Parabolic and elliptic boundary value and free boundary problems

抛物线和椭圆边值以及自由边界问题

• 批准号: 2349846
• 负责人: Steven Hofmann
• 金额: $ 24.72万
• 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: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349846&HistoricalAwards=false
• 关键词: Parabolic; elliptic; boundary; value; free

This project is concerned with the theory of boundary value problems and free boundary problems for elliptic and parabolic partial differential equations. Such equations arise, for example, in the mathematical theory of heat conduction: an equation of elliptic type describes steady state (equilibrium) temperature distributions and a related parabolic equation governs heat conduction in the time-evolutive case. In a boundary value problem, one uses mathematical knowledge of either 1) the temperature distribution on the boundary (i.e., perimeter) of some region in space (or of some evolving region in space-time) or 2) the heat flux (the rate at which heat flows across the boundary), to deduce information about the internal temperature distribution inside the region. In free boundary problems, one uses simultaneous knowledge of both the boundary temperature distribution and the heat flux to deduce information about the geometry of the region and its boundary. A central goal of this project is to understand the interplay between analytic information and geometry. This project provides research training opportunities for graduate students.The project has three main areas of focus: 1) to find a geometric characterization of the space-time domains for which the Dirichlet (or initial-Dirichlet) problem is solvable for the heat equation with singular (p-integrable) data, and to study related free boundary problems. The PI and coauthors have previously treated such problems in the steady state (elliptic) setting; in the present project, the PI seeks to treat the more difficult time-evolutive case. 2) to solve the Kato square root problem for elliptic equations in non-divergence form. The solution of the Kato problem for divergence form elliptic operators has led to 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 nondivergence setting, the PI plans to treat the Kato problem for non-divergence elliptic operators. 3) to solve the Dirichlet problem in Lipschitz domains for non-symmetric divergence from elliptic equations with periodic coefficients. A primary motivation for the study of operators with periodic coefficients is their applicability to the theory of homogenization, which in turn provides a mathematical model for materials with periodic microstructure.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.

这个项目是关于椭圆型和抛物型偏微分方程边值问题和自由边界问题的理论。例如，这样的方程出现在热传导的数学理论中：椭圆型方程描述稳态(平衡)温度分布，而相关的抛物线方程在时间演化的情况下支配热传导。在边值问题中，人们使用1)空间中某一区域(或时空中某一演化区域)边界(即周长)上的温度分布或2)热通量(热流通过边界的速率)的数学知识来推断该区域内部温度分布的信息。在自由边界问题中，人们使用边界温度分布和热通量的同时知识来推导关于区域及其边界的几何信息。这个项目的一个中心目标是了解解析信息和几何之间的相互作用。该项目为研究生提供了研究培训的机会。该项目有三个主要领域：1)寻找具有奇异(p-可积)数据的热方程的Dirichlet(或初始-Dirichlet)问题可解的时空区域的几何特征，并研究相关的自由边界问题。PI和合著者以前在稳态(椭圆)环境中处理过这样的问题；在本项目中，PI试图处理更困难的时间演化情况。2)求解无散度形式的椭圆型方程的Kato平方根问题。散度型椭圆算子Kato问题的求解是散度型方程边值问题理论的重大进展。作为在非散度环境中打开类比理论的第一步，PI计划处理非散度椭圆算子的加藤问题。3)求解周期系数椭圆型方程非对称发散的Lipschitz域上的Dirichlet问题。研究具有周期系数的运算符的一个主要动机是它们对均化理论的适用性，这反过来又为具有周期性微结构的材料提供了一个数学模型。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。