Existence and non existence results for minimizers of the Ginzburg-Landau energy with prescribed degrees

TL;DR

This paper investigates the existence and non-existence of minimizers for a simplified Ginzburg-Landau energy in annular domains with prescribed boundary degrees, revealing conditions under which minimizers exist or do not exist, especially in thin domains.

Contribution

It provides new existence results for minimizers when degrees are balanced and domain is thin, and establishes non-existence for unbalanced degrees in large capacity annuli.

Findings

Existence of minimizers for balanced degrees in thin domains.

Non-existence of minimizers for unbalanced degrees with large capacity.

Results extend understanding of Ginzburg-Landau energy minimization in annular geometries.

Abstract

Let $\mathcal{D} =\Omega \setminus\bar{\omega} \subset \mathbb{R}^2$ be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy $E_\epsilon(u)=\frac{1}{2}\int_{\mathcal{D}} |\nabla u|^2 +\frac{1}{4\epsilon^2}\int_{\mathcal{D}} (1-|u|^2)^2$, where $u: \mathcal{D} \rightarrow \mathbb{C}$, and look for minimizers of $E_\epsilon$ with prescribed degrees $deg(u,\partial \Omega)=p$, $deg(u,\partial \omega)=q$ on the boundaries of the domain. For large $\epsilon$ and for balanced degrees, i.e., $p=q$, we obtain existence of minimizers for {\it thin} domain. We also prove non-existence of minimizers of $E_\epsilon$, for large $\epsilon$, in the case $p\neq q$, $pq>0$ and $\mathcal{D}$ is a circular annulus with large capacity (corresponding to "thin" annulus). Our approach relies on similar results obtained for the Dirichlet energy $E_\infty(u)=\frac{1}{2}\int_{\mathcal{D}}|\nabla u|^2$, the existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.