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

TL;DR

This paper proves the existence of constant mean curvature hypersurfaces as radial graphs over spherical domains with prescribed boundaries, extending classical results to positive mean curvature cases under certain conditions.

Contribution

It extends Serrin's classical boundary value problem results to include hypersurfaces with positive constant mean curvature as radial graphs over spherical domains.

Findings

Existence of hypersurfaces with prescribed boundary and positive mean curvature

Extension of Serrin's classical results to new curvature cases

Conditions under which radial graph solutions exist

Abstract

We establish the existence of hypersurfaces with constant mean curvature and a prescribed boundary in Euclidean space, represented as radial graphs over domains of the unit sphere. Under the assumptions that the mean curvature of the domain's boundary is positive and that a subsolution exists for the associated Dirichlet problem, we extend Serrin's classical result to include the case of positive constant mean curvature.