Existence and non-existence of minimizers for Poincar\'e-Sobolev inequalities

TL;DR

This paper investigates conditions under which minimizers exist or do not exist for critical Poincaré-Sobolev inequalities, highlighting domain geometry's influence on these properties.

Contribution

It establishes existence results for smooth and certain polyhedral domains and non-existence for specific triangular domains, advancing understanding of domain-dependent minimizer behavior.

Findings

Minimizers exist for smooth domains in  and some polyhedral domains.

No minimizers exist for the rectangular isosceles triangle in .

Domain shape critically affects the existence of minimizers.

Abstract

In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincar\'{e}-Sobolev inequalities. We show that minimizers do exist for smooth domains in $\mathbb{R}^d$, an also for some polyhedral domains. On the other hand, we prove the non-existence of minimizers in the rectangular isosceles triangle in $\mathbb{R}^2$.