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

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