Characterization of Sobolev-Slobodeckij Spaces Using Curvature Energies

Abstract

We give a new characterization of Sobolev-Slobodeckij spaces $W^{1+s,p}$ for $n/p< 1+s$, where $n$ is the dimension of the domain. To achieve this we introduce a family of curvature energies inspired by the classical concept of integral Menger curvature. We prove that a function belongs to a Sobolev-Slobodeckij space if and only if it is in $L^p$ and the appropriate energy is finite.