Characterization of Sobolev-Slobodeckij spaces using curvature energies. (English) Zbl 1436.46036

Dąbrowski, Damian

Publ. Mat., Barc. 63, No. 2, 663-677 (2019).

Summary: 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.

MSC:

46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces

Keywords:

Sobolev-Slobodeckij spaces; geometric curvature energies; Menger curvature