论文标题
Monge-ampère型四阶方程的内部估计
Interior estimates for Monge-Ampère type fourth order equations
论文作者
论文摘要
在本文中，我们提供了几种新方法来研究Monge-Ampère类型的第四阶方程的内部估计。首先，我们通过使用部分legendre变换证明了维度二的均匀方程的内部估计。作为应用程序，我们在不使用caffarelli-gutiérrez的估计值的情况下获得了伯恩斯坦定理的新证明，包括Chern构想的仿射最大表面。对于不均匀方程式，我们还通过依靠Monge-AmpèreSobolev不平等的积分方法获得了第二个维度的新证明。即使右手侧是单数，此证明也有效。在较高的维度中，我们根据第二个衍生物的积分界限和决定因素的倒数来获得内部规律性。
In this paper, we give several new approaches to study interior estimates for a class of fourth order equations of Monge-Ampère 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-Gutiérrez'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-Ampère 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.
