Partial Differential Equations and Variational Problems

9801250研究项目摘要首席研究员：韩青首席研究员希望继续他在偏微分方程和变分问题方面的工作。这个项目的主要主题，也是过去几年PI研究的主题，是研究与微分方程组和变分问题的解相关的特殊集。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。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。在研究液晶的埃里克森模型中的光学指向矢轴时，出现了这样的集合。这些集合还与Ginzburg-Landau模型、超导电性和其他几何变分问题中的奇点有关。这项研究的部分动机是希望了解在多大程度上可以用多项式或齐次解来定量描述这些解。这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。奇异集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。准确的定义根据问题的出现而有所不同。在现实中，要消除奇异集，也就是所谓的“坏集”是不可能的。因此，研究奇异集在什么条件下可以被控制，在什么条件下奇异集是小的，是研究的中心任务之一。另一个应用涉及高性能计算，特别是图像处理。一个问题是从失真的副本中恢复图像，而差异是通过一些奇异集来精确测量的。通常情况下，这样的集合应该小到可以忽略。项目中的上述问题都是简化的数学模型。研究人员希望通过对这些数学问题的讨论，改进在各种应用中控制奇异集的方法。

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.