Author | Ziemer, William P. author |
---|---|
Title | Weakly Differentiable Functions [electronic resource] : Sobolev Spaces and Functions of Bounded Variation / by William P. Ziemer |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1989 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1015-3 |
Descript | XVI, 308 p. online resource |
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 Rademacherโs 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 Poincarรฉ InequalitiesโA Unified Approach -- 4.1 Inequalities in a General Setting -- 4.2 Applications to Sobolev Spaces -- 4.3 The Dual of WM,p(?) -- 4.4 Some Measures in (W0M,p(?))* -- 4.5 Poincarรฉ Inequalities -- 4.6 Another Version of Poincarรฉโs Inequality -- 4.7 More Measures in (WM,p(?))* -- 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 -- 5.3 Regularization 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