Interior C^{1,1} regularity of solutions to degenerate Monge-Amp\`{e}re type equations

TL;DR

This paper proves interior C^{1,1} regularity for viscosity solutions of a degenerate Monge-Ampère type equation under certain conditions, using auxiliary functions and Pogorelov estimates.

Contribution

It establishes C^{1,1} regularity for solutions to degenerate Monge-Ampère equations under A3 and A3w^+ conditions, extending regularity theory.

Findings

Solutions are in C^{1,1}( ext{interior}) under A3 and A3w^+ conditions.

Construction of auxiliary functions yields uniform a priori estimates.

Pogorelov type estimates are established for degenerate equations.

Abstract

In this paper, we study the interior C^{1,1} regularity of viscosity solutions for a degenerate Monge-Amp\`{e}re type equation \det[D^{2}u-A(x, u, Du)]=B(x, u, Du) when B \geq 0 and B^{\frac{1}{n-1}}\in C^{1,1}(\bar{\Omega}\times\mathbb{R}\times \mathbb{R}^n). We prove that u\in C^{1,1}(\Omega) under the A3 condition and A3w^+ condition respectively. In the former case, we construct a suitable auxiliary function to obtain uniform {\it a priori} estimates directly. In the latter case, the main argument is to establish the Pogorelov type estimates, which are interesting independently.