论文标题
跟踪带有切割及其应用的欧几里得空间的不平等现象
Trace Hardy inequality for the Euclidean space with a cut and its applications
论文作者
论文摘要
我们为欧几里得空间获得了痕迹不平等，并具有有限的切割$σ\ subset \ mathbb r^d $，$ d \ ge 2 $。在这个新颖的几何环境中，hardy型的不平等非平等也以$ d = 2 $而持有。各自的强力重量是根据$σ$边界的地球距离给出的。我们将其应用于$ \ Mathbb r^d $上的热量方程，并以$σ$的隔热削减以及$δ'$ - $σ$的交互作用的schrödinger运算符。我们还获得了一类无限削减的痕迹不平等的概括。
We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $Σ\subset\mathbb R^d$, $d \ge 2$. 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 $Σ$. We provide its applications to the heat equation on $\mathbb R^d$ 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.
