Curvature estimates for $p$-convex hypersurfaces of prescribed curvature

Abstract

In this paper, we establish curvature estimates for $p$-convex hypersurfaces in ${\mathbb{R}}^{n+1}$ of prescribed curvature with $p\ge n/2$. The existence of a star-shaped hypersurface of prescribed curvature is obtained. We also prove a type of interior $C^2$ estimates for solutions to the Dirichlet problem of the corresponding equation.