Local Minimizers of the Ginzburg-Landau Functional with Prescribed Degrees

Mickaël dos Santos (1)

1 ICJ - Institut Camille Jordan

We consider, in a smooth bounded multiply connected domain $\dom\subset\R^2$, the Ginzburg-Landau energy $\d E_\v(u)=\frac{1}{2}\int_\dom{|\n u|^2}+\frac{1}{4\v^2}\int_\dom{(1-|u|^2)^2}$ subject to prescribed degree conditions on each component of $\p\dom$. In general, minimal energy maps do not exist \cite{BeMi1}. When $\dom$ has a single hole, Berlyand and Rybalko \cite{BeRy1} proved that for small $\v$ local minimizers do exist. We extend the result in \cite{BeRy1}: $\d E_\v(u)$ has, in domains $\dom$ with $2,3,...$ holes and for small $\v$, local minimizers. Our approach is very similar to the one in \cite{BeRy1}; the main difference stems in the construction of test functions with energy control.

Mots clés: Ginzburg-Landau functional; prescribed degrees; local minimizers

Domaines: Equations aux dérivées partielles [math.AP]

Mickaël dos Santos. Local Minimizers of the Ginzburg-Landau Functional with Prescribed Degrees. 2009. ⟨hal-00368162v1⟩