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2018 | Book

Preliminaries Read chapter Read first chapter

Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations

Author: Messoud Efendiev

Publisher: Springer International Publishing

Book Series : Fields Institute Monographs

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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

Frontmatter
Chapter 1. Preliminaries
Abstract
In this chapter, we present notation and generally known facts (mostly without proofs) that we use to state and derive the results of the subsequent chapters.
Messoud Efendiev
Chapter 2. Trajectory Dynamical Systems and Their Attractors

We start with the definition of Kolmogorov ε-entropy, via which we define fractal dimension of the compact set in the metric space. We will use these two concepts in the sequel.

Messoud Efendiev
Chapter 3. Symmetry and Attractors: The Case N ≤ 3
Abstract
As it was shown in Chap. 1 , positive solutions of semilinear second order elliptic problems have symmetry and monotonicity properties which reflect the symmetry of the operator and of the domain.
Messoud Efendiev
Chapter 4. Symmetry and Attractors: The Case N ≤ 4
Abstract
In the previous chapter, we showed that nonnegative solutions of elliptic equations in “asymptotically symmetric” domains are “asymptotically symmetric” as well (see Theorem 3.3 ). However, in order to prove Theorem 3.3 , we imposed a restriction on the dimension (less or equal 3) of the underlying domain, which was crucial for our proof. The goal of this chapter is to extend Theorem 3.3 for higher dimensions.
Messoud Efendiev
Chapter 5. Symmetry and Attractors
Abstract
In this chapter, symmetry results in the half-space and in \(\mathbb {R}^N\) will be used towards the characterization of the asymptotic profiles of solutions in the quarter-space and in the half-space, respectively. As we have seen in the previous Chaps. 3 and 4 , here the dimension of the underlying domain plays an important role and to extend the results on the symmetrization and stabilization of solutions of semilinear elliptic equations for dimensions less or equal 3 to the case of dimensions less or equal 4 requires nontrivial arguments and assumptions on the nonlinearities. The goal of this chapter is to extend the results from Chaps. 3 and 4 to the case of dimensions less or equal 5. As we will see below, to this end we need new arguments and we cannot use the techniques from Chaps. 3 and 4 . Similar to the previous chapters, we will apply the trajectory dynamical systems approach in order to study the asymptotic profiles of solutions for this new case of dimension 5 or higher. Moreover, in contrast to the previous chapters, we will also study the case when the asymptotic profile is a constant.
Messoud Efendiev
Chapter 6. Symmetry and Attractors: Arbitrary Dimension
Abstract
Let Ω be the domain of \(\mathbb {R}^N\) (N ≥ 2) defined by
Messoud Efendiev
Chapter 7. The Case of p-Laplacian Operator
Abstract
We are interested in quasilinear elliptic problems over a half-space of the form
$$\displaystyle\left \{ \begin {array}{l}\Delta _p u+f(u)=0 \mbox{ in } \mathbb {R}_+\times \mathbb {R}^{N-1},\\ u(0, x_2,\ldots , x_N)=u_0(x_2,\ldots , x_N), \end {array} \right .$$
and similar problems over a quarter-space
$$\displaystyle\left \{ \begin {array}{l} \Delta _p u+f(u)=0 \mbox{ in } \mathbb {R}_+\times \mathbb {R}^{N-2}\times \mathbb {R}_+,\\ u(0, x_2,\ldots , x_N)=u_0(x_2,\ldots , x_N),\\ u(x_1,x_2,\ldots , x_{N-1}, 0)=0. \end {array} \right .$$
Messoud Efendiev
Backmatter
Metadata
Title
Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations
Author
Messoud Efendiev
Copyright Year
2018
Electronic ISBN
978-3-319-98407-0
Print ISBN
978-3-319-98406-3
DOI
https://doi.org/10.1007/978-3-319-98407-0

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