Asymptotical compliance optimization for connected networks

We consider the problem of the optimal location of a Dirichlet region in a two-dimensional domain Ω subject to a force f in order to minimize the compliance of the configuration. The class of admissible Dirichlet regions among which we look for the optimum consists of all one-dimensional connected sets (networks) of a given length L. Then we let L tend to infinity and look for the Γ-limit of suitably rescaled functionals, in order to identify the asymptotical distribution of the optimal networks. The asymptotically optimal shapes are discussed as well and links with average distance problems are provided.