Preprint

Inserted: 16 may 2025
Last Updated: 16 may 2025

Year: 2025

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.

Keywords: decay estimates, Inradius, Lane-Emden equation, Poincaré-Sobolev inequality

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• brabripri_steiner_final.pdf