Properties of Solutions to Degenerate Elliptic Equations and Applications

该研究计划侧重于偏微分方程（PDE）的数学分析，偏微分方程是描述在时间和/或空间中演变的各种物理过程的基本数学设备。众所周知的偏微分方程的例子包括描述电和磁之间联系的麦克斯韦方程、描述流体流动和空气动力学的纳维-斯托克斯方程、描述量子系统演化的薛定谔方程和爱因斯坦广义相对论的引力方程。我们的主要兴趣是椭圆和抛物方程，其中包括模拟导体和半导体内部和周围电场的偏微分方程，自然表面的形状，如肥皂泡形成的形状，运输商品的最佳方式，人体中热量的扩散，或污染物在大气中的传播。

This research program focuses on the mathematical analysis of partial differential equations (PDEs), which are the fundamental mathematical devices that describe diverse physical processes evolving in time and/or space. Examples of well-known PDEs include Maxwell’s equations describing the connection between electricity and magnetism, Navier-Stokes equation describing fluid flow and aerodynamics, Schrödinger’s equation describing the evolution of a quantum system, and Einstein’s gravitational equations of general relativity. Our main interest is in elliptic and parabolic equations, which include PDEs that model the electric field in and around conductors and semiconductors, the shape of natural surfaces like those formed by soap bubbles, the optimal way to transport merchandize, the diffusion of heat in the human body, or the spreading of pollutants in the atmosphere.