Isolated singularities of the capillary equation with negative gravity
Abstract
We classify isolated singularities of classical solutions to the capillary equation with negative gravity, divDu/(√(1+|Du|2))=−u in BR∖{0}⊂ℝn, n≥2, 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 +∞ or −∞. In the bounded case, the solution extends across the puncture as a W1,1loc distributional solution; this extension has a unique continuous representative when 2≤ n≤7. In the unbounded case, for some ε∈{−1,1}, u(x)=ε((n−1)/|x| −(n+3)/(2(n−1)2)|x|3)+O(|x|5), uniformly in the angular variable. In every fixed smaller ball, all sufficiently high level sets of ε u are connected, smooth, strictly convex hypersurfaces enclosing the puncture. The corresponding pressure-rescaled graphs converge smoothly with multiplicity one to the round cylinder ∂ Bn−1×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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BibTeX
@misc{deng2026isolated,
title = {{Isolated singularities of the capillary equation with negative gravity}},
author = {Bin Deng and Jiahuan Li and Yilu Liu and Xi-nan Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18832},
eprint = {2609.18832},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18832},
}
RIS
TY - UNPB
TI - Isolated singularities of the capillary equation with negative gravity
AU - Bin Deng
AU - Jiahuan Li
AU - Yilu Liu
AU - Xi-nan Ma
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18832
UR - https://arxiv.org/abs/2609.18832
AB - We classify isolated singularities of classical solutions to the capillary equation with negative gravity, divDu/(√(1+|Du|^2))=−u in B_R∖{0}⊂ℝ^n, n≥2, 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 +∞ or −∞. In the bounded case, the solution extends across the puncture as a W^1,1_loc distributional solution; this extension has a unique continuous representative when 2≤ n≤7. In the unbounded case, for some ε∈{−1,1}, u(x)=ε((n−1)/|x| −(n+3)/(2(n−1)^2)|x|^3)+O(|x|^5), uniformly in the angular variable. In every fixed smaller ball, all sufficiently high level sets of ε u are connected, smooth, strictly convex hypersurfaces enclosing the puncture. The corresponding pressure-rescaled graphs converge smoothly with multiplicity one to the round cylinder ∂ B_n−1×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.
ER -