Book Details :
Title : Lebesgue and Sobolev Spaces with Variable Exponents (Lecture Notes in Mathematics)
Author : Lars Diening, Petteri Harjulehto, Peter Hästö, Michael Ruzicka
Paperback : 518 pages
Publisher : Springer; 1st Edition. edition (April 12, 2011)
Language : English
ISBN-10 : 364218362X
ISBN-13 : 978-3642183621
Size : 3.70 MB.
Type : Pdf.
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Book Description :

The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.

In the past few years the subject of variable exponent spaces has undergone a vast development. Nevertheless, the standard reference is still the article by Kov´ aˇ cik and R´ akosn´ ık from 1991. This paper covers only basic properties, such as reflexivity, separability, duality and first results concerning embeddings and density of smooth functions. In particular, the boundedness of the maximal operator, proved by Diening in 2002, and its consequences are missing.

Naturally, progress on more advanced properties is scattered in a large number of articles. The need to introduce students and colleagues to the main results led around 2005 to some short survey articles. Moreover, Diening gave lectures at the University of Freiburg in 2005 and R˚ uˇ ziˇ cka gave a course in 2006 at the Spring School NAFSA 8 in Prague. The usefulness of a more comprehensive treatment was clear, and so we decided in the summer of 2006 to write a book containing both basic and advanced properties, with improved assumptions. Two further lecture courses were given by H¨ ast¨ o based on our material in progress (2008 in Oulu and 2009 at the Spring School in Paseky);
another summary is Diening’s 2007 habilitation thesis.

It has been our goal to make the book accessible to graduate students as well as a valuable resource for researchers.We present the basic and advanced theory of function spaces with variable exponents and applications to partial differential equations. Not only do we summarize much of the existing literature but we also present new results of our most recent research, including unifying approaches generated while writing the book.