ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

doi:10.1016/S0252-9602(11)60207-5

数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (1): 49-67.doi: 10.1016/S0252-9602(11)60207-5

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

Veli Shakhmurov

Department of Electronics Engineering and Communication, Okan University, Akfirat Beldesi, Tuzla, 34959, Istanbul, Turkey

收稿日期:2007-06-19 修回日期:2009-09-27 出版日期:2011-01-20 发布日期:2011-01-20

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

Veli Shakhmurov

Department of Electronics Engineering and Communication, Okan University, Akfirat Beldesi, Tuzla, 34959, Istanbul, Turkey

Received:2007-06-19 Revised:2009-09-27 Online:2011-01-20 Published:2011-01-20

摘要/Abstract

摘要：

The weighted Sobolev-Lions type spaces W ^l_p, _γ(Ω; E₀, E) = W ^l_p, _γ (Ω; E) \ L_p_, _γ(Ω; E₀) are two Banach spaces and E₀ is continuously and densely embedded on E. A new concept of capacity of region Ω∈Rⁿ in W ^l_p, _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 E_α between E₀ and E, depending of α and l, are found such that mixed differential operators D^α are bounded and compact from W ^l_p, γ(Ω; E₀, E) to E_α-valued L_p, 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

Abstract:

Key words: Capacity of regions, embedding theorems, Banach-valued function spaces, differential-operator equations, Semigroups of operators, operator-valued Fourier multipliers, interpolation of Banach spaces

中图分类号:

34G10

Veli Shakhmurov. ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS[J]. 数学物理学报(英文版), 2011, 31(1): 49-67.

Veli Shakhmurov. ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS[J]. Acta mathematica scientia,Series B, 2011, 31(1): 49-67.

参考文献

[1] Adams R A. Sobolev spaces. New York, San Francisco, London: Academic Press, 1975

[2] Adams R A. Compact imbedding of weighted Sobolev spaces on unbounded domain. Journal of Differential Equations, 1971, 9:   325--334

[3] Adams R A. Capacity and Compact Imbedding. Journal of Mathematics and Mechanica, 1970

[4] Amann H. Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math Nachr, 1997, 186: 5--56

[5] Amann H. Linear and quasi-linear equations, 1. Basel: Birkhauser, 1995

[6] Aubin J P. Abstract boundary-value operators and their adjoint. Rend Sem Mat Univ Padova, 1970, 43: 1--33

[7] Agarwal R, Shakhmurov V B. Mutipoint problems for degenerate abstract differential equations. Acta Math Hungar,   2009, 123(1/2): 65--89

[8] Besov O V, P Ilin, V P, Nikolskii S M. Integral representations of functions and embedding theorems. Moscow: Nauka, 1975

[9] Burkholder D L. A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions. Proc conf Harmonic analysis in honor of Antonu Zigmund, Chicago, 1981, Wads Worth, Belmont, 1983: 270--286

[10] Bourgain J. Some remarks on Banach spaces in which martingale differences are unconditional. Arkiv Math, 1983, 21: 163--168

[11] Dore G, Yakubov S. Semigroup estimates and non coercive boundary value problems. Semigroup Form, 2000, 60: 93--121

[12] Denk R, Hieber M, Prüss J. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem Amer Math Soc, 2003, 166: 788

[13] Guliev V S. Embedding theorems for spaces of UMD-valued function spaces. (Russion) Dokl Akad Nauk, 1993, 328(4):   408--410

[14] Guliev V S. Embedding theorems for weighted Sobolev spaces of B-valued functions. (Russion) Dokl Akad Nauk, 1994, 338(4):   440--443

[15] Hütönen T. Estimates for partial derivatives of vector-valued functions. Illinois J Math, 2007, 51(3):   731--742

[16] Krein S G. Linear differential equations in Banach space. Providence: American Mathematical Society, 1971

[17] Kree P. Sur les multiplicateurs dans FL aves poids. Grenoble: Annales Ins Fourier, 1966, 16(2): 191--121

[18] Lions J L, Peetre J. Sur une classe d'espaces d'interpolation. Inst Hautes Etudes Sci Publ Math, 1964, 19: 5--68

[19] Lizorkin P I. (L_p, L_q)-Multiplicators of Fourier Integrals. Doklady Akademii Nauk SSSR, 1963, 152(4): 808--811

[20] Lamberton D. Equations d'evalution lineaires associeees a'des semigroupes de contractions dans less espaces L_p. J Funct Anal, 1987, 72: 252--262

[21] Maz'ya V G. Sobolev spaces. Berlin: Springer, 1985

[22] McConnell Terry R. On Fourier Multiplier Transformations of Banach-Valued Functions. Trans Amer Mat Soc, 1984, 285(2): 739--757

[23] Pisier G. Les inegalites de Khintchine-Kahane d'apres C. Borel, Seminare sur la geometrie des espaces de Banach. Paris: Ecole Polytechnique, 1977/1978, 7

[24] Sobolev S L.Certain applications of functional analysis to mathematical physics. Moscow: Nauka, 1988

[25] Shklyar A Ya. Complate second order linear differential equations in Hilbert spaces. Basel: Birkhauser Verlak, 1997

[26] Shakhmurov V B. Theorems about of compact embedding and applications. Doklady Akademii Nauk SSSR, 1978, 241(6): 1285--1288

[27] Shakhmurov V B. Imbedding theorems and their applications to degenerate equations. Differential equations, 1988, 24(4): 475--482

[28] Shakhmurov V B. Imbedding theorems for abstract function- spaces and their applications. Mathematics of the USSR-Sbornik, 1987, 134(1/2): 261--276

[29] Shakhmurov V B. Embedding and separable differential operators in Sobolev-Lions type spaces. Mathematical Notes, 2008, 84(6): 906--926

[30] Shakhmurov V B. Embedding and maximal regular differential operators in Sobolev-Lions spaces. Acta Mathematica Sinica, 2006, 22(5): 1493--1508

[31] Shakhmurov V B. Embedding theorems and\ maximal regular differential operator equations in Banach-valued function spaces, Journal of Inequalities and Applications, 2005, 2(4): 329--345

[32] Shakhmurov V B. Coercive boundary value problems for regular degenerate differential-operator equations. J Math Anal Appl, 2004, 292(2): 605--620

[33] Triebel H. Interpolation theory. Function spaces. Differential operators. Amsterdam: North-Holland, 1978

[34] Triebel H. Spaces of distributions with weights. Multipliers in L_p-spaces with weights. Math Nachr, 1977, 78: 339--356

[35] Weis L. Operator-valued Fourier multiplier theorems and maximal L_pregularity. Math Ann, 2001, 319:  735--75

[36] Yakubov S. A nonlocal boundary value problem for elliptic differential-operator equations and applications. Integr Equ Oper Theory, 1999, 35: 485--506

[37] Yakubov S, Yakubov Ya. Differential-operator Equations. Ordinary and Partial Differential Equations. Boca Raton: Chapmen and Nall/CRC, 2000

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

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