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IMA Journal of Applied Mathematics 2003 68(4):355-408; doi:10.1093/imamat/68.4.355
© 2003 by Institute of Mathematics and its Applications
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On a transform method for the Laplace equation in a polygon

A. S. Fokas1 and A. A. Kapaev2

1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK 2 St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011, Russia

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 {rho}j(k), j = 1, ..., n. The function {rho}j consists of an integral over the jth side involving both qx and qy , thus each {rho}j involves one unknown boundary value. The functions {rho}j 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 {rho}j. 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 {rho}j 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.

Keywords: boundary value problems; Laplace equation; Poincaré boundary conditions; Riemann–Hilbert; Wiener–Hopf.

Received 26 March 2002.


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