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
M.Zuhair Nashed1
and Otmar Scherzer2
1Department of Mathematical Sciences, University of Delaware, Newark 19716, DE, USA
2Institut für Industriemathematik, Johannes-Kepler-Universität, öSTERREICH A-4040 Linz, Australia
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.