Regularity for Fully Nonlinear Equations

摘要：建立了具有凸性条件的全非线性方程的先验估计和可解性理论。而其他具体的方程，如随机最优控制理论中的艾萨克斯方程和标定几何中的特殊拉格朗日方程，则不具有通常的凸性条件。近年来，在这个方向上只进行了初步的尝试。复monge - ampere方程虽然具有一般的凸性条件，但其全纯不变性太大。对于具有非光滑边的复杂蒙日-安培方程，没有优先级估计的理论。本项目主要包括以下四个部分。在第一部分中，目标是通过在最近的工作中进一步使用这些思想来推导有限分段线性艾萨克方程的先验估计。在第二部分中，根据最近的研究表明，三维完全非线性椭圆方程的任何齐次二阶解都必须是线性的，尝试研究三维中一般完全非线性椭圆方程的正则性。第三部分的目的是回答四维或四维以上的特殊拉格朗日方程的齐次二阶解是否平凡。第四部分的目的是研究复蒙日-安培方程的Bernstein问题。微分方程和微分几何是牛顿微积分在研究自然规律和形状，甚至是我们现实世界中的一些现象方面的进一步应用。本课题研究了一些特殊的方程，如最优随机控制理论中的艾萨克斯方程、特殊拉格朗日方程和微分几何中的复蒙日-安培方程。理解这些方程不仅会对相关的数学领域产生影响，还会对数学以外的领域产生影响，比如物理学。

PI: Yu Yuan, University of WashingtonDMS-0200784Abstract:The theory of a priori estimates and solvability for fully nonlinearequations with the convexity condition are well developed. While otherconcrete equations like Isaacs equations from the stochastic optimalcontrol theory and special Lagrangian equations from calibration geometrydo not have the usual convexity condition. Only preliminary attempts weremade toward this direction in recent years. Though the complexMonge-Ampere equations have the usual convexity condition, theholomorphic invariance is too large. There is no theory of a prioriestimates for the complex Monge-Ampere equations with non-smoothright hand side. This project concentrates on the following four parts. Inpart one, the objective is to derive Holder a priori estimates forfinitely piecewise linear Isaacs equations by further employment of theideas in recent work. In part two, the attempt is to study the regularityfor general fully nonlinear elliptic equations in 3-d in the light ofrecent work that any homogeneous order two solution to fully nonlinearelliptic equation in 3-d must be linear. In part three, the purpose is toanswer whether any homogeneous order two solution to special Lagrangianequation of dimension four or higher is trivial. In part four, the aim isto study the Bernstein problem for complex Monge-Ampere equations.Differential equations and differential geometry are further applicationsof Newton's calculus to the investigation of laws and shapes of nature,and even some phenomena of our real world. This project deals with someparticular equations, like Isaacs equations from optimal stochasticcontrol theory, special Lagrangian equations and complex Monge-Ampereequations from differential geometry. Understanding those equations wouldhave impacts on not only the related mathematical fields, but also fieldsoutside mathematics like physics.