A fractional Korn-type inequality

James Scott; Tadele Mengesha

doi:10.3934/dcds.2019137

Article Contents

2019, Volume 39, Issue 6: 3315-3343. Doi: 10.3934/dcds.2019137

This issue Previous Article Existence of time-periodic strong solutions to a fluid–structure system Next Article Hardy-Sobolev type inequality and supercritical extremal problem

A fractional Korn-type inequality

James Scott and
Tadele Mengesha^,

The University of Tennessee, Knoxville, TN 37996, USA

^* Corresponding author
^* Corresponding author

Received: June 30, 2018

Revised: November 30, 2018

Published: February 2019

The second author is supported by NSF grant DMS-1615726

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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.

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Mathematics Subject Classification: Primary: 46E35, 46E40, 45G15; Secondary: 35B65, 74B99.

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