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

B. A. Aliev, Ya Yakubov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)269-300
Number of pages32
JournalIntegral Equations and Operator Theory
Volume69
Issue number2
DOIs
StatePublished - Feb 2011

Keywords

  • Differential-operator equations
  • completeness
  • elliptic equations
  • isomorphism
  • root functions
  • spectral parameter

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