Robert Finn and Jianan Lu

Some Remarkable Properties of h-Graphs

It is proved that if $H(u)$ is non-decreasing and if $H(-\infty)\linebreak \neq H(+\infty)$, then if $u(x)$ describes a graph over a disk $B_R(0)$, with (upward oriented) mean curvature $H(u)$, there is a bound on the gradient $|D\,u(0)|$ 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 that a) they are false if $H$ is constant, and b) the dependences indicated are essential (If $H(-\infty)=-\infty,\ H(+\infty)=\infty$, then the dependence on $u(0)$ can be deleted). The demonstrations are based on an existence theorem for a nonlinear boundary problem with singular data, which is of independent interest.

Mathematics Subject Classification: 53A10, 35J60, 35B45.

Key words and phrases: Mean curvature, $H$-graph, gradient estimate, Harnack inequality, moon surface, capillarity.