A remark on the radial minimizer of the Ginzburg-Landau functional

Abstract

Denote by $E_{\epsilon }$ the Ginzburg-Landau functional in the plane and let ${\tilde{u}}_{\epsilon }$ be the radial solution to the Euler equation associated to the problem $\min \left\{E_{\epsilon }(u,B_1):{u|}_{\partial B_1}=(\cos \vartheta ,\sin \vartheta )\right\}$. Let $\Omega \subset {\mathbb{R}}^2$ be a smooth, bounded domain with the same area as $B_1$. Denoted by $$\mathcal{K}=\left\{v=(v_1,v_2)\in H^1(\Omega ;{\mathbb{R}}^2):{\int }_{\Omega }{v}_{1}dx={\int }_{\Omega }{v}_{2}dx=0,{\int }_{\Omega }|v{|}^{2}dx\ge {\int }_{B_1}|{\tilde{u}}_{\epsilon }{|}^{2}dx\right\},$$ we prove $$\underset{v\in \mathcal{K}}{\min }{E}_{\epsilon }(v,\Omega )\le E_{\epsilon }({\tilde{u}}_{\epsilon },B_1).$$