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Contributions to Degree theory, and Applications
- Author / Creator
- Niksirat, Mohammad Ali
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.
- Graduation date
- Fall 2014
- Type of Item
- Doctor of Philosophy
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