**Robert Finn and Jianan Lu**

## Some Remarkable Properties of *h*-Graphs

**abstract:**

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.