Nonlinear Partial Differential Equations

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Proposal DMS-0354948 Title: Nonlinear partial differential equations PI: Qing Han, University of Notre Dame ABSTRACT The principal investigator plans to continue his work in partial differential equations. A common theme of this project, and also of the research of the PI over the past several years, is the important role of the nodal sets of solutions to differential equations, or in general the level sets of the functions involved. The problems discussed in the project involve differential equations of various types, elliptic, hyperbolic and of mixed type. Some problems have close connections with other fields in mathematics, including several complex variables and algebraic geometry. One class of problems that the PI would continue to work on includes the geometric structure of level sets of solutions, in particular the nodal sets, the singular sets and the branch sets. An important part of the study is the investigation of the asymptotic behavior of solutions near these sets or the asymptotic behavior of these sets themselves. Another class of problems that the PI would continue to work on involves the effect of level sets of known functions in the equations on the properties of solutions. The problems involving singular sets originate from the material science and the 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'. One of the central tasks is to identify the conditions under which the singular sets can be controlled and the conditions under which the singular sets are small. The proposed problems concerning singular sets in the project are in their simplest forms. They are related to the Erickson's model for liquid crystals and the Ginzburg-Landau equation in the superconductivity. The PI believes that the discussion of these mathematical problems would improve the methods to control the singular sets in various applications.