A gradient estimate for nonlocal minimal graphs

TL;DR

This paper proves a gradient estimate for nonlocal minimal graphs, leading to their smoothness in higher dimensions, by analyzing the fractional Jacobi operator and establishing new inequalities on nonlocal surfaces.

Contribution

It introduces a novel gradient bound for nonlocal minimal graphs, extending classical results to the fractional setting and providing tools for higher-dimensional regularity.

Findings

Gradient of nonlocal minimal graphs is bounded by oscillation power.

Normal vector is a supersolution of a fractional Jacobi operator.

Established a new fractional Sobolev inequality on nonlocal minimal surfaces.

Abstract

We consider the class of measurable functions defined in all of $\mathbb{R}^n$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^n$. 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 $C^\infty$ 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 & Miranda for solutions of the classical mean curvature equation.