Regularity properties of H-graphs

Abstract

It is proved that if H(u) is non-decreasing and if $H(-\infty )\ne H(+\infty )$ , then if u (x) describes a graph over a disk BR (0), with (upward oriented) mean curvature H(u), there is a bound on the gradient $|Du(0)|$ that depends only on R, on u (0), and on the particular function H (u). As a consequence a form of Harnack's inequality is obtained, in which no positivity hypothesis appears. The results are qualitatively best possible, in the senses a) that they are false if H is constant, and b) the dependences indicated are essential. The demonstrations are based on an existence theorem for a nonlinear boundary problem with singular data, which is of independent interest.