Fully nonlinear elliptic and parabolic equations

完全非线性椭圆和抛物线方程

• 批准号: 1100966
• 负责人: Yu Yuan
• 金额: $ 24万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1100966&HistoricalAwards=false
• 关键词: Fully; nonlinear; elliptic; parabolic; equations

This project concentrates on the study of special Lagrangian equations, symmetric Hessian equations, Isaacs equations, complex Monge-Ampere equations, and their parabolic versions (e.g., Lagrangian mean curvature flows). The theory of regularity and solvability for fully nonlinear uniformly elliptic and parabolic equations (with the convexity condition in general dimensions and without the convexity hypothesis in dimension two) is well developed. The concrete equations just listed either do not satisfy the convexity condition or do not exhibit uniform ellipticity or parabolicity. Only preliminary attempts have been made in the general saddle cases. Substantial advances have been achieved for the symmetric Hessian equations and the complex Monge-Ampere equations, yet there is still no Schauder or Calderon-Zygmund theory for these equations; and surprisingly the regularity problem for the quadratic symmetric Hessian equations in general dimension still remains open. This project seeks to address these fundamental issues.Investigations into the aforementioned equations will further our knowledge of two closely related mathematical fields, partial differential equations and differential geometry. Moreover, the project will also have impact on the areas where these equations arise. Special Lagrangian equations and complex Monge-Ampere equations provide the mathematical foundation for mirror symmetry in the string theory of modern physics, which is a unified way to describe our physical universe. Solutions to Isaacs equations lead to the optimal strategy for certain random processes, for example, in engineering and finance. Hessian equations are also related to nonlinear elasticity theory in mechanics, which studies the mechanisms whereby a material that is stretched returns to its original size and shape. Part of the research also involves participation of graduate students.

该项目集中于特殊拉格朗日方程，对称的黑森方程，以撒方程，复杂的蒙格 - 安培方程及其抛物线版本（例如，拉格朗日平均曲率流）。完全非线性椭圆形和抛物线方程的规律性和可溶性理论（具有一般维度的凸状条件，而没有凸度假设在二维的情况下）。仅列出的混凝土方程要么不满足凸状条件，要么不表现出均匀的椭圆形或抛物线。在一般的马鞍案件中，仅进行了初步尝试。对称的Hessian方程和复杂的Monge-Ampere方程已经取得了重大进步，但是对于这些方程式，仍然没有Schauder或Calderon-Zygmund理论。令人惊讶的是，对于一般维度中的二次对称Hessian方程的规律性问题仍然保持开放。该项目旨在解决这些基本问题。对上述方程式进行评估将进一步我们对两个密切相关的数学领域，部分微分方程和差异几何形状的了解。此外，该项目还将对这些方程式出现的区域产生影响。特殊的Lagrangian方程和复杂的Monge-Ampere方程为现代物理弦理论中的镜像对称性提供了数学基础，这是描述我们物理宇宙的统一方法。 ISAACS方程的解决方案为某些随机过程（例如工程和金融方面）提供了最佳策略。 Hessian方程也与力学中的非线性弹性理论有关，该理论研究了机制，该机制将伸展的材料恢复到其原始大小和形状。研究的一部分还涉及研究生的参与。