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arXiv · 2511.20260

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

Lorenzo Brasco·Luca Briani·Francesca Prinari
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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.

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Lorenzo Brasco, Luca Briani, Francesca Prinari. 2025-11-25. Extremals for sharp Poincar\'e-Sobolev inequalities: periodically perforated sets and beyond. https://arxiv.org/abs/2511.20260

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