Regularity Questions in Linear and Nonlinear Partial Differential Equations

该项目侧重于某些偏微分方程（PDE）的数学研究，这些偏微分方程模拟了物理，工程，材料科学和经济学中的现象。由于工业界需要设计改进的性能，在复合材料，特别是纤维增强材料的模型中产生的PDE的研究越来越重要。蒙赫-安培方程和相关方程的数学研究在微分几何和最优质量输运中有着特别重要的应用，例如，构造具有规定高斯曲率的表面和反射器/折射器设计。主要研究者（PI）将开展与这些主题密切相关的研究，并将尝试解决这些领域的一些开放性问题。他将吸引研究生和博士后研究人员参与项目的工作，主要研究者将集中精力研究三个主要领域的几个问题。首先，他将开发新的方法来研究椭圆和抛物方程的混合边界条件和粗糙系数在非光滑域使用工具从调和分析和保形映射，抛物方程与非本地时间导数或更一般的非本地衍生物在空间和时间，和Kolmogorov方程的超抛物（或hypoelliptical）型可测系数。第二，PI将研究退化完全非线性方程的正则性理论，特别是具有最佳幂的m-Hessian方程，退化商Hessian方程以及具有初等对称多项式的更一般类型的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Lipschitz夹杂物的复合材料以及此类准线性或奇异/退化方程特别感兴趣，并将开发分析这些偏微分方程的新方法。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.