Harmonic Analysis and Partial Differential Equations

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

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 of the square root problem of Kato, its ramifications, and related questions in 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 in other so-called "diffusion" processes, including those which occur in fields as 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 the surface 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 over time. Certainly, this is often the case when objects are heated or 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 to show 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 the regularity 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".