Maxwell's Equations in a Quasi-static Electromagnetic Field and the Generalized Ginzburg-Landau Functional: Regularity and Asymptotic Behavior

Yassin Alzubaidi (2011). Maxwell's Equations in a Quasi-static Electromagnetic Field and the Generalized Ginzburg-Landau Functional: Regularity and Asymptotic Behavior PhD Thesis, School of Mathematics and Physics, The University of Queensland.

       
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Author Yassin Alzubaidi
Thesis Title Maxwell's Equations in a Quasi-static Electromagnetic Field and the Generalized Ginzburg-Landau Functional: Regularity and Asymptotic Behavior
School, Centre or Institute School of Mathematics and Physics
Institution The University of Queensland
Publication date 2011-05
Thesis type PhD Thesis
Supervisor Dr. Min-Chun Hong
A/prof. Joseph Grotowski
Total pages 84
Total black and white pages 84
Subjects 01 Mathematical Sciences
Abstract/Summary This thesis consists of two main parts. In the first part, we present our work concerning the partial regularity of the weak solutions for Maxwell's equations in a quasi-static electromagnetic field in higher dimensions. We redefine Maxwell's equation in arbitrary dimensions by replacing the vectors on R^3 with 1-forms on R^n for all n >= 3. We study two systems: the time-independent case (elliptic system) and the time-dependent case (parabolic system). The method used here relies on deriving a Caccioppoli-type inequality and a reverse Holder's inequality in order to obtain an improved Lp-estimate. Then we use some variational methods and iteration techniques to prove the regularity of the weak solutions outside a very small closed set. The second part of this thesis is comprised of three main sections. In the first section, we give a generalization of the well-known Ginzburg-Landau functional which depends on a parameter ϵ>0. We show the existence of a minimizer for every ϵ>0, and we prove the strong convergence of the minimizers in a Sobolev space. In the second section we prove the partial regularity of the limit pair. We start the proof of the partial regularity of the limit pair by obtaining a monotonicity formula. Then we use the method employed in the first part of this thesis. In the last section, we prove the convergence of a subsequence of the minimizers of the generalized Ginzburg-Landau functional in some Holder space.
Keyword Maxwell's equations, Caccioppoli's inequality,
reverse Hölder's inequality, partial regularity,
Ginzburg-Landau functional, monotonicity

 
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Created: Tue, 06 Sep 2011, 09:31:04 EST by Mr Yassin Alzubaidi on behalf of Library - Information Access Service