A Riemann-Hilbert approach to the Laplace equation

Let q(x, y) satisfy the Laplace equation in an arbitrary convex polygon. By performing the spectral analysis of the equation mu (z) - ik mu = q(x) - i q(y), z = x + iy, which involves solving a scalar Riemann-Hilbert (RH) prob lem, we construct an integral representation in the complex k-plane of q(x, y) in terms of a function rho (k). It has been recently shown that the fun ction p(k) can be expressed in terms of the given boundary conditions by so lving a matrix RH problem. Here we show that this method is also useful for solving problems in a non-convex polygon. We also recall that for simple p olygons it is possible to bypass the above integral representation and to s olve the Laplace equation by formulating a RH problem in the complex z-plan e. (C) 2000 Academic Press.