Second order estimates for convex solutions of degenerate $k$-Hessian equations

TL;DR

This paper addresses the open problem of establishing second order estimates for convex solutions to degenerate $k$-Hessian equations with non-homogeneous boundary conditions, under minimal regularity assumptions on the right-hand side.

Contribution

It provides a solution to the open problem of second order estimates for convex solutions of degenerate $k$-Hessian equations with minimal regularity assumptions.

Findings

Established second order estimates for convex solutions

Solved the open problem for degenerate $k$-Hessian equations

Applicable to strictly convex bounded domains

Abstract

The $C^{1,1}$ estimate of the Dirichlet problem for degenerate $k$-Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function $f$ is only assumed to satisfy $f^{1/(k-1)} \in C^{1,1}$. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain.