## Rocky Mountain Journal of Mathematics

### Separability properties of singular degenerate abstract differential operators and applications

Veli B. Shakhmurov

#### Abstract

We study separability and spectral properties of singular degenerate elliptic equations in vector-valued $L_{p}$ spaces. We prove that a realization operator according to this equation with some boundary conditions is separable and Fredholm in $L_{p}$. The leading part of the associated differential operator is not self-adjoint. The sharp estimate of the resolvent, discreteness of spectrum and completeness of root elements of this operator is obtained. Moreover, we show that this operator is positive and generates a holomorphic $C_{0}$-semigroups on $L_{p}$. In application, we examine the regularity properties of nonlocal boundary value problem for degenerate elliptic equation and for the system of degenerate elliptic equations of either finite or infinite number.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 5 (2019), 1647-1666.

Dates
First available in Project Euclid: 19 September 2019

https://projecteuclid.org/euclid.rmjm/1568880095

Digital Object Identifier
doi:10.1216/RMJ-2019-49-5-1647

Mathematical Reviews number (MathSciNet)
MR4010577

Zentralblatt MATH identifier
07113703

#### Citation

Shakhmurov, Veli B. Separability properties of singular degenerate abstract differential operators and applications. Rocky Mountain J. Math. 49 (2019), no. 5, 1647--1666. doi:10.1216/RMJ-2019-49-5-1647. https://projecteuclid.org/euclid.rmjm/1568880095

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