Compliance estimates for two-dimensional problems with Dirichlet region of prescribed length

In this paper we prove some lower bounds for the compliance functional, in terms of the 1-dimensional Hausdorff measure of the Dirichlet region and the number of its connected components. When the measure of the Dirichlet region is large, these estimates are asymptotically optimal and yield a proof of a conjecture by Buttazzo and Santambrogi