The Ginzburg–Landau Functional
Søren Fournais, Bernard Helffer · Progress in nonlinear differential equations and their applications · 2009
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.