M. Zuhair Nashed, Otmar Scherzer, "Stable Approximations of a Minimal Surface Problem with Variational Inequalities", Abstract and Applied Analysis, vol. 2, Article ID 383457, 25 pages, 1997. https://doi.org/10.1155/S1085337597000316
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 of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional on defined by , where is the “area integral” of with respect to is the “trace operator” from into , and is the prescribed data on the boundary of . 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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