Singular Integrals and Complex Analysis in One and Several Variables

University Of Arkansas, Fayetteville AR

Investigators

Abstract

Abstract Lanzani This project concerns several questions about integral representations for holomorphic functions of one and several complex variables, related (singular) integral boundary operators, and elliptic boundary value problems. The main underlying theme consists of estimates of Cauchy singular integral operators with focus on the case of domains with rough boundaries. The crux of this project is the basic observation that it is often possible to determine the values a certain function, say, f, takes inside, say, a ball, by only measuring the values f takes on the surface of the ball, via the computation of a (surface) integral involving the datum f and an auxiliary function K ( the "kernel"). It may help to think of the value f at a point P as the temperature at this point: by (easily) measuring temperature on the surface of a ball (think of the ball as been made of a solid material, such as metal), one can then find out the value of the temperature inside the ball without having to reach the inside, simply by plotting the surface temperature data in an integral formula, and then computing the integral. Because of the microscopic nature of matter, it is much more natural to make these measurements and calculations on a "rough" body, say a cube (which has corners and edges), than a "smooth" ball. In the mathematical model, this new situation translates into additional constraints on the kernel K which make it much harder to effectively use K in the computation of the integrals. In this project we study certain ("complex")analogues of these mathematical models in the context of "rough" domains.