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 $\Gamma\subset\mathbb{R}^3$ and a function $\mathcal{H}$, in this paper we investigate the geometry of compact $\mathcal{H}$-surfaces spanning $\Gamma$ in terms of $\Gamma$. 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 $\Gamma$ 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.