Abstract and Applied Analysis

Abstract and Applied Analysis / 1997 / Article

Open Access

Volume 2 |Article ID 383457 | https://doi.org/10.1155/S1085337597000316

M. Zuhair Nashed, Otmar Scherzer, "Stable Approximations of a Minimal Surface Problem with Variational Inequalities", Abstract and Applied Analysis, vol. 2, Article ID 383457, 25 pages, 1997. https://doi.org/10.1155/S1085337597000316

Stable Approximations of a Minimal Surface Problem with Variational Inequalities

Received19 Feb 1997

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|TuΦ|, where 𝒜(u) is the “area integral” of u with respect to Ω,T is the “trace operator” from BV(Ω) into Li(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.

Copyright © 1997 Hindawi Publishing Corporation. 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.


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