Approximation of minimal surfaces with free boundaries

In this paper we develop a penalty method to approximate solutions of the free boundary problem for minimal surfaces. To this end we study the problem of finding minimizers of a functional Fλ which is defined as the sum of the Dirichlet integral and an appropriate penalty term weighted by a parameter λ. We prove existence of a solution for λ large enough as well as convergence to a solution of the free boundary problem as λ tends to infinity. Additionally regularity at the boundary of these solutions is shown, which is crucial for deriving numerical error estimates. Since every solution is harmonic, the analysis may be largely simplified by considering boundary values only and using harmonic extensions. In a subsequent paper we develop a fully discrete finite element procedure for approximating solutions to this one-dimensional problem and prove an error estimate which includes an order of convergence with respect to the grid size.