Partial Differential Equations and Variational Problems

项目简介项目负责人：韩青。主要从事偏微分方程和变分问题的研究。这个项目的主题，也是过去几年PI研究的主题，是研究与微分方程和变分问题解相关的特殊集。研究的一个重要部分是研究这些集附近解的渐近行为。PI将继续研究的问题包括水平集的几何结构，特别是节点集和奇异集。这种集合出现在对液晶埃里克森模型中光学导向器轴线的研究中。这些集合也与超导和其他几何变分问题中的金兹堡-朗道模型中的奇点有关。这项研究的部分动机是想要了解解在多大程度上可以用多项式或齐次解来定量描述。这些问题与数学的其他领域有密切的联系，包括几个复变量和代数几何。奇异集问题起源于材料科学和控制理论。奇点集合，顾名思义，就是那些有奇点的集合。精确的定义因问题的产生而异。在现实中，不可能消除奇异集，即所谓的“坏集”。因此，研究奇异集在什么条件下是可控制的以及在什么条件下奇异集是小的是中心任务之一。另一个应用涉及高性能计算，特别是图像处理。一个问题是如何从一个扭曲的副本中恢复图像，并且用一些奇异集精确地测量差值。通常期望这样的集合小到可以忽略不计。工程中上述问题均为简化的数学模型。研究者希望通过对这些数学问题的讨论，可以改进各种应用中的奇异集控制方法。

DMS-9801250 ABSTRACT OF RESEARCH PROJECT Principal Investigator: Qing Han The principal investigator would like to continue his work on partial differential equations and variational problems. The main theme of this project, and also of the research of the PI over the past several years, is to study special sets associated to solutions to differential equations and variational problems. An important part of the study is the investigation of the asymptotic behavior of the solutions near these sets. The problems that the PI would continue to work on include the geometric structure of level sets, in particular the nodal sets and the singular sets. Such sets arise in the study of the axis of the optical director in the Erickson model of liquid crystals. These sets are also related to the singularities in the Ginzburg-Landau model in the superconductivity and in other geometric variational problems. This study is partly motivated by the desire to understand to what extent the solutions can be described quantitatively by polynomials or by homogeneous solutions. These problems have a close connection with other fields in mathematics, including several complex variables and algebraic geometry. The problems of the singular sets originate from material sciences and control theory. Singular sets, as the name suggests, are those sets where singularities occur. Precise definitions vary according to problems where they arise. In reality it is impossible to eliminate the singular sets, the so-called "bad sets". Hence one of the central tasks is to investigate under what condition the singular sets can be controlled and under what condition such sets are small. Another application involves high-performance computing, in particular image processing. One problem is to recover the image from a distorted copy and the difference is measured exactly by some singular sets. It is usually expected that such sets should be small enough to be neglected. T he problems mentioned above in the project are simplified mathematical models. It is the hope of the investigator that the discussion of these mathematical problems will improve the methods to control the singular sets in various applications.