Problems on Partial Differential Equations (Problem Books in Mathematics)

Preface
Introduction
	The Content of the Chapters
	About the Authors
Notation
Contents
1 Preliminaries
	1.1 Integration by Parts
		1.1.1 Theoretical Background
			Basic Formulas
			Surface Integrals
			A Quick Review of Differential Forms
		1.1.2 Worked-Out Problems
		1.1.3 Problems
	1.2 Convolutions
		1.2.1 Theoretical Background
		1.2.2 Worked-Out Problems
		1.2.3 Problems
	1.3 Bibliographical Remarks
2 Distributions, Sobolev Spaces and the Fourier Transform
	2.1 The Fourier Transform
		2.1.1 Theoretical Background
			The Schwartz Space
			The Fourier Transform
			Properties of the Fourier Transform
			Parseval's Identity and Plancherel's Theorem
		2.1.2 Worked-Out Problems
		2.1.3 Problems
	2.2 The Theory of Distributions
		2.2.1 Theoretical Background
			Distributional Derivatives
			Tempered Distributions
			Convolutions
			Fundamental Solutions
		2.2.2 Worked-Out Problems
		2.2.3 Problems
	2.3 Sobolev Spaces
		2.3.1 Theoretical Background
			Sobolev Spaces Wm,p
			Spaces Hs(Rn)
			Spaces Hs(Ω)
			Duality and the Spaces W-m,p'(Ω)
			Trace of a Sobolev Function
			Embedding Theorems
		2.3.2 Worked-Out Problems
		2.3.3 Problems
	2.4 Bibliographical Remarks
3 Common Methods
	3.1 Weak Convergence
		3.1.1 Theoretical Background
			Weak Convergence in Hilbert Spaces
			Dual Spaces
			Weak and Weak* Convergence in Banach Spaces
			Banach Space-Valued Functions
		3.1.2 Worked-Out Problems
		3.1.3 Problems
	3.2 The Separation of Variables Method
		3.2.1 Theoretical Background
			Hyperbolic Problems
			Comments on the Regularity of u Given by Formula (3.21)
			Hyperbolic Problems: A Non-homogeneous Case
			Parabolic Problems
		3.2.2 Worked-Out Problems
		3.2.3 Problems
			Hyperbolic Equations
			Parabolic Equations
	3.3 Galerkin Method
		3.3.1 Theoretical Background
		3.3.2 Worked-Out Problems
		3.3.3 Problems
	3.4 Bibliographical Remarks
4 Elliptic Equations
	4.1 Classical Theory of Harmonic Functions
		4.1.1 Theoretical Background
			Definitions
			Green's Functions
			Holomorphic Functions and Harmonic Functions
			Subharmonic Functions
			General Elliptic Operators
		4.1.2 Worked-Out Problems
			The Laplace Operator in Different Coordinate Systems
			Green's Functions
			Harmonic and Holomorphic Functions
		4.1.3 Problems
			Harmonic Functions: Basic Properties
			The Laplace Operator in Different Coordinate Systems
			Green's Functions
			Harmonic and Holomorphic Functions
			Subharmonic Functions
	4.2 Weak Solutions
		4.2.1 Theoretical Background
			Physical Motivation
			Weak Formulation
			Variational Approach on Hilbert Spaces
			Lax–Milgram's Lemma
			Regularity of Weak Solutions
		4.2.2 Worked-Out Problems
		4.2.3 Problems
	4.3 Bibliographical Remarks
5 Evolution Equations
	5.1 First-Order Equations and the Method of Characteristics
		5.1.1 Theoretical Background
			The Characteristic System
			Admissible Boundary Conditions
			Noncharacteristic Initial Conditions
			Local Solutions
		5.1.2 Worked-Out Problems
		5.1.3 Problems
	5.2 Hyperbolic Problems
		5.2.1 Theoretical Background
			Duhamel's Formula
		5.2.2 Worked-Out Problems
			The Wave Equation
			Duhamel's Formula
			Hyperbolic Systems of the First Order
			The Use of the Fourier Transform
			Energy Estimates and Uniqueness
		5.2.3 Problems
			The Wave Equation
			Hyperbolic Systems of the First Order
			Hyperbolic Equations of the Second Order, Considered More Generally
			The Use of the Fourier Transform
			Energy Estimates and Uniqueness
	5.3 Parabolic Equations
		5.3.1 Theoretical Background
			Heat Equation
			Linear Second-Order Parabolic Equations
			Weak Solutions
		5.3.2 Worked-Out Problems
		5.3.3 Problems
	5.4 Bibliographical Remarks
Bibliography