Weakly differentiable functions: Sobolev spaces and functions of bounded variation

by Ziemer, William P.

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Title Statements

Full title :Weakly differentiable functions : Sobolev spaces and functions of bounded variation / William P. Ziemer.

Summary

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.

Edition Statements

Publication Date:©1989

Publication Statements

Publisher(s):Springer Science+Business Media
Place of Publication:New York
Language:eng
Publication Date:©1989

Notes

Bibliographic Level Mode of Issuance: Monograph.

The Physical Object

Description:1 online resource (XVI, 308 pages)

Edition Identifiers

ISBN:1461210151, 9781461210153
DOI:10.1007/978-1-4612-1015-3.
oclc:on1120933522
LC Classification:QA825
Dewey Decimal:515.96

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 Differentiabilit…