Poincar\'e inequalities in quasihyperbolic boundary condition domains

TL;DR

This paper investigates the conditions under which (q,p)-Poincaré inequalities hold in domains with quasihyperbolic boundary conditions, establishing explicit criteria based on domain geometry and boundary growth.

Contribution

It provides explicit criteria for the validity of (q,p)-Poincaré inequalities in quasihyperbolic boundary condition domains, linking geometric properties to functional inequalities.

Findings

Domains support (q,p)-Poincaré inequalities if p exceeds a specific constant p_0.

The constant p_0 depends explicitly on q, the boundary's growth condition, and the domain boundary.

The results connect geometric boundary conditions with functional inequality support.

Abstract

We study the validity of (q,p)-Poincar\'e inequalities, q<p, on domains in R^n which satisfy a quasihyperbolic boundary condition, i.e. domains whose quasihyperbolic metric satisfies a logarithmic growth condition. In the present paper, we show that the quasihyperbolic boundary condition domains support a (q,p)-Poincar\'e inequality whenever p>p_0, where p_0 is an explicit constant depending on q, on the logarithmic growth condition, and on the boundary of the domain.