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

本研究要处理的问题范围 涵盖了广泛的基本调查的性质， 偏微分方程的解 方程出现了 从物理和几何的来源， 部分是非线性的。 要处理的一类方程是 称为完全非线性方程。 他们描述了基本的 几何、弹性和物理现象。 典型的这些 蒙日-安培方程有效地描述了 域的曲率。 越有趣越困难 问题是决定一个给定的函数是否可以是 域的曲率。 计划开展工作， 的内光滑性质和相关的稳定性性质 解决方案 另一条调查线将继续免费工作 边界问题表示的物理量 属性在未知的某个水平上不连续地变化， 例如在多相系统中。 在这里，阶段之间的边界 是问题的一个动态部分，必须与 解伴随的微分方程。 这里工作 将集中于界面的几何性质， 将继续努力寻求一个统一的理论， 最小曲面正则性理论 奇异摄动 方法将被采用。 将研究本项目的两个主题 广义曲面的初等对称 曲率的函数是规定的。 这种性质的结果 深入了解与任意 集，如凸壳或最小面积壳。 这项工作的副产品预计将有广泛的 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.