Title: A gradient estimate for nonlocal minimal graphs,
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.