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Title:  Existence of Critical Points for the GinzburgLandau Functional on Riemannian Manifolds 
Authors:  Mesaric, Jeffrey Alan 
Advisor:  Jerrard, Robert 
Department:  Mathematics 
Keywords:  partial differential equations GinzburgLandau 
Issue Date:  19Feb2010 
Abstract:  In this dissertation, we employ variational methods to obtain a new existence result for solutions of a GinzburgLandau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3dimensional Riemannian manifold without boundary and $\gamma$ is a simple, smooth, connected, closed geodesic in $N$ satisfying a natural nondegeneracy condition, then for every $\ep>0$ sufficiently small, $\exists$ a
critical point $u^\ep\in H^1(N;\mathbb{C})$ of the GinzburgLandau functional \bd\ds E^\ep(u):=\frac{1}{2\pi \ln\ep}\int_N \nabla u^2+\frac{(u^21)^2}{2\ep^2}\ed
and these critical points have the property that $E^\ep(u^\ep)\rightarrow\tx{length}(\gamma)$ as $\ep\rightarrow 0$.
To accomplish this, we appeal to a recent general asymptotic minmax theorem which basically says that if $E^\ep$ $\Gamma$converges to $E$ (not necessarily defined on the same Banach space as $E^\ep$), $v$ is a saddle point of $E$ and some additional mild hypotheses are met, then there exists $\ep_0>0$ such that for every $\ep\in(0,\ep_0),E^\ep$ possesses a critical point $u^\ep$ and $\lim_{\ep\rightarrow 0}E^\ep(u^\ep)=E(v)$.
Typically, $E$ is only lower semicontinuous, therefore a suitable notion of saddle point is needed.
Using known results on $\mathbb{R}^3$, we show the GinzburgLandau functional $E^\ep$ defined above $\Gamma$converges to a functional $E$ which can be thought of as measuring the arclength of a limiting singular set. Also, we verify using regularity theory for almostminimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense. 
URI:  http://hdl.handle.net/1807/19062 
Appears in Collections:  Doctoral Department of Mathematics  Doctoral theses

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