Characterization of Sobolev-Slobodeckij spaces using curvature energies

Abstract

We give a new characterization of Sobolev-Slobodeckij spaces W1+s,p(Ω) for p>n and np<s<1, where n is the dimension of domain Ω. 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.

Publication
Publ. Mat. 63 (2019), no. 2, 663–677.