This award supports three investigators working on various aspects of problems combining several complex variables and partial differential equations. One primary focus will be continuing research on the regularity of two first order differential operators, the complex derivative d-bar which measures the degree to which a function is holomorphic and the associated operator restricted to manifolds or domain boundaries called the d-bar-b operator. The study of regularity is governed by sub-elliptic inequalities. Regularity is analyzed both locally and globally through these inequalities, and will be considered both in the analytic sense as well as that of infinite differentiability. Related to regularity, and possibly one of the strongest motivations for studying it, is the question of how smooth the boundary values of a function remain after projection onto the space of holomorphic functions on a domain. A concrete expression for the projection is given by integration against the Bergman kernel. A second thrust of this project is to understand the boundary behavior of the Bergman kernel on various kinds of domains, especially those of finite type. A third, somewhat more geometric line of work concerns the regularity theory of level surfaces, problems of crystal growth and degenerate equations. Level surfaces of solutions of differential equations, expecially those which analyze the level sets of viscosity solutions of heat equations will be studied. Among the most fundamental questions to be considered is one due to De Giorgi, which seeks the regularity of the surfaces as a function of a parameter in the differential equation. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.
