Regularity for Fully Nonlinear Equations

完全非线性方程的正则性

• 批准号: 0200784
• 负责人: Yu Yuan
• 金额: $ 8.37万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200784&HistoricalAwards=false
• 关键词: Regularity; Fully; Nonlinear; Equations

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.

PI：Yu Yuan，华盛顿大学-0200784Abstract：先验估计的理论和对具有凸状条件的完全非线性方程式的理论。尽管从随机的最佳控制理论和校准Gemomemitrydo的特殊Lagrangian方程中的其他连接方程（例如ISAACS方程）没有通常的凸条件。近年来，只有初步尝试朝着这一方向发展。尽管复合物 - 安培方程具有通常的凸条件，但晶状体的不变性太大了。对于具有非平滑右手方的复杂的蒙格 - 安培方程，没有一个优先的理论。该项目集中在以下四个部分上。 Intart One的目的是通过在最近的工作中进一步使用TheIdeas，从而获得持有人的先验估计。在第二部分中，尝试研究规律性的3-D中的一般完全非线性椭圆方程，以呈现的工作，任何同质阶的两个解决方案都必须在3-D中进行完全非线性的方程式，必须是线性的。在第三部分中，目的是toanswer是否有任何统一的秩序解决方案四个或更高的尺寸特殊lagrangianequation tose tosswer。在第四部分中，目的是研究复杂的蒙格 - 安培方程的伯恩斯坦问题。差异方程和差异几何形状进一步应用了牛顿的演算的应用，用于研究自然法律和形状，甚至是我们现实世界中的某些现象。该项目涉及某些特定方程，例如来自最佳随机控制理论的ISAACS方程，特殊的拉格朗日方程和来自差分几何的复杂的蒙格 - 室外方程。理解这些方程式不仅会影响相关的数学领域，还会影响诸如物理等实地的数学数学。