A note on Hardy's inequalities with boundary singularities

TL;DR

This paper investigates Hardy-Poincaré inequalities with boundary-singular weights in smooth bounded domains, providing conditions for the existence of optimal constants.

Contribution

It offers new sufficient conditions ensuring the attainment of the best constant in Hardy-Poincaré inequalities with boundary singularities.

Findings

Established conditions for the existence of extremal functions.

Identified when the best constant is achieved.

Extended understanding of boundary singularity effects.

Abstract

Let $\O$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincar\'e inequalities with weight function singular at the boundary of $\O$. In particular we give sufficient conditions so that the best constant is achieved.