Nonlinear Partial Differential Equations

建议DMS-0354948题目：非线性偏微分方程组PI：韩青，圣母大学摘要首席研究员计划继续他在偏微分方程组方面的工作。这一项目的一个共同主题，以及过去几年来和平研究所研究的一个共同主题是，微分方程解的节点集或所涉及的函数的水平集的重要作用。项目中讨论的问题涉及各种类型的微分方程组，椭圆型、双曲型和混合型。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将继续研究的一类问题包括解的水平集的几何结构，特别是节点集、奇异集和分支集。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。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性质的影响。涉及奇异集的问题起源于材料科学和控制理论。奇异集，顾名思义，就是奇点出现的集合。准确的定义根据问题的出现而有所不同。在现实中，消除奇异集是不可能的，也就是所谓的“坏集”。其核心任务之一是确定奇异集可控的条件和奇异集小的条件。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Erickson的液晶模型和超导性的Ginzburg-Landau方程有关。PI相信，这些数学问题的讨论将改进各种应用中控制奇异集的方法。

Proposal DMS-0354948Title: Nonlinear partial differential equationsPI: Qing Han, University of Notre DameABSTRACTThe principal investigator plans to continue his work in partial differentialequations. A common theme of this project, and also of the research of the PIover the past several years, is the important role of the nodal sets ofsolutions to differential equations, or in general the level sets of thefunctions involved. The problems discussed in the project involve differentialequations of various types, elliptic, hyperbolic and of mixed type. Someproblems have close connections with other fields in mathematics, includingseveral complex variables and algebraic geometry. One class of problems thatthe PI would continue to work on includes the geometric structure of level setsof solutions, in particular the nodal sets, the singular sets and the branchsets. An important part of the study is the investigation of the asymptoticbehavior of solutions near these sets or the asymptotic behavior of these setsthemselves. Another class of problems that the PI would continue to work oninvolves the effect of level sets of known functions in the equations on theproperties of solutions.The problems involving singular sets originate from the material science and thecontrol theory. Singular sets, as the name suggests, are those sets wheresingularities occur. Precise definitions vary according to problems where theyarise. In reality it is impossible to eliminate the singular sets, theso-called `bad sets'. One of the central tasks is to identify the conditionsunder which the singular sets can be controlled and the conditions under whichthe singular sets are small. The proposed problems concerning singular sets inthe project are in their simplest forms. They are related to the Erickson'smodel for liquid crystals and the Ginzburg-Landau equation in thesuperconductivity. The PI believes that the discussion of these mathematicalproblems would improve the methods to control the singular sets in variousapplications.