Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A dimension-free Hermite-Hadamard inequality via gradient estimates for the torsion function


Authors: Jianfeng Lu and Stefan Steinerberger
Journal: Proc. Amer. Math. Soc. 148 (2020), 673-679
MSC (2010): Primary 26B25, 28A75, 31A05, 31B05, 35B50
DOI: https://doi.org/10.1090/proc/14843
Published electronically: November 4, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Omega \subset \mathbb{R}^n$ be a convex domain, and let $ f:\Omega \rightarrow \mathbb{R}$ be a subharmonic function, $ \Delta f \geq 0$, which satisfies $ f \geq 0$ on the boundary $ \partial \Omega $. Then

$\displaystyle \int _{\Omega }{f ~dx} \leq \vert\Omega \vert^{\frac {1}{n}} \int _{\partial \Omega }{f ~d\sigma }.$    

Our proof is based on a new gradient estimate for the torsion function, $ \Delta u = -1$ with Dirichlet boundary conditions, which is of independent interest.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 26B25, 28A75, 31A05, 31B05, 35B50

Retrieve articles in all journals with MSC (2010): 26B25, 28A75, 31A05, 31B05, 35B50


Additional Information

Jianfeng Lu
Affiliation: Department of Mathematics, Department of Physics, and Department of Chemistry, Duke University, Box 90320, Durham, North Carolina 27708
Email: jianfeng@math.duke.edu

Stefan Steinerberger
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06510
Email: stefan.steinerberger@yale.edu

DOI: https://doi.org/10.1090/proc/14843
Keywords: Hermite--Hadamard, subharmonicity, Brownian motion
Received by editor(s): May 9, 2019
Published electronically: November 4, 2019
Additional Notes: The first author was supported in part by the National Science Foundation via grant DMS-1454939.
The second author was supported in part by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.
Communicated by: Deane Yang
Article copyright: © Copyright 2019 American Mathematical Society