Regularity for Fully Nonlinear Equations

主要研究者：Yu Yuan，University of Washington DMS-0200784摘要：具有凸性条件的完全非线性方程组的先验估计和可解性理论已经得到了很好的发展.而其它的具体方程，如随机最优控制理论中的Isaacs方程和标定几何中的特殊Lagrange方程，则不具有通常的凸性条件.近年来，我们只是朝着这个方向进行了初步的尝试。复Monge-Ampere方程虽然具有通常的凸性条件，但全纯不变性太大。对于具有非光滑右手边的复杂Monge-Ampere方程，目前还没有一个先验估计的理论。本课题主要包括以下四个部分。在第一部分中，我们的目标是通过进一步应用最近工作中的思想，得到非线性分段线性Isaacs方程的保持器先验估计。在第二部分中，根据最近的工作，即三维完全非线性椭圆型方程的任何齐次二阶解必是线性的，尝试研究一般三维完全非线性椭圆型方程的正则性。第三部分的目的是回答四维或更高维的特殊拉格朗日方程的齐次二阶解是否平凡。第四部分研究复Monge-Ampere方程的伯恩斯坦问题，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，甚至是我们真实的世界的某些现象方面的进一步应用。本项目研究的是一些特殊的方程，如最优随机控制理论中的Isaacs方程，微分几何中的特殊拉格朗日方程和复Monge-Amper方程。理解这些方程不仅会对相关的数学领域产生影响，而且会对物理学等数学以外的领域产生影响。

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.