Weakly Differentiable Functions

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 Differentiability. - 3. 10 A Lusin-Type Approximation for Sobolev Functions. - 3. 11 The Main Approximation. - Exercises. - Historical Notes. - 4 Poincaré InequalitiesA Unified Approach. - 4. 1 Inequalities in a General Setting. - 4. 2 Applications to Sobolev Spaces. - 4. 3 The Dual of WMp(?). - 4. 4 Some Measures in (W0Mp(?))*. - 4. 5 Poincaré Inequalities. - 4. 6 Another Version of Poincaré's Inequality. - 4. 7 More Measures in (WMp(?))*. - 4. 8 Other Inequalities Involving Measures in (WMp)*. - 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. 3Regularization 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. Language: English