Mathematical Sciences: Nonlinear Differential Equations and Variational Problems

点击翻译按钮获取中文摘要

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.