On a Fourth Order PDE - Some Analytic and Geometric Aspects

点击翻译按钮获取中文摘要

9706864 Chang In this project, Chang proposes to continue her study of a fourth order differential operator called the Paneitz operator. This operator, which arises naturally from consideration in conformal geometry, is the standard bi-Laplacian operator when restricted to domains in the Euclidean spaces. Four problems are proposed to study the geometric and analytic properties of the operator. The first is to investigate the geometric content of some best constant in the sharp Sobolev imbedding result associated with the operator, the second and the third are to study uniqueness and regularity of some non-linear PDE naturally related to the operator. The fourth is to study general conformal covariant operators which Paneitz operator is a special example. Mooers proposes to continue her work on analysis on incomplete manifolds, specifically to investigate the behavior of the heat kernel on manifolds with isolated metric singularities and to describe the relationship between the heat kernel of these spaces and their geometry. One of the most important topic in mathematics today is the study of 4-dimensional metric spaces (3-dimensional space plus one-dimensional time variable). Such spaces arise naturally as the state space of physical systems, and these systems can often be described mathematically as solutions to differential equations. The behavior of solutions is used to study and classify the geometry of metric spaces, and thus possible physical systems. Chang propose to study the geometry of 4-manifolds, metric spaces that look locally like Euclidean space, via some analytic tool--namely via a fourth order differential operator which leads to interesting and challenging open questions both in geometry and analysis. Mooers proposes to use the heat equation to study metric spaces with isolated metric singularities, spaces that are manifolds away from an isolated point, and with behavior at the point similar to a black hole type anoma ly in space.