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 of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional on defined by , where is the “area integral” of with respect to is the “trace operator” from into , and is the prescribed data on the boundary of . 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.
"Stable approximations of a minimal surface problem with variational inequalities." Abstr. Appl. Anal. 2 (1-2) 137 - 161, 1997. https://doi.org/10.1155/S1085337597000316