Regularity properties of \(H\)-graphs (Q1266441)

The surfaces described as a graph over a disk \(B_R(0)\) in the three-dimensional Euclidean space with the prescribed mean curvature given by a non-decreasing function \(H(u)\), \(H(-\infty)\neq H(+\infty)\), where \(x_3=u(x_1,x_2)\), are considered. In other words, \(u(x)\) is a solution to the following problem: \(\text{div } T u=2H(u)\), where \(Tu= Du/W\), \(W=\sqrt{1+| Du| ^2}\) and \(H'(u)\geq 0\). Namely, the upper bound on the length of the gradient \(| D u(0)| \) in terms of only \(R\), the value \(u(0)\) and the prescribed function \(H\) is obtained. An interesting issue is that the previous result fails if \(H\) is constant. To prove the main result, the authors prove an interesting special existence theorem for a nonlinear boundary problem with singular data. Other properties of solutions over the domains with singularities are also discussed.