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This study focuses on nonlocal boundary value problems (BVP) for degenerate elliptic differentialoperator equations (DOE), that are defined in Banach-valued function spaces, where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients. Several conditions obtained, that guarantee the maximal L p regularity and Fredholmness. These results are also applied to nonlocal BVP for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties.  2004 Elsevier Inc. All rights reserved.

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This study focuses on anisotropic Sobolev type spaces associated with Banach spaces E 0 , 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 E 0 and E. In particular, the most regular class of interpolation spaces E α between E 0 , E, depending of α and order of spaces are found that mixed derivatives D α belong with values; the boundedness and compactness of differential operators D α from this space to E α -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.

This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banachvalued weighted L p spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties.