Harmonic Analysis, Boundary Value Problems, and Parabolic Rectifiability

谐波分析、边值问题和抛物线可整流性

• 批准号: 2000048
• 负责人: Steven Hofmann
• 金额: $ 24.99万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000048&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Boundary; Value; Problems

This project is primarily concerned with the mathematical theory of heat conduction. The subject matter lies at the intersection of geometric measure theory, partial differential equations, and harmonic analysis. In geometric measure theory, one studies geometric properties of sets via the behavior of some measure on that set, where the concept of "measure" generalizes the notions of length, area and volume. In this project, the kind of set that we consider is typically the boundary of some region in space, or of some evolving region in space-time. Partial differential equations describe mathematically the conduction of heat, the propagation of waves, and many other physical phenomena. Harmonic analysis is a mathematical tool with which one extracts information by decomposing mathematical functions into fundamental constituent pieces. A principal goal of this project is to understand, in a quantitative way, how the geometry of a region influences the conduction of heat through the region, and across its boundary. This project will contribute to the development of the US workforce through the training of graduate students.The project has three main areas of focus: 1) the Neumann Problem. The PI plans to solve the Neumann problem for Laplace's equation, with p-integrable data, in domains satisfying a quantitative, scale invariant version of a measure theoretic condition which is equivalent to the existence of a measure-theoretic outer unit normal at almost every boundary point; thus, the condition is natural for the Neumann problem, and may be sharp. Eventually, the PI seeks to characterize geometrically the domains in which such solvability is possible. 2) Parabolic quantitative rectifiability. At present, the theory of quantitative rectifiability in the time-evolutive parabolic setting is quite rudimentary in comparison to the rich theory available in the steady-state elliptic setting. In recent work of the PI and co-authors, elliptic quantitative rectifiability has played a central role in the characterization of those domains for which the Dirichlet problem for Laplace's equation is solvable. The PI expects that the proposed work would be a first step towards an analogous characterization in the parabolic case. 3) The Kato square root problem for non-divergence form elliptic operators. The solution of the square root problem for divergence form elliptic operators has enabled significant progress in the theory of boundary value problems for divergence form equations. As a first step towards opening up the analogous theory in the non-divergence setting, the PI proposes to treat the Kato problem for non-divergence elliptic operators.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.

该项目主要与热传导的数学理论有关。该主题在于几何测量理论，部分微分方程和谐波分析的交集。在几何测量理论中，一个人通过某种度量的行为研究集合的几何特性，其中“测量”的概念概括了长度，面积和体积的概念。在这个项目中，我们认为的那种集合通常是空间中某些区域的边界，或时空中某些不断发展的区域的边界。 部分微分方程在数学上描述了热的传播，波的传播以及许多其他物理现象。谐波分析是一种数学工具，通过将数学函数分解为基本组成部分来提取信息。该项目的主要目标是以定量的方式了解一个区域的几何形状如何影响通过该地区的热量传导以及跨越其边界。该项目将通过研究生的培训为美国劳动力的发展做出贡献。该项目具有三个主要重点领域：1）Neumann问题。 PI计划在满足度量理论条件的定量，规模不变版本的域中解决Laplace方程的Neumann问题，并具有可依赖的数据，这几乎等于在几乎每个边界点上都存在量度理论外部单元正常；因此，这种条件对于诺伊曼问题是自然的，并且可能是尖锐的。最终，PI试图以几何表征这种可溶性的域。 2）抛物线定量可区分性。目前，与在稳态椭圆形设置中可用的丰富理论相比，定量可重新可相关性的理论是基本的。 在PI和合着者的最新工作中，椭圆定量的重新可相关性在表征那些域的表征中起着核心作用，而Laplace方程的Dirichlet问题是可以解决的。 PI期望拟议的工作将是抛物线案例中类似表征的第一步。 3）非差异形式的椭圆算子的Kato Square root问题。 椭圆形算子的平方根问题的解决方案在差异形式方程的边界价值问题理论中取得了重大进展。 作为在非差异环境中开放类似理论的第一步，PI提议为非差异椭圆操作员处理加藤问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准通过评估来通过评估来支持的。

项目成果

The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

具有 BMO 反对称部分的椭圆算子的狄利克雷问题

• DOI: 10.1007/s00208-021-02219-1
• 发表时间: 2021
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Hofmann, Steve;Li, Linhan;Mayboroda, Svitlana;Pipher, Jill
• 通讯作者: Pipher, Jill

On big pieces approximations of parabolic hypersurfaces

关于抛物线超曲面的大块近似

• DOI: 10.54330/afm.115417
• 发表时间: 2021
• 期刊: Annales Fennici Mathematici
• 作者: Bortz, Simon;Hoffman, John;Hofmann, Steve;Luna-Garcia, Jose Luis;Nyström, Kaj
• 通讯作者: Nyström, Kaj

Regularity and Neumann problems for operators with real coefficients satisfying Carleson conditions

• DOI: 10.1016/j.jfa.2023.110024
• 发表时间: 2022-07
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: M. Dindoš;S. Hofmann;J. Pipher

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

满足容量密度条件的1边NTA域中椭圆算子的平方函数和非切向极大函数估计

• DOI: 10.1515/acv-2021-0053
• 发表时间: 2022
• 期刊: Advances in Calculus of Variations
• 影响因子: 1.7
• 作者: Akman, Murat;Hofmann, Steve;Martell, José María;Toro, Tatiana
• 通讯作者: Toro, Tatiana

Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets

卡勒森测量热量函数和抛物线一致可整流集的估计

• DOI: 10.2140/apde.2023.16.1061
• 发表时间: 2023
• 期刊: Analysis & PDE
• 影响因子: 2.2
• 作者: Bortz, Simon;Hoffman, John;Hofmann, Steve;Luna García, José Luis;Nyström, Kaj
• 通讯作者: Nyström, Kaj