# Regularity properties of H-graphs

### Robert Finn

Stanford University, USA### J. Lu

IBM Corporation, Mountain View, USA

## Abstract

It is proved that if H(u) is non-decreasing and if $H (-\infty) \neq H (+\infty)$ , 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.

## Cite this article

Robert Finn, J. Lu, Regularity properties of H-graphs. Comment. Math. Helv. 73 (1998), no. 3, pp. 379–399

DOI 10.1007/S000140050060