Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations (Fields Institute Monographs)

Preface
Contents
1 Preliminaries
	1.1 Functional Spaces and Their Properties
		1.1.1 Lp Spaces
		1.1.2 Sobolev Spaces
	1.2 Linear Elliptic Boundary Value Problems
	1.3 Nemytskii Operator
	1.4 Maximum Principles and Their Applications
		1.4.1 Classical Maximum Principles
	1.5 Uniform Estimates and Boundedness of the Solutions of Semilinear Elliptic Equations
	1.6 The Sweeping Principle and the Moving Plane Method in a Bounded Domain
	1.7 The Sliding and the Moving Plane Method in General Domains
	1.8 Variational Solutions of Elliptic Equations
	1.9 Elliptic Regularity for the Neumann Problem for the Laplace Operator on an Infinite Edge
2 Trajectory Dynamical Systems and Their Attractors
	2.1 Kolmogorov epsilon-Entropy and Its Asymptotics in FunctionalSpaces
	2.2 Global Attractors and Finite-Dimensional Reduction
	2.3 Classification of Positive Solutions of Semilinear Elliptic Equations in a Rectangle: Two Dimensional Case
		2.3.1 Sketch of the Proof of Theorem 2.4
	2.4 Existence of Solutions of Nonlinear Elliptic Systems
	2.5 Regularity of Solutions
	2.6 Boundedness of Solutions as
	2.7 Basic Definitions: Trajectory Attractor
	2.8 Trajectory Attractor of Nonlinear Elliptic System
	2.9 Dependence of the Trajectory Attractor on the UnderlyingDomain
	2.10 Regularity of Attractor
	2.11 Trajectory Attractor of an Elliptic Equation with a Nonlinearity That Depends on x
	2.12 Examples of Trajectory Attractors
	2.13 The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems in Non-smooth Domains
		2.13.1 Existence of Solutions
		2.13.2 Trajectory Attractor for the Nonlinear Elliptic System
		2.13.3 Stabilization of Solutions in the Potential Case
		2.13.4 Regular and Singular Part of the Trajectory Attractor
	2.14 The Dynamics of Fast Nonautonomous Travelling Waves and Homogenization
3 Symmetry and Attractors: The Case N<=3
	3.1 Introduction
	3.2 A Priori Estimates and Solvability Results
	3.3 The Attractor
	3.4 Symmetry and Stabilization
4 Symmetry and Attractors: The Case N<=4
	4.1 Introduction
	4.2 A Priori Estimates and Solvability Results
	4.3 The Attractor
	4.4 Symmetry and Stabilization
5 Symmetry and Attractors
	5.1 Introduction
		5.1.1 Statement of Results
	5.2 The Dynamical System Approach
	5.3 Proof of Theorem 5.1
	5.4 Proof of Theorem 5.2
		5.4.1 Symmetry of the Profiles
		5.4.2 Completion of the Proof of Theorem 5.2
	5.5 Proof of Theorem 5.3
		5.5.1 Positivity of Solutions
		5.5.2 Completion of the Proof of Theorem 5.3
6 Symmetry and Attractors: Arbitrary Dimension
	6.1 Introduction
	6.2 The PDE Approach
		6.2.1 Problem  in the Quarter-Space
		6.2.2 Problem  in the Half-Space
	6.3 Classification Results in the Whole Space RN or in the Half-Space RN-1x(0,+infty) with Dirichlet BoundaryConditions
	6.4 The Dynamical Systems\' Approach
7 The Case of p-Laplacian Operator
	7.1 Introduction
	7.2 Some Basic Results
		7.2.1 The Weak Sweeping Principle
		7.2.2 Classification of One-Dimensional Solutions
		7.2.3 A Liouville Type Result
	7.3 Half- and Quarter-Space Problems
		7.3.1 Asymptotic Convergence in Half-Spaces
		7.3.2 One-Dimensional Symmetry in Half-Spaces
		7.3.3 Asymptotic Convergence in Quarter-Spaces
	7.4 Comparison with Standard Laplacian (p=2)
	7.5 Existence Result
Bibliography