1 April 2019 A gradient estimate for nonlocal minimal graphs
Xavier Cabré, Matteo Cozzi
Duke Math. J. 168(5): 775-848 (1 April 2019). DOI: 10.1215/00127094-2018-0052


We consider the class of measurable functions defined in all of Rn that give rise to a nonlocal minimal graph over a ball of Rn. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known results, leads to the C regularity of the function in the ball. While the smoothness of nonlocal minimal graphs was known for n=1,2—but without a quantitative bound—in higher dimensions only their continuity had been established. To prove the gradient bound, we show that the normal to a nonlocal minimal graph is a supersolution of a truncated fractional Jacobi operator, for which we prove a weak Harnack inequality. To this end, we establish a new universal fractional Sobolev inequality on nonlocal minimal surfaces. Our estimate provides an extension to the fractional setting of the celebrated gradient bounds of Finn and of Bombieri, De Giorgi, and Miranda for solutions of the classical mean curvature equation.


Download Citation

Xavier Cabré. Matteo Cozzi. "A gradient estimate for nonlocal minimal graphs." Duke Math. J. 168 (5) 775 - 848, 1 April 2019. https://doi.org/10.1215/00127094-2018-0052


Received: 7 December 2017; Revised: 23 October 2018; Published: 1 April 2019
First available in Project Euclid: 6 March 2019

zbMATH: 07055193
MathSciNet: MR3934589
Digital Object Identifier: 10.1215/00127094-2018-0052

Primary: 53A10
Secondary: 28A75 , 35J60 , 47G20 , 49Q05 , 58J05

Keywords: fractional Sobolev inequalities , Gradient estimates , nonlocal minimal graphs , nonlocal minimal surfaces , regularity results , rigidity theorems , weak Harnack inequalities

Rights: Copyright © 2019 Duke University Press

Vol.168 • No. 5 • 1 April 2019
Back to Top