Abstract and Applied Analysis

Stable approximations of a minimal surface problem with variational inequalities

M. Zuhair Nashed and Otmar Scherzer

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 7 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1049737247

Digital Object Identifier
doi:10.1155/S1085337597000316

Mathematical Reviews number (MathSciNet)
MR1604173

Zentralblatt MATH identifier
0937.49020

Subjects
Primary: 49J40: Variational methods including variational inequalities [See also 47J20] 49Q05: Minimal surfaces [See also 53A10, 58E12] 49N60: Regularity of solutions 49J45: Methods involving semicontinuity and convergence; relaxation 65K10: Optimization and variational techniques [See also 49Mxx, 93B40]
Secondary: 26A45: Functions of bounded variation, generalizations 65J15: Equations with nonlinear operators (do not use 65Hxx) 65J20: Improperly posed problems; regularization

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

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


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