Measure theory and fine properties of functions 1st Edition by Ronald Gariepy, Lawrence Evans 1482242389 9781482242386
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Product details:
ISBN 10: 1482242389
ISBN 13: 9781482242386
Author: Ronald F. Gariepy; Lawrence C. Evans
This book emphasizes the roles of Hausdorff measure and the capacity in characterizing the fine properties of sets and functions. The book covers theorems and differentiation in Rn , Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions and functions of bounded variation. This second edition includes countless improvements in notation, format, and clarity of exposition. Also new are several sections describing the p- theorem, weak compactness criteria in L1, and Young measure methods for weak convergence. In addition, the bibliography has been updated.
Table of contents:
Chapter 1 General Measure Theory
1.1 Measures and measurable functions
1.1.1 Measures
1.1.2 Systems of sets
1.1.3 Approximation by open and compact sets
1.1.4 Measurable functions
1.2 Lusin’s and Egoroff’s Theorems
1.3 Integrals and limit theorems
1.4 Product measures, Fubini’s Theorem, Lebesgue measure
1.5.2 Besicovitch’s Covering Theorem
1.6 Differentiation of Radon measures
1.6.1 Derivatives
1.6.2 Integration of derivatives; Lebesgue decomposition
1.7 Lebesgue points, approximate continuity
1.7.1 Differentiation Theorem
1.7.2 Approximate limits, approximate continuity
1.8 Riesz Representation Theorem
1.9 Weak convergence
1.9.1 Weak convergence of measures
1.9.2 Weak convergence of functions
1.9.3 Weak convergence in L1
1.9.4 Measures of oscillation
1.10 References and notes
Chapter 2 Hausdorff Measures
2.1 Definitions and elemen
tary properties DEFINITION 2.1.
2.2 Isodiametric inequality, Hn=Ln
2.3 Densities
2.4 Functions and Hausdorff measure
2.4.1 Hausdorff measure and Lipschitz mappings DEFINITION 2.4.
2.4.2 Graphs of Lipschitz functions
2.4.3 Integrals over balls
2.5 References and notes
Chapter 3 Area and Coarea Formulas
3.1 Lipschitz functions, Rademacher’s Theorem
3.1.1 Lipschitz continuous functions
3.1.2 Rademacher’s Theorem
3.2 Linear maps and Jacobians
3.2.1 Linear mappings
3.2.2 Jacobians
3.3 The area formula
3.3.1 Preliminaries
3.3.2 Proof of the area formula
3.3.3 Change of variables formula
3.3.4 Applications
3.4 The coarea formula
3.4.1 Preliminaries
3.4.2 Proof of the coarea formula
3.4.3 Change of variables formula
3.4.4 Applications
3.5 References and notes
Chapter 4 Sobolev Functions
4.1 Definitions and elementary properties
4.2 Approximation
4.2.1 Approximation by smooth functions
4.2.2 Product and chain rules
4.2.3 W1,∞ and Lipschitz continuous functions
4.3 Traces
4.4 Extensions
4.5 Sobolev inequalities
4.5.1 Gagliardo–Nirenberg–Sobolev inequality
4.5.2 Poincaré’s inequality on balls
4.5.3 Morrey’s inequality
4.6 Compactness
4.7 Capacity
4.7.1 Definitions and elementary properties DEFINITION 4.9.
4.7.2 Capacity and Hausdorff dimension
4.8 Quasicontinuity, precise representatives of Sobolev functions
4.9 Differentiability on lines
4.9.1 Sobolev functions of one variable
4.9.2 Differentiability on a.e. line
4.10 References and notes
Chapter 5 Functions of Bounded Variation, Sets of Finite Perimeter
5.1 Definitions, Structure Theorem
5.2 Approximation and compactness
5.2.1 Lower semicontinuity
5.2.2 Approximation by smooth functions
5.2.3 Compactness
5.3 Traces
5.4 Extensions
5.5 Coarea formula for BV functions
5.6 Isoperimetric inequalities
5.6.1 Sobolev’s and Poincaré’s inequalities for BV THEOREM 5.10 (Inequalities for BV functions).
5.6.2 Isoperimetric inequalities
5.6.3 Hn-1and Cap1
5.7 The reduced boundary
5.7.1 Estimates
5.7.2 Blow-up
5.7.3 Structure Theorem for sets of finite perimeter
5.8 Gauss–Green Theorem
5.9 Pointwise properties of BV functions
5.10 Essential variation on lines
5.10.1 BV functions of one variable
5.10.2 Essential variation on almost all lines
5.11 A criterion for finite perimeter
5.12 References and notes
Chapter 6 Differentiability, Approximation by C1 Functions
6.1 Lp differentiability, approximate differentiability
6.1.1 L1* differentiability for BV
6.1.2 Lp* differentiability a.e. for W1,p
6.1.3 Approximate differentiability
6.2 Differentiability a.e. for W1,p(p>n)
6.3 Convex functions
6.4 Second derivatives a.e. for convex functions
6.5 Whitney’s Extension Theorem
6.6 Approximation by C1 functions
6.6.1 Approximation of Lipschitz continuous functions
6.6.2 Approximation of BV functions
6.6.3 Approximation of Sobolev functions
6.7 References and notes
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