3D-2D asymptotic analysis of an optimal design problem for thin films

The Gamma-limit of a rescaled version of an optimal material distribution p roblem for a cylindrical two-phase elastic mixture in a thin three-dimensio nal domain is explicitly computed. Its limit is a two-dimensional optimal d esign problem on the cross-section of the thin domain; it involves optimal energy bounds on two-dimensional mixtures of a related two-phase bulk mater ial. Thus, it is shown in essence that 3D-2D asymptotics and optimal design commute from a variational standpoint.