Radial graphs over domains of $\mathbb{S}^{n}$ with prescribed mean curvature

TL;DR

This paper establishes the existence and uniqueness of radial graphs over domains in the sphere with prescribed mean curvature, under certain boundary and monotonicity conditions, advancing geometric analysis on spherical surfaces.

Contribution

It introduces new existence and uniqueness results for radial graphs with prescribed mean curvature on spherical domains, under specific boundary and curvature conditions.

Findings

Proves existence of radial graphs with prescribed mean curvature.

Establishes uniqueness of such graphs under given conditions.

Provides conditions ensuring boundary and curvature compatibility.

Abstract

We prove the existence and uniqueness of radial graphs over a given domain of $\mathbb{S}^{n}$ having boundary on the sphere $\mathbb{S}^{n}$ and whose mean curvature at every point equals a prescribed positive function satisfying suitable barrier-type and monotonicity conditions.