On the numerical solution of Plateau's problem

The present paper is dedicated to the numerical computation of minimal surfaces by the boundary element method. Having a parametrization gamma of the boundary curve over the unit circle at hand, the problem is reduced to seeking a reparametrization kappa of the unit circle. The Dirichlet energy of the harmonic extension of gamma circle kappa has to be minimized among all reparametrizations. The energy functional is calculated as boundary integral that involves the Dirichlet-to-Neumann map. First and second order necessary optimality conditions of the underlying minimization problem are formulated. Existence and convergence of approximate solutions is proven. An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented. (C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.