$C^{2,\alpha}$ estimates and existence results for certain nonconcave PDE

TL;DR

This paper extends regularity estimates to a class of nonconcave PDEs involving convex and weakly concave Hessian functions, and proves an existence theorem for a generalized Monge-Ampère equation.

Contribution

It generalizes $C^{2,eta}$ estimates to nonconcave PDEs and establishes existence results for a broader class of Monge-Ampère type equations.

Findings

Established $C^{2,eta}$ estimates for certain nonconcave PDEs.

Proved an existence theorem for a generalized Monge-Ampère PDE.

Extended previous results by Collins, inspired by Caffarelli and Yuan.

Abstract

We establish $C^{2,\alpha}$ estimates for PDE of the form convex $+$ a sum of weakly concave functions of the Hessian, thus generalising a recent result of Collins which is in turn inspired by a theorem of Caffarelli and Yuan. Independently, we also prove an existence result for a certain generalised Monge-Amp\`ere PDE.