Front cover image for Sobolev Spaces

Sobolev Spaces

Sobolev Spaces presents an introduction to the theory of Sobolev Spaces and other related spaces of function, also to the imbedding characteristics of these spaces. This theory is widely used in pure and Applied Mathematics and in the Physical Sciences. This second edition of Adam's 'classic' reference text contains many additions and much modernizing and refining of material. The basic premise of the book remains unchanged: Sobolev Spaces is intended to provide a solid foundation in these spaces for graduate students and researchers alike. * Self-contained and acc
eBook, English, 2014
Elsevier Science, Oxford, 2014
1 online resource (321 pages).
9780080541297, 0080541291
Front Cover; SOBOLEV SPACES; Copyright Page; CONTENTS; Preface; List of Spaces and Norms; CHAPTER 1. PRELIMINARIES; Notation; Topological Vector Spaces; Normed Spaces; Spaces of Continuous Functions; The Lebesgue Measure in Rn; The Lebesgue Integral; Distributions and Weak Derivatives; CHAPTER 2. THE LEBESGUE SPACES Lp(); Definition and Basic Properties; Completeness of LP (); Approximation by Continuous Functions; Convolutions and Young's Theorem; Mollifiers and Approximation by Smooth Functions; Precompact Sets in LP (); Uniform Convexity; The Normed Dual of LP (); Mixed-Norm LP Spaces. The Marcinkiewicz Interpolation TheoremCHAPTER 3. THE SOBOLEV SPACES Wm, P (); Definitions and Basic Properties; Duality and the Spaces W -m, p' (); Approximation by Smooth Functions on ; Approximation by Smooth Functions on Rn; Approximation by Functions in C0 (); Coordinate Transformations; CHAPTER 4. THE SOBOLEV IMBEDDING THEOREM; Geometric Properties of Domains; Imbeddings by Potential Arguments; Imbeddings by Averaging; Imbeddings into Lipschitz Spaces; Sobolev's Inequality; Variations of Sobolev's Inequality; W m, p () as a Banach Algebra; Optimality of the Imbedding Theorem. Nonimbedding Theorems for Irregular DomainsImbedding Theorems for Domains with Cusps; Imbedding Inequalities Involving Weighted Norms; Proofs of Theorems 4.51-4.53; CHAPTER 5. INTERPOLATION, EXTENSION, AND APPROXIMATION THEOREMS; Interpolation on Order of Smoothness; Interpolation on Degree of Sumability; Interpolation Involving Compact Subdomains; Extension Theorems; An Approximation Theorem; Boundary Traces; CHAPTER 6. COMPACT IMBEDDINGS OF SOBOLEV SPACES; The Rellich-Kondrachov Theorem; Two Counterexamples; Unbounded Domains
Compact Imbeddings of Wom'p (). An Equivalent Norm for Wom'p ()Unbounded Domains m Decay at Infinity; Unbounded Domains
Compact Imbeddings of W m, p (); Hilbert-Schmidt Imbeddings; CHAPTER 7. FRACTIONAL ORDER SPACES; Introduction; The Bochner Integral; Intermediate Spaces and Interpolation-The Real Method; The Lorentz Spaces; Besov Spaces; Generalized Spaces of Hölder Continuous Functions; Characterization of Traces; Direct Characterizations of Besov Spaces; Other Scales of Intermediate Spaces; Wavelet Characterizations; CHAPTER 8. ORLICZ SPACES AND ORLICZ-SOBOLEV SPACES; Introduction; N-Functions; Orlicz Spaces. Duality in Orlicz SpacesSeparability and Compactness Theorems; A Limiting Case of the Sobolev Imbedding Theorem; Orlicz-Sobolev Spaces; Imbedding Theorems for Orlicz-Sobolev Spaces; References; Index