Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations

✍ Scribed by Messoud Efendiev

• Publisher: Springer
• Year: 2018
• Tongue: English
• Leaves: 273
• Series: Fields Institute Monographs 36
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter ....Pages i-xvii
Preliminaries (Messoud Efendiev)....Pages 1-70
Trajectory Dynamical Systems and Their Attractors (Messoud Efendiev)....Pages 71-162
Symmetry and Attractors: The Case N ≤ 3 (Messoud Efendiev)....Pages 163-175
Symmetry and Attractors: The Case N ≤ 4 (Messoud Efendiev)....Pages 177-186
Symmetry and Attractors (Messoud Efendiev)....Pages 187-198
Symmetry and Attractors: Arbitrary Dimension (Messoud Efendiev)....Pages 199-224
The Case of p-Laplacian Operator (Messoud Efendiev)....Pages 225-252
Back Matter ....Pages 253-258

✦ Subjects

Sobolev spaces, Nemytskii operator, trajectory attractor, Laplace operator, symmetry and attractors, stabilization properties, reduction of dynamics,