Partial differential equations for manifolds with boundary

这个项目将研究几何定义的线性算子，当基本几何结构经历扭曲曲线长度而不是它们之间的角度的变化时，这些算子具有一定的不变性。所研究的空间称为带边界的流形，通常用作具有特殊结构(如共形几何、柯西-黎曼几何)的背景空间。前述线性运算符规定了空间形状的变化，该变化借助于通常是非线性的微分方程式，根据特定的曲率测量来量化。这项研究的一个长期目标是找到这些曲率方程可解的条件，并研究解的质量。拟议项目的成功结果将通过提供一大类新的特殊爱因斯坦空间的例子来丰富保角几何的学科，这些例子引起了广大数学家和数学物理学家的兴趣。除了直接为保形几何领域做出贡献外，该项目还将帮助培养一批研究生和博士后，他们将具有下一代数学家的研究人员和教育者的双重角色。由主要研究人员主持的广泛的讲座课程对几个地理领域的科学家的数学教育做出了贡献。这个项目的一个基本工具是一个著名的不等式的版本，索博列夫不等式，它通过测量函数的变化量来控制函数的大小。事实上，通常的情况是，这种不平等的尖锐版本是必要的。主要研究人员打算利用变分法中的传统方法，找出某些空间中的这种尖锐的不等式。该项目的次要目标是改进主要研究人员之前的工作，以便获得共形紧致爱因斯坦空间族的界限，这些界限可能最终允许人们像物理学家预测的那样，构造出一大族这样的空间族。第三个目标是扩大人们可以断言四阶偏微分方程强最大值原理有效性的条件，这是一种只有由于主要调查人员先前的工作才能获得的技术手段。这样的极大值原理不仅可以应用于几何问题，还可以应用于有关板的弹性和弯曲的一般解析问题。

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.