On a transform method for the Laplace equation in a polygon
ATHANASSIOS S. FOKAS · IMA Journal of Applied Mathematics · 2003
Let q(x, y) satisfy a boundary value problem for the Laplace equation in an arbitrary convex polygon with n sides. An integral representation in the complex k‐plane is given for q(x, y) in terms of n functions ρj(k), j = 1, …, n. The function ρj consists of an integral over the jth side involving both qx and qy , thus each ρj involves one unknown boundary value. The functions ρj are not independent but they satisfy the important global relation that their sum vanishes. The solution of a given boundary value problem reduces to the analysis of this single relation for the n unknown ρj. For a general polygon with general Poincaré boundary conditions, this gives rise to a matrix Riemann–Hilbert problem. In this paper it is shown that for simple polygons and for a large class of boundary conditions, the above Riemann–Hilbert problem (a) can either be reduced to a triangular RH problem which can be solved in closed form or (b) can be bypassed, and the ρj can be obtained using only algebraic manipulations. As an illustration of these ‘triangular’ and ‘algebraic’ cases we solve the Laplace equation in the quarter‐plane, the semi‐infinite strip and the right isosceles triangle with certain Poincaré boundary conditions. These boundary value problems, which include the Dirichlet and the Neumann problems as particular cases, cannot be solved by conformal mappings.