Trace Hardy inequality for the Euclidean space with a cut and its applications

We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $Sigmasubsetmathbb R^d$, $d ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy weight is given in terms of the geodesic distance to the boundary of $Sigma$. We provide its applications to the heat equation on $mathbb R^d$ with an insulating cut at $Sigma$ and to the Schrodinger operator with a $delta$-interaction supported on $Sigma$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.