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

Abstract

We prove that every smooth entire solution $ u\colon\mathbb{R}^2\to\mathbb{R}^2 $ of the Ginzburg--Landau equation $ -Δu=u(1-|u|^2) $ with $ |u(x)|\to1 $ 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.

Disclosure

“UMDF and MYRG- GRG2025-00051-FST), and the University of Macau Development Foundation (Grant No. TISF/2025/006/FST). Data availability statement: There are no new data associated with this article. AI assistance statement: The authors used OpenAI models to assist with language polishing and manuscript editing. R EFERENCES [1] F. Bethuel, H. Brezis, and F. Hélein, Ginzburg–Landau Vortices, Progress in Nonlinear Differen- tial Equatio”

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2026-08-23T18:58:11.712342Z

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