- 弱可微函数(影印版)(英文版)
- - 作者：(美)W.P.齐默尔|责编:高蓉//李黎
  - 出版社：世界图书出版公司
  - ISBN：9787506210225
  - 出版日期：2001/01/01
  - 页数：308
- 售价：35.6

内容大纲
索伯列夫函数和有界变差函数均具有弱收敛甚至不连续性质。这类函数在逼近理论、变分学、微分方程、非线性位势理论等诸领域占有很重要的地位。
本书的讨论是建立在实分析的框架上，重点放在以实变函数方法为手段的实序空间的分析，讨论了上述两函数的点态特征。全书论述清晰、易于入门，是该方面较好的研究生教材。
作者介绍
目录
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, H?lder continuous, H?lder 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 Lipschitzian way
     Besicovitch covering lemma
     Besicovitch differentiation theorem
  1.4  Hausdorff Measure
    Equivalence 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 local behavior
    Convolution of distributions
    Differentiation of distributions
  1.8  Lorentz Spaces
    Non-increasing rearrangement of a function
    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
2  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
    Rademacher'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 Rellich-Kondrachov Compactness Theorem
    Extension domains
  2.6  Bessel Potentials and Capacity
    Riesz and Bessel kernels
    Bessel potentials
    Bessel capacity
    Basic properties of Bessel capacity
    Capacitability of Suslin sets
    Minimax theorem and alternate formulation of Bessel capacity
    Metric properties of Bessel capacity
  2.7  The Best Constant in the Soboley Inequality Co-area formula
    Sobolev's inequality and isoperimetric inequality
  2.8  Alternate Proofs of the Fundamental Inequalities
    Hardy-Littlewood-Wiener maximal theorem
    Sobolev's inequality for Riesz potentials
  2.9  Limiting Cases of the Sobolev Inequality
    The case kp = n by infinite series
    The best constant in the case kp = n
    An L∞-bound in the limiting case
  2.10  Lorentz Spaces, A Slight Improvement
    Young's inequality in the context of Lorentz spaces
    Sobolev's inequality in Lorentz spaces
    The limiting case
  Exercises
  Historical Notes
3  Pointwise Behavior of Sobolev Functions
  3.1  Limits of Integral Averages of Sobolev Functions
    Limiting values of integral averages except for capacity null set
  3.2  Densities of Measures
  3.3  Lebesgue Points for Sobolev Functions
    Existence of Lebesgue points except for capacity null set
    Approximate continuity
    Fine continuity everywhere except for capacity null set
  3.4  LP-Derivatives for Sobolev Functions
    Existence of Taylor expansions LP
  3.5  Properties of LP-Derivatives
    The spaces Tk, tk, Tk,p,tk,p
    The implication of a function being in Tk,p at all points of a closed set

  3.6  An LP-Version of the Whitney Extension Theorem
    Existence of a Coo function comparable to the distance function to a closed set
    The Whitney extension theorem for functions in Tk,p and tk,p
  3.7  An Observation on Differentiation
  3.8  Rademacher's Theorem in the IP-Context
    A function in Tk,p everywhere implies it is in tk,p almost everywhere
  3.9  The Implications of Pointwise Differentiability
    Comparison of LP-derivatives and distributional derivatives
    If u ∈ tk,p(x) for every x,and if the
    LP-derivatives are in Lp,then u ∈ Wk,p
  3.10  A Lusin-Type Approximation for Sobolev Functions
    Integral averages of Sobolev functions are uniformly close to their limits on the complement of sets of small capacity
    Existence of smooth functions that agree with Sobolev functions on the complement of sets of small capacity
  3.11  The Main Approximation
    Existence of smooth functions that agree with Sobolev functions on the complement of sets of small capacity and are close in norm
  Exercises
  Historical Notes
4  Poincaré Inequalities-A Unified Approach
  4.1  Inequalities in a General Setting
    An abstract version of the Poincaré inequality
  4.2  Applications to Sobolev Spaces
    An interpolation inequality
  4.3  The Dual of Wm,p(Ω)
    The representation of (Wom,P(Ω))*
  4.4  Some Measures in (Wom,P(Ω))*
    Poincaré inequalities derived from the abstract version by identifying Lebesgue and Hausdorff measure with elements in (Wom,P(Ω))*
    The trace of Sobolev functions on the boundary of Lipschitz domains
    Poincaré inequalities involving the trace of a Sobolev function
  4.5  Poincaré Inequalities
    Inequalities involving the capacity of the set on which a function vanishes
  4.6  Another Version of Poincaré's Inequality
    An inequality involving dependence on the set on which the function vanishes, not merely on its capacity
  4.7  More Measures in(Wom,P(Ω))*
    Sobolev's inequality for Riesz potentials involving measures other than Lebesgue measure
    Characterization of measures in (Wm,p(Rn))*
  4.8  Other Inequalities Involving Measures in (Wk,P)*
    Inequalities involving the restriction of Hausdorff measure to lower dimensional manifolds
  4.9  The Case p =1
    Inequalities involving the L1-norm of the gradient
  Exercises
  Historical Notes
5  Functions of Bounded Variation
  5.1  Definitions
    Definition of BV functions
    The total variation measure ||Du||
  5.2  Elementary Properties of BV Functions
    Lower semicontinuity of the total variation measure
    A condition ensuring continuity of the total variation measure
    Regularization of BV Functions
  5.3  Regularization does not increase the BV norm

    Approximation of BV functions by smooth functions
    Compactness in L1 of the unit ball in BV
  5.4  Sets of Finite Perimeter
    Definition of sets of finite perimeter
    The perimeter of domains with smooth boundaries
    Isoperimetric and relative isoperimetric inequality for sets of finite perimeter
  5.5  The Generalized Exterior Normal
    A preliminary version of the Gauss-Green theorem
    Density results at points of the reduced boundary
  5.6  Tangential Properties of the Reduced Boundary and the Measure-Theoretic Normal
    Blow-up at a point of the reduced boundary
    The measure-theoretic normal
    The reduced boundary is contained in the measure-theoretic boundary
    A lower bound for the density of ||DXE||
    Hausdorff measure restricted to the reduced boundary is bounded above by ||DXE||
  5.7  Rectifiability of the Reduced Boundary
    Countably (n-1)-rectifiable sets
    Countable (n-1)-rectifiability of the measure-theoretic boundary
  5.8  The Gauss-Green Theorem
    The equivalence of the restriction of Hausdorff measure to the measure-theoretic boundary and ||DXE||
    The Gauss-Green theorem for sets of finite perimeter
  5.9  Pointwise Behavior of BV Functions
    Upper and lower approximate limits
    The Boxing inequality
    The set of approximate jump discontinuities
  5.10  The Trace of a BV Function
    The bounded extension of BV functions
    Trace of a BV function defined in terms of the upper and lower approximate limits of theextended function
    The integrability of the trace over the measure-theoretic boundary
  5.11  Sobolev-Type Inequalities for BV Functions
    Inequalities involving elements in (BV(Ω))*
  5.12  Inequalities Involving Capacity
    Characterization of measure in(BV(Ω))*
    Poincaré inequality for BV functions
  5.13  Generalizations to the Case p>1
  5.14  Trace Defined in Terms of Integral Averages
  Exercises
  Historical Notes
Bibliography
List of Symbols
Index

内容大纲

作者介绍

目录

同类热销排行榜

推荐书目