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

• 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

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.

该项目的重点是某些偏微分方程 (PDE) 的数学研究，这些偏微分方程对物理、工程、材料科学和经济学中的现象进行建模，复合材料（特别是纤维增强材料）模型中出现的偏微分方程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具有特别重要的应用，例如构造具有规定的高斯曲率和反射器/折射器设计。首席研究员（PI）将开展与这些主题密切相关的研究，并尝试解决这些领域的一些悬而未决的问题。他将让研究生和博士后研究人员参与该项目的工作。 PI 将重点关注三个主要主题领域的几个问题，首先，他将使用调和分析和共形映射、抛物线方程等工具，开发新方法来研究非光滑域中具有混合边界条件和粗糙系数的椭圆方程和抛物线方程。具有非局部时间导数或更一般地具有空间和时间上的非局部导数，以及具有可测量系数的超抛物线（或亚椭圆）型柯尔莫哥洛夫方程。 其次，PI 将研究退化完全非线性方程的正则理论，特别是 m-。具有最优幂的 Hessian 方程、简并商 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 对具有 Lipschitz 夹杂物的复合材料以及此类拟线性或奇异/简并方程特别感兴趣，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优点和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Weighted mixed norm estimates for fractional wave equations with VMO coefficients

具有 VMO 系数的分数波动方程的加权混合范数估计

• DOI: 10.1016/j.jde.2022.07.040
• 发表时间: 2021-02-01
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Hongjie Dong;Yanze Liu
• 通讯作者: Yanze Liu

Gradient estimates for singular parabolic p-Laplace type equations with measure data

具有测量数据的奇异抛物线 p-拉普拉斯型方程的梯度估计

• DOI: 10.1007/s00526-022-02189-5
• 发表时间: 2022-06
• 期刊: Calculus of Variations and Partial Differential Equations
• 影响因子: 2.1
• 作者: Dong, Hongjie;Zhu, Hanye
• 通讯作者: Zhu, Hanye

Time Fractional Parabolic Equations with Measurable Coefficients and Embeddings for Fractional Parabolic Sobolev Spaces

具有可测系数的时间分数抛物型方程和分数抛物型Sobolev空间的嵌入

• DOI: 10.1093/imrn/rnab229
• 发表时间: 2021-08
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Dong, Hongjie;Kim, Doyoon
• 通讯作者: Kim, Doyoon

The Dirichlet-conormal problem for the heat equation with inhomogeneous boundary conditions

非齐次边界条件热方程的狄利克雷协整问题

• DOI: 10.1016/j.aim.2022.108777
• 发表时间: 2022-12
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Dong, Hongjie;Li, Zongyuan
• 通讯作者: Li, Zongyuan

Global $${L}_{p}$$ Estimates for Kinetic Kolmogorov–Fokker–Planck Equations in Nondivergence Form

非散度形式的动力学 Kolmogorov→Fokker→Planck 方程的全局 $${L}_{p}$$ 估计

• DOI: 10.1007/s00205-022-01786-0
• 发表时间: 2022-07
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Dong, Hongjie;Yastrzhembskiy, Timur
• 通讯作者: Yastrzhembskiy, Timur