Properties of Solutions to Degenerate Elliptic Equations and Applications

这项研究计划专注于偏微分方程(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工具。著名的偏微分方程组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，以及爱因斯坦的广义相对论引力方程。我们主要对椭圆型和抛物型方程感兴趣，其中包括对导体和半导体内部和周围的电场进行建模的偏微分方程组、由肥皂泡形成的自然表面的形状、运输商品的最佳方式、热量在人体内的扩散，或污染物在大气中的扩散。

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.