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

TL;DR

This paper establishes a new trace Hardy inequality for Euclidean spaces with bounded cuts, including the case of two dimensions, and explores its applications to heat equations and Schrödinger operators with delta-prime interactions.

Contribution

It introduces a novel geometric trace Hardy inequality applicable to spaces with cuts, extending to unbounded cuts and including the two-dimensional case.

Findings

Hardy inequality holds for $d=2$ in this setting

Application to heat equation with insulating cut

Application to Schrödinger operator with $ abla$-interaction

Abstract

We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $\Sigma\subset\mathbb R^d$, $d \ge 2$. 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 Schr\"odinger operator with a $\delta'$-interaction supported on $\Sigma$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.