Partial Differential Equations and Variational Problems

DMS-9801250研究项目摘要 主要研究者：韩青 首席研究员想继续他的工作， 微分方程和变分问题。的主旋律 这个项目，以及PI在过去几年的研究 多年来，是研究特殊的集相关的解决方案，微分 方程和变分问题。这项研究的一个重要部分是 研究解的渐近性态， 这些集。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， 奇异集这样的集合出现在研究的轴的 埃里克森液晶模型中的光学指向矢。这些集合 也与金兹伯格-朗道模型中的奇点有关， 超导性和其他几何变分问题。 这项研究的部分动机是为了了解 在某种程度上，解决方案可以用多项式或 均匀的解决方案。这些问题与 数学中的其他领域，包括几个复杂的变量， 代数几何 奇异集问题起源于材料科学 控制理论。奇异集，顾名思义，就是那些 奇点发生的地方。精确的定义因 问题出现的地方。在现实中， 奇异集合，也就是所谓的“坏集合”因此， 任务是研究在什么条件下奇异集可以 控制和在什么条件下这样的集合是小的。另一 应用程序涉及高性能计算，特别是 图像处理一个问题是从失真的图像中恢复图像。 复制和差异是由一些奇异集精确测量。它 通常认为，这样的集合应该足够小，可以忽略不计。 本课题中的上述问题均为简化数学问题， 模型调查人员希望，这些问题的讨论 数学问题将改善方法来控制奇异 在各种应用中设置。

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.