A gradient estimate for nonlocal minimal graphs

Matteo Cozzi, Xavier Cabré

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

We consider the class of measurable functions defined in all of ℝ n that give rise to a nonlocal minimal graph over a ball of ℝ n . 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.

Original languageEnglish
Pages (from-to)775-848
Number of pages74
JournalDuke Mathematical Journal
Volume168
Issue number5
DOIs
Publication statusPublished - 1 Apr 2019

ASJC Scopus subject areas

  • Mathematics(all)

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