Harmonic Analysis and Partial Differential Equations

摘要：在本课题中，我们利用调和分析技术研究了偏微分方程理论中的两组独立问题。在第一组问题的情况下，我们将与John L. Lewis和Kaj Nystrom合作处理，我们将试图理解非圆柱形（即时变）域边界的几何关系，抛物线测度相对于该域内不动点的规律性，以及在Lebesgue空间上定义在“抛物线边界”上的抛物线奇异积分的有界性。第二组问题涉及研究加藤的平方根问题及其分支，以及包括复系数的椭圆算子在内的散度形式的微扰理论中的相关问题。有趣的是，我们最近在第二组问题上取得了重大进展，部分地使用了霍夫曼和刘易斯在第一组问题上的一些技术。“抛物线”型偏微分方程是上面提到的第一组问题的研究主题，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，包括那些发生在经济学、人口生物学和地下水流动等不同领域的过程。例如，在之前的工作中，Hofmann和Lewis已经解决了新情况下的经典热传导问题（在本项目所考虑的问题中，这种情况确实是问题的关键）。问题是确定物体内部任何一点的温度，因为我们可以测量物体表面任何地方的温度。我们考虑的“新情况”是采取现实的观点，即物体的形状可能随着时间的推移而改变。当然，当物体被加热或冷却时，这种情况经常发生。第一段提到的第二组问题，即所谓的“加藤问题”（或“平方根”问题）及相关问题，起源于加藤俊雄于1953年和1961年撰写的两篇论文。加藤的工作涉及某些双曲（即波状）偏微分方程解的“规律性”。粗略地说，他试图证明这些广义波的平滑性（或规律性）与引起波的初始扰动的平滑性具有数学上的相关性。加藤在他1961年的论文中指出，他所寻求的这些双曲方程解的规律性，可以从与原方程相关的偏微分算子的“平方根”的某种技术性质中推导出来。这个技术性质，如果是真的，将使人们能够把问题归结为他在以前的论文中已经得到的结果。要证明这一技术性质实际上是非常困难的，而这样做的追求（对于比加藤原来的波问题实际需要的更一般的算子类）已经被称为“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".