.

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Aliev, B.A., Yakubov, Y. Second Order Elliptic Differential-Operator Equations with Unbounded Operator Boundary Conditions in UMD Banach Spaces. Integr. Equ. Oper. Theory 69, 269–300 (2011). https://doi.org/10.1007/s00020-010-1832-5

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  • DOI: https://doi.org/10.1007/s00020-010-1832-5

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