Functional Analysis, Sobolev Spaces, and Calculus of Variations (UNITEXT, 157)

Preface
Contents
1 Motivation and Perspective
	1.1 Some Finite-Dimensional Examples
	1.2 Basic Examples
	1.3 More Advanced Examples
		1.3.1 Transit Problems
		1.3.2 Geodesics
		1.3.3 Dirichlet\'s Principle
		1.3.4 Minimal Surfaces
		1.3.5 Isoperimetric Problems
		1.3.6 Hamiltonian Mechanics
	1.4 The Model Problem, and Some Variants
	1.5 The Fundamental Issues for a Variational Problem
	1.6 Additional Reasons to Care About Classes of Functions
	1.7 Finite Versus Infinite Dimension
	1.8 Brief Historical Background
	1.9 Exercises
Part I Basic Functional Analysis and Calculus of Variations
	2 A First Exposure to Functional Analysis
		2.1 Overview
		2.2 Metric, Normed and Banach Spaces
		2.3 Completion of Normed Spaces
		2.4 Lp-Spaces
		2.5 Weak Derivatives
		2.6 One-Dimensional Sobolev Spaces
			2.6.1 Basic Properties
			2.6.2 Weak Convergence
		2.7 The Dual Space
		2.8 Compactness and Weak Topologies
		2.9 Approximation
		2.10 Completion of Spaces of Smooth Functions with Respect to Integral Norms
		2.11 Hilbert Spaces
			2.11.1 Orthogonal Projection
			2.11.2 Orthogonality
			2.11.3 The Dual of a Hilbert Space
			2.11.4 Basic Calculus in a Hilbert Space
		2.12 Some Other Important Spaces of Functions
		2.13 Exercises
	3 Introduction to Convex Analysis: The Hahn-Banach and Lax-Milgram Theorems
		3.1 Overview
		3.2 The Lax-Milgram Lemma
		3.3 The Hahn-Banach Theorem: Analytic Form
		3.4 The Hahn-Banach Theorem: Geometric Form
		3.5 Some Applications
		3.6 Convex Functionals, and the Direct Method
		3.7 Convex Functionals, and the Indirect Method
		3.8 Stampacchia\'s Theorem: Variational Inequalities
		3.9 Exercises
	4 The Calculus of Variations for One-dimensional Problems
		4.1 Overview
		4.2 Convexity
		4.3 Weak Lower Semicontinuity for Integral Functionals
		4.4 An Existence Result
		4.5 Some Examples
			4.5.1 Existence Under Constraints
		4.6 Optimality Conditions
		4.7 Some Explicit Examples
		4.8 Non-existence
		4.9 Exercises
Part II Basic Operator Theory
	5 Continuous Operators
		5.1 Preliminaries
		5.2 The Banach-Steinhaus Principle
		5.3 The Open Mapping and Closed Graph Theorems
		5.4 Adjoint Operators
		5.5 Spectral Concepts
		5.6 Self-Adjoint Operators
		5.7 The Fourier Transform
		5.8 Exercises
	6 Compact Operators
		6.1 Preliminaries
		6.2 The Fredholm Alternative
		6.3 Spectral Analysis
		6.4 Spectral Decomposition of Compact, Self-Adjoint Operators
		6.5 Exercises
Part III Multidimensional Sobolev Spaces and Scalar Variational Problems
	7 Multidimensional Sobolev Spaces
		7.1 Overview
		7.2 Weak Derivatives and Sobolev Spaces
		7.3 Completion of Spaces of Smooth Functions of Several Variables with Respect to Integral Norms
		7.4 Some Important Examples
		7.5 Domains for Sobolev Spaces
		7.6 Traces of Sobolev Functions: The Space W1, p0(Ω)
		7.7 Poincaré\'s Inequality
		7.8 Weak and Strong Convergence
		7.9 Higher-Order Sobolev Spaces
		7.10 Exercises
	8 Scalar, Multidimensional Variational Problems
		8.1 Preliminaries
		8.2 Abstract, Quadratic Variational Problems
		8.3 Scalar, Multidimensional Variational Problems
		8.4 A Main Existence Theorem
		8.5 Optimality Conditions: Weak Solutions for PDEs
		8.6 Variational Problems in Action
		8.7 Some Examples
		8.8 Higher-Order Variational Principles
		8.9 Non-existence and Relaxation
		8.10 Exercises
	9 Finer Results in Sobolev Spaces and the Calculus of Variations
		9.1 Overview
		9.2 Variational Problems Under Integral Constraints
		9.3 Sobolev Inequalities
			9.3.1 The Case of Vanishing Boundary Data
				9.3.1.1 The Subcritical Case
				9.3.1.2 The Critical Case
				9.3.1.3 The Supercritical Case
			9.3.2 The General Case
			9.3.3 Higher-Order Sobolev Spaces
		9.4 Regularity of Domains, Extension, and Density
		9.5 An Existence Theorem Under More General Coercivity Conditions
		9.6 Critical Point Theory
		9.7 Regularity. Strong Solutions for PDEs
		9.8 Eigenvalues and Eigenfunctions
		9.9 Duality for Sobolev Spaces
		9.10 Exercises
A Hints and Solutions to Exercises
	A.1 Chapter 1
	A.2 Chapter 2
	A.3 Chapter 3
	A.4 Chapter 4
	A.5 Chapter 5
	A.6 Chapter 6
	A.7 Chapter 7
	A.8 Chapter 8
	A.9 Chapter 9
B So Much to Learn
	B.1 Variational Methods and Related Fields
		B.1.1 Some Additional Sources for the Calculus of Variations
		B.1.2 Introductory Courses
		B.1.3 Indirect Methods
		B.1.4 Convex and Non-smooth Analysis
		B.1.5 Lagrangian and Hamiltonian Formalism
		B.1.6 Variational Inequalities
		B.1.7 Non-existence and Young Measures
		B.1.8 Optimal Control
		B.1.9 -Convergence
		B.1.10 Other Areas
	B.2 Partial Differential Equations
		B.2.1 Non-linear PDEs
		B.2.2 Regularity for PDEs: Regularity of Ω Is Necessary
		B.2.3 Numerical Approximation
	B.3 Sobolev Spaces
		B.3.1 Spaces of Bounded Variation, and More General Spaces of Derivatives
	B.4 Functional Analysis
References
Index