On the Hardy constant of some non-convex planar domains

TL;DR

This paper investigates the Hardy constant for various non-convex planar domains, extending previous work by relating these constants to those of infinite sectorial regions studied by Davies.

Contribution

It computes the Hardy constant for new classes of non-convex planar domains, building on prior results for quadrilaterals and connecting to sectorial regions.

Findings

Hardy constant for specific non-convex domains determined

Relation established between domain Hardy constants and sectorial regions

Extends previous results on quadrilaterals to broader classes of domains

Abstract

The Hardy constant of a simply connected domain $\Omega\subset\mathbf{R}^2$ is the best constant for the inequality \[ \int_{\Omega}|\nabla u|^2dx \geq c\int_{\Omega} \frac{u^2}{{\rm dist}(x,\partial\Omega)^2}\, dx \; , \;\;\quad u\in C^{\infty}_c(\Omega). \] After the work of Ancona where the universal lower bound 1/16 was obtained, there has been a substantial interest on computing or estimating the Hardy constant of planar domains. In \cite{BT} we have determined the Hardy constant of an arbitrary quadrilateral in the plane. In this work we continue our investigation and we compute the Hardy constant for other non-convex planar domains. In all cases the Hardy constant is related to that of a certain infinite sectorial region which has been studied by E.B. Davies.