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

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 Schrödinger operator with a ${\delta }^{\prime }$-interaction supported on $\Sigma$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.

References in corpus (3)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Sobolev spaces on non-Lipschitz subsets of ${\mathbb{R}}^n$ with application to boundary integral equations on fractal screens
• Gap control by singular Schrödinger operators in a periodically structured metamaterial