Extremals for sharp Poincaré-Sobolev inequalities: periodically perforated sets and beyond
Authors
Lorenzo Brasco
Luca Briani
Francesca Prinari
Abstract
We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincaré-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is sufficient that the latter is a compact set with positive capacity. We also show how to apply the main result in order to get a similar existence statement, for sets which are periodic in some directions and bounded in all the others.
