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Carleman estimates and unique continuation property for the anisotropic differential-operator equations

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Science China Mathematics 2008年第7期51卷 1215-1231页

作者： veli b shakhmurov Department of Mathematics Okan UniversityAkfirat34959 TuzlaIstanbulTurkey

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coefficients in banach-valued L p -spaces are studied. To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations, the sufficient conditions are founded. by using these facts, the unique continuation properties are established. In the application part, the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.

关键词： Carleman estimates unique continuation embedding theorems banach-valued function spaces differential operator equations maximal L p -regularity operator-valued Fourier multipliers interpolation of banach spaces 34G10 35J25 35J70

Embedding Theorems in J5-Spaces and Applications

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Chinese Annals of Mathematics,Series b 2008年第1期29卷 95-112页

作者： veli b.shakhmurov Department of Mathematics Okan University

This study focuses on the anisotropic besov-Lions type spaces b^lp,θ（Ω;E0,E） associated with banach spaces E0 and E. Under certain conditions, depending on l =（l1,l2,…,ln）and α=（α1,α2,…,αn）,the most regular class of interpolation space Eα between E0 and E are found so that the mixed differential operators D^α are bounded and compact, from b^l＋s p,θ（Ω;E0,E） to b^s p,θ（Ω;Eα）.These results are applied to concrete vector-valued function spaces and to anisotropic differential-operator equations with parameters to obtain conditions that guarantee the uniform b separability with respect to these parameters. by these results the maximal b-regularity for parabolic Cauchy problem is obtained. These results are also applied to infinite systems of the quasi-elliptic partial differential equations and parabolic Cauchy problems with parameters to obtain sufficient conditions that ensure the same properties.

关键词： Embedding theorems banach-valued function spaces Differential operator equations b-Separability Operator-valued Fourier multipliers Interpolation of banach spaces

The Quality Properties of Integral Type Problems for Wave Equations and Applications

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Journal of Applied Mathematics and Physics 2022年第4期10卷 1217-1239页

作者： veli b. shakhmurov Rishad Shahmurov Department of Industrial Engineering Antalya Bilim University Dosemealti Antalya Turkey Center of Analytical-Information Resource Azerbaijan State Economic University Baku Azerbaijan University of Alabama Tuscaloosa AL USA

In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data and the operators, the existence, uniqueness, regularity properties of solutions are established. by choosing the space H and A, the regularity properties of solutions of a wide class of wave equations in the field of physics are obtained.

关键词： Abstract Differential Equations boussinesq Equations Wave Equations Regularity Property of Solutions Fourier Multipliers

Estimates of Approximation Numbers and Applications

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Acta Mathematica Sinica,English Series 2012年第9期28卷 1883-1896页

作者： veli shakhmurov Department of Electronics Engineering and Communication Okan University

In this study, the estimates of approximation numbers of embedding operators in weighted spaces have been analyzed. These estimates depend on orders of differential operators, dimensions of function spaces and weighted functions. This fact implies that the associated embedding operators belong to Schatten class of compact operators. by using these estimates, the discreetness of spectrum and completion of root elements relating to principal nonselfedjoint degenerate differential operators is obtained.

关键词： Approximation numbers Kolmogorov numbers separable boundary value problems differential-operator equations banach-valued function spaces operator-vMued multipliers interpo- lation of banach spaces

Abstract Elliptic Equations with Integral boundary Conditons

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Chinese Annals of Mathematics,Series b 2016年第4期37卷 625-642页

作者： veli shakhmurov Department of Mechanical Engineering Okan University Department of Mathematics Azerbaycan Khazar University

This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in L_p spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed L_p norm.

关键词： boundary value problems Integral boundary conditions Differential-operator equations Maximal Lp regularity Abstract parabolic equation,Operator valued multipliers Interpolation of banach spaces Semigroupsof operators

Free boundary Value Problems for Abstract Elliptic Equations and Applications

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Chinese Annals of Mathematics,Series b 2011年第5期32卷 749-770页

作者： veli shakhmurov Department of Electronics Engineering and Communication Okan University

The free boundary value problems for elliptic differential-operator equations are *** conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L p-spaces are *** application,the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.

关键词： Free boundary value problems Sobolev-Lions type spaces Differential-operator equations Maximal Lp regularity banach spaces Operator-

Compact embedding in besov spaces and b-separable elleptic operators

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Science China Mathematics 2010年第4期53卷 1067-1084页

作者： shakhmurov veli b. Department of Electronics Engineering and Communication Okan University

Necessary and sufficient conditions for compactness of sets in banach space valued besov class bp,q s(Ω;E) is *** embedding theorems in besov-Lions type spaces b l,s p,q(Ω;E0,E) are studied,where E0,E are two banach spaces and E 0  *** most regular class of interpolation space E α,between E 0 and E are found such that the mixed differential operator D α is bounded and compact from b p,q l,s (Ω;E 0,E) to b p,q s (Ω;E α) and Ehrling-Nirenberg-Gagliardo type sharp estimates *** using these results the separability of differential operators with variable coefficients and the maximal b-regularity of parabolic Cauchy problem are *** applications,the infinite systems of the elliptic partial differential equations and parabolic Cauchy problems are studied.

关键词： embedding theorems banach-valued function spaces differential-operator equations maximal b-regularity operator-valued Fourier multipliers interpolation of banach spaces abstract parabolic Cauchy problem

Maximal b-Regular Integro-Differential Equation

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Chinese Annals of Mathematics,Series b 2009年第1期30卷 39-50页

作者： veli shakhmurov Rishad SHAHMUROV Department of Electronics Engineering and Communication Okan University Vocational High School Okan University

by using Fourier multiplier theorems, the maximal b-regularity of ordinary integro-differential operator equations is investigated. It is shown that the corresponding differential operator is positive and satisfies coercive estimate. Moreover, these results are used to establish maximal regularity for infinite systems of integro-differential equations.

关键词： banach-valued besov spaces Operator-valued multipliers,boundary value problems Integro-differential equations

AbSTRACT CAPACITY OF REGIONS AND COMPACT EMbEDDING WITH APPLICATIONS

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Acta Mathematica Scientia 2011年第1期31卷 49-67页

作者： veli shakhmurov Department of Electronics Engineering and Communication Okan University

The weighted Sobolev-Lions type spaces W pl,γ（Ω; E0, E） = W pl,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 W pl,γ（; 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 W pl,γ（Ω; E0, E） to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.

关键词： Capacity of regions embedding theorems banach-valued function spaces differential-operator equations Semigroups of operators operator-valued Fourier multipliers interpolation of banach spaces