A remark on the radial minimizer of the Ginzburg-Landau functional (1305.4028v1)

Abstract: Denote by $E_\epsilon$ 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 $\Omega\subset \R^2$ be a smooth, bounded domain with the same area as $B_1$. Denoted by $$\mathcal{K}=\left{v=(v_1,v_2) \in H^1(\Omega;\R^2):> \int_\Omega v_1\,dx=\int_\Omega v_2\,dx=0,> \int_\Omega |v|^2\,dx\ge \int_{B_1} |\tilde u_\varepsilon|^2\,dx\right},$$ we prove $$ \min_{v \in \mathcal{K}} E_\varepsilon (v,\Omega)\le E_\varepsilon (\tilde u_\varepsilon,B_1). $$