Topological and capacitary methods for Poincaré inequalities

This thesis is devoted to prove characterizations of the validity of Poincaré-type inequalities on general open sets in the euclidean space. In the super-conformal case, i.e. when points are not removable sets, the finiteness of the inradius of an open set turns out to be alone a necessary and sufficient condition for the Poincarè inequality to hold. In the planar case, this condition is sufficient for open sets with prescribed topology. A similar characterization is still valid in arbitrary dimension and for a general open set, when the points are removable sets, by using the capacitary inradius, in place of the usual one. In the first two situations, we prove a geometric lower bound on the sharp Poincaré-Sobolev embedding constants associated to an open set, in terms of its inradius. In the sub-conformal case, we prove a two--sided estimate on the sharp Poincaré-Sobolev constants of a general open set, in terms of its capacitary inradius. This extends a result by Maz'ya and Shubin, originally proved for the case p=2.