A central focus of Ginzburg-Landau theory is the understanding and characterization of vortex configurations. On a bounded domain $\mathrm{\Omega }\subseteq {ℝ}^2,$ global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit $ϵ\to 0,$ where $ϵ>0$ is the inverse of the Ginzburg-Landau parameter. A notable open problem is whether there are solutions of the Ginzburg-Landau equation for any number of vortices below $h_{\mathrm{e}x}\mid \mathrm{\Omega }\mid /2\pi ,$ for external fields of up to superheating field strength.

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二维 Ginzburg-Landau 泛函的无界涡度局部极小化器

Ginzburg-Landau 理论的一个中心焦点是涡旋构型的理解和表征。在有界域上$\mathrm{\Omega }\subseteq {ℝ}^2,$相应能量泛函的全局最小化器和一般的临界状态已在极限中进行了彻底研究$\epsilon \to 0,$在哪里$\epsilon >0$是 Ginzburg-Landau 参数的倒数。一个值得注意的悬而未决的问题是，对于以下任意数量的涡旋，Ginzburg-Landau 方程是否存在解$H_{\mathrm{e}X}\mid \mathrm{\Omega }\mid /2\pi ,$用于高达过热场强的外部场。