Characterization of Sobolev-Slobodeckij spaces using curvature energies

Abstract

We give a new characterization of Sobolev-Slobodeckij spaces $W^{1+s,p}(\mathrm{\Omega })$ for $p>n$ and $\frac{n}{p}<s<1$, where $n$ is the dimension of domain $\mathrm{\Omega }$. To achieve this we introduce a family of geometric curvature energies – functionals on the space of surfaces 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 Lipschitz continuous and its graph has finite geometric curvature energy of appropriate type.