The Fractional Korn Inequality on Uniform Domains and New Korn Inequalities for Truncated Seminorms

TL;DR

This paper establishes a fractional Korn inequality for uniform and John domains, introducing a new approach using truncated seminorms and weighted estimates related to boundary properties.

Contribution

It presents a novel fractional Korn inequality based on truncated seminorms applicable to broader classes of domains, including John domains.

Findings

Proved fractional Korn inequality for uniform domains.

Extended results to John domains with weighted estimates.

Connected boundary geometry to fractional inequalities.

Abstract

We prove the so-called second case of the fractional Korn inequality for uniform domains. We obtain this result as an application of a novel fractional Korn-type inequality formulated in terms of truncated seminorms, which turns out to be valid for the broader class of John domains. We also obtain weighted estimates in which the weights are certain powers of the distance to the boundary that depend on the fractional exponent and the Assouad codimension of the boundary of the domain.