Extremals for Poincaré-Sobolev sharp constants in Steiner symmetric sets

2025/05/16 by Lorenzo Brasco, Brasco, Lorenzo, Luca Briani +3 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.11084

openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove existence of minimizers for the sharp Poincaré-Sobolev constant in general Steiner symmetric sets, in the subcritical and superhomogeneous regime. The sets considered are not necessarily bounded, thus the relevant embeddings may suffer from a lack of compactness. We prove existence by means of an elementary compactness method. We also prove an exponential decay at infinity for minimizers, showing that in the case of Steiner symmetric sets the relevant estimates only depend on the underlying geometry. Finally, we illustrate the optimality of the existence result, by means of some examples.

Citations

• Variations on the capacitary inradius
• Capacitary inradius and Poincaré-Sobolev inequalities
• Beyond Cheeger's constant
• The Cauchy-Dirichlet Problem for the Fast Diffusion Equation on Bounded Domains
• On the spectrum of sets made of cores and tubes
• Sobolev embeddings and distance functions
• Uniqueness of extremals for some sharp Poincaré-Sobolev constants
• 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
• Stability estimates for the lowest eigenvalue of a Schrödinger operator
• Nonlinear scalar field equations, I existence of a ground state

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