Dirichlet conditions in Poincar\'e-Sobolev inequalities: the sub-homogeneous case

We investigate the dependence of optimal constants in Poincar\'e- Sobolev inequalities of planar domains on the region where the Dirichlet condition is imposed. More precisely, we look for the best Dirichlet regions, among closed and connected sets with prescribed total length $L$ (one-dimensional Hausdorff measure), that make these constants as small as possible. We study their limiting behaviour, showing, in particular, that Dirichler regions homogenize inside the domain with comb-shaped structures, periodically distribuited at different scales and with different orientations. To keep track of these information we rely on a $\Gamma$-convergence result in the class of varifolds. This also permits applications to reinforcements of anisotropic elastic membranes. At last, we provide some evidences for a conjecture.