Existence and non-existence of minimizers for Poincaré–Sobolev inequalities

Abstract

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