Open Access
1997 Stable approximations of a minimal surface problem with variational inequalities
M. Zuhair Nashed, Otmar Scherzer
Abstr. Appl. Anal. 2(1-2): 137-161 (1997). DOI: 10.1155/S1085337597000316


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.


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M. Zuhair Nashed. Otmar Scherzer. "Stable approximations of a minimal surface problem with variational inequalities." Abstr. Appl. Anal. 2 (1-2) 137 - 161, 1997.


Published: 1997
First available in Project Euclid: 7 April 2003

zbMATH: 0937.49020
MathSciNet: MR1604173
Digital Object Identifier: 10.1155/S1085337597000316

Primary: 49J40 , 49J45 , 49N60 , 49Q05 , 65K10
Secondary: 26A45 , 65J15 , 65J20

Keywords: bounded variation norm , Minimal surface problem , nondifferentiable optimization in nonreflexive spaces , relaxed Dirichlet problem , Uzawa′s algorithm , variational inequalities

Rights: Copyright © 1997 Hindawi

Vol.2 • No. 1-2 • 1997
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