Interior estimates for Monge-Amp\`ere type fourth order equations

TL;DR

This paper introduces new methods for interior estimates of Monge-Ampère type fourth order equations, providing simplified proofs and extending regularity results to higher dimensions with integral bounds.

Contribution

It presents novel approaches for interior estimates, including new proofs of classical results and regularity criteria in higher dimensions for Monge-Ampère type equations.

Findings

Interior estimates for homogeneous equations in 2D via partial Legendre transform

New proof of Bernstein theorem and Chern conjecture without Caffarelli-Gutiérrez's estimate

Interior regularity in higher dimensions based on integral bounds

Abstract

In this paper, we give several new approaches to study interior estimates for a class of fourth order equations of Monge-Amp\`ere type. First, we prove interior estimates for the homogeneous equation in dimension two by using the partial Legendre transform. As an application, we obtain a new proof of the Bernstein theorem without using Caffarelli-Guti\'errez's estimate, including the Chern conjecture on affine maximal surfaces. For the inhomogeneous equation, we also obtain a new proof in dimension two by an integral method relying on the Monge-Amp\`ere Sobolev inequality. This proof works even when the right hand side is singular. In higher dimensions, we obtain the interior regularity in terms of integral bounds on the second derivatives and the inverse of the determinant.