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 R3and 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↦ λH( 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.