## Abstract and Applied Analysis

### Stable approximations of a minimal surface problem with variational inequalities

#### 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 $BV(\Omega)$ of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional $\mathcal{J}$ on $BV(\Omega)$ defined by $\mathcal{J}(u) =\mathcal{A}(u)+\int_{\partial\Omega}|Tu-\phi|$, where $\mathcal{A}(u)$ is the “area integral” of $u$ with respect to $\Omega,T$ is the “trace operator” from $BV(\Omega)$ into $L^1(\partial\Omega)$, and $\phi$ is the prescribed data on the boundary of $\Omega$. 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2, Number 1-2 (1997), 137-161.

Dates
First available in Project Euclid: 7 April 2003

https://projecteuclid.org/euclid.aaa/1049737247

Digital Object Identifier
doi:10.1155/S1085337597000316

Mathematical Reviews number (MathSciNet)
MR1604173

Zentralblatt MATH identifier
0937.49020

#### Citation

Nashed, M. Zuhair; Scherzer, Otmar. Stable approximations of a minimal surface problem with variational inequalities. Abstr. Appl. Anal. 2 (1997), no. 1-2, 137--161. doi:10.1155/S1085337597000316. https://projecteuclid.org/euclid.aaa/1049737247