PreprintCC BY-NC-SA 4.0arXiv
Mean curvature bounds for the obstacle problem
Abstract
We prove optimal (n−1)-semiconvexity estimates for solutions to the classical obstacle problem, possibly with a smooth right-hand side. We deduce that the mean curvature of the free boundary is universally bounded above in all dimensions. Existing examples show that full curvature bounds fail in dimensions three and higher. Noticeably, these results are novel even for a constant right-hand side in the n=2 case.
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BibTeX
@misc{colombo2026mean,
title = {{Mean curvature bounds for the obstacle problem}},
author = {Giacomo Colombo and Federico Franceschini},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.19991},
eprint = {2608.19991},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19991},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Mean curvature bounds for the obstacle problem
AU - Giacomo Colombo
AU - Federico Franceschini
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.19991
UR - https://arxiv.org/abs/2608.19991
AB - We prove optimal (n−1)-semiconvexity estimates for solutions to the classical obstacle problem, possibly with a smooth right-hand side. We deduce that the mean curvature of the free boundary is universally bounded above in all dimensions. Existing examples show that full curvature bounds fail in dimensions three and higher. Noticeably, these results are novel even for a constant right-hand side in the n=2 case.
ER -