Efendiev M. Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations

Cham: Springer, 2018. — 273 p.This book deals with a systematic study of a dynamical system approach to investigate the symmetrization and stabilization properties of nonnegative solutions of nonlinear elliptic problems in asymptotically symmetric unbounded domains. The usage of infinite dimensional dynamical systems methods for elliptic problems in unbounded domains as well as finite dimensional reduction of their dynamics requires new ideas and tools. To this end, both a trajectory dynamical systems approach and new Liouville type results for the solutions of some class of elliptic equations are used. The work also uses symmetry and monotonicity results for nonnegative solutions in order to characterize an asymptotic profile of solutions and compares a pure elliptic partial differential equations approach and a dynamical systems approach. The new results obtained will be particularly useful for mathematical biologists.Preliminaries
Functional Spaces and Their Properties
Lp Spaces
Sobolev Spaces
Linear Elliptic Boundary Value Problems
Nemytskii Operator
Maximum Principles and Their Applications
Classical Maximum Principles
Uniform Estimates and Boundedness of the Solutions of Semilinear Elliptic Equations
The Sweeping Principle and the Moving Plane Method in a Bounded Domain
The Sliding and the Moving Plane Method in General Domains
Variational Solutions of Elliptic Equations
Elliptic Regularity for the Neumann Problem for the Laplace Operator on an Infinite Edge
Kolmogorov epsilon-Entropy and Its Asymptotics in FunctionalSpaces
Global Attractors and Finite-Dimensional Reduction
Classification of Positive Solutions of Semilinear Elliptic Equations in a Rectangle: Two Dimensional Case
Sketch of the Proof of Theorem
Existence of Solutions of Nonlinear Elliptic Systems
Regularity of Solutions
Boundedness of Solutions as
Basic Definitions: Trajectory Attractor
Trajectory Attractor of Nonlinear Elliptic System
Dependence of the Trajectory Attractor on the UnderlyingDomain
Regularity of Attractor
Trajectory Attractor of an Elliptic Equation with a Nonlinearity That Depends on x
Examples of Trajectory Attractors
The Trajectory Dynamical Approach for the Nonlinear Elliptic Systems in Non-smooth Domains
Existence of Solutions
Trajectory Attractor for the Nonlinear Elliptic System
Stabilization of Solutions in the Potential Case
Regular and Singular Part of the Trajectory Attractor
The Dynamics of Fast Nonautonomous Travelling Waves and HomogenizationA Priori Estimates and Solvability Results
The Attractor
Symmetry and StabilizationA Priori Estimates and Solvability Results
The Attractor
Symmetry and Stabilization
Statement of Results
The Dynamical System Approach
Proof of Theorem
Symmetry of the Profiles
Completion of the Proof of Theorem
Positivity of Solutions
Completion of the Proof of TheoremThe PDE Approach
Problem in the Quarter-Space
Problem in the Half-Space
Classification Results in the Whole Space RN or in the Half-Space RN-x(,+infty) with Dirichlet BoundaryConditions
The Dynamical Systems' ApproachThe Weak Sweeping Principle
Classification of One-Dimensional Solutions
A Liouville Type Result
Asymptotic Convergence in Half-Spaces
One-Dimensional Symmetry in Half-Spaces
Asymptotic Convergence in Quarter-Spaces
Comparison with Standard Laplacian (p=2)
Existence Result