Communications in Analysis and Geometry

Volume 29 (2021)

Number 4

Bernstein–Moser-type results for nonlocal minimal graphs

Pages: 761 – 777



Matteo Cozzi (Dipartimento di Matematica, Università degli Studi di Milano, Italy)

Alberto Farina (Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, Université de Picardie, Amiens, France)

Luca Lombardini (Dipartimento di Matematica, Università degli Studi di Padova, Italy)


We prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. This result generalizes fractional versions of classical theorems due to Bernstein and Moser. Our arguments rely on a general splitting result for blow-downs of nonlocal minimal graphs.

Employing similar ideas, we establish that entire nonlocal minimal graphs bounded on one side by a cone are affine.

Moreover, we show that entire graphs having constant nonlocal mean curvature are minimal, thus extending a celebrated result of Chern on classical CMC graphs.

The first author acknowledges support from a Royal Society Newton International Fellowship, from the MINECO grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P, and from the María de Maeztu Programme for Units of Excellence in R&D with project code MDM-2014-0445.

Received 26 July 2018

Accepted 19 November 2018

Published 22 July 2021