Partial differential equations for manifolds with boundary

该项目将研究几何定义的线性算子，当底层几何结构发生变化时，这些线性算子具有某些不变性，这些变化会扭曲曲线的长度，但不会扭曲曲线之间的角度。被研究的空间，称为有边界的流形，通常用作具有特殊结构的背景空间（例如，共形几何，柯西-黎曼几何）。上述线性算子规定了空间形状的变化，通过通常是非线性的微分方程根据一定的曲率测量来量化变化。本研究的一个长期目标是找到这些曲率方程可解的条件，并研究解决方案的质量。成功的结果，在拟议的项目将丰富保形几何的主题，提供了一个大的新家庭的特殊爱因斯坦空间的例子，是感兴趣的广大观众的数学家，以及数学物理学家。除了直接贡献于共形几何领域，该项目将有助于培养一些研究生和博士后，他们将具有下一代数学家的研究人员和教育工作者的双重角色。由主要研究人员进行的广泛的讲座课程有助于在几个地理区域的科学家的数学教育。这个项目的一个基本工具是一个著名的不等式的版本，Sobolev不等式，控制的大小的功能方面的测量其变化。事实上，我们常常需要一个尖锐的不平等版本。主要研究人员打算利用变分法的传统方法来寻找某些空间的尖锐不等式。该项目的第二个目标是改进主要研究人员以前的工作，以获得共形紧致爱因斯坦空间族的边界，这些边界最终可能允许人们构建一个大的此类空间族，正如物理学家所预测的那样。 第三个目标是拓宽条件，在此条件下，人们可以断言四阶偏微分方程的强极大值原理的有效性，这是一种技术手段，只有成为可用，因为先前的工作由principal调查。这样一个极大值原理不仅可以应用于几何问题，而且可以应用于板的弹性和弯曲的一般分析问题。

This project will study geometrically-defined linear operators that enjoy certain invariance properties when the underlying geometric structure undergoes changes that distort the lengths of curves but not the angles between them. The spaces under investigation, called manifolds with boundaries, serve often as background spaces with special structures (e.g., conformal geometry, Cauchy-Riemann geometry). The aforementioned linear operators prescribe the change in the shape of space, change quantified in terms of certain curvature measurement by means of a differential equation that is typically nonlinear. A long-term goal of this study is to find conditions under which these curvature equations are solvable and to study the quality of the solution. Successful results in the proposed project will enrich the subject of conformal geometry by providing a large new family of examples of special Einstein spaces that are of interest to a wide audience of mathematicians, as well as to mathematical physicists. In addition to contributing directly to the field of conformal geometry, this project will help train a number of graduate students and postdocs, who will have the dual role of researchers and educators of the next generation of mathematicians. The extensive lecture course conducted by the principal investigators contributes to the mathematical education of scientists in several geographic areas.A basic tool for this project is a version of a well-known inequality, the Sobolev inequality, that controls the size of a function in terms of a measurement of its variation. Indeed, it is often the case that a sharp version of such an inequality is what is needed. The principal investigators intend to find such sharp inequalities for certain spaces by making use of traditional methods from the calculus of variations. A secondary objective of the project is to improve upon previous work of the principal investigators in order to obtain bounds for a family of conformally compact Einstein spaces, bounds that might eventually allow one to construct of a large family of such spaces, as predicted by physicists. A third objective is to broaden the conditions under which one can assert the validity of a strong maximum principle for fourth-order partial differential equations, a technical device that only became available because of prior work by the prinicipal investigators. Such a maximum principle will allow for a number of applications not only to geometric questions but also to general analytic questions about elasticity and the bending of plates.