Abstract and Applied Analysis

Stable approximations of a minimal surface problem with variational inequalities

M. Zuhair Nashed and Otmar Scherzer

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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

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

First available in Project Euclid: 7 April 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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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