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

Abstract

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 \(\Gamma \) taken in the cone. Then we address the problem of describing such surface X as radial graph when the Jordan curve \(\Gamma \) admits a radial representation. Assuming a suitable monotonicity condition on the mapping \(\lambda \mapsto \lambda H(\lambda p)\) and some strong convexity-type condition on the radial projection of the Jordan curve \(\Gamma \), we show that the H-surface X can be represented as a radial graph.