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 spaceBV(Ω)of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional𝒥onBV(Ω)defined by𝒥(u)=𝒜(u)+∫∂Ω|Tu−Φ|, where𝒜(u)is the “area integral” ofuwith respect toΩ,Tis the “trace operator” fromBV(Ω)intoL i(∂Ω), andϕis the prescribed data on the boundary ofΩ. 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.