弱可微函數(shù)

內(nèi)容概要

The term "weakly differentiable functions" in the title refers to those inte grable functions defined on an open subset of Rn whose partial derivatives in the sense of distributions are either Lr 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.

書(shū)籍目錄

Preface 1 Preliminaries  1.1 Notation   Inner product of vectors   Support of a function   Boundary of a set   Distance from a point to a set   Characteristic function of a set   Multi-indices   Partial derivative operators   Function spaces--continuous， HSlder continuous，   HSlder continuous derivatives  1.2 Measures on Rn   Lebesgue measurable sets   Lebesgue measurability of Borel sets   Suslin sets  1.3 Covering Theorems   Hausdorff maximal principle   General covering theorem   Vitali covering theorem   Covering lemma, with n-balls whose radii vary in Lips hitzian way  Besicovitch  covering lemma  Besicovitch differentiation theorem 1.4  Hausdorff Measure  Equivalen e of Hausdorff and Lebesgue measures  Hausdorff dimension 1.5  LP-Spaces  Integration of a function via its distribution  function  Young's inequality  Holder's and Jensen's inequality 1.6  Regularization  LP-spaces and regularization 1.7  Distributions  Functions and measures, as distributions  Positive distributions  Distributions determined by their lo al behavior  Convolution of distributions  Differentiation of distributions 1.8 Lorentz Spaces  Non-in reasing rearrangement of a fun tion  Elementary properties of rearranged functions  Lorentz spaces  O'Neil's inequality, for rearranged functions  Equivalence of LP-norm and (p,p)-norm  Hardy's inequality  Inclusion relations of Lorentz spaces  Exercises  Historical Notes  Sobolev Spaces and Their Basic  Properties 2.1 Weak Derivatives  Sobolev spaces  Absolute  continuity on lines  LP-norm of difference quotients  Truncation of Sobolev functions  Composition of Sobolev functions 2.2  Change of Variables for Sobolev functions  Radema her's theorem  Bi-Lipschitzian  change of variables 2.3  Approximation of Sobolev functions by Smooth functions  Partition of unity  Smooth functions are dense in Wk'p 2.4 Sobolev Inequalities  Sobolev's inequality 2.5 The Relli h-Kondrachov  compactness Theorem  Extension domains 2.6  Bessel Potentials and  apacity  Riesz and Bessel kernels  Bessel potentials  Bessel  apacity  Basic  properties of Bessel  apacity  Capa itability of Suslin sets  Minimax theorem and alternate formulation of  Bessel  apacity  ……3 Pointwise Behavior of Sobolev Functions4 Poincare Inequalities5 Functions of Bounded VariationBibliographyList of SymbolsIndex

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