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

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ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

Veli Shakhmurov   

  1. 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   

  1. 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:

The weighted Sobolev-Lions type spaces W lpγ(Ω; E0, E) = W lpγ (Ω; E) \ Lp, 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 lp, 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 W lp, 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

Abstract:

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

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