Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary

Abstract

We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. Purpose: We wish to construct explicit formulas for solutions of these problems when the boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. Methods: To study the problem, we apply the familiar Vekua-Muskhelishvili method which consists in the use of conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This in turn later reduces to a Toeplitz operator equation on the unit circle with symbol-bearing discontinuities of the second kind. Results: We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions. Conclusions: Our results raise Fredholm theory for boundary value problems in domains with singularities which are not necessarily rectifiable.