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

Mickaël dos Santos (1) , Remy Rodiac (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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

Mots clés: Ginzburg-Landau energy; prescribed degrees; lack of compactness.

Domaines: Equations aux dérivées partielles [math.AP]

Mickaël dos Santos, Remy Rodiac. Existence and non existence results for minimizers of the Ginzburg-Landau energy with prescribed degrees. 2015. ⟨hal-01139179⟩