Brezis关于平面Ginzburg--Landau方程问题2.5的一个解答
A Solution of Problem 2.5 by Brezis on the Planar Ginzburg--Landau Equation
AI总结：
本文肯定回答了Brezis提出的公开问题2.5：证明平面Ginzburg--Landau方程解的渐近性质蕴含有限势能估计，方法结合Kelvin反演、Morrey型能量衰减、相位形式$L^4$可积性及线性化振幅方程。
AI中文摘要：
Brezis, Merle和Riviére [Arch. Rational Mech. Anal. 1994]证明了：若平面全空间Ginzburg--Landau方程 $$ -\Delta u = u(1-|u|^2)\quad \text{in}\quad \mathbb{R}^2 $$ 的光滑解 $u:\mathbb{R}^2\to\mathbb{C}$ 满足有限势能估计 $$ \int_{\mathbb{R}^2}\left[1-|u(x)|^2\right]^2\\,dx < \infty, $$ 则它具有渐近性质 $$ |u(x)|\to 1\quad \text{as}\quad |x|\to\infty. $$ 其逆问题——渐近性质是否蕴含有限势能估计——最初在他们的工作中提出，后来由Brezis在[Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 2023]中将其表述为公开问题2.5。在本文中，我们对该问题给出肯定回答。我们的证明依赖于Kelvin反演、平移Jacobi系统的Morrey型能量衰减、相位形式的$L^4$可积性以及线性化振幅方程；所有这些工具共同蕴含了函数$1-|u|$在外区域上的$L^2$可积性，且无需任何先验可积性假设。
英文摘要：
Brezis, Merle, and Riviére [Arch. Rational Mech. Anal. 1994] proved that, if a smooth solution $u:\mathbb{R}^2\to\mathbb{C}$ of the planar entire Ginzburg--Landau equation $$ -Δu = u(1-|u|^2)\quad \text{in}\quad \mathbb{R}^2 $$ satisfies the finite potential energy estimate that $$ \int_{\mathbb{R}^2}\left[1-|u(x)|^2\right]^2\,dx < \infty, $$ then it has the asymptotic property that $$ |u(x)|\to 1\quad \text{as}\quad |x|\to\infty. $$ 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.
