Characterization of Sobolev-Slobodeckij spaces using geometric curvature energies

We give a new characterization of Sobolev–Slobodeckij spaces W1+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 Lp and the appropriate energy is finite.