On Beurling's boundary value problem in circle packing

Abstract

We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions in the framework of circle packing. In the case of discrete analytic functions modelled on arbitrary combinatorically closed disks existence of solutions is shown under rather general assumptions. As in the nondiscrete case finitely many branch circles can be prescribed, so the solutions include locally univalent as well as branched packings. The proof of existence rests on an application of Brouwer's fixed point theorem and a global parameterization of the differentiable manifold of circle packings. We also present some first results on the uniqueness of solutions. In the last two sections we propose an algorithm for the numerical solution of the problem, which is based on an embedded Newton method, and report on some test calculations.