In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3-dimensional 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 Ginzburg-Landau functional \bd\ds E^\ep(u):=\frac{1}{2\pi |\ln\ep|}\int_N |\nabla u|^2+\frac{(|u|^2-1)^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 Ginzburg-Landau 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 almost-minimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OTU.1807/19062 |
| Date | 19 February 2010 |
| Creators | Mesaric, Jeffrey Alan |
| Contributors | Jerrard, Robert |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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