دانلود کتاب Beyond Sobolev and Besov: Regularity of Solutions of PDEs and Their Traces in Function Spaces (Lecture Notes in Mathematics)

فهرست مطالب :

Preface
Contents
Quick Guide for Function Spaces
Spaces of Continuous Functions
Lebesgue Spaces
Sobolev Spaces
Besov Spaces (B-Spaces)
Triebel-Lizorkin Spaces (F-Spaces)
Kondratiev Spaces (Weighted Sobolev Spaces)
Smoothness Morrey Spaces
Further Function Spaces
Sequence Spaces
1 Introduction
2 Function Spaces and General Concepts
2.1 Preliminaries
2.1.1 Notation
2.1.2 Lipschitz Domains and Their Boundaries
2.2 Classical Function Spaces
2.2.1 Sobolev and Fractional Sobolev Spaces
2.2.2 Banach-Space Valued Hölder, Lebesgue, and Sobolev Spaces
2.3 Besov and Triebel-Lizorkin Spaces
2.3.1 Spaces of Type Bsp,q, Fsp,q
2.3.2 Wavelet Characterization for Bsp,q
2.3.3 Spaces of Type Bsp,q
2.3.4 Smooth and Non-smooth Atomic Decompositions for Bsp,q
2.3.5 Besov Spaces Bsp,q on Boundaries of Lipschitz Domains
2.3.6 Properties of Besov Spaces Bsp,q on Lipschitz Domains
2.3.7 Spaces of Type Fsp,q
2.4 Kondratiev Spaces
2.4.1 Definition and Basic Properties
2.4.2 Kondratiev Spaces on Domains of Polyhedral Type
2.4.3 Embeddings and Pointwise Multiplication
2.4.4 Relations Between Kondratiev and Other Function Spaces
2.4.5 Fractional Kondratiev Spaces
2.5 Smoothness Morrey Spaces and Their Generalizations
2.5.1 Spaces of Type AsMpφ,q
2.5.2 Properties
2.5.3 Spaces of Type Bs,τp,q
2.6 Function Spaces on Manifolds
2.6.1 Sobolev Spaces on Manifolds
2.6.2 Besov and Triebel-Lizorkin Spaces on Manifolds
Part I Besov and Fractional Sobolev Regularity of PDEs
3 Theory and Background Material for PDEs
3.1 Definitions and Classifications
3.2 Elliptic Boundary Value Problems
3.2.1 Bilinear Forms and Lax-Milgram Theorem
3.2.2 Existence of Weak Solutions
3.2.3 Fractional Sobolev Regularity on Smooth, Convex and Lipschitz Domains
3.3 Parabolic Boundary Value Problems
3.3.1 Existence of Weak Solutions
3.4 Hyperbolic Boundary Value Problems
3.5 Operator Pencils
4 Regularity Theory for Elliptic PDEs
4.1 The Fundamental Semilinear Elliptic Problem
4.2 Regularity of Semilinear Elliptic PDEs in Kondratiev Spaces
4.2.1 Composition Operators in Kondratiev Spaces
4.2.2 Fixed Points of Nonlinear Operators in Banach Spaces
Admissibility of Kondratiev Spaces
Complete Continuity of LN
4.2.3 Existence of Solutions in Kondratiev Spaces
4.2.4 Uniqueness of Solutions
4.3 Regularity of Semilinear Elliptic PDEs in Besov Spaces
5 Regularity Theory for Parabolic PDEs
5.1 The Fundamental Parabolic Problems
5.1.1 Problems I and II: Parabolic PDEs on Domains of Polyhedral Type
5.1.2 Problem III: Linear Parabolic PDEs of Second Order on Generalized Wedges
5.1.3 Problem IV: Linear Parabolic PDEs of Second Order on General Lipschitz Domains
5.2 Regularity Results in Sobolev and Kondratiev Spaces
5.2.1 Regularity Results in Sobolev Spaces for Problem I
5.2.2 Regularity Results in Kondratiev Spaces for Problem I
5.2.3 Regularity Results in Sobolev and Kondratiev Spaces for Problem II
5.2.4 Regularity Results in Kondratiev Spaces for Problem III
5.2.5 Regularity Results in Kondratiev Spaces for Problem IV
5.3 Regularity Results in Fractional Sobolev Spaces
5.3.1 Fractional Sobolev Regularity of Problem III
5.3.2 Fractional Sobolev Regularity of the Heat Equation on Specific Smooth Cones
Eigenvalues of the Laplacian on a Spherical Cap
5.4 Regularity Results in Besov Spaces
5.4.1 Besov Regularity of Problem I
5.4.2 Besov Regularity of Problem II
5.4.3 Hölder-Besov Regularity of Problem I
5.4.4 Besov Regularity of Problem IV
6 Regularity Theory for Hyperbolic PDEs
6.1 The Fundamental Hyperbolic Problem
6.2 Regularity Results in Kondratiev Spaces
6.3 Regularity Results in Besov Spaces
7 Applications to Adaptive Approximation Schemes
7.1 N-Term Approximation and Adaptive Wavelet Algorithms
7.2 Finite Element Approximation
7.3 Time-Marching Adaptive Algorithms
Part II Traces in Function Spaces
8 Traces on Lipschitz Domains
8.1 Boundedness of the Trace Operator
8.2 Construction of an Extension Operator
8.2.1 Extension of Atoms
8.2.2 Boundedness of the Extension Operator
8.3 Trace Results for Spaces Bsp,q(Ω), Fsp,q(Ω)
8.3.1 The General Case 0 8.3.2 The Limiting Case s=0
8.4 Application: Pointwise Multipliers in Function Spaces
9 Traces of Generalized Smoothness Morrey Spaces on Domains
9.1 A Lift Operator
9.2 Construction of an Extension Operator
9.3 Trace Results
9.3.1 Trace Theorem for Spaces AsMpφ,q(Ω)
9.3.2 Trace Theorem for Spaces Bs,τp,q(Ω)
9.4 Application: A Priori Estimates for Solutions of Elliptic Boundary Value Problems
10 Traces on Riemannian Manifolds
10.1 Fermi Coordinates on Submanifolds
10.2 Trace Results
10.2.1 Trace Theorem for Spaces Hsp(M)
10.2.2 Trace Theorem for Spaces Fsp,q(M), Bsp,q(M)
10.3 Application: Spaces with Symmetries
Bibliography
Index