Properties of Solutions to Degenerate Elliptic Equations and Applications

该研究项目侧重于偏微分方程（PDEs）的数学分析，偏微分方程是描述在时间和/或空间中发展的各种物理过程的基本数学设备。著名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Schrödinger方程，以及爱因斯坦广义相对论的引力方程。我们的主要兴趣是椭圆方程和抛物线方程，其中包括对导体和半导体内部和周围的电场、自然表面（如肥皂泡形成的表面）的形状、运输商品的最佳方式、人体热量的扩散或大气中污染物的扩散进行建模的偏微分方程。

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. The expected outcomes of this program are two-folded: the scientific training of personnel and the advancement of knowledge available on key applicable problems in differential equations. The proposed areas of research are rich in challenges at all levels, offering all participants the opportunity to grow together and develop as scientists. This program contains clear provisions for the education of undergraduate and graduate students, and for synergizing training and advanced research with postdoctoral collaborations. Previous trainees are now contributing to society in industrial and academics jobs. This research program will support the continuing contribution to the training of specialized personnel and facilitate their progression into productive careers and lives. We concentrate on equations with degenerate structure, coefficients with limited smoothness, or within rough domains; as such, their treatments fall outside the reach of classical or currently availably theories. These elliptic and parabolic PDEs are deeply interesting both physically and mathematically because of the tight interplay between properties of the coefficients in the equation (are they bounded? continuous? smooth? etc.), geometric properties of the region where the equation lives, and the resulting properties of the solution to the equation. For instance, the electric field near a sharp metallic point will likely be singular and cause physical breakdown of the surrounding material a proper analysis of the corresponding elliptic PDE can describe the details of this singularity. This program concentrates on perspectives on degenerate equations which are the result of a long history, literally centuries long, of important developments in applied differential equations, and the proposed research stands at the edge of critical revelations in the subject. The advances this program pursues may have significant applications to energy exploration, transport networks, medicine, and physics in general.