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

刘易斯9531642本研究涉及抛物型测度和勒贝格测度的绝对连续性以及相关的Dirichlet-Neumann问题。在某些时变区域上，Lewis和Murray证明了热方程的抛物型测度关于某一射影Lebesgue型测度是相互绝对连续的。本研究将从一个模型PDE的研究开始，该模型的原型是由热方程通过一定的映射到上述时变域上得到的回撤PDE。由于拉回PDE具有相对于勒贝格测度相互绝对连续的抛物线测度，因此调查的目的将是确定实际上需要PDE模型的什么性质来保证上述测度的相互绝对连续性。至于Dirichlet和Neumann问题，它们现在被很好地理解为定义在上述时变域边界上的平方可积函数。下一步是考虑p次方可积函数的Neumann问题，当p在1和2之间时。同样，上述拉回偏移量需要仔细分析。许多物理过程可以用偏微分方程式来分析。最著名的经典偏微分方程是拉普拉斯方程、热方程和波动方程，它们中的每一个都起源于18世纪和19世纪，并在重力、电学、流体流动、电磁波的研究中得到了应用，仅举几个例子。我的研究涉及以下类型的问题：给定房间墙壁上的温度，找出房间中任何地方的温度在以后的任何时间？这个问题称为热方程的狄里克莱特问题。诺依曼问题可以用从房间墙壁中流出的热量的速率来表述。从数学上讲，如果房间墙壁上的温度是固定的，并且足够好(连续)，那么狄里克莱特问题可以被证明有唯一的解。我的部分研究关注的是，当房间的墙壁和墙壁上的温度允许变化时，这个问题是否有唯一的解决方案。这个问题现在已经基本上完全解决了，相应的Neumann问题正在研究中。这项研究的可能应用是自由边界问题，其中对象的大小和温度不断变化(冰融化、气体膨胀、核废料固化)。

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).