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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 Euclideanspace $\mathbb{R}^3$ 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 aclosed simple curve $\Gamma\subset\mathbb{R}^3$ and a function $\mathcal{H}$,in this paper we investigate the geometry of compact $\mathcal{H}$-surfacesspanning $\Gamma$ in terms of $\Gamma$. Under mild assumptions on$\mathcal{H}$, we prove non-existence of closed $\mathcal{H}$-surfaces, incontrast with the classical case of constant mean curvature. We give conditionson $\mathcal{H}$ that ensure that if $\Gamma$ is a circle, then $M$ is arotational surface. We also establish the existence of estimates of the area of$\mathcal{H}$-surfaces in terms of the height of the surface.

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