Mathematical Sciences: Absolute Continuity of Parabolic Measure and Regularity of PDE's

数学科学：抛物线测度的绝对连续性和偏微分方程的正则性

• 批准号: 9531642
• 负责人: John Lewis
• 金额: $ 6.58万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-15 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531642&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Absolute; Continuity; Parabolic

Abstract Lewis 9531642 This research is concerned with mutual absolute continuity of parabolic and Lebesgue measure as well as related Dirichlet-Neumann problems. In certain time varying domains Lewis and Murray have shown that parabolic measure for the heat equation is mutually absolutely continuous with respect to a certain projective Lebesgue measure. This investigation will begin with the study of a model pde whose prototype is the pullback pde obtained from the heat equation by way of a certain mapping onto the above time varying domain. Since the pullback pde has a parabolic measure that is mutually absolutely continuous with respect to Lebesgue measure, the object of the investigation will be to determine what properties of the model pde are actually needed to guarantee mutual absolute continuity of the above measures. As for the Dirichlet and Neumann problems they are now well understood for square integrable functions defined on the boundary of the above time varying domains. The next step is to consider the Neumann problem for p th power integrable functions when p is between 1 and 2. Again the above pullback pde needs to be analyzed closely. Many phyical processes can be analyzed using partial differential equations. The most famous classical partial differential equations are Laplace's equation, the heat equation, and the wave equation each of which originated in the 18 th and 19 th centuries and found uses in the study of gravity, electricity, fluid flow, electromagnetic waves, to mention only a few topics. My research concerns problems of the following type : Given the temperature on the walls of a room, find the temperature at any place in the room at any later time ? This problem is called the Dirichlet problem for the heat equation. The Neumann problem can be stated similarly in terms of the rate at which heat is flowing out of the walls of the room. Mathematically if the temperature on the walls of the room is fixed and nice enough (cont inuous), then the Dirichlet problem can be shown to have a unique solution. Part of my research has been concerned with whether this problem has a unique solution when the walls of the room and temperature on the walls is allowed to vary. This problem has now been essentially completely solved and the corresponding Neumann problem is being studied. Possible applications of this research are to free boundary problems where the size and temperature of an object are constantly changing (ice melting, gases expanding, nuclear waste solidifying).

摘要刘易斯9531642这项研究与抛物线派和列布斯格度量的相互绝对连续性以及相关的Dirichlet-Neumann问题有关。在某些时候，刘易斯和默里的域变化，表明，相对于某些投射的lebesgue度量，热方程式的抛物线度量是相互连续的。 这项研究将从对模型PDE的研究开始，该模型PDE的原型是通过在上述时间变化域上映射到上述时间变化的域中从热方程中获得的回溯PDE。 由于回调PDE具有相对于Lebesgue度量绝对连续的抛物线度量，因此研究的目的是确定实际上需要使用哪些模型PDE的属性来保证上述测量的相互绝对连续性。至于Dirichlet和Neumann问题，他们现在对在上述时间变化域的边界上定义的平方集成函数进行了充分的理解。下一步是考虑到p在1到2之间的努力集合功能的诺伊曼问题。同样，需要对上述回调PDE进行仔细分析。 可以使用部分微分方程分析许多植物过程。最著名的经典部分偏微分方程是拉普拉斯的方程，热方程式和波动方程，每个方程都起源于18世纪和19世纪，并在重力，电力，流体流，电磁波研究中发现了用途，仅提及几个主题。 我的研究涉及以下类型的问题：鉴于房间的墙壁上的温度在以后的任何时候在房间的任何位置找到温度吗？ 这个问题称为热方程式的Dirichlet问题。诺伊曼的问题可以在热量从房间的墙壁上流出的速度相似。 在数学上，如果房间的墙壁上的温度固定并且足够好（连续），则可以证明Dirichlet问题具有独特的解决方案。 我的一部分研究一直关注当房间的墙壁和墙壁上的温度变化时，这个问题是否具有独特的解决方案。现在，这个问题已经完全解决了，并且正在研究相应的Neumann问题。 这项研究的可能应用是在物体的大小和温度不断变化的地方自由边界问题（冰熔化，气体扩展，核废料固化）。