Harmonic Analysis, Boundary Value Problems, and Parabolic Rectifiability

这个项目主要涉及热传导的数学理论。其主题是几何测度论、偏微分方程式和调和分析的交集。在几何测度论中，人们通过集合上的某种度量的行为来研究集合的几何性质，其中“度量”的概念推广了长度、面积和体积的概念。在这个项目中，我们考虑的那种集合通常是空间中某个区域的边界，或者是时空中某个演化区域的边界。偏微分方程式在数学上描述了热的传导、波的传播和许多其他物理现象。调和分析是一种通过将数学函数分解成基本组成部分来提取信息的数学工具。这个项目的一个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区域和跨越其边界的热传导。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。该项目有三个主要重点领域：1)诺伊曼问题。PI计划解决拉普拉斯方程的Neumann问题，具有p-可积数据，在满足测度论条件的定量的、尺度不变的版本的区域内，该测度论条件等价于在几乎每个边界点上存在测度论外单位法线；因此，该条件对于Neumann问题是自然的，并且可能是尖锐的。最后，PI试图从几何上刻画这样的可解性可能存在的区域。2)抛物线定量整流性。目前，与稳态椭圆背景下的丰富理论相比，时间演化抛物线背景下的定量可纠正性理论是相当初级的。在PI和合著者最近的工作中，椭圆定量可纠正性在刻画拉普拉斯方程的Dirichlet问题可解的区域中发挥了核心作用。PI预计，拟议的工作将是朝着抛物线情况下的类似特征迈出的第一步。3)非散度椭圆算子的Kato平方根问题。散度型椭圆算子平方根问题的求解使散度型方程边值问题的理论取得了重大进展。作为在非散度环境中开放类比理论的第一步，PI建议为非散度椭圆算子处理加藤问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.