On the Hardy constant of non-convex planar domains: the case of the quadrilateral

TL;DR

This paper determines the Hardy constant for any planar quadrilateral, linking it to a known constant for an infinite sector, thus advancing understanding of Hardy inequalities in non-convex domains.

Contribution

It uniquely computes the Hardy constant for all planar quadrilaterals, connecting it to existing results for sectorial regions.

Findings

Hardy constant for quadrilaterals is explicitly determined.

The Hardy constant matches that of a related infinite sector.

Provides a method to compute Hardy constants for non-convex domains.

Abstract

The Hardy constant of a simply connected domain $\Omega\subset\R^2$ is the best constant for the inequality \[ \int_{\Omega}|\nabla u|^2dx \geq c\int_{\Omega} \frac{u^2}{{\rm dist}(x,\partial\Omega)^2}\, dx \;, u\in C^{\infty}_c(\Omega). \] 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 this work we determine the Hardy constant of an arbitrary quadrilateral in the plane. In particular we show that the Hardy constant is the same as that of a certain infinite sectorial region which has been studied by E.B. Davies.