PrefaceIn the past few years the subject of variable exponent spaces has undergone a vast development. Nevertheless, the standard reference is still the article by Kováčik and Rákosník 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 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.Writing such a book would not have been possible without various sources of support. We thank our universities for their hospitality and the Academy of Finland and the DFG research unit "Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis" for financial support. We also wish to express our appreciation of our fellow researchers whose results are presented and ask for understanding for the lapses, omissions and misattributions that may have entered the text. Thanks are also in order to Springer Verlag for their cooperation and assistance in publishing the book.We thank our friends, colleagues and especially our families for their continuous support and patience during the preparation of this book. v vi PrefaceFinally, we hope that you find this book useful in your journey into the world of variable exponent Lebesgue and Sobolev spaces.

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Abstract. We develop a method to decompose functions with mean value zero that are defined on a (possibly unbounded) John domain into a countable sum of functions with mean value zero and support in cubes or balls. This method enables us to generalize results known for simple domains to the class of John domains and domains satisfying a certain chain condition. As applications we present the solvability of the divergence equation div u = f , the negative norm theorem, Korn's inequality, Poincaré's inequality and a localized version of the Fefferman-Stein inequality. We present the results for weighted Lebesgue spaces and Orlicz spaces.

This paper deals with a derivation (using a perturbation technique) of an approximation, due to Oberbeck8,9 and Boussinesq,1 to describe the thermal response of linearly viscous fluids that are mechanically incompressible but thermally compressible. The present approach uses a nondimensionalization suggested by Chandrasekhar2 and utilizing the ratio of two characteristic velocities as a measure of smallness, systematically derives the Oberbeck-Boussinesq approximation as a third-order perturbation. In the present approach, the material is subjected to the constraint that the volume change is determined solely by the temperature change in the body and uses a novel approach in deriving the thermodynamical restrictions. Consequently, it is free from the additional assumptions usually added on in earlier works in order to obtain the correct equations.