A Solution of Problem 2.5 by Brezis on the Planar Ginzburg--Landau Equation

@misc{lin2026solution,
  title = {{A Solution of Problem 2.5 by Brezis on the Planar Ginzburg--Landau Equation}},
  author = {Xiaosheng Lin and Dachun Yang and Sibei Yang and Wen Yuan and Yangyang Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19020},
  eprint = {2609.19020},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19020},
}

TY  - UNPB
TI  - A Solution of Problem 2.5 by Brezis on the Planar Ginzburg--Landau Equation
AU  - Xiaosheng Lin
AU  - Dachun Yang
AU  - Sibei Yang
AU  - Wen Yuan
AU  - Yangyang Zhang
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.19020
UR  - https://arxiv.org/abs/2609.19020
AB  - Brezis, Merle, and Riviére [Arch. Rational Mech. Anal. 1994] proved that, if a smooth solution u:ℝ^2→ℂ of the planar entire Ginzburg--Landau equation −Δu = u(1−|u|^2) in ℝ^2 satisfies the finite potential energy estimate that ∫_ℝ^2[1−|u(x)|^2]^2 dx < ∞, then it has the asymptotic property that |u(x)|→ 1 as |x|→∞. The converse problem whether the asymptotic property implies the finite potential energy estimate was originally posed in their work and later formulated by Brezis as \emph{Open Problem 2.5} in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 2023]. In this article, we give an affirmative answer to this question. Our proof relies on the Kelvin inversion, a Morrey-type energy decay for the translation Jacobi system, the L^4-integrability of the phase form, and a linearised amplitude equation; all these tools together imply the L^2-integrability of the function 1−|u| on an exterior domain, without any a priori integrability assumption.
ER  -