Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map

Given a C-1 function H defined in the unit sphere S-2, an H-surface M is a surface in the Euclidean space R-3 whose mean curvature H-M satisfies H-M( p) = H(N-p), p is an element of M, where N is the Gauss map of M. Given a closed simple curve Gamma subset of R-3 and a function H, in this paper we investigate the geometry of compact H-surfaces spanning Gamma in terms of Gamma. Under mild assumptions on H, we prove non-existence of closed H-surfaces, in contrast with the classical case of constant mean curvature. We give conditions on H that ensure that if Gamma is a circle, then M is a rotational surface. We also establish the existence of estimates of the area of H-surfaces in terms of the height of the surface.