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.
