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 Σ⊂Rd, d≥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 Σ. We provide its applications to the heat equation on Rd with an insulating cut at Σ and to the Schrödinger operator with a δ′-interaction supported on Σ. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.