Non-Linear PDEs from Spectral Theory and Conformal Geometry

Gursky Proposal DMS-9801046涉及谱理论中出现的四阶问题的研究，涉及：非线性偏微分方程解的存在性和正则性问题，保角几何中椭圆型方程的渐近性，以及它与确定Sobolev嵌入的最佳常数的关系。需要研究偏微分方程式是因为它们在各种“真实世界”问题中的核心重要性：从CAT扫描仪到疾病的传播，从喷气发动机设计到计算机可视化。这项拨款支持的工作是关于“四阶”方程的一些基本问题，这是一个有许多开放和非常有趣的问题的领域，以及这些方程与其他数学领域的关系，特别是几何。

GurskyProposal DMS-9801046 The research proposed is concerned with the study of fourth orderproblems which arise in spectral theory, and involve: existence and regularity questions for non-linear partial differential equations, asymptotics of an elliptic equation which arises in conformal geometry, and its relation to the determination of the best constant in aSobolev imbedding. The need for the study of partial differential equations is due totheir central importance in every manner of "real world" problems: from CAT scan machines to the spread of diseases, from jet engine design to computer visualization. The work supported by this grant is concerned with some basic questions about "fourth order" equations, a field that has a lot of open and very interesting questions, and the relation of these equations to other fields of mathematics, especially geometry.