The Dirichlet problem for a class of prescribed curvature equations

TL;DR

This paper investigates the Dirichlet problem for certain prescribed curvature equations, establishing existence results for regular hypersurfaces with specified curvature and boundary conditions in both degenerate and non-degenerate cases.

Contribution

It provides new existence theorems for $C^{1,1}$ hypersurfaces solving prescribed curvature equations, including degenerate cases.

Findings

Existence of $C^{1,1}$ solutions in degenerate cases

Solutions with prescribed curvature and constant boundary

Analysis of both degenerate and non-degenerate cases

Abstract

In this paper, we consider the Dirichlet problem for a class of prescribed curvature equations. Both degenerate and non-degenerate cases are considered. The existence of the $C^{1,1}$ regular graphic hypersurfaces with prescribing a class of curvatures and constant boundary is proved for the degenerate case.