Regularity Questions in Linear and Nonlinear Partial Differential Equations

线性和非线性偏微分方程的正则性问题

基本信息

批准号：
2055244
负责人：
Hongjie Dong
金额：
$ 30万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055244&HistoricalAwards=false
关键词：
Regularity Questions Linear Nonlinear Partial

项目摘要

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.

该项目着重于某些部分偏微分方程（PDE）的数学研究，以模拟物理，工程，材料科学和经济学中的现象。在复合材料模型中引起的PDE的研究，特别是纤维增强材料，由于行业需要设计以提高性能的需求而提高了重要性。 Monge-Ampére方程和相关方程的数学研究在差异几何和最佳质量传输中特别有用，例如，例如使用处方的高斯货币和反射器/折射率设计构建表面。首席研究员（PI）将进行与这些主题密切相关的研究，并试图解决这些领域的一些开放问题。他将与研究生和博士后研究人员一起参与项目的工作。PI将把注意力集中在三个主要主题领域的几个问题上。 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.其次，PI将研究归化完全非线性方程的规则性理论，特别是具有最佳幂的M-Hessian方程，退化引号Hessian方程以及具有基本对称多项式的Hessian方程的更多一般类型。 Finally, regarding PDE arising in the study of composite materials, the PI is particularly interested in compositions 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 precious of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.