Capacity of a multiply-connected domain and nonexistence of Ginzburg-Landau minimizers with prescribed degrees on the boundary

Abstract

Suppose that $ω\subsetΩ\subset R^2$. In the annular domain $A=Ω\setminus\barω$ we consider the class $J$ of complex valued maps having degree 1 on $\partial Ω$ and on $\partialω$. It was conjectured by Berlyand and Mironescu (&#39;04), that he existence of minimizers of the Ginzburg-Landau energy $E_κ$ in $J$ is completely determined by the value of the $H^1$-capacity $cap(A)$ of the domain and the value of the Ginzburg-Landau parameter $κ$. The existence of minimizers of $E_κ$ for all $κ$ when $cap(A)\geqπ$ (domain $A$ is ``thin&#39;&#39;) and for small $κ$ when $cap(A)<π$ (domain $A$ is ``thick&#39;&#39;) was established by Berlyand and Mironescu (&#39;04). Here we provide the answer for the remaining case of large $κ$ when $cap(A)<π$. We prove that, when $cap(A)<π$, there exists a finite threshold value $κ_1$ of the Ginzburg-Landau parameter $κ$ such that the minimum of the Ginzburg-Landau energy $E_κ$ is not attained in $J$ when $κ>κ_1$ while it is attained when $κ<κ_1$.