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Contributions to Degree theory, and Applications Open Access


Other title
Dynamical Systems
Periodic Solutions
Onsager Model
Fully nonlinear equations
Degree Theory
Monotone Maps
Liquid Crystal
Type of item
Degree grantor
University of Alberta
Author or creator
Niksirat, Mohammad Ali
Supervisor and department
Yu, Xinwei (Mathematicsl Sciences)
Examining committee member and department
Minev, Peter (Mathematicsl Sciences)
Dai, Feng (Mathematicsl Sciences)
Runde, Vilker (Mathematicsl Sciences)
Flynn, Morris (Mechanical Engineering)
Liu, Hailiang (Mathematicsl Sciences)
Department of Mathematical and Statistical Sciences
Applied Mathematics
Date accepted
Graduation date
Doctor of Philosophy
Degree level
This thesis is dedicated to the study of topological degree for different classes of monotone maps and applications to nonlinear problems in mathematical analysis and the applied mathematics. Employing topological methods for nonlinear problems in mathematics goes back to the pioneering work of H. Poincaré on the three body problem. The generalization of Brouwer degree for mappings in finite dimensional spaces to compact perturbations of the identity in arbitrary Banach spaces, by J. Schauder and J. Leray, opened up a way to apply such a powerful method to a broad class of complicated nonlinear problems. Further generalizations, including the degree for classes of monotone maps as well as the degree for multi-valued maps have been carried out by several authors in recent decades. This thesis is divided into two parts. The first part, consisting of Chapters 1 and 2, is about the theoretical aspects of topological degree. The second part, Chapters 3--5, is devoted to the applications of the topological degree in three fields: an integral equation coming from the Doi-Onsager model for liquid crystals; dynamical systems governed by nonlinear ordinary differential equations and finally fully nonlinear elliptic and parabolic partial differential equations. After a detailed introduction on various topological methods for linear and quasi-linear elliptic problems and presentation of some of its implications for monotone maps and variational problems in Chapter 1, we systematically introduce the concept of finite rank approximation of a map in Chapter 2. This concept enables us to prove the stability of the homotopy class of finite rank approximations for different types of monotone maps including (S)+, pseudo-monotone and maximal monotone maps in separable, locally uniformly convex Banach spaces. Furthermore, we generalize the degree for mappings that are only demi-continuous in a subspace, not necessarily dense, of the focal space. We use this generalization for the Doi-Onsager problem presented in Chapter 3. The Doi-Onsager problem is a mathematical formulation to model the behaviour of the liquid crystals in terms of the interaction potential field and the temperature of the liquid. In our work on this problem, we solve the problem in dimension D=2 and also prove the uniqueness of the isotropic solution for high temperature and the bifurcation of nematic solutions for low temperature in general dimension. For a classical application of degree theory, we return in Chapter 4 to the problem of periodic solutions for dynamical systems described in ordinary differential equations. The method that we employ in this chapter is based on the continuation method. For this, we consider a one-parameter family of dynamical systems and then prove (under certain conditions) that periodic orbits survive when the parameter increases. The last chapter of the thesis, Chapter 5 is dedicated to defining a degree for fully nonlinear elliptic and parabolic equations. Even though it is not novel to define a degree for fully nonlinear elliptic equations, our construction can be employed to define a degree for fully nonlinear parabolic equations which, to our knowledge is new.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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