Elliptic partial differential equations: theory and applications

椭圆偏微分方程：理论与应用

• 批准号: RGPIN-2015-05713
• 负责人: Cowan, Craig
• 金额: $ 1.17万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708669
• 关键词: Elliptic; partial; differential; equations; theory

Partial differential equations (pde's) lie at the heart of many mathematical models which attempt to describe various physical phenomena in the real world.    Because of this understanding pde's is of the utmost importance.   Mathematically pde's is a very broad area and generally one concentrates on certain classes of pde's.   My area of expertise is in the area of elliptic partial differential equations and in particular in the area of semilinear and quasilinear elliptic pde's. I propose to examine various elliptic  equations which are not variational in structure.  In the case of variational pde's one can attach an energy functional whose critical points correspond to solutions of the pde.  Often this allows one to use the very well developed machinery of variational calculus, for instance min-max techniques, to obtain existence and multiplicity results.  When the equation is no longer variational one does not have this variational machinery available and one must utilize other methods, for instance one can try topological methods.   In addition I propose to study some elliptic problems which are variational in nature but are supercritical, which in general, does not allow for a variational approach.    These problems range in difficulty from fairly straightforward to quite challenging.    Some of these problems are very  suitable for a masters student and do not require a lot of background.

偏微分方程（pde）是许多试图描述真实的世界中各种物理现象的数学模型的核心。 由于这种理解，偏微分方程是最重要的。 在数学上偏微分方程是一个非常广泛的领域，一般集中在某些类的偏微分方程。 我的专长领域是椭圆型偏微分方程，特别是半线性和拟线性椭圆型偏微分方程。 我建议检查各种椭圆方程是不是变分的结构。在变分PDE的情况下，可以附加一个能量泛函的临界点对应的PDE的解决方案。这往往允许一个使用非常发达的变分微积分的机器，例如最小最大技术，当方程不再是变分的时候，人们就没有这种变分机制，必须利用其他方法，例如可以尝试拓扑方法。 此外，我建议研究一些椭圆形问题的性质是变分的，但超临界的，这在一般情况下，不允许变分的方法。 这些问题的难度从相当简单到相当具有挑战性不等。 其中一些问题非常适合硕士生，不需要很多背景知识。