Finite Potential Energy for Entire Solutions of the Planar Ginzburg--Landau Equation

Abstract

We prove that every smooth entire solution $u:{\mathbb{R}}^2\to {\mathbb{R}}^2$ of the Ginzburg--Landau equation $-\Delta u=u(1-|u{|}^2)$ with $|u(x)|\to 1$ as $|x|\to \infty$ has finite potential energy, i.e., \begin{equation*} \int_{\mathbb{R}^2}(1-|u|^2)^2 \mathrm{d}x<+\infty, \end{equation*} thereby resolving Brezis' Open Problem 2.5 in [4]. The main difficulty stems from the possible presence of a curl-free mode that carries nonzero circulation and decays only like $|x{|}^{-1}$; such a mode lies outside $L^2$ and does not admit a single-valued potential. By minimizing over $L^2$ gradient corrections, we construct a comparison field that solves the homogeneous equation and inherits the same circulation. The Kelvin inversion, combined with the De Giorgi--Nash--Moser theory for quasilinear elliptic equations, then produces the optimal decay $O(|x{|}^{-1})$. For a Ginzburg--Landau solution, the Bernstein estimate and the coercivity of the Jacobi form produce an $L^2$ forcing term in the exterior phase equation. The resulting $L^4$ bound on the phase field implies $1-|u{|}^2\in L^2({\mathbb{R}}^2)$, and therefore the potential energy is finite.

28 pages