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.

该项目主要涉及热传导的数学理论。主题在于几何测度理论、偏微分方程和调和分析的交叉点。在几何测度论中，人们通过集合上某种测度的行为来研究集合的几何性质，其中“测度”的概念概括了长度、面积和体积的概念。在这个项目中，我们考虑的集合类型通常是空间中某个区域的边界，或者是时空中某个不断演化的区域的边界。 偏微分方程以数学方式描述热传导、波的传播和许多其他物理现象。调和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区域的几何形状如何影响通过该区域及其边界的热传导。该项目将通过研究生培训为美国劳动力的发展做出贡献。该项目主要关注三个领域：1）诺伊曼问题。 PI 计划使用 p 可积数据，在满足测度理论条件的定量、尺度不变版本的域中求解拉普拉斯方程的诺依曼问题，该条件相当于几乎每个边界都存在测度理论外部单位法线观点;因此，对于诺伊曼问题来说，该条件是自然的，并且可能是尖锐的。最终，PI 寻求以几何方式描述可能存在这种可解性的域。 2) 抛物线定量整流性。目前，与稳态椭圆设置中可用的丰富理论相比，时间演化抛物线设置中的定量可整流性理论还相当初级。 在 PI 和合著者最近的工作中，椭圆定量可整流性在拉普拉斯方程的狄利克雷问题可解的那些域的表征中发挥了核心作用。 PI 预计所提议的工作将是在抛物线情况下进行类似表征的第一步。 3）非散度椭圆算子的加藤平方根问题。 散度型椭圆算子的平方根问题的求解使得散度型方程的边值问题理论取得了重大进展。 作为在非发散环境中开放类似理论的第一步，PI 提议处理非发散椭圆算子的 Kato 问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的评估进行评估，认为值得支持。智力价值和更广泛的影响审查标准。

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