Degenerate Elliptic Equations: Regularity of weak solutions with applications

我的研究计划涉及使用介于经典分析、泛函分析和调和分析之间的方法研究二阶拟线性和非线性偏微分方程式。经典分析是数学的三个主要分支，在物理科学和工程中有应用。偏微分方程(PDE)与所有科学(重点是物理和数学)密切相关，因为它们通过牛顿定律与物理系统的行为联系在一起。我学习与几何和物理有关的非椭圆形或退化椭圆型偏微分方程组，这些偏微分方程组研究的是具有规定曲率和流体动力学的曲面。一些熟悉的方程属于我的工作范围，例如著名的拉普拉斯方程和p-拉普拉斯方程，它们被用来模拟与物理学中的非牛顿流体相关的流体流动。我研究这些方程，以找出方程的解存在和/或正则(有界、连续、可微)的结构条件。我这样做是从德乔吉、纳什、莫泽、塞林和特鲁丁格等数学家的深奥技巧中获得灵感。我的研究计划还涉及新的规律性结果的应用，特别是在数学方面。我特别感兴趣的两个方面：我对探索退化椭圆型偏微分方程边值问题正则解的存在性与重要的范数不等式(如Poincare和Soblev估计)的有效性之间的深层联系非常感兴趣。其次，我感兴趣的是将经典的Myers-Serrin H=W结果进一步推广到关于可能不是Lipschitz连续的向量场类定义的Soblev空间。这一研究项目为本科生和研究生提供了机会，同时也促进了与在美国和欧洲大学工作的数学家的国内和国际合作。

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.