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

Denote by $E_ε$ the Ginzburg-Landau functional in the plane and let $\tilde u_\varepsilon$ be the radial solution to the Euler equation associated to the problem $\min \left\{E_\varepsilon(u,B_1): \>\left. u\right\vert _{\partial B_{1}}=(\cos \vartheta,\sin \vartheta)\right\}$. Let $Ω\subset \R^2$ be a smooth, bounded domain with the same area as $B_1$. Denoted by $$\mathcal{K}=\left\{v=(v_1,v_2) \in H^1(Ω;\R^2):\> \int_Ωv_1\,dx=\int_Ωv_2\,dx=0,\> \int_Ω|v|^2\,dx\ge \int_{B_1} |\tilde u_\varepsilon|^2\,dx\right\},$$ we prove $$ \min_{v \in \mathcal{K}} E_\varepsilon (v,Ω)\le E_\varepsilon (\tilde u_\varepsilon,B_1). $$