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

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.