Ebook: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation

The term "weakly differentiable functions" in the title refers to those inte­ n grable functions defined on an open subset of R whose partial derivatives in the sense of distributions are either LP functions or (signed) measures with finite total variation. The former class of functions comprises what is now known as Sobolev spaces, though its origin, traceable to the early 1900s, predates the contributions by Sobolev. Both classes of functions, Sobolev spaces and the space of functions of bounded variation (BV func­ tions), have undergone considerable development during the past 20 years. From this development a rather complete theory has emerged and thus has provided the main impetus for the writing of this book. Since these classes of functions play a significant role in many fields, such as approximation theory, calculus of variations, partial differential equations, and non-linear potential theory, it is hoped that this monograph will be of assistance to a wide range of graduate students and researchers in these and perhaps other related areas. Some of the material in Chapters 1-4 has been presented in a graduate course at Indiana University during the 1987-88 academic year, and I am indebted to the students and colleagues in attendance for their helpful comments and suggestions.

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The major thrust of this book is the analysis of pointwise behavior of Sobolev functions of integer order and BV functions (functions whose partial derivatives are measures with finite total variation). The development of Sobolev functions includes an analysis of their continuity properties in terms of Lebesgue points, approximate continuity, and fine continuity as well as a discussion of their higher order regularity properties in terms of Lp-derivatives. This provides the foundation for further results such as a strong approximation theorem and the comparison of Lp and distributional derivatives. Also included is a treatment of Sobolev-Poincaré type inequalities which unifies virtually all inequalities of this type. Although the techniques required for the discussion of BV functions are completely different from those required for Sobolev functions, there are similarities between their developments such as a unifying treatment of Poincaré-type inequalities for BV functions. This book is intended for graduate students and researchers whose interests may include aspects of approximation theory, the calculus of variations, partial differential equations, potential theory and related areas. The only prerequisite is a standard graduate course in real analysis since almost all of the material is accessible through real variable techniques.