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。*该计划的预期结果是双重的：人员的科学培训和关于微分方程关键应用问题的现有知识的进步。*拟议的研究领域充满了各个层面的挑战，为所有参与者提供了作为科学家共同成长和发展的机会。该计划包含对本科生和研究生的教育，以及将培训和高级研究与博士后合作相结合的明确规定。以前的学员现在在工业和学术界的工作岗位上为社会做出贡献。这项研究计划将支持对专业人员培训的持续贡献，并促进他们进入生产性职业和生活。*我们专注于具有退化结构的方程、有限光滑的系数或在粗糙区域内；因此，它们的处理超出了经典或当前有效的理论的范围。这些椭圆和抛物型偏微分方程组在物理和数学上都非常有趣，因为方程中系数的性质之间存在紧密的相互作用(它们是有界的吗？连续不断？很顺利？等)、方程所在区域的几何性质以及方程解的结果性质。例如，尖锐的金属点附近的电场可能是奇异的，并导致周围材料的物理击穿。对相应的椭圆偏微分方程的适当分析可以描述这种奇异性的细节。*本课程专注于简并方程的观点，这是应用微分方程重要发展的长期历史的结果，确切地说是数百年之久，拟议的研究处于该学科关键启示的边缘。该计划所追求的进展可能在能源勘探、交通网络、医学和一般物理方面有重要的应用。

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.