Embedding theorems related to torsional rigidity and principal frequency

We study criteria for the finiteness of the constants $C$ in integral inequalities generalizing the Poincaré–Friedrichs inequality and Saint-Venant's variational definition of torsional rigidity. The Rayleigh–Faber–Krahn isoperimetric inequality and the Saint-Venant–Pólya inequality guarantee the existence of finite constants $C$ for domains of finite volume. Criteria for the existence of finite constants $C$ for unbounded domains of infinite volume were known only in the cases of planar simply connected and spatial convex domains. We generalize and strengthen some known results and extend them to the case when $1<p<2$. Here is one of our results.
Suppose that $1\leqslant p <2$ and $\Omega=\Omega^0\setminus K$, where $K\subset \Omega^0$ is a compact set and $\Omega^0$ is either a planar domain with uniformly perfect boundary or a spatial domain satisfying the exterior sphere condition. Under these assumptions, a finite constant $\Lambda_{p-1}(\Omega)$ exists if and only if the integral $\int_\Omega\rho^{{2p}/{(2-p)}}(x,\Omega)\, dx$ is finite, where $\rho(x,\Omega)$ is the distance from the point $x$ to the boundary of $\Omega$.