Veli Shakhmurov

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal Lp-regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces Wp,q2,2.

Degenerate Differential Operators with Parameters

The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valued Lp− spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.

Embedding operators and maximal regular differential-operator equations in Banach-valued function spaces

This study focuses on anisotropic Sobolev type spaces associated with Banach spaces,. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of and. In particular, the most regular class of interpolation spaces between,, depending of and order of spaces are found that mixed derivatives belong with values; the boundedness and compactness of differential operators from this space to-valued spaces are proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal regularity uniformly with respect to these parameters.

Linear and nonlinear nonlocal boundary value problems for differential-operator equations

This study focuses on nonlocal boundary value problems (BVPs) for linear and nonlinear elliptic differential-operator equations (DOEs) that are defined in Banach-valued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal Lp -regularity and Fredholmness of linear BVPs, uniformly with respect to this parameter, are presented. This fact implies that the appropriate differential operator is a generator of an analytic semigroup. Then, by using these results, the existence, uniqueness and maximal smoothness of solutions of nonlocal BVPs for nonlinear DOEs are shown. These results are applied to nonlocal BVPs for regular elliptic partial differential equations, finite and infinite systems of differential equations on cylindrical domains, in order to obtain the algebraic conditions that guarantee the same properties.

A fractional Gabor transform

We present a fractional Gabor expansion on a general, non-rectangular time-frequency lattice. The traditional Gabor expansion represents a signal in terms of time- and frequency-shifted basis functions, called Gabor logons. This constant-bandwidth analysis results in a fixed, rectangular time frequency plane tiling. Many of the practical signals require a more flexible, non-rectangular time-frequency lattice for a compact representation. The proposed fractional Gabor expansion uses a set of basis functions that are related to the fractional Fourier basis and generate a non-rectangular tiling. The completeness and bi-orthogonality conditions of the new Gabor basis are discussed.

Abstract parabolic problems with parameter and application

In this work, the uniform well-posedenes of singular perturbation problems for parameter dependent parabolic differential-operator equations is established. These problems occur in phytoremediation modelling.

Embedding theorems in Banach-valued -spaces and maximal -regular differential-operator equations

The embedding theorems in anisotropic Besov-Lions type spaces are studied; here and are two Banach spaces. The most regular spaces are found such that the mixed differential operators are bounded from to, where are interpolation spaces between and depending on and. By using these results the separability of anisotropic differential-operator equations with dependent coefficients in principal part and the maximal-regularity of parabolic Cauchy problem are obtained. In applications, the infinite systems of the quasielliptic partial differential equations and the parabolic Cauchy problems are studied.

Parabolic problems with parameter occurring in environmental engineering

In this work, the uniform well possedenes of initial value problems for parameter dependent parabolic differential operator equations are obtained. These problems occur in phytoremediation modelling.

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

The weighted Sobolev-Lions type spaces Wlp,γ(Ω;E0,E) =Wlp,γ(Ω;E)∩ Lp,γ(Ω;E0) are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω ∈ Rn in Wlp,γ(Ω;E0,E) is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from Wlp,γ(Ω;E0,E) to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.

Uniform separable differential operators with parameters

Abstract In this paper we study boundary value problems for anisotropic partial differential-operator equations with parameters. The principal part of the appropriate differential operators are not self-adjoint. Several conditions for the uniform separability in weighted Banach-valued L p -spaces are given. Sharp estimates for the resolvent of the corresponding differential operator are obtained. In particular the positivity and R-positivity of these operators are established. As an application we study the separability of degenerate DOEs, maximal regularity for degenerate abstract parabolic problem with parameters, the uniform separability of finite and infinite systems for degenerate anisotropic partial differential equations with parameters.