Bounds for Sobolev Embedding Constants in Non-simply Connected Planar Domains

In a bounded non-simply connected planar domain Ω, with a boundary split in an interior part and an exterior part, we obtain bounds for the embedding constants of some subspaces of H1(Ω) into Lp(Ω) for any p > 1, p ≠ 2. The subspaces contain functions which vanish on the interior boundary and are constant (possibly zero) on the exterior boundary. We also evaluate the precision of the obtained bounds in the limit situation where the interior part tends to disappear and we show that it does not depend on p. Moreover, we emphasize the failure of symmetrization techniques in these functional spaces. In simple situations, a new phenomenon appears: the existence of a break even surface separating masses for which symmetrization increases/decreases the Dirichlet norm. The question whether a similar phenomenon occurs in more general situations is left open.