A Note on Muskhelishvili-Vekua Reduction

Abstract

We focus on classical boundary value problems for the Laplace equation in a plane domain bounded by a nonsmooth curve which has a finite number of singular points. Using a conformal mapping of the unit disk onto the domain, we pull back the problem to the unit disk, which is usually referred to as the Muskhelishvili-Vekua method. The problem in the unit disk reduces to a Toeplitz equation with symbol having discontinuity of second kind. We develop a constructive invertibility theory for Toeplitz operators in the unit disk to derive solvability conditions and explicit formulas for solutions of the boundary value problem. 1. Statement of the problem Elliptic partial differential equations are known to appear in many applied areas of mathematical physics. To name but a few, we mention mechanics of solid medium, diffraction theory, hydrodynamics, gravity theory, quantum field theory, and many others. In this paper, we focus on boundary value problems for the Laplace equation in plane domains bounded by nonsmooth curves C. We are primarily interested in domains whose boundaries have a finite number of singular points of oscillating type. By this is meant that the curve may be parametrised in a neighbourhood of a singular point z0 by z(r )= z0 + r exp(iϕ(r)) for r ∈ (0 ,r 0), where r is the distance of z and z0 and ϕ(r) is a real-valued function which is bounded while its derivative in general is unbounded at r =0 . There is a vast literature devoted to boundary value problems for elliptic equa- tions in domains with nonsmooth boundary, cf. (KL91), (MNP00), (KMR00) and the references given there. In most of these papers, piecewise smooth curves with corner points or cusps are treated, cf. (DS00), (KKP98), (KP03), (MS89), (Rab99). The paper (RST04) is of particular importance, for it gives a character- isation of Fredholm boundary value problems in domains with weakly oscillating cuspidal edges on the boundary. There are significantly fewer works dealing with more complicated curves C. They mostly focus on qualitative properties, such as existence, uniqueness and