Open Access

Embedding operators and maximal regular differential-operator equations in Banach-valued function spaces

Journal of Inequalities and Applications20052005:826085

DOI: 10.1155/JIA.2005.329

Received: 11 November 2003

Published: 31 August 2005

Abstract

This study focuses on anisotropic Sobolev type spaces associated with Banach spaces https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq2_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq4_HTML.gif . In particular, the most regular class of interpolation spaces https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq5_HTML.gif between https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq7_HTML.gif , depending of https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq8_HTML.gif and order of spaces are found that mixed derivatives https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq9_HTML.gif belong with values; the boundedness and compactness of differential operators https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq10_HTML.gif from this space to https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq11_HTML.gif -valued https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq12_HTML.gif spaces are proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.329/MediaObjects/13660_2003_Article_1554_IEq13_HTML.gif regularity uniformly with respect to these parameters.

Authors’ Affiliations

(1)
Department of Electric-Electronics Engineering, Faculty of Engineering, Istanbul University

Copyright

© Shakhmurov. 2005