Domain identification problem for elliptic hemivariational inequalities (Q5936106)

The author considers the following optimal control problem: Find an admissible set \(\Omega^*\) and a function \(u^*\) such that NEWLINE\[NEWLINEJ(\Omega^*, u^*)= \min_{\Omega\in \Pi_0} \min_{u\in S(\Omega)} J(\Omega, u),NEWLINE\]NEWLINE where the cost functional \(J\) is of the integral form, \(\Pi_0\) is a set of admissible shapes satisfying the cone property, \(S(\Omega)\) is a set of solutions of the following hemivariational inequality within the domain \(\Omega\): NEWLINE\[NEWLINE\begin{cases} a_\Omega(u,v)+ (\chi,v)_{L^2(\Omega)}= (f,v)_{L^2(\Omega)}\quad &\text{for all }v\in H^1(\Omega),\\ \chi(x)\in \partial j(x,u(x))\quad &\text{for a.a. }x\in\Omega.\end{cases}NEWLINE\]NEWLINE In the above-mentioned hemivariational inequality we look for a function \(u\in H^1(\Omega)\) and a selection \(\chi\in L^2(\Omega)\) of the Clarke subdifferential of the locally Lipschitz potential \(j\). By \(a_\Omega\) we denote an integral form also depending on the set \(\Omega\).NEWLINENEWLINENEWLINEThe existence of the optimal control problem is obtained by the direct method of calculus of variations for lower semicontinuous cost functionals. As for the above-mentioned hemivariational inequality, its existence can be obtained as a consequence of Theorem 4.25 of [\textit{Z. Naniewicz} and \textit{P. D. Panagiotopoulos}, ``Mathematical theory of hemivariational inequalities and applications'' (1995; Zbl 0968.49008)].