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

TL;DR

This paper derives curvature bounds for p-convex hypersurfaces with prescribed curvature in Euclidean space, proving existence and interior regularity results for related geometric PDEs.

Contribution

It establishes new curvature estimates for p-convex hypersurfaces with prescribed curvature and proves existence and interior C^2 estimates for the associated Dirichlet problem.

Findings

Curvature estimates for p-convex hypersurfaces with p ≥ n/2.

Existence of star-shaped hypersurfaces with prescribed curvature.

Interior C^2 estimates for solutions to the Dirichlet problem.

Abstract

In this paper, we establish the curvature estimates for $p$-convex hypersurfaces in $\mathbb{R}^{n+1}$ of prescribed curvature with $p\geq \frac{n}{2}$. 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.