Partial Differential Equations in Geometry

根据这一奖项，首席研究员计划继续他在几何偏微分方程式方面的工作。特别是，他将研究微分方程解的水平集的作用。水平集的特例由节点集、奇异集和分支集给出。研究的一个重要部分是研究解在这些集合附近的渐近行为或这些集合本身的渐近行为。微分方程可以是椭圆型、双曲型或混合型。有些问题与数学中的其他领域有着密切的联系，包括多个复变量和代数几何。PI将继续研究的另一类问题涉及方程中已知函数的水平集对解的存在和性质的影响。一个特殊的问题是当高斯曲率的零点集表现良好时，二维黎曼流形在三维空间中的等距嵌入问题。这个问题是退化的Monge-Ampere方程的形式。涉及节点集或奇异集的问题起源于材料科学和控制理论。奇异集合，顾名思义，就是奇点出现的集合。准确的定义取决于它们出现的问题。要消除奇异集，即所谓的“坏集”，往往是不可能的。中心任务之一是确定奇异集可以控制的条件和奇异集较小的条件。项目中提出的关于奇异集的问题都是最简单的形式。它们与液晶的Erickson模型和超导中的Ginzburg-Landau方程有关。PI相信，对这些数学问题的讨论将有助于科学家在各种应用中处理奇异集。

Under this award, the principal investigator plans to continue his work in geometric partial differential equations. In particular he will investigate the role of level sets of solutions of differential equations. Special cases of level sets are given by the nodal sets, the singular sets and the branch sets. An important part of the study is the investigation of the asymptotic behavior of solutions near these sets or the asymptotic behavior of these sets themselves. The differential equations may be elliptic, hyperbolic or of mixed type. Some problems have close connections with other fields in mathematics, including several complex variables and algebraic geometry. Another class of problems that the PI will continue to work on involves the effect of level sets of known functions in the equations on the existence and properties of solutions. A particular problem is the isometric embedding of 2-dim Riemannian manifolds in 3-space when the zero set of Gauss curvature is well behaved. This problem is in the form of degenerate Monge-Ampere equations. The problems involving nodal sets or the singular sets originate from materials science and control theory. Singular sets, as the name suggests, are those sets where singularities occur. Precise definitions vary depending on the problems where they arise. It often is impossible to eliminate the singular sets, the so-called "bad sets". One of the central tasks is to identify the conditions under which the singular sets can be controlled and the conditions under which the singular sets are small. The proposed problems concerning singular sets in the project are in their simplest forms. They are related to the Erickson's model for liquid crystals and the Ginzburg-Landau equation in the superconductivity. The PI believes that the discussion of these mathematical problems will help scientists work with singular sets in various applications.