Abstract and Applied Analysis
Volume 2 (1997), Issue 1-2, Pages 137-161
doi:10.1155/S1085337597000316
Stable approximations of a minimal surface problem with
variational inequalities
1Department of Mathematical Sciences, University of Delaware, Newark 19716, DE, USA
2Institut für Industriemathematik, Johannes-Kepler-Universität, öSTERREICH A-4040 Linz, Australia
Received 19 February 1997
Copyright © 1997 M. Zuhair Nashed and Otmar Scherzer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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 i(∂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.