Mathematical Sciences: Nonlinear Differential Equations and Variational Problems

项目摘要青汉将继续他在微分方程式和几何测度论及其应用方面的工作。这些问题包括变分问题解的正则性、局部结构和微分方程解的唯一连续性。他将继续研究的核心问题之一是自由界面的变分问题。它与极小曲面的一般正则性理论密切相关。所考虑的泛函包括修正的Dirichlet积分和表面积。由于泛函中的两个项具有不同的比例因子，因此出现了困难。将研究溶液在这种标度下的行为。他还将研究椭圆型和抛物型微分方程解的奇异集的结构。这是微分方程组理论中的一个基本问题。他将研究系数的光滑性对奇异集结构的影响。这些问题都与解决方案的独特延续性有关。另一个相关的问题是奇异集的大小。期望这类集合的适当Hausdorff维度应由解的频率控制。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，要消除奇异集，即所谓的“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，而差异是通过自由界面及其奇异集来精确测量的。预计这样的集合应该小到可以忽略。项目中提到的问题都是简化的数学模型。期望通过对这些数学问题的讨论，改进各种应用中控制奇异集的方法。

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.