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

Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with the same area as the unit disk $B_1$ and let $$ E_\varepsilon(u,\Omega)=\frac{1}{2}\int_\Omega |\nabla u|^2\,dx +\frac{1}{4\varepsilon^2}\int_\Omega (|u|^2-1)^2\,dx $$ be the Ginzburg-Landau functional. Denote by $\tilde u_\varepsilon$ the radial solution to the Euler equation associated to the problem $\min \{E_\varepsilon(u,B_1): \> u\big| _{\partial B_{1}}=x\}$ and by $$\eqalign{ \mathcal{K}=\Big\{v=(v_1,v_2) \in H^1(\Omega;\mathbb{R}^2): \int_\Omega v_1\,dx=\int_\Omega v_2\,dx=0,\cr \int_\Omega |v|^2\,dx\ge \int_{B_1} |\tilde u_\varepsilon|^2\,dx\Big\}. \cr}$$ In this note we prove that $$ \min_{v \in \mathcal{K}} E_\varepsilon (v,\Omega) \le E_\varepsilon (\tilde u_\varepsilon,B_1). $