Interior estimates for the Monge-Ampre type fourth order equations

In this paper, we give several new approaches to study interior estim-ates for a class of fourth order equations of Monge-Ampere type. First, we prove interior estimates for the homogeneous equation in dimension two by using the par-tial Legendre transform. As an application, we obtain a new proof of the Bernstein theorem without using Caffarelli-Gutierrez's estimate, including the Chern conjec-ture 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-Ampere 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.