Interior estimates for the Monge–Ampère type fourth order equations

Abstract

. In this paper, we give several new approaches to study the interior estimates for a class of fourth order equations of Monge-Amp`ere type. First, we prove the 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 the 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 the integral bounds on the second derivatives and the inverse of the determinant.