The Ginzburg–Landau Functional

Abstract

Let us describe the mathematical problem. It is naturally posed for domains in $\mathbb{R}^{3}$, but for cylindrical domains in $\mathbb{R}^{3}$, it is natural (though not completely justified mathematically) to consider a functional defined in a domain Ω ⊂ $\mathbb{R}^{3}$, where Ω is the cross-section of the cylinder. This explains why we also consider models in $\mathbb{R}^{3}$. The behavior of a sample of material can be read off from the properties of the minimizers (ψ,A) of the Ginzburg–Landau functional (free energy) $\mathcal{G}$ to be defined below.