PreprintarXiv licencearXiv
Concavity Properties of Robin Solutions on C3,1 Uniformly Convex Domains
Abstract
We prove that, on a bounded uniformly convex domain of class C3,1, the first Robin eigenfunction is strictly log-concave and the Robin torsion function is strictly 1/2-concave for all sufficiently large Robin parameters. This means that Conjecture 1.1 of Andrews, Clutterbuck and Hauer holds for uniformly convex C3,1 domains in any dimension, and settles an open problem posed by Crasta and Fragalà when the domain has C3,1 regularity. Our proof derives suitable uniform C2-decay estimates by maximum principle arguments, thereby removing the imposed higher-order boundary regularity assumption required previously.
Related
- Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets
- Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains
- Concavity of the Ornstein--Uhlenbeck torsion function on centered ellipses and its failure on convex domains
- A remark on the compactness of manifolds with non-negative Ricci curvature and uniformly convex boundary
- Uniformly hyperbolic strictly convex billiards
BibTeX
@misc{ye2026concavity,
title = {{Concavity Properties of Robin Solutions on $C^3,1$ Uniformly Convex Domains}},
author = {Dong Ye and Dekai Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06223},
eprint = {2609.06223},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06223},
}
RIS
TY - UNPB
TI - Concavity Properties of Robin Solutions on C^3,1 Uniformly Convex Domains
AU - Dong Ye
AU - Dekai Zhang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06223
UR - https://arxiv.org/abs/2609.06223
AB - We prove that, on a bounded uniformly convex domain of class C^3,1, the first Robin eigenfunction is strictly log-concave and the Robin torsion function is strictly 1/2-concave for all sufficiently large Robin parameters. This means that Conjecture 1.1 of Andrews, Clutterbuck and Hauer holds for uniformly convex C^3,1 domains in any dimension, and settles an open problem posed by Crasta and Fragalà when the domain has C^3,1 regularity. Our proof derives suitable uniform C^2-decay estimates by maximum principle arguments, thereby removing the imposed higher-order boundary regularity assumption required previously.
ER -