Stable Approximations of a Minimal Surface Problem with Variational Inequalities
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) defi...
| المؤلفون الرئيسيون: | , |
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| التنسيق: | مقال |
| اللغة: | English |
| منشور في: |
Hindawi Limited
1997-01-01
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| سلاسل: | Abstract and Applied Analysis |
| الموضوعات: | |
| الوصول للمادة أونلاين: | http://dx.doi.org/10.1155/S1085337597000316 |
| _version_ | 1827080894509219840 |
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| author | M. Zuhair Nashed Otmar Scherzer |
| author_facet | M. Zuhair Nashed Otmar Scherzer |
| author_sort | M. Zuhair Nashed |
| collection | DOAJ |
| description | 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|Tu−Φ|, where 𝒜(u) is the “area integral” of u with respect to Ω,T is the “trace operator” from BV(Ω) into L i(∂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. |
| first_indexed | 2025-03-20T03:08:43Z |
| format | Article |
| id | doaj.art-3fd1a50cf78a4426a17f737e3a92078c |
| institution | Directory Open Access Journal |
| issn | 1085-3375 |
| language | English |
| last_indexed | 2025-03-20T03:08:43Z |
| publishDate | 1997-01-01 |
| publisher | Hindawi Limited |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj.art-3fd1a50cf78a4426a17f737e3a92078c2024-10-03T07:21:22ZengHindawi LimitedAbstract and Applied Analysis1085-33751997-01-0121-213716110.1155/S1085337597000316Stable Approximations of a Minimal Surface Problem with Variational InequalitiesM. Zuhair Nashed0Otmar Scherzer1Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USAInstitut für Industriemathematik, Johannes-Kepler-Universität, öSTERREICH A-4040 Linz, AustriaIn 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|Tu−Φ|, where 𝒜(u) is the “area integral” of u with respect to Ω,T is the “trace operator” from BV(Ω) into L i(∂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.http://dx.doi.org/10.1155/S1085337597000316Minimal surface problemrelaxed Dirichlet problemnondifferentiable optimization in nonreflexive spacesvariational inequalitiesbounded variation normUzawa's algorithm. |
| spellingShingle | M. Zuhair Nashed Otmar Scherzer Stable Approximations of a Minimal Surface Problem with Variational Inequalities Abstract and Applied Analysis Minimal surface problem relaxed Dirichlet problem nondifferentiable optimization in nonreflexive spaces variational inequalities bounded variation norm Uzawa's algorithm. |
| title | Stable Approximations of a Minimal Surface Problem with Variational Inequalities |
| title_full | Stable Approximations of a Minimal Surface Problem with Variational Inequalities |
| title_fullStr | Stable Approximations of a Minimal Surface Problem with Variational Inequalities |
| title_full_unstemmed | Stable Approximations of a Minimal Surface Problem with Variational Inequalities |
| title_short | Stable Approximations of a Minimal Surface Problem with Variational Inequalities |
| title_sort | stable approximations of a minimal surface problem with variational inequalities |
| topic | Minimal surface problem relaxed Dirichlet problem nondifferentiable optimization in nonreflexive spaces variational inequalities bounded variation norm Uzawa's algorithm. |
| url | http://dx.doi.org/10.1155/S1085337597000316 |
| work_keys_str_mv | AT mzuhairnashed stableapproximationsofaminimalsurfaceproblemwithvariationalinequalities AT otmarscherzer stableapproximationsofaminimalsurfaceproblemwithvariationalinequalities |