Radial graphs of constant curvature and prescribed boundary

Abstract

In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with constant curvature and that realizes the prescribed boundary. As an application we prove that if $\Omega \subset {\mathbb{S}}^n$ is a mean convex domain whose closure is contained in an open hemisphere of ${\mathbb{S}}^n$ then, for $0<R<n(n-1),$ there exists a radial graph of constant scalar curvature $R$ and boundary $\partial \Omega .$

19 pages. Revised version. To appear in Calculus of Variations and Partial Differential Equations

References in corpus (1)

• Starshaped locally convex hypersurfaces with prescribed curvature and boundary

Cited by in corpus (5)

• Convex Hypersurfaces with Prescribed Scalar Curvature and Asymptotic Boundary in Hyperbolic Space
• Strictly Locally Convex Hypersurfaces with Prescribed Curvature and Boundary in Space Forms
• Lipschitz Continuous Hypersurfaces with Prescribed Curvature and Asymptotic Boundary in Hyperbolic Space
• The boundary value Minkowski problem for Weingarten curvatures
• Strictly Locally Convex Radial Graphs of Prescribed Curvature and Boundary in Space Forms