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

Rafael D. Benguria; Cristóbal Vallejos; Hanne Van Den Bosch

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

Mathematical Physics 2018-10-16 v1 Analysis of PDEs math.MP

Authors: Rafael D. Benguria , Cristóbal Vallejos , Hanne Van Den Bosch

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 $\mathbb{R}^2$ .

Keywords

poincaré-sobolev inequalities minimizers existence non-existence smooth domains polyhedral domains isosceles triangle

Cite

@article{arxiv.1810.05698,
  title  = {Existence and non-existence of minimizers for Poincar\'e-Sobolev inequalities},
  author = {Rafael D. Benguria and Cristóbal Vallejos and Hanne Van Den Bosch},
  journal= {arXiv preprint arXiv:1810.05698},
  year   = {2018}
}

Comments

20 pages, 3 figures

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