Stable approximations of a minimal surface problem with variational inequalities

Nashed, M. Zuhair ; Scherzer, Otmar

Abstr. Appl. Anal., Tome 2 (1997) no. 1-2, p. 137-161 / Harvested from Project Euclid

Résumé

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 $BV(\Omega)$ of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional $\mathcal{J}$ on $BV(\Omega)$ defined by $\mathcal{J}(u) =\mathcal{A}(u)+\int_{\partial\Omega}|Tu-\phi|$ , where $\mathcal{A}(u)$ is the “area integral” of $u$ with respect to $\Omega,T$ is the “trace operator” from $BV(\Omega)$ into $L^1(\partial\Omega)$ , and $\phi$ is the prescribed data on the boundary of $\Omega$ . 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.

Détail

Publié le : 1997-05-14
Classification:  Minimal surface problem,  relaxed Dirichlet problem,  nondifferentiable optimization in nonreflexive spaces,  variational inequalities,  bounded variation norm,  Uzawa′s algorithm,  49J40,  49Q05,  49N60,  49J45,  65K10,  26A45,  65J15,  65J20