  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.
