Extremals for Poincar\'e-Sobolev sharp constants in Steiner symmetric sets

TL;DR

This paper establishes the existence and decay properties of minimizers for the sharp Poincaré-Sobolev constant in Steiner symmetric sets, even when these sets are unbounded, highlighting the influence of geometry on these extremals.

Contribution

It proves the existence of minimizers in unbounded Steiner symmetric sets for the sharp Poincaré-Sobolev constant, using an elementary compactness approach and analyzing decay at infinity.

Findings

Existence of minimizers in unbounded Steiner symmetric sets.

Exponential decay of minimizers at infinity.

Dependence of estimates on the geometry of the sets.

Abstract

We prove existence of minimizers for the sharp Poincar\'e-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.