Extremals for sharp Poincar\'e-Sobolev inequalities: periodically perforated sets and beyond

TL;DR

This paper proves the existence of extremals for sharp Poincaré-Sobolev inequalities in periodically perforated sets, regardless of hole shape, with applications to mixed periodic and bounded domains.

Contribution

It establishes the existence of extremals in unbounded perforated sets with minimal regularity assumptions and extends results to mixed periodic-bounded domains.

Findings

Existence of extremals for sharp Poincaré-Sobolev inequalities in perforated sets.

Applicability to sets with arbitrary hole shapes and regularity.

Extension to periodically bounded-in-all-but-some-directions domains.

Abstract

We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincar\'e-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.