Boundary integral equation approach for conformal mapping, complex boundary value problems, and reproducing kernels.

Abstract

Conformal mapping has been a familiar tool of science and engineering for generations. Its ability to map one planar region onto another via analytic function proves invaluable in applied mathematics. For numerical purposes in conformal mapping, the integral equation methods are more preferable and effective. Of special interest is the Riemann map which maps a simply connected region onto a unit disk. The Riemann map is closely to complex boundary value problem and some reproducing kernels known as the Szego and the Bergman kernels. The fact that there is an efficient numerical method, based on the Kerzman-Stein-Trummer integral equation, for computing the Szego kernel has been known since 1986. The study of the Kerzman-Stein-Trummer integral equation has led to our discovery of a new integral equation for the Bergman kernel which can be used effectively for numerical conformal mapping. This discovery also motivated a general formulation of integral equations associated to certain boundary relationships which can rise to various integral equations (classical and new) related to conformal mapping and boundary value problem of interior, exterior and doubly connected regions. This paper presents some of our past discoveries as well as ongoing research activities regarding the integral equation approach for conformal mapping, complex boundary value problem and reproducing kernels.