Radial Powered Mean Curvature Flow with Robin Boundary Conditions
Abstract
We study a positive radial graph evolving by V=Hα+b, with α>0 and b<0, in the unit cylinder and with ur(1,t)=u(1,t). We separate finite-time continuation estimates from the time-uniform interior estimates needed for convergence of normalized translates. On existing classical intervals we prove height and gradient bounds and an explicit conversion of velocity bounds into estimates for H, ur/r, and urr. We derive the correct boundary defect for comparisons with finite-slope translators. Under independent interior velocity bounds and two-sided slope trapping, we prove compactness and identify every translated limit with the cup translator. We identify the radial zero-number comparison still required to establish that trapping. Stationary caps and power-law Robin laws are included.
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BibTeX
@misc{pu2026radial,
title = {{Radial Powered Mean Curvature Flow with Robin Boundary Conditions}},
author = {Tianhong Pu and Xiaoqian Xin and Lixia Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08283},
eprint = {2610.08283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08283},
}
RIS
TY - UNPB
TI - Radial Powered Mean Curvature Flow with Robin Boundary Conditions
AU - Tianhong Pu
AU - Xiaoqian Xin
AU - Lixia Yuan
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08283
UR - https://arxiv.org/abs/2610.08283
AB - We study a positive radial graph evolving by V=H^α+b, with α>0 and b<0, in the unit cylinder and with u_r(1,t)=u(1,t). We separate finite-time continuation estimates from the time-uniform interior estimates needed for convergence of normalized translates. On existing classical intervals we prove height and gradient bounds and an explicit conversion of velocity bounds into estimates for H, u_r/r, and u_rr. We derive the correct boundary defect for comparisons with finite-slope translators. Under independent interior velocity bounds and two-sided slope trapping, we prove compactness and identify every translated limit with the cup translator. We identify the radial zero-number comparison still required to establish that trapping. Stationary caps and power-law Robin laws are included.
ER -