Intro; 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 Edge2 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 System2.9 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 Homogenization3 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 5.25.5 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
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ISBN
9783319984063
3319984063
LCCN
2018955495
OCLC
1077772500
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