Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation / Edition 1

by William P. Ziemer

Overview

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 functions), 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.

Product Details

ISBN-13:	9780387970172
Publisher:	Springer New York
Publication date:	09/18/1989
Series:	Graduate Texts in Mathematics , #120
Edition description:	1989
Pages:	308
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

1 Preliminaries.- 1.1 Notation.- 1.2 Measures on Rn.- 1.3 Covering Theorems.- 1.4 Hausdorff Measure.- 1.5 Lp-Spaces.- 1.6 Regularization.- 1.7 Distributions.- 1.8 Lorentz Spaces.- Exercises.- Historical Notes.- 2 Sobolev Spaces and Their Basic Properties.- 2.1 Weak Derivatives.- 2.2 Change of Variables for Sobolev Functions.- 2.3 Approximation of Sobolev Functions by Smooth Functions.- 2.4 Sobolev Inequalities.- 2.5 The Rellich-Kondrachov Compactness Theorem.- 2.6 Bessel Potentials and Capacity.- 2.7 The Best Constant in the Sobolev Inequality.- 2.8 Alternate Proofs of the Fundamental Inequalities.- 2.9 Limiting Cases of the Sobolev Inequality.- 2.10 Lorentz Spaces, A Slight Improvement.- Exercises.- Historical Notes.- 3 Pointwise Behavior of Sobolev Functions.- 3.1 Limits of Integral Averages of Sobolev Functions.- 3.2 Densities of Measures.- 3.3 Lebesgue Points for Sobolev Functions.- 3.4 LP-Derivatives for Sobolev Functions.- 3.5 Properties of Lp-Derivatives.- 3.6 An Lp-Version of the Whitney Extension Theorem.- 3.7 An Observation on Differentiation.- 3.8 Rademacher’s Theorem in the Lp-Context.- 3.9 The Implications of Pointwise Differentiability.- 3.10 A Lusin-Type Approximation for Sobolev Functions.- 3.11 The Main Approximation.- Exercises.- Historical Notes.- 4 Poincaré Inequalities—A Unified Approach.- 4.1 Inequalities in a General Setting.- 4.2 Applications to Sobolev Spaces.- 4.3 The Dual of WM,p(Ω).- 4.4 Some Measures in (W0M,p(?))*.- 4.5 Poincaré Inequalities.- 4.6 Another Version of Poincaré’s Inequality.- 4.7 More Measures in (WM,p(Ω))*.- 4.8 Other Inequalities Involving Measures in (WM,p)*.- 4.9 The Case p= 1.- Exercises.- Historical Notes.- 5 Functions of Bounded Variation.- 5.1 Definitions.- 5.2 Elementary Properties of BV Functions.- 5.3Regularization of BV Functions.- 5.4 Sets of Finite Perimeter.- 5.5 The Generalized Exterior Normal.- 5.6 Tangential Properties of the Reduced Boundary and the Measure-Theoretic Normal.- 5.7 Rectifiability of the Reduced Boundary.- 5.8 The Gauss-Green Theorem.- 5.9 Pointwise Behavior of BV Functions.- 5.10 The Trace of a BV Function.- 5.11 Sobolev-Type Inequalities for BV Functions.- 5.12 Inequalities Involving Capacity.- 5.13 Generalizations to the Case p> 1.- 5.14 Trace Defined in Terms of Integral Averages.- Exercises.- Historical Notes.- List of Symbols.

Editorial Reviews

Demanding text for well-prepared graduate students of mathematics; derives from a course taught 1987-88 at Indiana University (Bloomington) and treats the fairly esoteric topic indicated by the title. Five chapters, with exercises and historical notes. Nicely composed and produced. (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com)

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