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

摘要刘易斯9531642 本文研究了抛物测度与Lebesgue测度的相互绝对连续性以及相关的Dirichlet-Neumann问题。刘易斯和Murray证明了在某些时变区域上热方程的抛物测度关于某个射影Lebesgue测度是相互绝对连续的。 本研究将开始于一个模型偏微分方程的研究，其原型是通过在上述时变域上的某种映射从热方程得到的拉回偏微分方程。 由于拉回偏微分方程有一个抛物线测度， 关于勒贝格测度的连续性，研究的对象将是确定模型PDE的哪些性质实际上需要保证上述测度的相互绝对连续性。至于狄利克雷和诺依曼问题，他们现在很好地理解为平方可积函数的边界上定义的上述时变域。下一步是考虑当p在1和2之间时p次幂可积函数的Neumann问题。同样，上述回调PDE 需要仔细分析。 许多物理过程可以用偏微分方程来分析。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，它们都起源于18世纪和19世纪，在重力、电、流体流动、电磁波等方面都有应用。 我的研究涉及以下类型的问题： 温度的墙壁上的一个房间，找到温度在任何地方的房间在任何以后的时间？ 这个问题被称为热方程的狄利克雷问题。诺依曼问题也可以用类似的方式表述，即热量从房间墙壁流出的速率。 在数学上，如果房间墙壁上的温度是固定的，并且足够好（连续），那么狄利克雷问题可以被证明有一个唯一的解。 我的研究的一部分一直关注这个问题是否有一个独特的解决方案时，房间的墙壁和墙壁上的温度允许变化。这个问题现在基本上已经完全解决， Neumann问题正在研究中。 这项研究的可能应用是自由边界问题，其中物体的尺寸和温度不断变化（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).