Fourth Order Equations Involoving Critical Sobolev Exponentsand Systems of PDE's From Geometry

摘要古尔斯基9623048这项资助的研究涉及从微分几何中研究四阶微分方程组和方程组。所考虑的方程是泛函的欧拉方程，它与某些尖锐的索博列夫不等式有关。在研究相对论中引入的泛函的曲率和临界点的存在性的问题时，微分几何中出现了方程组。从历史上看，纯数学在非常真实的世界问题上最有成效的应用往往是学科内两个不同领域相互作用的结果。这里要支持的工作试图使用数学分析的方法来研究微分几何领域中的某些问题。这些问题中至少有一个与物理学有关，但通常情况是，数学比应用早了很多年，所以很难评估这里提出的问题的即时使用情况：除了说，它们是非常基本的问题，有望增加我们对一个在过去有许多应用的学科的理解深度。

Abstract Gursky 9623048 The research to be supported by this grant is concerned with the study of fourth order differential equations and systems of equations from differential geometry. The equations under consideration are Euler equations for functionals that are connected to certain sharp Sobolev inequalities. The systems of equations arise in differential geometry when studying the problem of prescribing curvature and the existence of critical points of a functional introduced in relativity. Historically, the most fruitful applications of pure mathematics to very real world problems are often the result of the interaction of two different fields within the subject. The work to be supported here attempts to use methods of mathematical analysis to study certain problems in the field of differential geometry. At least one of these problems is relevant to physics, but it is often the case that the mathematics precedes the application by many years, so that it is difficult to assess the immediate use of the problems posed here: except to say that they are very basic questions that hopefully will increase the depth of our understanding of a subject that has seen many applications in the past.