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.

我的研究计划关注本身与二阶拟线性和非线性偏微分方程的研究，使用的方法，奠定了在经典，功能和谐波分析，数学的三个主要分支与应用在物理科学和工程之间的横截面。偏微分方程（PDE）与所有科学（重点是物理学和数学）密切相关，因为它们通过牛顿定律与物理系统的行为联系在一起。我研究类的非椭圆形或退化椭圆偏微分方程连接到几何和物理学领域研究表面规定的曲率和流体动力学。 一些熟悉的方程，属于我的工作范围是著名的拉普拉斯和p-拉普拉斯方程，用于，例如，在物理学中与非牛顿流体相关的流体流动模型。 我研究这些方程，以找到其结构的条件下，方程的解决方案存在和/或定期（有界，连续，可微）。 我这样做的灵感来自DeGiorgi，Nash，Moser，Serrin和Trudinger等数学家的深层技术。 我的研究计划还涉及新的规律性结果的应用，特别是在数学中。 特别感兴趣的两个方面：我热衷于探索退化椭圆偏微分方程边值问题的正则解的存在性与重要的规范不等式（如Poincare和Sobolev估计）的有效性之间的深层联系。 其次，我感兴趣的是进一步扩展经典的迈尔斯-Serrin H=W结果的Sobolev空间定义的类向量场，可能不是Lipschitz连续的。这项研究计划为本科生和研究生提供了机会，同时也促进了与美国和欧洲大学数学家的国家和国际合作。