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

TL;DR

This paper investigates the properties of radial minimizers of the Ginzburg-Landau functional, demonstrating a comparison result between the energy of certain vector fields in a domain and the radial solution in a unit ball.

Contribution

It establishes an inequality comparing the energy of divergence-free vector fields with prescribed mean in a domain to the energy of the radial minimizer in the unit ball.

Findings

The radial minimizer's energy serves as an upper bound for a class of vector fields in a domain.

The result provides insight into the structure of minimizers of the Ginzburg-Landau functional in planar domains.

The proof involves variational techniques and properties of the radial solution.

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 $$ \min_{v \in \mathcal{K}} E_\varepsilon (v,\Omega)\le E_\varepsilon (\tilde u_\varepsilon,B_1). $$