Communications in Mathematical Sciences

Volume 20 (2022)

Number 2

A fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations

Pages: 405 – 423



Tadele Mengesha (Department of Mathematics, University of Tennessee, Knoxville, Tn., U.S.A.)

James M. Scott (Department of Mathematics, University of Pittsburgh, Pennsylvania, U.S.A.)


In this paper we prove a fractional analogue of the classical Korn’s first inequality. The inequality makes it possible to show the equivalence of a function space of vector field characterized by a Gagliardo-type seminorm with projected difference with that of a corresponding fractional Sobolev space. As an application, we will use it to obtain a Caccioppoli-type inequality for a nonlinear system of nonlocal equations, which in turn is a key ingredient in applying known results to prove a higher fractional differentiability result for weak solutions of the nonlinear system of nonlocal equations. The regularity result we prove will demonstrate that a well-known self-improving property of scalar nonlocal equations will hold for strongly coupled systems of nonlocal equations as well.


fractional Korn-type inequality, fractional Sobolev spaces, self-improving property

2010 Mathematics Subject Classification

35R11, 46E35, 46E40

TM’s research is supported by NSF DMS-1910180.

Received 4 January 2021

Received revised 13 July 2021

Accepted 14 July 2021

Published 28 January 2022