Communications in Mathematical Sciences
Volume 20 (2022)
A fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations
Pages: 405 – 423
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