The paper deals with optimal design problems of the form first considered in [\((*)\) \textit{L. Ambrosio} and \textit{G. Buttazzo}, Calc. Var. Paratial Differ. Equa. 1, No. 1, 55--69 (1993; Zbl 0794.49040)]. More precisely, the energy \[ E(u,A)= \int_\Omega \sigma_A|\nabla u|^2 dx+ \int_\Omega fu\,dx+ \text{Per}_\Omega(A)+ C|A| \] is considered, where \(f\in L^\infty(\Omega)\), \(C> 0\) and \[ \sigma_A(x)= \alpha 1_A(x)+ \beta 1_{\Omega\setminus A}(x), \] being \(0<\alpha<\beta\). The goal is to minimize \(E(u, A)\) and to study the regularity properties of minimizers. It is already known that the optimal \(A\) are (essentially) open sets [\((*)\)] verifying a partial regularity property [\((**)\) \textit{F. H. Lin}, Calc. Var. Partial Differ. Equ. 1, No. 2, 149--168 (1993; Zbl 0794.49038)], and that the optimal states \(u\) are Hölder continuous functions [\((**)\)]. In the present paper it is shown that in the case \(N= 2\) every component of an optimal domain \(A\) is of class \(C^1\) away from \(\partial\Omega\). This would imply the \(C^1\) regularity of \(A\) away from \(\partial\Omega\) as soon as the components of \(A\) are shown to be finite. This seems to be expected, even if no proof is yet available. The optimal state \(u\) is conjectured to be Lipschitz continuous.