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

We prove that every smooth entire solution $u\text{\hspace{0.17em}⁣}:{\mathbb{R}}^2\to {\mathbb{R}}^2$ of the Ginzburg–Landau equation $-\mathrm{\Delta }u=u(1-\mathrm{\mid }u{\mathrm{\mid }}^2)$ with $\mathrm{\mid }u(x)\mathrm{\mid }\to 1$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ 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 $\mathrm{\mid }x{\mathrm{\mid }}^{-1}$; 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 $O(\mathrm{\mid }x{\mathrm{\mid }}^{-1})$. 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 $1-\mathrm{\mid }u{\mathrm{\mid }}^2\in L^2({\mathbb{R}}^2)$, and therefore the potential energy is finite.