AI 中文总结
本文对布雷齐斯公开问题2.1给出部分解答，证明了单位圆盘及ℝ²到ℝ⁴的有界连通C¹,¹区域上金兹堡-朗道方程（系统）解在ε接近ε₊的区间内具有唯一性，完善了该问题的相关结果。
AI 中文摘要
本文针对布雷齐斯公开问题2.1给出部分解答，该问题涉及单位圆盘上带次数1边界条件的金兹堡-朗道方程解的唯一性。设λ₁为单位圆盘上-Δ的第一狄利克雷特征值，令ε₊:=λ₁⁻¹/²，我们证明存在δ>0，使得对任意ε∈(ε₊-δ,∞)，径向解是唯一的弱解。更一般地，我们对ℝᴺ（2≤N≤4）中带非平凡边界数据的有界连通C¹,¹区域上的金兹堡-朗道系统，建立了类似的唯一性结果。特别地，在凸性阈值ε₊以下的小范围内，严格凸性论证不再适用时，唯一性依然成立。证明结合了ε≥ε₊时的严格凸性、紧性论证、ε=ε₊处解的非退化性以及隐函数定理。
英文摘要
In this paper, we give a partial answer to Brezis' Open Problem~2.1, which concerns the uniqueness of solutions to the Ginzburg--Landau equation in the unit disc with the degree-one boundary condition. Let $λ_1$ be the first Dirichlet eigenvalue of $-Δ$ in the unit disc and set $\varepsilon_*:=λ_1^{-1/2}$. We prove that there exists $δ>0$ such that the radial solution is the unique weak solution for every $\varepsilon\in(\varepsilon_*-δ,\infty)$. More generally, we establish the analogous uniqueness result for the Ginzburg--Landau system on bounded connected $C^{1,1}$ domains in $\mathbb R^N$, $2\leq N\leq4$, with nontrivial boundary data. In particular, uniqueness persists slightly below the convexity threshold $\varepsilon_*$, where the strict convexity argument is no longer available. The proof combines strict convexity for $\varepsilon\geq\varepsilon_*$ with a compactness argument, nondegeneracy of the solution at $\varepsilon=\varepsilon_*$ and the implicit function theorem.
Comments12 pages. Comments and suggestions are most welcome
