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.

该项目涉及边界价值问题的理论和椭圆形和抛物线部分微分方程的自由边界问题。例如，在热传导的数学理论中出现了这样的方程：椭圆类型的方程描述了稳态（平衡）温度分布，而相关的抛物线方程则控制了及时的情况下的热传导。在边界价值问题中，人们使用数学知识1）1）在空间中某些区域（或时空中某些不断发展的区域）边界上的温度分布（即周围）或2）热通量（速率在其中热流过边界），以推断有关该地区内部温度分布的信息。在自由边界问题中，一个人同时了解边界温度分布和热通量的知识来推断有关该地区几何形状及其边界的信息。该项目的一个核心目标是了解分析信息和几何形状之间的相互作用。该项目为研究生提供了研究培训机会。该项目具有三个主要的重点领域：1）找到对时空域的几何表征，该域（或初始 - 迪里奇莱特）问题可用于使用该域，可用于加热方程。单数（可P综合）数据，并研究相关的自由边界问题。 PI和合着者以前在稳态（椭圆形）设置中处理过此类问题。在本项目中，PI试图治疗更困难的时地案例。 2）以非差异形式解决Kato Square Root问题。 Kato问题的解决方案的解决方案椭圆形操作员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。作为在非散发环境中开放类似理论的第一步，PI计划为非差异椭圆运算符治疗Kato问题。 3）解决Lipschitz结构域中的Dirichlet问题，以与具有周期性系数的椭圆方程相对差异。 具有周期性系数的运营商研究的主要动机是它们适用于均质理论，这又为具有定期微观结构的材料提供了数学模型。该奖项反映了NSF的法定任务，并被认为是通过使用评估来支持的，基金会的智力优点和更广泛的影响评论标准。