Control problems in the coefficients and the domain for linear elliptic equations (Q6619540)

A mixture of two conductive materials, each limited in amount, with diffusion coefficients \(\alpha\) and \(\beta\) is placed in measurable subsets \(\omega^{\alpha}\) and \(\omega^{\beta}\) of \(\Omega\), respectively. The goal is to choose the sets \(\omega^{\alpha}\) and \(\omega^{\beta}\) such that the potential \(u\) in \(\omega^{\alpha} \cup \omega^{\beta}\), which solves a linear elliptic state equation with homogeneous boundary conditions, minimizes the functional \(\int_{\Omega} F(x,u)\,dx\).\N\NFor the formulated optimal design problem, optimality conditions are derived, and a numerical algorithm is presented along with some numerical simulations.