Degenerate Elliptic Equations: Regularity of weak solutions with applications

简并椭圆方程：弱解的正则性及其应用

• 批准号: RGPIN-2018-06229
• 负责人: Rodney, Scott
• 金额: $ 1.17万
• 依托单位: Cape Breton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675722
• 关键词: Degenerate; Elliptic; Equations; Regularity; weak

My research programme concerns itself with the study of second order quasilinear and nonlinear partial differential equations using methods that lay at a cross section between classical, functional and harmonic analysis, three major branches of mathematics with applications in the physical sciences and engineering. Partial differential equations (PDEs) are closely connected to all of the sciences (with emphasis on physics and mathematics) because of their connection to the behavior of physical systems via Newton's laws. I study classes of non-elliptic or degenerate elliptic PDEs connected to Geometry and Physics in areas studying surfaces with prescribed curvature and fluid dynamics. Some familiar equations that fall in the scope of my work are the famous Laplace and p-Laplace equations that are used, for example, to model fluid flow associated to non-Newtonian fluids in Physics. I study these equations to find conditions on their structure under which a solution of the equation exists and/or is regular (bounded, continuous, differentiable). I do this taking inspiration from the deep techniques of mathematicians like DeGiorgi, Nash, Moser, Serrin, and Trudinger. My research programme is also concerned with applications of new regularity results, particularly in mathematics. Two aspects of particular interest: I am keenly interested in the exploration of deep connections between the existence of regular solutions of boundary value problems for degenerate elliptic PDEs and the validity of important norm-inequalities like Poincare and Sobolev estimates. Secondly, I am interested in further extending the classical Myers-Serrin H=W result to Sobolev spaces defined with respect to classes of vector fields that may not be Lipschitz continuous. This research progamme provides opportunities for both undergraduate and graduate students while also promoting both national and international collaborations with mathematicians working in universities in the United States and Europe.

我的研究计划涉及对二阶准线性和非线性偏微分方程的研究，该方法使用了经典，功能和谐波分析之间的横截面的方法，这是数学的三个主要分支，以及在物理科学和工程中的应用。部分微分方程（PDE）与所有科学（重点介绍了物理和数学）密切相关，因为它们通过牛顿定律与物理系统的行为有联系。我研究了与规定曲率和流体动力学的表面相关的区域中与几何和物理相关的非椭圆形PDE的类别。 我作品范围的一些熟悉的方程是著名的拉普拉斯和p拉普拉斯方程，例如用于建模与物理学中非牛顿流体相关的流体流动。 我研究这些方程式以找到其结构的条件，在这些条件下，方程解决方案存在和/或是规则的（有限，连续，可区分的）。 我这样做从数学家，纳什，摩泽，塞林和特鲁丁格等数学家的深入技术中汲取了灵感。 我的研究计划还关注新的规律性结果的应用，尤其是在数学中。 特别感兴趣的两个方面：我对探索简并椭圆形PDE的定期解决方案的存在与重要规范质量（如Poincare和Sobolev估计）的有效性之间的深入联系感兴趣。 其次，我有兴趣将经典的Myers-Serrin H = W结果扩展到相对于可能不是Lipschitz连续的矢量场定义的Sobolev空间。这项研究进程为本科和研究生提供了机会，同时还促进了与在美国和欧洲大学工作的数学家的国家和国际合作。