Stable Approximations of a Minimal Surface Problem with Variational Inequalities

In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the space BV(Ω) of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional J on BV(Ω) defined by J (u) = A(u) + ∫ ∂Ω |Tu − φ|, where A(u) is the “area integral” of u with respect to Ω, T is the “trace operator” from BV(Ω) into L(∂Ω), and φ is the prescribed data on the boundary of Ω. We establish convergence and stability of approximate regularized solutions which are solutions of a family of variational inequalities. We also prove convergence of an iterative method based on Uzawa’s algorithm for implementation of our regularization procedure.