A height estimate for \(H\)-surfaces and existence of \(H\)-graphs (Q2731392)

The author contributes to the challenging subject of determining the existence of smooth solutions to the equation of nonparametric surfaces of constant mean curvature \(H\) NEWLINE\[NEWLINE\operatorname {div}\frac{\nabla u}{\sqrt{1+| \nabla u| ^2}}=-2H,\quad H\in{\mathbb R}NEWLINE\]NEWLINE on a domain \(\Omega\) in \({\mathbb R}^2.\) Let \(\Omega\) be a convex bounded domain in \({\mathbb R}^2\) with area \(A.\) Then it is shown that, for any real number \(H\) satisfying \(AH^2<\rho^2\pi\) with \(\rho=(\sqrt5 -1)/2,\) there exists a graph over \(\Omega\) with constant mean curvature \(H\) and boundary \(\partial\Omega.\) The proof uses an \(L^{\infty}\) estimate for compact constant mean curvature surfaces with planar boundary in terms of the \(L^1\) norm of a component of its Gauss map.