Embedding operators and maximal regular differential-operator equations in Banach-valued function spaces
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This study focuses on anisotropic Sobolev type 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 E0 and E. In particular, the most regular class of interpolation spaces Eα between E0 , E, depending of α and order of spaces are found that mixed derivatives Dα belong with values; the boundedness and compactness of differential operators Dα from this space to Eα -valued L p spaces are proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal L p regularity uniformly with respect to these parameters. 1. Introduction Embedding theorems in function spaces have been elaborated in detail by [5, 28]. A comprehensive introduction to the theory of embedding of function spaces and historical references may be also found in [28]. In abstract function spaces embedding theorems have been studied by [3, 18, 22, 23, 24, 25, 26]. Lions-Peetre [18] showed that, if u ∈ L2 (0,T;H0 ), u(m) ∈ L2 (0,T;H) then u(i) ∈ L2 (0,T;[H,H0 ]i/m ), i = 1,2,...,m − 1, where H0 , H are Hilbert spaces, H0 is continuously and densely embedded in H and [H0 ,H]θ are interpolation spaces between H0 , H for 0 ≤ θ ≤ 1. In [22, 23, 24, 25, 26] the similar questions were investigated for anisotropic Sobolev spaces W pl (Ω;H0 ,H), Ω ⊂ Rn . Moreover, boundary value problems for differential-operator equations have been studied in detail by [16, 27, 30, 32]. The solvability and the spectrum of boundary value problems for elliptic differential-operator equations have also been refined by [1, 2, 4, 8, 10, 11, 13, 22, 23, 24, 25, 26]. A comprehensive introduction to the differential-operator equations and historical references may be found in [16, 32]. In these works Hilbert-valued function spaces essentially have been considered. In the present paper, are to be introduced a Banach-valued function spaces W pl (Ω;E0 ,E), where l = (l1 ,l2 ,...,ln ) and E0 , E are Banach spaces such that E0 is continuously and densely embedded in E. The properties of continuity and compactness of embedding operators in these spaces are obtained. We prove that the generalized derivative operator Dα is continuous from these Banach-valued Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:4 (2005) 329–345 DOI: 10.1155/JIA.2005.329
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Embedding operators and maximal regular equations
Sobolev spaces to Eα -valued L p spaces, where Eα are interpolation spaces between E0 and E depending on the order of differentiations Dα . By applying these results, the maximal L p -regularity of certain class of anisotropic partial differential-operator equations are derived. Let α1 ,α2 ,...,αn be nonnegative integer numbers and ∂α
Dα = D1α1 D2α2 · · · Dnαn =
∂x1α1 ∂x2α2 · · · ∂xnαn
.
(1.1)
Under certain assumptions to be stated later, we prove that the operators u → Dα u are bounded from space W pl (Ω;E(A),E) to space Lq
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