Extremals for sharp Poincaré-Sobolev inequalities: periodically perforated sets and beyond

2025/11/25 by Brasco, Lorenzo, Briani, Luca, Prinari, Francesca
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2511.20260

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.

Citations

• Extremals for Poincaré-Sobolev sharp constants in Steiner symmetric sets
• Variations on the capacitary inradius
• Capacitary inradius and Poincaré-Sobolev inequalities
• The role of topology and capacity in some bounds for principal frequencies
• Sobolev embeddings and distance functions
• A comparison principle for the Lane-Emden equation and applications to geometric estimates
• An overview on constrained critical points of Dirichlet integrals
• Extremal functions for Morrey's inequality in convex domains
• Principal frequency of the p-Laplacian and the inradius of Euclidean\n domains

Related

• Extremals for Poincaré-Sobolev sharp constants in Steiner symmetric sets
• The role of topology and capacity in some bounds for principal frequencies
• Capacitary inradius and Poincaré-Sobolev inequalities
• On Morrey's inequality in Sobolev-Slobodecki\uı spaces
• Extremal maps in best constants vector theory - Part I: Duality and Compactness