A Riemann Hilbert Approach to the Laplace Equation

Let qŽx, y. satisfy the Laplace equation in an arbitrary convex polygon. By performing the spectral analysis of the equation z ik qx iqy , z x iy, which involves solving a scalar Riemann HilbertŽRH.problem, we construct an integral representation in the complex k-plane of qŽx, y.in terms of a function Žk.. It has been recently shown that the function Žk. can be expressed in terms of the given boundary conditions by solving 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 polygons it is possible to bypass the above integral representation and to solve the Laplace equation by formulating a RH problem in the complex z-plane.