On a Fourth Order PDE - Some Analytic and Geometric Aspects

9706864 Chang在这个项目中，Chang建议继续她的四阶微分算子Paneitz算子的研究。这个算子是在保形几何中自然产生的，当限定在欧几里德空间的域上时，它是标准的双拉普拉斯算子。提出了四个问题来研究算子的几何性质和解析性质。第一个是研究与算子相关的尖锐Sobolev嵌入结果中某些最佳常数的几何内容，第二个和第三个是研究与算子自然相关的非线性偏微分方程的唯一性和正则性。四是研究一般共形协变算子，其中Paneitz算子是一个特殊的例子。Mooers建议继续她对不完全流形的分析工作，特别是研究具有孤立度量奇点的流形上的热核的行为，并描述这些空间的热核与其几何之间的关系。四维度量空间（三维空间加一维时间变量）是当今数学中最重要的课题之一。这样的空间作为物理系统的状态空间自然出现，这些系统通常可以用数学方法描述为微分方程的解。解的行为是用来研究和分类度量空间的几何，从而可能的物理系统。Chang建议通过一些解析工具——即通过四阶微分算子——来研究局部看起来像欧几里德空间的4流形的几何，这将导致几何和分析中有趣且具有挑战性的开放问题。Mooers建议使用热方程来研究具有孤立度量奇点的度量空间，这些空间是远离孤立点的流形，并且在点处的行为类似于空间中的黑洞型异常。

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.