AI 中文总结
本文对负重力毛细方程经典解的孤立奇点进行了完整分类，证明解必有界或趋于正负无穷，并给出了无界解的精确渐近展开及几何结构，且非径向解族具有无穷维自由度。
AI 中文摘要
我们在没有对称性、符号、单侧有界性或爆破速率等先验假设的情况下，对负重力毛细方程 \\(\operatorname{div}\frac{Du}{\sqrt{1+|Du|^2}}=-u\\)（在 \\(B_R\setminus\{0\}\subset\mathbb R^n\\) 中，\\(n\ge2\\)）的经典解的孤立奇点进行了分类。每个这样的解要么在奇点附近有界，要么一致趋于 \\(+\infty\\) 或 \\(-\infty\\)。在有界情形下，该解可延拓穿过奇点，成为 \\(W^{1,1}_{\mathrm{loc}}\\) 分布解；当 \\(2\le n\le7\\) 时，此延拓具有唯一的连续代表元。在无界情形下，对于某个 \\(\varepsilon\in\{-1,1\}\\)，有 \\(u(x)=\varepsilon\left(\frac{n-1}{|x|} -\frac{n+3}{2(n-1)^2}|x|^3\right)+O(|x|^5)\\)，且在角变量上一致成立。在每个固定的更小球内，\\(\varepsilon u\\) 的所有足够高的水平集都是连通、光滑、严格凸的超曲面，并包围奇点。相应的压力重标度图形以重数一光滑收敛到圆柱面 \\(\partial B_{n-1}\times\mathbb R\\)。非径向例子构成无穷维族，它们与径向极点在每个代数阶上都一致，因此完整的渐近展开并不能确定奇异解的芽。分类的证明结合了临界尾部估计和对数 \\(BV\\) 紧性，以及水平集刚性和毛细方程的平移恒等式。临界二维情形需要额外的有限高度分析来排除平移缺陷。
英文摘要
We classify isolated singularities of classical solutions to the capillary equation with negative gravity, \[ \operatorname{div}\frac{Du}{\sqrt{1+|Du|^2}}=-u \qquad\text{in }B_R\setminus\{0\}\subset\mathbb R^n, \qquad n\ge2, \] without a priori assumptions on symmetry, sign, one-sided boundedness, or blow-up rate. Every such solution is either bounded near the puncture or tends uniformly to $+\infty$ or $-\infty$. In the bounded case, the solution extends across the puncture as a $W^{1,1}_{\mathrm{loc}}$ distributional solution; this extension has a unique continuous representative when $2\le n\le7$. In the unbounded case, for some $\varepsilon\in\{-1,1\}$, \[ u(x)=\varepsilon\left(\frac{n-1}{|x|} -\frac{n+3}{2(n-1)^2}|x|^3\right)+O(|x|^5), \] uniformly in the angular variable. In every fixed smaller ball, all sufficiently high level sets of $\varepsilon u$ are connected, smooth, strictly convex hypersurfaces enclosing the puncture. The corresponding pressure-rescaled graphs converge smoothly with multiplicity one to the round cylinder $\partial B_{n-1}\times\mathbb R$. Nonradial examples form infinite-dimensional families that agree with the radial pole to every algebraic order, so the complete asymptotic expansion does not determine the singular solution germ. The proof of the classification combines critical tail estimates and logarithmic $BV$ compactness with level-set rigidity and the translation identities of the capillary equation. The critical two-dimensional case requires an additional finite-height analysis to exclude a translation defect.
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