Laplace’s equation in the exterior of a convex polygon. The equilateral triangle

A general method for studying boundary value problems for linear and for integrable nonlinear partial differential equations in two dimensions was introduced in Fokas, 1997. For linear equations in a convex polygon (Fokas and Kapaev (2000) and (2003), and Fokas (2001)), this method: (a) expresses the solution $q(x,y)$ in the form of an integral (generalized inverse Fourier transform) in the complex $k$-plane involving a certain function $\hat q(k)$ (generalized direct Fourier transform) that is defined as an integral along the boundary of the polygon, and (b) characterizes a generalized Dirichlet-to-Neumann map by analyzing the so-called global relation. For simple polygons and simple boundary conditions, this characterization is explicit. Here, we extend the above method to the case of elliptic partial differential equations in the exterior of a convex polygon and we illustrate the main ideas by studying the Laplace equation in the exterior of an equilateral triangle.

Regarding (a), we show that whereas $\hat q(k)$ is identical with that of the interior problem, the contour of integration in the complex $k$-plane appearing in the formula for $q(x,y)$ depends on $(x,y)$. Regarding (b), we show that the global relation is now replaced by a set of appropriate relations which, in addition to the boundary values, also involve certain additional unknown functions. In spite of this significant complication we show that, for certain simple boundary conditions, the exterior problem for the Laplace equation can be mapped to the solution of a Dirichlet problem formulated in the interior of a convex polygon formed by three sides.