# Fractional Korn's inequalities without boundary conditions

## TL;DR

This paper develops fractional Korn's inequalities for vector fields in bounded domains without boundary conditions, extending previous results to more general domains and providing proofs for convex planar cases.

## Contribution

It introduces boundary-condition-free fractional Korn's inequalities for bounded domains with Lipschitz boundaries, broadening their applicability in fractional Sobolev spaces.

## Key findings

- Established fractional Korn's inequalities without boundary conditions.
- Proved inequalities for planar convex domains.
- Extended validity to Lipschitz domains with small Lipschitz constant.

## Abstract

This work establishes fractional analogues of Korn's first and second inequalities for vector fields in fractional Sobolev spaces defined over a bounded domain. The validity of the inequalities require no additional boundary condition, extending existing fractional Korn's inequalities that are only applicable for Sobolev vector fields satisfying zero Dirichlet boundary conditions. The domain of definition is required to have a $C^1$-boundary or, more generally, a Lipschitz boundary with small Lipschitz constant. We conjecture that the inequalities remain valid for vector fields defined over any Lipschitz domain. We support this claim by presenting a proof of the inequalities for vector fields defined over planar convex domains.