AI 中文总结
该研究构造反例以否定边界唯一延拓问题的光滑情形，揭示非局部椭圆方程的灵活性原理，证明调和函数边界唯一延拓在光滑范畴中失效。
AI 中文摘要
对于每个 $n\geq3$，我们构造了一个非恒定实值函数 $U\in C^\infty(\overline{\mathbb R^n_+})$，该函数在上半空间中调和，其完整边界喷流在 $\partial\mathbb R^n_+$ 的一个紧无处稠密子集上消失，该子集具有正的 $(n-1)$ 维测度。核心构造在二维空间中进行，得到 $n=3$ 的情况；更高维的例子通过柱形提升得到。在每个维度中，例外集可占据固定边界立方体的任意大比例。这以否定方式解决了 Bourgain 和 Wolff 在 1990 年留下的边界唯一延拓问题的光滑情形 \cite[p.~260]{BourgainWolff1990}。在三维空间中，通过 Möbius–Kelvin 变换得到单位球中的相应反例。它否定了 Nadirashvili 关于边界奇异集的光滑单位球猜想 \cite[Conjecture~4, p.~232]{Nadirashvili1997}，进而也否定了 Logunov 和 Malinnikova \cite[Section~7.4]{LogunovMalinnikova2020} 以及 Lin \cite[Conjecture~3, pp.~15--16]{Lin2020Current} 随后记录的仅梯度形式。该证明揭示了非局部椭圆方程的一个隐藏灵活性原理：微观修改可对外部数据施加宏观控制。半拉普拉斯算子的定量校正机制跨尺度迭代，在正测度集上产生平坦的非局部柯西数据。因此非局部性具有显著的双重性质：驱动唯一延拓刚性的相同长程相互作用，也能提供使该刚性在光滑范畴中失效的灵活性。
英文摘要
For every $n\geq 3$, we construct a nonconstant real-valued function $U\in C^\infty(\overline{\mathbb R^n_+})$, harmonic in the upper half-space, whose complete boundary jet vanishes on a compact nowhere dense subset of $\partial\mathbb R^n_+$ of positive $(n-1)$-dimensional measure. The essential construction takes place in two dimensions and yields the case $n=3$; higher-dimensional examples follow by cylindrical lifting. In every dimension, the exceptional set may occupy an arbitrarily large proportion of a fixed boundary cube. This resolves, in the negative, the smooth case of the boundary unique-continuation problem left open by Bourgain and Wolff in 1990 \cite[p.~260]{BourgainWolff1990}. In dimension three, a Möbius--Kelvin transfer gives the corresponding counterexample in the unit ball. It disproves Nadirashvili's smooth unit-ball conjecture on boundary singular sets \cite[Conjecture~4, p.~232]{Nadirashvili1997} and, a fortiori, disproves the gradient-only formulation subsequently recorded by Logunov and Malinnikova \cite[Section~7.4]{LogunovMalinnikova2020} and by Lin \cite[Conjecture~3, pp.~15--16]{Lin2020Current}. The proof uncovers a hidden flexibility principle for nonlocal elliptic equations: microscopic modifications can exert macroscopic control over exterior data. A quantitative correction mechanism for the half-Laplacian, iterated across scales, produces flat nonlocal Cauchy data on a set of positive measure. Thus nonlocality has a striking dual character: the same long-range interaction that drives unique-continuation rigidity can also furnish the flexibility through which that rigidity fails in the smooth category.
