Electron. J. Diff. Equ., Vol. 2014 (2014), No. 224, pp. 1-4.

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

Barbara Brandolini, Francesco Chiacchio

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

Submitted September 15, 2014. Published October 21, 2014.
Math Subject Classifications: 35Q56, 35J15.
Key Words: Ginzburg-Landau functional, Szego-Weinberger inequality.

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Barbara Brandolini
Dipartimento di Matematica e Applicazioni "R. Caccioppoli"
Universitá degli Studi di Napoli "Federico II"
Complesso Monte S. Angelo, via Cintia - 80126 Napoli, Italy
email: brandolini@unina.it
Francesco Chiacchio
Dipartimento di Matematica e Applicazioni "R. Caccioppoli"
Universitá degli Studi di Napoli "Federico II"
Complesso Monte S. Angelo, via Cintia - 80126 Napoli, Italy
email: francesco.chiacchio@unina.it

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