Properties of Solutions to Degenerate Elliptic Equations and Applications

简并椭圆方程解的性质及应用

• 批准号: RGPIN-2017-04872
• 负责人: Rios, Cristian
• 金额: $ 1.02万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=663386
• 关键词: Properties; Solutions; Degenerate; Elliptic; Equations

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.

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。** 该计划的预期成果有两个方面：人员的科学培训和微分方程关键应用问题的知识进步。** 拟议的研究领域在各个层面都充满挑战，为所有参与者提供了共同成长和发展的机会。该计划包含明确的规定，为本科生和研究生的教育，并协同培训和先进的研究与博士后合作。以前的受训人员现在在工业和学术工作中为社会作出贡献。 该研究计划将支持对专业人员培训的持续贡献，并促进他们进入富有成效的职业生涯和生活。我们专注于退化结构的方程，有限的光滑系数，或在粗糙的域;因此，他们的治疗范围之外的经典或目前可用的理论。这些椭圆和抛物偏微分方程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（它们有界吗？连续？光滑？等等），方程所在区域的几何性质，以及方程解的结果性质。例如，一个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-对相应椭圆PDE的适当分析可以描述这种奇异性的细节。该计划集中在退化方程的观点，这是一个漫长的历史，字面上几个世纪之久，在应用微分方程的重要发展的结果，拟议的研究站在关键启示的边缘。该计划所追求的进步可能对能源勘探，运输网络，医学和物理学有重要的应用。