Capacity of a multiply-connected domain and nonexistence of Ginzburg-Landau minimizers with prescribed degrees on the boundary
Leonid V. Berlyand, Dmitry Golovaty, Volodymyr Rybalko · arXiv (Cornell University) · 2006
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 ('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'') and for small $κ$ when $cap(A)κ_1$ while it is attained when $κ