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Embedding and Maximal Regular Differential operators in Sobolev-Lions Spaces

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Acta Mathematica Sinica,English Series 2006年第5期22卷 1493-1508页

作者： Veli B.SHAKHMUROV Istanbul University Engineering Faculty Department of Electric-Electronics EngineeringAvcilar 34850 Istanbul Turkey

This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. 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 spaces E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.

关键词： embedding operators Banach-valued function spaces differential operator equations (DOE) maximal regularity operator-valued fourier multipliers interpolation of Banach spaces

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

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

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