Mathematical Sciences: Free Boundary Problems, Fully Nonlinear Equations, Nonlinear Parabolic Equations, and the Navier-Stokes Equation

在本研究中要处理的问题范围涵盖了对偏微分方程解的性质的广泛的基本调查。这些方程既有物理来源，也有几何来源，而且大部分都是非线性的。要处理的一类方程称为全非线性方程。它们描述了几何、弹性和物理的基本现象。其中典型的是蒙日-安培方程，它有效地描述了一个域的曲率。更有趣和困难的问题是决定一个给定的函数是否可以是一个定义域的曲率。计划对解的内部平滑特性和相关稳定性特性进行表征。另一条研究路线将继续研究自由边界问题，这些问题表示在某些未知水平上性质不连续变化的物理量，例如在多相系统中。在这里，相间的边界是问题的一个动态部分，必须与伴随的微分方程的解一起找到。这里的工作将集中于界面的几何性质，并将继续努力寻求与最小表面规则理论相结合的统一理论。将采用奇异摄动方法。连接这个项目的两个主要主题将是对广义曲面的研究，其中曲率的初等对称函数是规定的。这种性质的结果提供了洞察与任意集，如凸或最小面积船体相关的各种包络。这项工作的副产品有望广泛应用于物理和数学科学。

The range of questions to be dealt with in this research covers broad fundamental investigations into the nature of solutions of partial differential equations. The equations arise from physical as well as geometrical sources and, for the most part, are nonlinear. One class of equations to be treated are known as fully nonlinear equations. They describe basic phenomena of geometry, elasticity and physics. Typical of these is the Monge-Ampere equation which effectively describes the curvature of a domain. The more interesting, and difficult question is that of deciding whether a given function can be the curvature of a domain. Work is planned on characterizing interior smoothing properties and related stability properties of solutions. Another line of investigation will continue work on free boundary problems which represent physical quantities whose properties change discontinuously at some level of the unknown, such as in multi-phase systems. Here the boundary between phases is a dynamic part of the problem and must be found along with solutions of the attendant differential equations. Work here will concentrate on geometric properties of the interphase and efforts will continue in seeking a unified theory coupled with minimal surface regularity theory. A singular perturbation approach will be employed. Bridging the two main themes of this project will be a study of generalized surfaces for which an elementary symmetric function of the curvature is prescribed. Results of this nature provide insight into various envelopes associated with arbitrary sets such as the convex or minimal area hulls. By-products of this work are expected to have extensive applications to the physical and mathematical sciences.