Existence of stable H-surfaces in cones and their representation as radial graphs

In this paper we study the Plateau problem for disk-type surfaces contained in conic regions of $\mathbb{R}^{3}$ and with prescribed mean curvature $H$. Assuming a suitable growth condition on $H$, we prove existence of a least energy $H$-surface $X$ spanning an arbitrary Jordan curve $Γ$ taken in the cone. Then we address the problem of describing such surface $X$ as radial graph when the Jordan curve $Γ$ admits a radial representation. Assuming a suitable monotonicity condition on the mapping $λ\mapstoλH(λp)$ and some strong convexity-type condition on the radial projection of the Jordan curve $Γ$, we show that the $H$-surface $X$ can be represented as a radial graph.

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