Harmonic Analysis and Partial Differential Equations

摘要：在这个项目中，我们利用调和分析技术，从偏微分方程组理论出发，研究了两组不同的问题。在与John L.Lewis和Kaj Nystrom合作处理的第一组问题中，我们将试图理解非圆柱(即时变)区域的边界的几何形状、关于区域中不动点的抛物线测度的正则性以及定义在“抛物线边界”上的抛物型奇异积分在勒贝格空间上的有界性之间的关系。第二组问题包括研究Kato的平方根问题，它的分支，以及散度椭圆算子的摄动理论中的相关问题，包括具有复系数的椭圆算子。有趣的是，我们最近在第二组问题上取得了重大进展，部分使用了Hofmann和Lewis在第一组问题上的一些技术。“抛物线型”偏微分方程组是上述第一组问题的研究对象，它出现在热传导的数学理论中，也出现在其他所谓的“扩散”过程中，包括在经济学、人口生物学和地下水流动等不同领域中出现的那些过程。例如，在以前的工作中，Hofmann和Lewis解决了一个新情况下的热传导经典问题(这些情况实际上是本项目所考虑的问题的症结所在)。问题是如何测量物体内部任何一点的温度，因为人们可以测量物体表面的任何地方的温度。我们考虑的“新情况”，是从现实的角度看，物体的形状可能会随着时间的推移而改变。当然，当物体被加热或冷却时，通常就是这种情况。第一段中提到的第二组问题，即所谓的“加藤问题”(或“平方根”问题)及其相关问题，起源于加藤敏夫1953年和1961年的两篇论文。加藤的工作是关于某些双曲(即，波状)偏微分方程解的“正则性”。粗略地说，他试图证明这些广义波的平稳性(或规律性)与引起该波的初始扰动的平稳性在数学上是相关的。加藤在他1961年的论文中指出，他寻求的这些双曲型方程的解的规则性可以从与原始方程有关的偏微分算子的“平方根”的某种技术性质中推导出来。如果这一技术性质为真，将使人们能够将问题归结为他在他早期的论文中已经获得的结果。事实证明，证明这种技术性质确实成立是极其困难的，而寻求这样做的努力(对于比加藤的原始波动问题实际需要的更一般的一类算子来说)已被称为‘加藤问题’。

Abstract:In this project, we study two separate sets of problems from the theory of Partial Differential Equations, using the techniques of Harmonic Analysis. In the case of the first set of problems, which will be treated in collaboration with John L. Lewis and Kaj Nystrom, we will attempt to understand the relationship between the geometry of the boundary of a non-cylindrical (i.e. time-varying) domain, the regularity of parabolic measure with respect to a fixed point in the domain, and the boundedness, on Lebesgue spaces, of parabolic singular integrals defined on the "parabolic boundary." The second set of problems involves the study ofthe square root problem of Kato, its ramifications, and related questionsin the perturbation theory for divergence form elliptic operators including those with complex coefficients. Interestingly, we have recently made significant progress on the second set of problems, using in part some techniques from the work of Hofmann and Lewis on the first set. Partial differential equations of "parabolic" type, which are the subject of study in first set of problems mentioned above, arise in the mathematical theory of heat conduction, and also inother so-called "diffusion" processes, including those which occur in fieldsas diverse as economics, population biology, and the flow of ground water.For example, in previous work, Hofmann and Lewis have solved a classical problem of heat conduction under new circumstances (which circumstances are really the crux of the matter in the problems under consideration in the present project). The problem is that of determing the temperature at any point inside an object, given that one can measure the temperature everywhere on thesurface of the object. The "new circumstance" which we consider,is to take the realistic point of view that the shape of the object may change overtime. Certainly, this is often the case when objects are heatedor cooled. The second set of problems alluded to in the first paragraph,namely, the so-called "Kato problem" (or "square root" problem) and related questions, has its origins in two papers written by Tosio Kato in 1953 and 1961.Kato's work concerned the "regularity" of solutions of certain hyperbolic (i.e., wave-like) partial differential equations. Roughly speaking, he was trying toshow that the smoothness (or regularity) of these generalized waves has a mathematical correlation with the smoothness of the initial disturbance which causes the wave. Kato observed, in his 1961 paper, that theregularity which he sought, for solutions of these hyperbolic equations, could be deduced from a certain technical property of the ``square root" of a partial differential operator related to the original equation. This technical property, if true, would enable one to reduce matters to results which he had already obtained in his earlier paper. To prove that this technical property actually did hold turned out to be extremely difficult, and the quest to do so (for a somewhat more general class of operators than Kato's original wave problem actually required) has become known as the ``Kato Problem".