On a Fourth Order PDE - Some Analytic and Geometric Aspects

9706864 常 在这个项目中，Chang建议继续她对四阶微分算子的研究，称为Paneitz算子。 这个算子是标准的双拉普拉斯算子，它是从共形几何中自然产生的 当被限制在欧几里德空间中的域时。四个问题 的几何和分析性质的研究。 操作符.第一个问题是研究与下列问题相关的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唯一性， 一些自然与算子相关的非线性偏微分方程的正则性。 四是研究了一般共形协变算子， Paneitz算子是一个特例。 穆尔斯建议继续她的工作分析不完整的流形，特别是调查的行为热核上 流形与孤立度量奇点，并描述这些空间的热核和它们的几何之间的关系。 今天数学中最重要的课题之一是研究四维度量空间（三维空间加上一维时间变量）。 这样的空间自然地产生为状态空间， 物理系统，这些系统通常可以 数学上描述为微分方程的解。 解的行为被用来研究和分类度量空间的几何，从而可能的物理系统。 Chang提出研究4-流形、度量空间的几何 通过某种分析工具， 通过一个四阶微分算子，导致有趣的， 在几何学和分析学中挑战开放性问题。 Mooers建议使用热方程来研究度量空间， 孤立的度量奇点，空间是流形远离 孤立点，并在该点的行为类似于黑洞型空间异常。

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.