Partial Differential Equations and Variational Problems

首席研究员想继续他在偏微分方程和变分问题上的工作。这个项目的第一个主题，也是过去几年PI研究的第一个主题，是研究与微分方程和变分问题的解相关的特殊集。PI将继续研究的问题包括水平集的几何结构，特别是节点集和奇异集。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。一个新的研究方向是研究节点集的生长与解本身的生长之间的关系。可以预料，它们之间是密切相关的。该项目的第二个主题是研究二维度量在三维欧几里得空间中的等距嵌入，包括局部和全局。一般来说，等距嵌入是基于复杂的纳什-莫泽迭代，这需要详细讨论线性化方程。此外，全局等距嵌入预计会遇到一些拓扑障碍。当高斯曲率改变其符号时，困难就出现了。即使嵌入存在，也可能出现奇异集。奇异集问题起源于材料科学和控制论。奇点集合，顾名思义，就是出现奇点的集合。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”。以建筑材料中的裂缝为例。因此，中心任务之一是研究奇异集可以被控制的条件，从而可以使其变小。另一个应用涉及高性能计算，特别是图像处理。其中一个问题是如何从一个扭曲的副本中恢复图像，并通过一些奇异集精确测量差值。人们高度期望这样的集合足够小，可以忽略不计。项目中所述的问题是简化的数学模型。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。

The principal investigator would like to continue his work on partialdifferential equations and variational problems. The first theme ofthis project, and also of the research of the PI over the past severalyears, is to study special sets associated to solutions to differentialequations and variational problems. The problems that the PI wouldcontinue to work on include the geometric structure of level sets, inparticular the nodal sets and the singular sets. The study is partlymotivated by the desire to understand to what extent the solutions canbe 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. A newresearch trend is the study of the relation between the growth of nodalsets and the growth of solutions themselves. It is expected that theyare closely related to each other. The second theme of this project isto study the isometric embedding of two-dimensional metrics in the threedimensional Euclidean space, both locally and globally. In general, theisometric embedding is based on the complicated Nash-Moser iteration,which requires detailed discussions of the linearized equations. Moreover,the global isometric embedding is expected to meet some topologicalobstacles. Difficulties arise when the Gaussian curvature changes itssign. Singular sets may appear even if the embedding exists.The problems of the singular sets originate from the material scienceand the control theory. Singular sets, as the name suggests, are thosesets where singularities occur. Precise definitions vary according toproblems where they arise. In reality it is impossible to eliminatesingular sets, the so-called "bad sets". An example is provided bycracks in the building material. Hence one of the central tasks is toinvestigate the conditions under which the singular sets can becontrolled and hence can be made small. Another application involvesthe high-performance computing, in particular the image processing. Oneproblem is to recover the image from a distorted copy and the differenceis measured exactly by some singular sets. It is highly expected thatsuch sets should be small enough to be neglected. The problems statedin the project are simplified mathematical models. It is the hope bythe investigator that the discussion of these mathematical problemswould improve the methods to control the singular sets in variousapplications.