Characterization of Sobolev-Slobodeckij spaces using curvature energies

TL;DR

This paper introduces a new way to characterize Sobolev-Slobodeckij spaces using curvature energies inspired by Menger curvature, providing a novel criterion based on finiteness of these energies.

Contribution

It offers a new characterization of Sobolev-Slobodeckij spaces through curvature energies, connecting geometric concepts with functional space properties.

Findings

Characterization of Sobolev-Slobodeckij spaces via curvature energies

Equivalence between membership in Sobolev-Slobodeckij spaces and finiteness of curvature energies

Extension of classical curvature concepts to functional space analysis

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.