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A First Course in Sobolev Spaces Second Edition by Giovanni Leoni ISBN 1470429217 9781470429218

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Author(s): Giovanni Leoni
ISBN(s): 9781470429218, 1470429217
File Details: PDF, 5.13 MB
Year: 2017
Language: english
SKU: EB-57412136 Category: Tag:

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Product details:

ISBN 10: 1470429217 

ISBN 13: 9781470429218

Author: Giovanni Leoni

This book is about differentiation of functions. It is divided into two parts, which can be used as different textbooks, one for an advanced undergraduate course in functions of one variable and one for a graduate course on Sobolev functions. The first part develops the theory of monotone, absolutely continuous, and bounded variation functions of one variable and their relationship with Lebesgue-Stieltjes measures and Sobolev functions. It also studies decreasing rearrangement and curves. The second edition includes a chapter on functions mapping time into Banach spaces.The second part of the book studies functions of several variables. It begins with an overview of classical results such as Rademacher’s and Stepanoff’s differentiability theorems, Whitney’s extension theorem, Brouwer’s fixed point theorem, and the divergence theorem for Lipschitz domains. It then moves to distributions, Fourier transforms and tempered distributions. The remaining chapters are a treatise on Sobolev functions. The second edition focuses more on higher order derivatives and it includes the interpolation theorems of Gagliardo and Nirenberg. It studies embedding theorems, extension domains, chain rule, superposition, Poincare’s inequalities and traces. A major change compared to the first edition is the chapter on Besov spaces, which are now treated using interpolation theory.

Table of contents:

PART 1. FUNCTIONS OF ONE VARIABLE

Chapter 1. Monotone Functions
§1.1. Continuity
§1.2. Differentiability

Chapter 2. Functions of Bounded Pointwise Variation
§2.1. Pointwise Variation
§2.2. Continuity
§2.3. Differentiability
§2.4. Monotone Versus BPV
§2.5. The Space BPV(I; Y)
§2.6. Composition in BPV(I; Y)
§2.7. Banach Indicatrix

Chapter 3. Absolutely Continuous Functions
§3.1. AC(I; Y) Versus BPV(I; Y)
§3.2. The Fundamental Theorem of Calculus
§3.3. Lusin (N) Property
§3.4. Superposition in AC(I; Y)
§3.5. Chain Rule
§3.6. Change of Variables
§3.7. Singular Functions

Chapter 4. Decreasing Rearrangement
§4.1. Definition and First Properties
§4.2. Function Spaces and Decreasing Rearrangement

Chapter 5. Curves
§5.1. Rectifiable Curves
§5.2. Arclength
§5.3. Length Distance
§5.4. Curves and Hausdorff Measure
§5.5. Jordan’s Curve Theorem

Chapter 6. Lebesgue–Stieltjes Measures
§6.1. Measures Versus Increasing Functions
§6.2. Vector-Valued Measures Versus BPV(I; Y)
§6.3. Decomposition of Measures

Chapter 7. Functions of Bounded Variation and Sobolev Functions
§7.1. BV(Ω) Versus BPV(Ω)
§7.2. Sobolev Functions Versus Absolutely Continuous Functions
§7.3. Interpolation Inequalities

Chapter 8. The Infinite-Dimensional Case
§8.1. The Bochner Integral
§8.2. Lp Spaces on Banach Spaces
§8.3. Functions of Bounded Pointwise Variation
§8.4. Absolutely Continuous Functions
§8.5. Sobolev Functions

PART 2. FUNCTIONS OF SEVERAL VARIABLES

Chapter 9. Change of Variables and the Divergence Theorem
§9.1. Directional Derivatives and Differentiability
§9.2. Lipschitz Continuous Functions
§9.3. The Area Formula: The C¹ Case
§9.4. The Area Formula: The Differentiable Case
§9.5. The Divergence Theorem

Chapter 10. Distributions
§10.1. The Spaces 𝒟ₖ(Ω), 𝒟(Ω), and 𝒟′(Ω)
§10.2. Order of a Distribution
§10.3. Derivatives of Distributions and Distributions as Derivatives
§10.4. Rapidly Decreasing Functions and Tempered Distributions
§10.5. Convolutions
§10.6. Convolution of Distributions
§10.7. Fourier Transforms
§10.8. Littlewood–Paley Decomposition

Chapter 11. Sobolev Spaces
§11.1. Definition and Main Properties
§11.2. Density of Smooth Functions
§11.3. Absolute Continuity on Lines
§11.4. Duals and Weak Convergence
§11.5. A Characterization of W¹,ᵖ(Ω)

Chapter 12. Sobolev Spaces: Embeddings
§12.1. Embeddings: mp < N
§12.2. Embeddings: mp = N
§12.3. Embeddings: mp > N
§12.4. Superposition
§12.5. Interpolation Inequalities in ℝᴺ

Chapter 13. Sobolev Spaces: Further Properties
§13.1. Extension Domains
§13.2. Poincaré Inequalities
§13.3. Interpolation Inequalities in Domains

Chapter 14. Functions of Bounded Variation
§14.1. Definition and Main Properties
§14.2. Approximation by Smooth Functions
§14.3. Bounded Pointwise Variation on Lines
§14.4. Coarea Formula for BV Functions
§14.5. Embeddings and Isoperimetric Inequalities
§14.6. Density of Smooth Sets
§14.7. A Characterization of BV(Ω)

Chapter 15. Sobolev Spaces: Symmetrization
§15.1. Symmetrization in Lp Spaces
§15.2. Lorentz Spaces
§15.3. Symmetrization of W¹,ᵖ and BV Functions
§15.4. Sobolev Embeddings Revisited

Chapter 16. Interpolation of Banach Spaces
§16.1. Interpolation: K-Method
§16.2. Interpolation: J-Method
§16.3. Duality
§16.4. Lorentz Spaces as Interpolation Spaces

Chapter 17. Besov Spaces
§17.1. Besov Spaces Bˢ,ᵖ_q
§17.2. Some Equivalent Seminorms
§17.3. Besov Spaces as Interpolation Spaces
§17.4. Sobolev Embeddings
§17.5. The Limit of Bˢ,ᵖ_q as s → 0⁺ and s → m⁻
§17.6. Besov Spaces and Derivatives
§17.7. Yet Another Equivalent Norm
§17.8. And More Embeddings

Chapter 18. Sobolev Spaces: Traces
§18.1. The Trace Operator
§18.2. Traces of Functions in W¹,¹(Ω)
§18.3. Traces of Functions in BV(Ω)
§18.4. Traces of Functions in W¹,ᵖ(Ω), p > 1
§18.5. Traces of Functions in Wᵐ,¹(Ω)
§18.6. Traces of Functions in Wᵐ,ᵖ(Ω), p > 1
§18.7. Besov Spaces and Weighted Sobolev Spaces

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