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.