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

TL;DR

This paper establishes a fractional Korn-type inequality for smooth domains, enabling the equivalence of certain fractional Sobolev spaces and leading to new regularity results for nonlinear nonlocal systems.

Contribution

It introduces a fractional Korn inequality for smooth domains and applies it to prove higher regularity of solutions for nonlinear nonlocal systems.

Findings

Fractional Korn inequality for smooth domains proved.

Equivalence of fractional Sobolev spaces established.

Higher fractional differentiability of solutions demonstrated.

Abstract

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.