Journal of Inequalities and Applications
Volume 2005 (2005), Issue 4, Pages 329-345
doi:10.1155/JIA.2005.329
Embedding operators and maximal regular differential-operator equations in
Banach-valued function spaces
Department of Electric-Electronics Engineering, Faculty of Engineering, Istanbul University, Avcilar, Istanbul 34850, Turkey
Received 11 November 2003; Revised 27 December 2004
Copyright © 2005 Veli B. Shakhmurov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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 Lp spaces are proved. These results are
applied to partial differential-operator equations with parameters
to obtain conditions that guarantee the maximal Lp regularity uniformly with respect to these parameters.