Stable approximations of a minimal surface problem with variational inequalities. (1997)

• Record Type: Journal Article
• Authors: Nashed, M. Zuhair; Scherzer, Otmar
• Is Part Of: Abstract and applied analysis. Volume 2:Number 1/2(1997)
• Page Start: 137
• Page End: 161
• Publication Date: 1997
• Subjects: Minimal surface problem -- relaxed Dirichlet problem -- nondifferentiable optimization in nonreflexive spaces -- variational inequalities -- bounded variation norm -- Uzawa's algorithm; Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/S1085337597000316
• Languages: English
• ISSNs: 1085-3375

Abstract

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 spaceB V ( Ω ) of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional𝒥 onB V ( Ω ) defined by𝒥 ( u ) = 𝒜 ( u ) + ∫ ∂ Ω | T u − Φ |, where𝒜 ( u ) is the "area integral" ofu with respect toΩ, T is the "trace operator" fromB V ( Ω ) intoL i ( ∂ Ω ), andϕ is the prescribed data on the boundary ofΩ . 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.