Regularity Questions in Linear and Nonlinear Partial Differential Equations

该项目专注于对某些偏微分方程式(PDE)的数学研究，这些偏微分方程会模拟物理、工程、材料科学和经济学中的现象。随着工业对性能改进设计的需要，对复合材料，特别是纤维增强材料模型中的偏振性的研究变得越来越重要。Monge-Ampére方程和相关方程的数学研究在微分几何和最佳质量传输方面有特别重要的应用，例如，构造具有指定高斯曲率的表面和反射器/折射器设计。首席调查员(PI)将开展与这些专题密切相关的研究，并将试图解决这些领域中的一些未决问题。他将邀请研究生和博士后研究人员参与该项目的工作。他将把注意力集中在三个主要主题领域的几个问题上。首先，他将发展新的方法来研究非光滑区域上具有混合边界条件和粗糙系数的椭圆和抛物型方程，利用调和分析和保角映射的工具，在空间和时间上具有非局部时间导数或更一般地具有非局部导数的抛物型方程，以及具有可测系数的超抛物型(或亚椭圆型)Kolmogorov方程。其次，PI将研究退化完全非线性方程的正则性理论，特别是具有最优功率的m-Hessian方程，简并商Hessian方程，以及更一般类型的具有初等对称多项式的Hessian方程。最后，关于复合材料研究中出现的偏微分方程，PI对含有Lipschitz夹杂的复合材料以及这类拟线性或奇异/退化方程特别感兴趣，并将开发分析这些PDE的新方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project focuses on mathematical research in certain partial differential equations (PDE) that model phenomena in physics, engineering, materials science, and economics. The study of PDE arising in models for composite materials, in particular fiber reinforced materials, is of increasing importance due to industry's need to design for improved performance. The mathematical research of the Monge-Ampére equation and related equations has particularly significant applications in differential geometry and optimal mass transport such as, for example, constructing surfaces with prescribed Gaussian curvature and reflector/refractor design. The principal investigator (PI) will carry out research closely related to these topics and will attempt to address some of the open questions in these areas. He will engage graduate students and postdoctoral researchers in the work of the project.The PI will focus his attention on several questions in three main topical areas. First, he will develop new methods to study elliptic and parabolic equations with mixed boundary conditions and rough coefficients in nonsmooth domains by using tools from harmonic analysis and conformal maps, parabolic equations with nonlocal time derivatives or more generally with nonlocal derivatives in both space and time, and Kolmogorov equations of ultraparabolic (or hypoelliptic) type with measurable coefficients. Second, the PI will study the regularity theory for degenerate fully nonlinear equations, in particular the m-Hessian equation with optimal power, the degenerate quotient Hessian equations, and more general types of Hessian equations with elementary symmetric polynomials. Finally, regarding PDE arising in the study of composite materials, the PI is particularly interested in composites with Lipschitz inclusions, and quasilinear or singular/degenerate equations of this type, and will develop new methods for the analysis of these PDE.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.