A fractional Korn-type inequality

TL;DR

This paper establishes an equivalence between certain vector field spaces involving directional difference quotients and fractional Sobolev spaces, providing a Korn-type characterization that enhances understanding of nonlocal equations and their solutions.

Contribution

It introduces a Korn-type inequality for fractional Sobolev spaces, linking difference quotient spaces to fractional Sobolev spaces and enabling improved analysis of nonlocal systems.

Findings

Equivalence between directional difference quotient spaces and fractional Sobolev spaces.

Application of the result to analyze energy spaces in nonlocal equations.

Proof that weak solutions exhibit better differentiability and integrability.

Abstract

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of "directional" difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to understand better the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability.