Regularization of an unilateral obstacle problem (Q2781513)

The authors consider the following unilateral obstacle problem: Find NEWLINE\[NEWLINEu\in K=\bigl\{v \in H^1(\Omega): v\geq\psi \text{ a.e. in }\Omega,\;v=g \text{ on }\partial \Omega\bigr\},NEWLINE\]NEWLINE such that NEWLINE\[NEWLINE\int_\Omega\nabla u\nabla(v-u) dx+\langle f,v-u \rangle\geq 0\quad \text{for all }v\in K,NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(g\in H^{1/2} (\partial \Omega)\), \(\psi\in H^1(\Omega)\) and \(f\in H^1(\Omega)\). An existence result as well as the uniqueness of the solution of the above problem comes from [\textit{D. Kinderlehrer} and \textit{G. Stampacchia}, An introduction to variational inequalities and their applications, Pure Appl. Math. 88 (1980; Zbl 0457.35001)]. The authors develop a regularization method for solving a non-differentiable minimization problem equivalent to the above one. To this end they approximate the non-differentiable term by a sequence of differentiable ones. The paper is a generalization of [\textit{H. Huang}, \textit{W. Han} and \textit{J. Zhou}, Numer. Math. 69, 155-166 (1994; Zbl 0817.65050)].