Stable approximations of a minimal surface problem with variational inequalities

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 space $BV\left(\Omega \right)$ of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional $𝒥$ on $BV\left(\Omega \right)$ defined by $𝒥\left(u\right)=𝒜\left(u\right)+{\int }_{\partial \Omega }\left|Tu-\Phi \right|$, where $𝒜\left(u\right)$ is the “area integral” of $u$ with respect to $\Omega ,T$ is the “trace operator” from $BV\left(\Omega \right)$ into $L^i\left(\partial \Omega \right)$, 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.