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

Suppose that $\omega \subset \mathrm{\Omega }\subset R^2$. In the annular domain $A=\mathrm{\Omega }\setminus \bar{\omega }$ we consider the class $J$ of complex valued maps having degree 1 on $\mathrm{\partial }\mathrm{\Omega }$ and on $\mathrm{\partial }\omega$. It was conjectured by Berlyand and Mironescu (‘04), that he existence of minimizers of the Ginzburg-Landau energy $E_{\kappa }$ 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 $\kappa$. The existence of minimizers of $E_{\kappa }$ for all $\kappa$ when $cap(A)\ge \pi$ (domain $A$ is thin'') and for small $\kappa$ when $cap(A)<\pi$ (domain $A$ is thick’') was established by Berlyand and Mironescu ('04). Here we provide the answer for the remaining case of large $\kappa$ when $cap(A)<\pi$. We prove that, when $cap(A)<\pi$, there exists a finite threshold value ${\kappa }_1$ of the Ginzburg-Landau parameter $\kappa$ such that the minimum of the Ginzburg-Landau energy $E_{\kappa }$ is not attained in $J$ when $\kappa >{\kappa }_1$ while it is attained when $\kappa <{\kappa }_1$.