Harmonic Analysis and Partial Differential Equations

翻译后摘要：在这个项目中，我们研究两个独立的偏微分方程理论的问题，使用调和分析的技术。在第一组问题的情况下，这将是与约翰L。刘易斯和Kaj Nystrom，我们将试图理解非圆柱形（即时变）区域的边界几何，关于区域中不动点的抛物测度的正则性，以及定义在“抛物边界”上的抛物奇异积分在Lebesgue空间上的有界性之间的关系。第二组问题涉及到Kato平方根问题的研究，它的分支，以及包括复系数在内的发散型椭圆算子的扰动理论中的相关问题。 有趣的是，我们最近在第二组问题上取得了重大进展，部分使用了霍夫曼和刘易斯在第一组问题上工作的一些技术。“抛物”型偏微分方程是上述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，包括那些出现在经济学、人口生物学和地下水流动等不同领域的过程。例如，在以前的工作中，霍夫曼和刘易斯已经解决了一个新情况下的经典热传导问题（这种情况实际上是本项目所考虑问题的关键）。问题是要确定物体内部任何一点的温度，因为人们可以测量物体表面任何地方的温度。我们所考虑的“新情况”，是从现实的角度来看，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，情况往往如此。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）及相关问题，起源于加藤托西奥在1953年和1961年写的两篇论文。加藤的工作涉及某些双曲（即，波动型偏微分方程。粗略地说，他试图表明，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相关性。加藤观察到，在他1961年的论文中，thegurgency，他寻求，解决这些双曲方程，可以推导出一定的技术性质的“平方根”的偏微分算子有关的原始方程。这个技术性质，如果是真的，将使一个减少问题的结果，他已经在他以前的文件。要证明这一技术性质确实成立是极其困难的，而要做到这一点（对于比加藤最初的波问题实际上需要的更一般的一类算子）的探索被称为“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".