Mathematical Sciences: Nonlinear Differential Equations and Variational Problems

项目摘要 青涵将继续从事微分方程和几何测度理论及其应用的研究。这些问题包括： 问题、局部结构和微分方程解的唯一延拓。其中一个中心问题，他将继续工作是变分问题的自由接口。它与极小曲面的一般正则性理论密切相关。所考虑的功能由修改后的Dirichlet积分和表面积。困难在于泛函中的两个项具有不同的比例因子。在这种标度下的解决方案的行为将被调查。他还将研究椭圆和抛物微分方程的奇异解集的结构 方程这是微分方程理论中的基本问题之一。他将调查的影响光滑系数的结构奇异集。这些问题与解的唯一连续性有关。另一个相关的问题是奇异集的大小。预计这些集合的适当Hausdorff维数应该由解的频率来控制。 自由界面和奇异集问题起源于材料科学和控制理论。在现实中，不可能消除奇异集，即所谓的“坏集”。“因此，中心任务之一是研究在什么条件下可以控制奇异集，以及在什么条件下这些集是小的。另一 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，并且通过自由界面及其奇异集精确地测量差异。预计这样的集合应该足够小以被忽略。本课题所涉及的问题均为简化的数学模型。希望通过对这些数学问题的讨论，能够改进奇异集控制方法在各种应用中的应用。

ABSTRACT OF PROJECT Qing Han will continue his work on the differential equations and the geometric measure theory and their applications. These include regularities of solutions to variational problems, local structure and unique continuation of solutions to differential equations. One of the central problems he will continue to work on is the variational problem with free interfaces. It is closely related to the general regularity theory of minimal surfaces. The functional under the consideration consists of the modified Dirichlet integral and surface area. The difficulty arises as the two terms in the functional have different scaling factors. Behaviors of solutions under such scalings would be investigated. He will also study the structure of the singular sets of solutions to elliptic and parabolic differential equations. This is one of the fundamental problems in the theory of differential equations. He will investigate the effect of the smoothness of coefficients on the structure of singular sets. Such problems are related to the unique continuation of solutions. Another related problem is the size of the singular sets. It is expected that the appropriate Hausdorff dimension of such sets should be controlled by the frequency of solutions. The problems of free interfaces and the singular sets originate from the material science and the control theory. 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 the 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 the free interfaces and their singular sets. It is expected that such sets should be small enough to be neglected. The problems mentioned abo ve in the project are simplified mathematical models. It is expected that the discussion of these mathematical problems will improve the methods to control the singular sets in various applications.