Elliptic Equations: An Introductory Course (Birkhäuser Advanced Texts Basler Lehrbücher)

Preface
Preface to the Second Edition
Contents
Part I Basic Techniques
	1 Hilbert Space Techniques
		1.1 The Projection on a Closed Convex Set
		1.2 The Riesz Representation Theorem
		1.3 The Lax–Milgram Theorem
		1.4 Orthonormal Sets
		1.5 Convergence Techniques
		Exercises
	2 A Survey of Essential Analysis
		2.1 Lp-Techniques
		2.2 Introduction to Distributions
		2.3 Sobolev Spaces
		Exercises
	3 Weak Formulation of Elliptic Problems
		3.1 Motivation
		3.2 The Weak Formulation
		Exercises
	4 Elliptic Problems in Divergence Form
		4.1 Weak Formulation
		4.2 The Weak Maximum Principle
		4.3 Inhomogeneous Problems
		Exercises
	5 Singular Perturbation Problems
		5.1 A Prototype of a Singular Perturbation Problem
		5.2 Anisotropic Singular Perturbation Problems
		Exercises
	6 Asymptotic Analysis for Problems in Large Cylinders
		6.1 A Model Problem
		6.2 Another Type of Convergence
		6.3 The General Case
		6.4 Global Convergence
		6.5 An Application
		Exercises
	7 Periodic Problems
		7.1 A General Theory
		7.2 Some Additional Remarks
		Exercises
	8 Homogenization
		8.1 More on Periodic Functions
		8.2 Homogenization of Elliptic Equations
			The One-Dimensional Case
			The n-Dimensional Case
		Exercises
	9 Eigenvalues
		9.1 The One-Dimensional Case
		9.2 The Higher Dimensional Case
		9.3 An Application
		Exercises
	10 Numerical Computations
		10.1 The Finite Difference Method
		10.2 The Finite Element Method
		Exercises
Part II More Advanced Theory
	11 Nonlinear Problems
		11.1 Monotone Methods
		11.2 Quasilinear Equations
		11.3 Nonlocal Problems
		11.4 Variational Inequalities
		Exercises
	12 L∞-Estimates
		12.1 Some Simple Cases
		12.2 A More Involved Estimate
		12.3 The Sobolev–Gagliardo–Nirenberg Inequality
		12.4 The Maximum Principle on Small Domains
		Exercises
	13 Linear Elliptic Systems
		13.1 The General Framework
		13.2 Some Examples
		Exercises
	14 The Stationary Navier–Stokes System
		14.1 Introduction and Notation
		14.2 Existence and Uniqueness Result
		14.3 The Poiseuille Flow
		Exercises
	15 Some More Spaces
		15.1 Motivation
		15.2 Essential Features of the Sobolev Spaces Wk,p
		15.3 An Application
		Exercises
	16 Regularity Theory
		16.1 Introduction
		16.2 The Translation Method
		16.3 Regularity of Functions in Sobolev Spaces
		16.4 The Bootstrap Technique
		Exercises
	17 p-Laplace-Type Equations
		17.1 A Minimization Technique
		17.2 A Weak Maximum Principle and Its Consequences
		17.3 A Generalization of the Lax–Milgram Theorem
		17.4 An Existence Result in a Nonmonotone Case
		17.5 Uniqueness Issues
		17.6 A Nonlocal Case
		Exercises
	18 The Strong Maximum Principle
		18.1 A First Version of the Maximum Principle
		18.2 The Hopf Maximum Principle
		18.3 Application: The Moving Plane Technique
		Exercises
	19 Problems in the Whole Space
		19.1 The Harmonic Functions: Liouville Theorem
		19.2 The Schrödinger Equation
		Exercises
	20 Large Solutions
		20.1 Preliminary Results
		20.2 Existence of Large Solutions
		Exercises
	21 Mountain Pass Techniques
		21.1 The Ekeland Variational Principle
		21.2 The Mountain Pass Theorem
		21.3 The Pohozaev Identity
		Exercises
A Fixed Point Theorems
	A.1 The Brouwer Fixed Point Theorem
	A.2 The Schauder Fixed Point Theorem
	Exercises
B The Equation div v = f
	B.1 Introduction
	B.2 A Constructive Method
References
Index