\section{Introduction and main results}
In this paper, we prove the fractional analogues of Korn's first and second inequalities in the so-called bounded Lipschitz domains with small Lipschitz constant. For $n\geq 2$, suppose that $\Omega\subset\mathbb{R}^{n}$ is a bounded domain with Lipschitz boundary. Classical Korn's inequalities give a means of controlling the $L^{p}$-norm of the gradient $\nabla \Bu$ of a Sobolev vector field $\Bu: \Omega\to \mathbb{R}^{n}$ by the symmetric part of its gradient, $e(\Bu)(\Bx) = \frac{1}{2} (\nabla \Bu (\Bx) + \nabla \Bu ^{\intercal}(\Bx))$ (and the vector field itself). For $1<p<\infty$,  a version of the classical Korn's first inequality \cite{Korn1,Korn2} 
states that there is a constant $C=C(\Omega)>0,$ such that 
\begin{equation}
\label{Intro-Korn1st}\inf_{\BA\in \text{Skew}(\mathbb{R}^{n})}\|\nabla \Bu - \BA\|_{L^{p}(\Omega)} \leq C \|e(\Bu)\|_{L^{p}(\Omega)}\quad \text{for all}\quad \Bu\in W^{1,p}(\Omega, \mathbb{R}^{n}).
\end{equation}
Here, $\text{Skew}(\mathbb{R}^n)$ represents the set of $n\times n$ skew symmetric matrices. 
Korn's second inequality reads as follows  \cite{Korn1,Korn2}: there is a constant $C=C(\Omega)>0,$ such that 
\begin{equation}\label{Intro-Korn2nd}
\|\nabla \Bu \|_{L^{p}(\Omega)} \leq C \left(\|e(\Bu)\|_{L^{p}(\Omega)}  + \|\Bu\|_{L^{p}}\right)\quad \text{for all}\quad \Bu\in W^{1,p}(\Omega, \mathbb{R}^{n}).
\end{equation}
These inequalities play a fundamental role in establishing the well posedness of the linear equations of elastostatics, a system of partial differential equations arising from linearized elasticity under various boundary conditions \cite{Korn1,Korn2,Kond-Oleinik}. 

By a fractional analogue of these inequalities we mean estimates of these type for vector fields in the fractional Sobolev spaces $W^{s,p}(\Omega;\mathbb{R}^{n}),$ for $s\in (0,1)$, where $\Bu\in L^{p}(\Omega;\mathbb{R}^{n})$ is in $W^{s,p}(\Omega;\mathbb{R}^{n})$ if and only if 
\[
|\Bu|_{W^{s,p}(\Omega;\mathbb{R}^{n})}^{p} =  \int_{\Omega}\int_{\Omega}\frac{|\Bu(\Bx)-\Bu(\By)|^p}{|\Bx-\By|^{n+ps}}d\Bx d\By <\infty. 
\]
The function space $W^{s,p}(\Omega;\mathbb{R}^{n})$ is a Banach space with the norm $\|\cdot\|_{W^{s,p}(\Omega)} = \|\cdot\|_{L^{p}(\Omega)}  + |\cdot |_{W^{s,p}(\Omega)}$. When writing inequalities analogous to \eqref{Intro-Korn1st} and \eqref{Intro-Korn2nd} for vector fields in $W^{s,p}(\Omega;\mathbb{R}^{n})$, one must find proper substitutes for the notion of gradient and symmetric part of the gradient. It is intuitively clear that the difference quotient $\frac{\Bu(\By) - \Bu(\Bx)}{|\By-\Bx|}$ could be used as a substitute for the gradient of $\nabla \Bu$ while the seminorm $|\Bu|_{W^{s,p}(\Omega)}$  replaces $\|\nabla \Bu\|_{L^{p}(\Omega)}$. Noting that for $\Bx\in \Omega$ and $\By$ close to $\Bx$, one has for suffficiently smooth vector fields $\Bu(\Bx)$ that
{\small$
\frac{(\Bu(\By) - \Bu(\Bx))}{|\By-\Bx|}\cdot \frac{(\By-\Bx)}{|\By - \Bx|} \approx e(\Bu)(\Bx)\frac{(\By-\Bx)}{|\By-\Bx|}\cdot \frac{(\By-\Bx)}{|\By - \Bx|},$}
we will use the projected difference quotient $\frac{(\Bu(\By) - \Bu(\Bx))}{|\By-\Bx|}\cdot \frac{(\By-\Bx)}{|\By - \Bx|}$ as the nonlocal analogue for $e(\Bu),$ and the seminorm
\[
[\Bu]^{p}_{\mathcal{X}^{s,p}(\Omega)} :=\int_{\Omega}\int_{\Omega}\frac{\left|(\Bu(\By) - \Bu(\Bx))\cdot \frac{(\By-\Bx)}{|\By - \Bx|}\right|^p}{|\By-\Bx|^{d+ps}} d\By d\Bx
\] to replace its norm. This weighted norm of the projected difference quotient not only approximates the norm of $e(\Bu)$  but also inherits its zero sets. Indeed, for $L^1_{loc}(\Omega)$ vector fields $\Bu$, the equality $(\Bu(\Bx)-\Bu(\By))\cdot(\Bx-\By)=0$ holds for a.e. $\Bx,\By\in\Omega,$ if and only if $\Bu$ is an infinitesimal rigid motion \cite[Proposition 1.2]{Temam-Miran}, i.e., $\Bu$ has the form 
\[
\Bu(\Bx) = \BA\Bx + \Bb,\quad\text{for some $\BA\in \text{Skew}(\mathbb{R}^{n})$ and $\Bb\in \mathbb{R}^n$}. 
\]
Those are exactly Sobolev vector fields that make $e(\Bu)= \boldsymbol{0}$  $a.e.$ in $\Omega,$ as can be seen from (\ref{Intro-Korn1st}). We denote this class of vector fields by $\mathcal{R}$. Before we are ready to state the desired fractional analogues of Korn's inequalities, we need to introduce the definition of bounded domains with Lipschitz constant not exceeding a number $L>0.$ 
\begin{definition}
 An open bounded Lipschitz domain 
 $\Omega\subset\mathbb R^n$ is said to have a local Lipschitz constant $\leq L,$ if the boundary of $\Omega$ can be covered by finitely many balls (or cylinders) $B_i,$ $i=1,2,\dots,m,$ so that each portion 
$\partial\Omega\cap B_i$ is the graph of a Lipschitz function with Lipschitz constant $\leq L,$ upon a rotation of the coordinate system. 
\end{definition}

\begin{remark}
An open bounded domain $C^1$-boundary has a local Lipschitz constant as small as any initially chosen positive constant $\epsilon>0.$
\end{remark}


\begin{theorem}\label{Intro:mainthm}
    Let $n \geq 2$, $s \in (0,1)$, $p \in (1,\infty)$. There exists a constant $M_0>0,$ depending only on $n,p,$ and $s,$ such that the following holds: for any open bounded Lipschitz set $\Omega \subset \mathbb{R}^n$ with Lipschitz constant $\leq M_0,$ there exist positive constants $C_1$ and $C_2,$ depending only on $d,s,p$ and $\Omega,$
    such that for all $\Bu \in W^{s,p}(\Omega;\mathbb{R}^n)$, one has
\begin{equation}
\label{thm:korn1st}
\inf_{\Br \in \mathcal{R}}|{\Bu}-{\Br}|^p_{W^{s,p}(\Omega)} \leq C_1 [\Bu]^{p}_{\mathcal{X}^{s,p}(\Omega)},
\end{equation}
and
\begin{equation}
|{\Bu}|^p_{W^{s,p}(\Omega)} \leq C_2([{\Bu}]^p_{\mathcal{X}^{s,p}(\Omega)} + \|\Bu\|^p_{L^{p}(\Omega)}).\label{thm:korn2nd} 
\end{equation}

\end{theorem}
Some remarks are in order. The same way the classical Korn's inequalities are linked to the linearized elasticity, so are their fractional analogues to some nonlocal models of elasticity. We discuss here one such model, peridynamics,  a continuum  nonlocal theory of mechanics of materials initially proposed by Stewart Silling \cite{Silling2001,Silling2007,Silling2010}. In bond-based linearized peridynamics, a material occupying a domain $\Omega$ is approximated to be a complex mass-spring system where material points interact, at a distance, with each other over a bond joining them. If the material is subject  to a deformation $\Bv(\Bx) = \Bx + \Bu(\Bx)$, then $\frac{(\Bu(\By) - \Bu(\Bx))}{|\By-\Bx|}\cdot \frac{(\By-\Bx)}{|\By - \Bx|}$ represents a (unit less) linearized nonlocal strain at $\Bx$ along the bond $\boldsymbol{\xi} =\By-\Bx$. The total strain energy is postulated to be proportional to 
\[
W_\rho(\Bu) = \int_{\Omega}\int_{\Omega} \rho(\By-\Bx) \left|\frac{(\Bu(\By) - \Bu(\Bx))}{|\By-\Bx|}\cdot \frac{(\By-\Bx)}{|\By - \Bx|}\right|^{2} d\By d \Bx,
\]
where $\rho(\boldsymbol{\xi})$ is locally integrable and serves as a weight for the long-range interactions.  Given an external force $\Bf \in L^{2}(\Omega;\mathbb{R}^n)$, the corresponding configuration can be found as a minimizer of the functional
\begin{equation}\label{pot-func}
\Bu\mapsto W_\rho(\Bu) - \int_{\Omega} \Bf(\Bx) \cdot \Bu(\Bx) d\Bx
\end{equation}
over an appropriate admissible subset of $L^{2}(\Omega;\mathbb{R}^n)$. In fact, existence of minimizers in some subsets of the energy space 
$
\mathcal{S}_{\rho}(\Omega) = \{\Bu\in L^{2}(\Omega;\mathbb{R}^n): W_\rho(\Bu) <\infty\}
$
is demonstrated in 
\cite{Mengesha-Du}. See also \cite{Mengesha-Du-non} for existence of solutions to more generalized models of linearized peridynamics. Except for $n=1$, the question that whether $\mathcal{S}_{\rho}(\Omega)$, which is based on the {\it projected difference-quotient}, 
is equal to the space 
\[X_\rho(\Omega) = \{\Bu\in L^{2}(\Omega;\mathbb{R}^n): \int_{\Omega}\int_{\Omega} \rho(\By-\Bx) \frac{\left|\Bu(\By) - \Bu(\Bx)\right|^2}{|\By-\Bx|^2} d \By d\Bx < \infty\}\]
based on the full difference-quotient remains open. The fractional Korn's inequalities proved in Theorem \ref{Intro:mainthm} address this question and establish equality of sets for a special case when $\rho(\boldsymbol{\xi}) = |\boldsymbol{\xi}|^{-n-2(s-1)}$, for $s\in (0, 1)$. In this case,  $W(\Bu) = [\Bu]^{2}_{\mathcal{X}^{s,2}(\Omega)}$ and 
minimizing the functional in \eqref{pot-func} over a weakly closed subset of the smaller $W^{s,2}(\Omega;\mathbb{R}^n)$, say,  $W^{s,2}_{\omega}(\Omega;\mathbb{R}^n)= \{\Bu \in W^{s,2}(\Omega;\mathbb{R}^n): \omega\subset\Omega,\,\, \Bu=\boldsymbol{0},\  \text{a.e. in} \ \omega\}$ is possible. To apply Hilbert space methods, inequality \eqref{thm:korn2nd} is now essential, along with a Poincar\'e-Korn inequality, see Lemma \ref{K-P} below, to show the coercivity of the functional. We leave the details to the interested readers, see \cite{Mengesha-Du, Mengesha-Du-non}. 



We emphasize that the main contribution of this work is proving inequalities \eqref{thm:korn1st} and \eqref{thm:korn2nd} for vector field in  $W^{s,p}(\Omega,\mathbb{R}^n)$ without any ``boundary conditions.'' While \eqref{thm:korn1st}
as stated appears to be new to our best knowledge, its special version with $\BA=0,$ and inequality \eqref{thm:korn2nd} have appeared in recent works albeit in restricted forms. Indeed, the variant of \eqref{thm:korn1st} with $\BA=0$ was proved first proven in \cite{M-half} for the case when $\Omega$ is the half-space, $p=2,$ and for vector fields $\Bu\in W_0^{s,2}(\Omega)$ which is the closure of $C^{1}_{c}(\Omega,\mathbb{R}^n)$ with respect to the norm $\|\cdot\|_{W^{s,2}(\Omega)}$ (roughly speaking for vector fields satisfying zero Dirichlet boundary conditions on $\partial\Omega$). The estimate for the half-space was then extended for any values $1<p<\infty$ in \cite{MScott2018Korn}. The estimate was then proven in \cite{MS2022} for the restricted class $W^{s,p}_0(\Omega,\mathbb{R}^n)$ for bounded $C^1$ domains, and the same result appeared in \cite{Rut.2022} significantly shortening the proof presented in \cite{MS2022}. A tighter version of estimates \eqref{thm:korn1st} and \eqref{thm:korn2nd}  have also been proved in \cite{HM2023} for the case when $ps>1$, where for some constant $C>0,$ 
\begin{align}\label{HM:korn1st}
|{\Bu}|_{W^{s,p}(\Omega)} \leq C [{\Bu}]_{\mathcal{X}^{s,p}(\Omega)}\quad \text{for all} \quad \Bu \in W^{s,p}_0(\Omega;\mathbb{R}^n).
\end{align}
Via  a counterexample \cite{Kirzbraun1934}, inequality \eqref{HM:korn1st} is shown to fail for any open bounded subset $\Omega\subset\mathbb R^n$ in the case $ps<1$. This is in stark contrast to the case when $\Omega=\mathbb{R}^{n}$ or $\Omega=\mathbb{R}^{n}_{+}$ (unbounded domains), where \eqref{HM:korn1st} is proved to hold for any $s\in(0,1)$, $p\in (1, \infty)$ such that $ps\neq 1,$ \cite{M-half, MS2022}.   In fact, in this case, our current work implies that the restriction $ps\neq 1$ is not even necessary.   
We note that \eqref{HM:korn1st} is the fractional analogue of another version of Korn's first inequality: 
\[
\|\nabla \Bu \|_{L^{p}(\Omega)} \leq C_p \|e(\Bu)\|_{L^{p}(\Omega)}\quad\text{for all} \quad \Bu\in W^{1,p}_0(\Omega, \mathbb{R}^{n}),
\]
for a constant $C_p>0$ that depends only on $p.$ This being said, a new phenomenon occurs in the fractional setting. 
\begin{remark}
It is well known that the range of exponent that validates the classical Korn inequalities \eqref{Intro-Korn1st} and \eqref{Intro-Korn2nd} is $1<p<\infty.$  Moreover, for Sobolev vector fields that satisfy zero Dirichlet boundary conditions, one can always choose $\BA=0$ 
in \eqref{Intro-Korn1st} in that range. However, this is no longer true in the fractional setting because despite the fact that the case $ps<1$ is included in the validity range for \eqref{thm:korn1st} and \eqref{thm:korn2nd}, the version of \eqref{thm:korn1st}
with $\BA=0$ fails in bounded domains in the case 
$ps<1.$
\end{remark}

As it is clear from the formulation, Theorem \ref{Intro:mainthm} has a limitation that inequalities \eqref{thm:korn1st} and \eqref{thm:korn2nd} are established for a class of vector fields defined over a domain with a boundary  that has a sufficiently small Lipschitz constant. Taking clues from the classical Korn's inequalities \cite{nitsche1981korn}, we conjecture that in fact the inequalities remain valid for any bounded Lipschitz domain. To support the claim, we establish the same inequalities for planar convex Lipschitz domains with no constraint on the size of Lipschitz constant of the boundary.  This will be demonstrated in Section \ref{sec:convex}.

As we will show in Section \ref{sec:epigraph}, inequality  \eqref{thm:korn1st} follows from \eqref{thm:korn2nd}. The main challenge is thus proving \eqref{thm:korn2nd}. Our method of proof is standard. We first establish \eqref{thm:korn2nd} for epigraphs supported by a Lipschitz function and then use a partition of unity to localize near the boundary of the domain. The later is successfully carried out in \cite{MS2022} and \cite{Rut.2022} and we will not repeat that part of the argument here. We would rather focus on obtaining the estimate for epigraphs. That will be accomplished after proving the existence
of an extension operator to extend the vector fields in the epigraph to be defined on $\mathbb{R}^n$. As in \cite{MS2022} we will use the extension introduced in \cite{nitsche1981korn} which allows us to control the seminorm of the extended vector fields by the seminorm over the epigraph. In this work, we use an improved Hardy-type inequality, Lemma \ref{lem:2.4} in Section \ref{sec:tools}, to overcome a technical difficulty that we encountered in \cite{MS2022} and restricted the validity \eqref{thm:korn2nd} to only vector fields that vanish on the boundary. 



\section{Korn-Poincar\'e and Hardy-type inequalities}
\label{sec:tools}
Given an open set $D\subset \mathbb{R}^{n},$ we define the spaces $\mathcal{X}^{s,p}_{0}(D)$ and $\mathcal{X}^{s,p}(D)$ to be the closure of $C_{c}^{1}(D;\mathbb{R}^n)$  and $C^{1}(D;\mathbb{R}^n)$ respectively with respect to the norm $\|\cdot\|_{\mathcal{X}^{s,p}(D)}.$ 
We begin with the following Korn-Poincar\'e inequalities that are compatible with the seminorm $[\cdot]_{\mathcal{X}^{s,p}(\Omega)}$.  
It is worth mentioning that the fractional Korn-Poincar\'e inequality is an important component in the proof of the first and second fractional Korn inequalities. This is in contrast to the classical local setting where the Korn-Poincar\'e inequality is derived as a consequence of Korn's first inequality after the latter has been established by other means.


\begin{lemma}[Korn-Poincar\'e inequalities]\label{K-P}
Suppose that $\Omega$ is a bounded Lipschitz domain. Then for any $s\in (0,1)$, $p\in (1, \infty)$,  there exists a positive constant $C$ depending only on $n,p,s,$ and $\Omega,$ such that 
\begin{align*}
\min_{\Br\in \mathcal{R}}\|{\bf u}-\Br\|_{L^{p}(\Omega)} &\leq C [{\bf u}]_{\mathcal{X}^{s,p}(\Omega)} \quad \text{for all} \quad {\bf u}\in \mathcal{X}^{s,p}(\Omega).
\end{align*}
Moreover, if $V\subset L^{p}(\Omega;\mathbb{R}^{n})$ is a weakly closed subset such that $V\cap \mathcal{R} = \{\boldsymbol{0}\}$, then there exists a constant $\tilde C>0,$ that in addition may depend on $V,$ such that  
\[
\|\Bu\|_{L^{p}(\Omega;\mathbb{R}^{n})} \leq \tilde C [{\bf u}]_{\mathcal{X}^{s,p}(\Omega)} \quad \text{for all}\quad {\bf u} \in V.
\]
 
\end{lemma}
\begin{proof}
We prove the first assertion. The proof of the second can be found in \cite{Mengesha-Du-non}. 
We will use a standard contradiction argument adopted by Kondratiev and Oleinik for the classical case in \cite{Kond-Oleinik}. Suppose that there is a sequence  $\Bu_k\in \mathcal{X}^{s,p}(\Omega)$ and the corresponding minimizers  $\BA_k\in \mathrm{skew}(\mathbb R^n)$ and ${\bf b}_k\in\mathbb R^n$ such that 
\begin{equation}
\label{seq-bound}
\|\Bu_k-\BA_k\cdot \Bx-{\bf b}_k\|_{L^{p}(\Omega)}=1\quad \text{and} \quad [\Bu_k]_{\mathcal{X}^{s,p}(\Omega)}\leq 1/k,\quad k=1,2,\dots
\end{equation}
Upon passing to the fields $\Bv_k=\Bu_k-\BA_k\cdot \Bx - {\bf b}_k$ we can assume without loss of generality that $\BA_k=0$ and ${\bf b}_k = 0$ in \eqref{seq-bound} for all $k$. Thus we have the minimality conditions 
\begin{equation}
\label{optimality}
\|\Bv_k\|_{L^{p}(\Omega)}\leq \|\Bv_k-\BA\cdot \Bx-{\bf b}\|_{L^{p}(\Omega)}\quad \text{for any}\quad \BA\in \mathrm{skew}(\mathbb R^n), \ \  {\bf b}\in \mathbb R^n, \ \ k=1,2,\dots
\end{equation}
We then have from \eqref{seq-bound} that the sequence ${\bf v}_k$ is bounded in $\mathcal{X}^{s,p}(\Omega)$. 
We can now apply the compactness theorem in \cite[Theorem~1.3]{DMT-compact}  to conclude that   
the sequence $\{\Bv_k\}$ is pre-compact in $L^p(\Omega),$ thus we can assume without loss of generality that 
\begin{equation}
\label{Lp-convergence}
\Bv_k\to \Bv \quad \text{in}\quad L^p(\Omega),
\end{equation}
for some field $\Bv\in L^p(\Omega).$ We have by  (\ref{seq-bound}) that
\begin{align*}
 \|\Bv_k-\Bv_m\|_{\mathcal{X}^{s,p}(\Omega)}&= ([\Bv_k-\Bv_m]_{\mathcal{X}^{s,p}(\Omega)}+\|\Bv_k-\Bv_m\|_{L^p(\Omega)})\\ \nonumber
 & \leq  C([\Bv_k]_{\mathcal{X}^{s,p}(\Omega)}+[\Bv_m]_{\mathcal{X}^{s,p}(\Omega)}+\|\Bv_k-\Bv_m\|_{L^p(\Omega)})\\ \nonumber
 &\leq C(1/k+1/m+\|\Bv_k-\Bv_m\|_{L^p(\Omega)}),
 \end{align*}
thus the condition (\ref{Lp-convergence}) implies that the sequence $\{\Bv_k\}$ is Cauchy and thus is convergent in $\mathcal{X}^{s,p}(\Omega)$ and the limit is ${\bf v}$. This gives $\Bv_k\to \Bv$ in  $\mathcal{X}^{s,p}(\Omega)$ as $k\to \infty$ as well. 
We thus have from (\ref{seq-bound}) that 
\begin{align*}
[\Bv]_{\mathcal{X}^{s,p}(\Omega)}&\leq [\Bv_k]_{\mathcal{X}^{s,p}(\Omega)}+[\Bv-\Bv_k]_{\mathcal{X}^{s,p}(\Omega)}\\
&\leq 1/k+[\Bv-\Bv_k]_{\mathcal{X}^{s,p}(\Omega)}\to0
\end{align*}
as $k\to\infty,$ thus  $[\Bv]_{\mathcal{X}^{s,p}(\Omega)}=0,$ which gives 
\begin{equation}
\label{rigid-bound}
\Bv(\Bx)=\BA\cdot \Bx+\Bb,\quad\text{for a.e.}\quad\Bx\in\Omega,
\end{equation}
for some constant skew-symmetric matrix $\BA\in \mathbb R^{n\times n}$ and some vector $\Bb\in\mathbb R^n$ \cite[Proposition 1.2]{Temam-Miran} (\cite[Theorem 3.1]{Korn-characterization}). 
We then have by (\ref{seq-bound}), (\ref{optimality}), and (\ref{rigid-bound}) that  
\begin{align*}
1=\|\Bv_k\|_{L^p(\Omega)}
 \leq \|\Bv_k-\BA\cdot \Bx-{\bf b}\|_{L^p(\Omega)} 
 = \|\Bv_k-\Bv\|_{L^{p}(\Omega)}\to 0, 
\end{align*}
as $k\to\infty,$ which is a contradiction.
\end{proof}
\begin{remark}
We remark that if, for a given $\tau>0$ and $\Bx_0 \in \mathbb{R}^n$, $Q(\Bx_0, \tau)$ represents a cube centered at $\Bx_0$ with radius $\tau$, then a simple scaling argument yields the estimate 
\begin{equation}
\label{Korn-Poincare}
\min_{\Br\in \mathcal{R}}\|{\bf u}-\Br\|_{L^{p}(Q(\Bx_0, \tau))} \leq C\, \tau^{s}\,[{\bf u}]_{\mathcal{X}^{s,p}(Q(\Bx_0, \tau)}
\end{equation}
for all ${\bf u}\in \mathcal{X}^{s,p}(Q(\Bx_0, \tau))$, 
where the constant $C$ is the constant for the unit cube $Q({\boldsymbol 0}, 1)$, which depends only on $n,$ $p$, and $s$.
\end{remark}
The following variant of the fractional Hardy-type 
inequality is key for proving the boundedness of the extension operator we will define in the next section. For notational convenience, we represent points $\Bx\in \mathbb{R}^n$ as $\Bx = (\Bx', x_n)\in \mathbb{R}^{n-1}\times \mathbb{R}$. 


\begin{definition}[Epigraph]
Let $f\colon \mathbb R^{n-1}\to\mathbb R$ be a continuous function. The set  
$$D=\{(\Bx',x_n) \ : \ \Bx'\in U,\ x_n>f(\Bx')\}$$ 
is called an epigraph supported by the function $f.$ In that case we also denote 
$$D_{-}=\{(\Bx',x_n) \ : \ \Bx'\in U,\ x_n<f(\Bx')\}.$$ 
\end{definition}

In what follows, $f$ will be a globally Lipschitz function with $\|\nabla f\|_{L^{\infty}(\mathbb R^{n-1})}\leq M.$ Also, capital letter $C$ will denote a constant that depends on $n,p,s$ and $M,$ while small letter $c$ will denote a constant that depends only on $n,p$ and $s.$ For any epigraph $D$ and any $\eta>0$, define the mapping $\Phi_\eta: {D_{-}} \to D$ given by
\begin{equation}
\label{Phi}
\Phi_\eta(\Bx)=(\Bx', f(\Bx') + \eta(f(\Bx')-x_n)),
\end{equation}
which is clearly a Lipschitz diffeomorphism with the inverse 
\begin{equation*}
(\Phi_\eta)^{-1}(\Bx) = (\Bx', f(\Bx') + \frac{1}{\eta}(f(\Bx')-x_n)),
\end{equation*}
and $\text{det}(\nabla \Phi_\eta) = -\eta$.
By direct calculation we get 
\begin{equation}
\label{NormPhi}
\|\nabla \Phi_{\eta}\|_{L^{\infty}(D_-)}=
\sqrt{n-1 + \eta^{2} + (1+\eta)^{2}\|\nabla f\|_{L^{\infty}}},
\end{equation}
and
\begin{equation}
\label{NormPhiInverse}
\|\nabla (\Phi_{\eta})^{-1}\|_{L^{\infty}(D)}=  \sqrt{n-1 + \frac{1}{\eta^{2}} + (1+\frac{1}{\eta})^{2}\|\nabla f\|_{L^{\infty}}}.
\end{equation}

Hence, in space dimensions $n\geq 2,$ the norms $\|\nabla\Phi_{\eta}\|_{L^{\infty}(D_-)}$ and 
$\|\nabla (\Phi_{\eta})^{-1}\|_{L^{\infty}(D)}$
are bounded from below by one (independent of the Lipschitz constant of $f$). Moreover, as proved in 
Lemma~\ref{lem:A.1}, there exists a constant $C>0,$
depending only on $n,\eta,$ and $M$ such that
\begin{equation}\label{flat-vs-curved}
|\Bx-\By| \leq C|(\Phi_\eta)^{-1}(\Bx) - \By|
\quad \text{for all}\quad \Bx, \By\in D.
\end{equation}


\vspace{0.3cm}

\begin{lemma}[Hardy-type inequality]
\label{lem:2.4}
Let $f\colon \mathbb R^{n-1}\to\mathbb R$ be a Lipschitz function with Lipschitz constant $\leq M,$ and let $D\subset \mathbb R^n$ be the epigraph supported by $f.$ There exist a constant $C=C(n,p,s,M)>0,$ depending only on $n,p,s,$ and $M,$ and a constant $c_2(n)>0$ depending only on $n$ (coming from the Whitney cover of $D$), such that for every $\lambda,\mu \in [1-\delta,1+\delta]$ with $\delta=\frac{1}{2c_2(n)\sqrt{n}(1+M)}$ and every vector field $\Bu\in W^{s,p}(D,\mathbb R^n)$ one has
{\small \begin{equation}
\label{Hardy-type}
\int_{D}\frac{|u_n(\Bx',f(\Bx')+\lambda(x_n-f(\Bx')))-u_n(\Bx',f(\Bx')+\mu(x_n-f(\Bx')))|^p}{|x_n-f(\Bx')|^{ps}}d\Bx \leq C[\Bu]_{\mathcal{X}^{s,p}(D)}^p.
\end{equation}}
\end{lemma}


\begin{proof} 
Consider a Whitney cover of $D$ with a sequence of cubes $\{Q_k\}_{k=1}^\infty$ with the following properties.
\begin{itemize}
\item[(i)] The doubled cubes $\hat Q_k=2\cdot Q_k$ satisfy the inclusion condition 
$$\hat Q_k\subset D\quad\text{ for all }\quad k\in\mathbb N.$$
\item[(ii)] The doubled cubes have the finite intersection property
\begin{equation*}
\sum_{k=1}^\infty\chi_{\hat Q_k}(\Bx) \leq c_1(n)\quad\text{for all}\quad \Bx\in D. 
\end{equation*}
\item[(iii)] There exists a constant $c_2(n)$ such that each of the $c_2(n)-$times enlarged cube $c_2(n)\cdot Q_k$ intersects with the graph of $f.$
\end{itemize}
Here, the constants $c_1(n)$ and $c_2(n)$
depend only on the space dimension $n.$ This is always possible, see \cite[Chapter VI]{Stein} for details. Let now $a_k>0$ be the side length of $Q_k.$ Observe that on one hand condition (i) in particular implies that 
\begin{equation}
\label{2.18}
|x_n-f(\Bx')|\geq \frac{a_k}{2},\quad\text{for all}\quad k\in\mathbb N, \ \Bx\in Q_k. 
\end{equation}
On the other hand for a fixed point $\Bx=(\Bx',x_n)\in Q_k,$ let $\mathrm{dist}(\Bx,\mathrm{Gr}(f))=|\Bx-\By|,$
where $\By=(\By',f(\By'))\in\mathrm{Gr}(f)$ and $\mathrm{Gr}(f)$ is the graph of $f.$ We have that 
$|f(\Bx')-f(\By')|\leq M|\Bx'-\By'|,$ thus we can estimate 
\begin{align*}
 \mathrm{dist}(\Bx,\mathrm{Gr}(f))&=|\Bx-\By|\\
 &\geq \frac{1}{\sqrt{2}}(|\Bx'-\By'|+|x_n-f(\By')|)\\
 &\geq \frac{1}{\sqrt{2}M}|f(\Bx')-f(\By')|
 +\frac{1}{\sqrt{2}}|x_n-f(\By')|\\
 &\geq \min(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}M})|x_n-f(\Bx')|,
\end{align*}
hence condition (iii) yields the bound 
\begin{equation}
\label{2.19}
|x_n-f(\Bx')|\leq c_2(n)\sqrt{n}(1+M)a_k\quad\text{for all}\quad k\in\mathbb N, \ \Bx\in Q_k. 
\end{equation}
 We can estimate 
\begin{align*}
\int_{D}&\frac{|u_n(\Bx',f(\Bx')+\lambda(x_n-f(\Bx')))-u_n(\Bx',f(\Bx')+\mu(x_n-f(\Bx')))|^p}{|x_n-f(\Bx')|^{ps}}d\Bx \\ \nonumber
&\leq \sum_{k=1}^\infty
\int_{Q_k}\frac{|u_n(\Bx',f(\Bx')+\lambda(x_n-f(\Bx')))-u_n(\Bx',f(\Bx')+\mu(x_n-f(\Bx')))|^p}{|x_n-f(\Bx')|^{ps}}d\Bx. 
\end{align*}
Setting 
$\Phi_\lambda^\ast(\Bx)=(\Bx',f(\Bx')+\lambda(x_n-f(\Bx')))$ for brevity, we aim to prove the inequality 
\begin{equation}
\label{2.20}
\int_{Q_k}\frac{|u_n(\Phi_\lambda^\ast(\Bx))-u_n(\Phi_{\mu}^\ast(\Bx))|^p}{|x_n-f(\Bx')|^{ps}}d\Bx 
\leq C [\Bu]_{\mathcal{X}^{s,p}(\hat Q_k)}^p,
\end{equation}
for each cube $Q_k.$ For every fixed $k\in\mathbb N$ we have by (\ref{2.18}), that
\begin{equation}
\label{2.21}
\int_{Q_k}\frac{|u_n(\Phi_\lambda^\ast(\Bx))-u_n(\Phi_{\mu}^\ast(\Bx))|^p}{|x_n-f(\Bx')|^{ps}}d\Bx
\leq 2^{-ps}a_k^{-ps}\int_{Q_k} |u_n(\Phi_\lambda^\ast(\Bx))-u_n(\Phi_{\mu}^\ast(\Bx))|^p d\Bx.
\end{equation}
Next we apply the Korn-Poincar\'e inequality (\ref{Korn-Poincare}) to the cube 
$\hat Q_k$ and the vector field $\Bu.$ Hence, there exists a constant $C>0,$ a skew-symmetric matrix $\BA_k\in\mathbb R^{n\times n},$ and a vector $\Bb_k\in\mathbb R^n$ such that 
\begin{equation}
\label{2.22}
\|\Bu(\cdot)-\BA_k(\cdot)-\Bb_k\|_{L^p(\hat Q_k)}\leq
 C a_k^{s}[\Bu]_{\mathcal{X}^{s,p}(\hat Q_k)}.
\end{equation} 
Observe that $\Phi_\lambda^\ast \colon Q_k\to \Phi_\lambda^\ast(Q_k)$ is a one-to-one diffeomorphism with the inverse $\Phi_{1/\lambda}^\ast$ and has Jacobian equal to $\lambda\leq 1+\delta.$ Also, due to the inequality (\ref{2.19}) we have for every 
$\Bx\in Q_k$ and every $\lambda\in [1-\delta,1+\delta],$ that 
\begin{align*}
|\Phi_\lambda^\ast(\Bx)-\Bx|&=|1-\lambda||x_n-f(\Bx')|\\ \nonumber
&\leq \delta c_2(n)\sqrt{n}(1+M)a_k\\ \nonumber
&\leq \frac{a_k}{2},
\end{align*}
provided $\delta=\frac{1}{2c_2(n)\sqrt{n}(1+M)}.$ This implies the   
inclusion conditions $\Phi_\lambda^\ast(Q_k),\Phi_\mu^\ast(Q_k)\subset \hat Q_k.$ Consequently noting that 
$(\BA_k\cdot\Phi_\lambda^\ast(\Bx)+\Bb_k)_n
 -(\BA_k\cdot\Phi_{\mu}^\ast(\Bx)+\Bb_k)_n=0,$ we can calculate bearing in mind the bound in (\ref{2.22}):

\begin{align}
\label{2.23}
 \int_{Q_k} &|u_n(\Phi_\lambda^\ast (\Bx))-u_n(\Phi_{\mu}^\ast(\Bx))|^p d\Bx \\ \nonumber
 &= \int_{Q_k} |u_n(\Phi_\lambda^\ast (\Bx))-(\BA_k\cdot\Phi_\lambda^\ast(\Bx)+\Bb_k)_n
 -(u_n(\Phi_{\mu}^\ast (\Bx))-(\BA_k\cdot\Phi_{\mu}^\ast(\Bx)+\Bb_k)_n)|^p d\Bx \\ \nonumber
 &\leq 2^{p-1}\int_{Q_k} |u_n(\Phi_\lambda^\ast (\Bx))-(\BA_k\cdot\Phi_\lambda^\ast(\Bx)+\Bb_k)_n|^pd\Bx \\ \nonumber 
&+2^{p-1}\int_{Q_k} |u_n(\Phi_{\mu}^\ast (\Bx))-(\BA_k\cdot\Phi_{\mu}^\ast(\Bx)+\Bb_k)_n|^p d\Bx \\ \nonumber 
&\leq C\|\Bu(\By)-\BA_k\cdot\By-\Bb_k\|_{L^p(\hat Q_k)}^p \\ \nonumber
&\leq C a_k^{ps}[\Bu(\By)]_{\mathcal{X}^{s,p}(\hat Q_k)}^p.
\end{align}
Putting together now (\ref{2.21}) and (\ref{2.23}) we discover 
\begin{equation*}
\int_{Q_k}\frac{|u_n(\Phi_\lambda^\ast(\Bx))-
u_n(\Phi_{\mu}^\ast(\Bx))|^p}{|x_n-f(\Bx')|^{ps}}d\Bx
\leq C[\Bu(\By)]_{\mathcal{X}^{s,p}(\hat Q_k)}^p.
\end{equation*}
In order to complete the proof of the lemma, one needs to sum (\ref{2.20}) over $k$ and keep in mind the finite intersection property in (ii). This completes the proof of Lemma~2.4.
\end{proof}
\begin{remark}
    In the special case of the half space, where $D= \mathbb{R}^{n}_{+}$ and $f\equiv 0$, inequality \eqref{Hardy-type} reduces to 
    \[
\int_{D}\frac{|u_n(\Bx',\lambda x_n)-u_n(\Bx',\mu x_n)|^p}{|x_n|^{ps}}d\Bx \leq C[\Bu]_{\mathcal{X}^{s,p}(D)}^p.
\]
This inequality was proved in \cite{M-half} for vector fields in $\mathcal{X}^{s,p}_{0}(D)$ and  under the extra assumption that $ps\neq1.$ It is now clear from Lemma \ref{lem:2.4} that this requirement is not necessary and that the fractional Korn's inequality proved in \cite{M-half} for vector fields in $\mathcal{X}^{s,p}_{0}(D)$ is also valid for all $s\in (0,1)$ and $1<p<\infty$ (even when $ps=1$). A consequence of this is that the Korn inequality proved in \cite{MS2022} for vector fields defined on bounded domains with smooth boundary will also be true for the full ranges of $s$ and $p$.  
\end{remark}





\section{Fractional Korn's inequalities}
\label{sec:epigraph}
\subsection{Korn's second inequality over epigraphs}


This section is devoted to the fractional Korn's second inequality in epigraphs. We prove the following theorem. 
\begin{theorem}[Korn's second inequality in epigraphs]
\label{thm:Korn.Epigraph}
Given $s\in (0,1)$ and $1\leq p<\infty$, there exists a universal constant $M_0>0$ and another constant $C_0>0$ depending only on $n, p,s$ and $M_0$ with the following property: For any epigraph $D$ supported by $f$ with $\|\nabla f\|_{L^{\infty}}\leq M_0$, one has for all $\Bu\in W^{s,p}(D;\mathbb{R}^n)$ the inequality
\begin{equation}
\label{Korn.sec.epi}
\|\Bu\|_{W^{s,p}(D)} \leq C_0\|\Bu\|_{\mathcal{X}^{s,p}(D)}.
\end{equation}

\end{theorem}


As we described in the introduction, to prove the fractional Korn's inequality (\ref{Korn.sec.epi}) for an epigraph $D$, we first prove the existence of an extension operator to extend the vector fields in $\mathcal{X}^{s,p}(D)$ to be in $\mathcal{X}^{s,p}(\mathbb{R}^n)$ in such a way that the seminorm of the extended vector fields is controlled by the seminorm over $D$. As in \cite{MS2022}, we will show that the extension operator that was used in \cite{nitsche1981korn} for the proof of the classical Korn's inequality will also be useful to prove the fractional case. 
\begin{proposition}[Extension operator]
\label{prop:extension}
Let $s\in (0,1)$ and $1\leq p<\infty$ and let $D$ be an epigraph supported by a Lipschitz function $f$ with 
$\|\nabla f\|_{L^{\infty}}=M.$ There exists a bounded extension operator $E: W^{s,p}(D;\mathbb{R}^n)\to W^{s, p}(\mathbb{R}^n;\mathbb{R}^n),$ a constant $C>0,$ depending only on $n, p, s,$ and $M,$ and a constant $c>0$ depending only on $n, p,$ and $s,$ with the property that for all $\Bu\in W^{s,p}(D;\mathbb{R}^n)$ one has
\begin{equation}\label{ext-norm-estimate}
\|E(\Bu)\|_{\mathcal{X}^{s,p}(\mathbb{R}^n)} \leq C\|\Bu\|_{\mathcal{X}^{s,p}(D)} + c(1+M)^{2+\frac{n}{2p}} M|\Bu|_{W^{s,p}(D)}.
\end{equation} 

\end{proposition}


\begin{proof}
By density, it suffices to show the inequality for $C^1$ vector fields. Following the approach in \cite{nitsche1981korn}, we define the extension operator $E$ as follows. 
For $\Bu = (\Bu',u_n) \in C^1(D;\mathbb{R}^n)$, and for constants $\lambda$, $\mu$, $k$, $\ell$, $m$, and $q$, set
\begin{equation}\label{ext-epigraph}
[ \mathrm{E}(\Bu)(\Bx) ]_i :=
\begin{cases}
u_i(\Bx)\,, & \qquad \Bx \in D\,, \quad i = 1, 2, \ldots n-1, n\,, \\
k \, u_i^{\lambda}(\Bx) + \ell \, u_i^{\mu}(\Bx)\,, & \qquad \Bx \in D_-\,, \quad i = 1, 2, \ldots n-1\,, \\
m \, u_n^{\lambda}(\Bx) + q \, u_n^{\mu}(\Bx)\,, & \qquad \Bx \in D_-\,,
\end{cases}
\end{equation}
where
\begin{equation}\label{ulambdamu}
\begin{split}
u_j^{\lambda}(\Bx) := u_j \big( \Bx',f(\Bx') + \lambda(f(\Bx')-x_d) \big) = u_j(\Phi_\lambda(\Bx)),\\
u_j^{\mu}(\Bx) := u_j \big( \Bx',f(\Bx') + \mu(f(\Bx')-x_d) \big)\,=u_j(\Phi_\mu(\Bx)).
\end{split}
\end{equation}
We choose constants $\lambda$, $\mu$, $k$, $\ell$, $m$, $q$, such that
\begin{equation}\label{eq:ReflectionConditions}
\begin{split}
\lambda > 0\,, \quad \mu > 0\,, \qquad k+ \ell = 1 = m+q\,, \qquad \lambda k = -m\,, \quad \mu \ell = -q\,.
\end{split}
\end{equation}
For $0\leq \lambda<\mu$ these constants are uniquely 
defined and are given by 
\begin{equation}\label{klmq}
k=\frac{1+\mu}{\mu-\lambda},\quad \ell=-\frac{1+\lambda}{\mu-\lambda},\quad
m=-\frac{\lambda(1+\mu)}{\mu-\lambda},\quad q=\frac{\mu(1+\lambda)}{\mu-\lambda}.
\end{equation}
Let now $\delta=\frac{1}{2c_2(n)\sqrt{n}(1+M)}$ be as in Lemma~2.4, and choose 
\begin{equation}
\label{Def.LambdaMu}
\lambda=1-\delta\quad\text{and}\quad \mu=1+\delta.
\end{equation}
Recalling that the boundary $\partial D$ is given by the equation $x_n=f(\Bx')$, it is clear that the operator $E$ takes continuous map defined on $D$ to continuous maps on $\mathbb{R}^n$. Moreover, for $\Bu\in C^{1}(D,\mathbb{R}^n)$, $E(\Bu) \in W^{s,p}(\mathbb{R}^{n}, \mathbb{R}^{n})$. This can be shown  following calculations similar to the ones that will be used below estimating $[E(\Bu)]_{\mathcal{X}^{s,p}(\mathbb{R}^{n)}}$ to demonstrate the inequality \eqref{ext-norm-estimate}. We split the domain of integration and write 
\begin{align}\label{Int-split}
[E(\Bu)]^p_{\mathcal{X}^{s,p}(\mathbb{R}^{n)}}  = [\Bu]^p_{\mathcal{X}^{s,p}(D)}& + [E(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}\nonumber\\
&+2\int_{D_{-}}\int_{D} \frac{\left|(E(\Bu)(\Bx) - E(\Bu)(\By)) \cdot(\Bx-\By)\right|^p}{|\Bx-\By|^{n + (s+1)p}}d\By d \Bx.
\end{align}
We need to estimate the second and the third terms. For $\Bx\in D_{-}$, we write $$E(\Bu)(\Bx)= E_{\lambda}(\Bu)(\Bx)  + E_{\mu}(\Bu)(\Bx) $$ 
where $E_{\lambda}(\Bu)(\Bx) = (k (\Bu')^{\lambda}, m\, u^{\lambda}_{n})$ and $E_{\mu}(\Bu)(\Bx) = (\ell (\Bu')^{\mu}, q\, u^{\mu}_{n})$. We have  $$[E(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})} \leq 2^{p-1}([E_\lambda(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}+[E_\mu(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}),$$
and will estimate each of the summands next. To estimate $[E_\lambda(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}$, we make the change of coordinates $\Bz=\Phi_\lambda(\Bx)$ and $\Bw=\Phi_\lambda(\By)$ and recall the discussion about the mapping $\Phi$ in (\ref{Phi})--(\ref{NormPhiInverse}) to write the integral as  
{\small \begin{align*}
&\lambda ^2\, [E_\lambda(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}\\
& = \int_{D}\int_{D}\frac{|k(\Bu'(\Bz) - \Bu'(\Bw))\cdot (\Bz'-\Bw') + m(u_n(\Bz)-u_n(\Bw))\cdot([(\Phi_\lambda)^{-1}(\Bz)]_n - [(\Phi_\lambda)^{-1}(\Bw)]_n)|^{p}}{|(\Phi_\lambda)^{-1}(\Bz) - (\Phi_\lambda)^{-1}(\Bw)|^{n+ (s+1)p}}d\Bz d\Bw.
\end{align*}}


Notice that 
$$[(\Phi_\lambda)^{-1}(\Bz)]_n - [(\Phi_\lambda)^{-1}(\Bw)]_n = -\frac{1}{\lambda}(z_n - w_n) + \frac{1+\lambda}{\lambda}(f(\Bz')-f(\Bw')),$$
and  
\[
|\Bz - \Bw| \leq \|\nabla \Phi_{\lambda}\|_{L^{\infty}(D_-)} \left| (\Phi_{\lambda})^{-1}(\Bz) - (\Phi_{\lambda})^{-1}(\Bw) \right|. 
\]
It then follows using the relation $\lambda \,k = -m,$ that
\begin{align*}
    [E_\lambda(&\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})} \leq \frac{2^{p-1}k^p }{\lambda^2} \|\nabla \Phi_{\lambda}\|^{n+(s+1)p}_{L^{\infty}(D_-)}\,[{\Bu}]^p_{\mathcal{X}^{s,p}(D)}\\ \nonumber
    &+ \frac{2^{p-1}m^p(1+ \lambda)^p}{\lambda^{2+p}} \|\nabla \Phi_{\lambda}\|^{n+(s+1)p}_{L^{\infty}(D_-)}\int_{D}\int_{D}{\frac{\left| \big( u_n(\Bz) - u_n(\Bw) \big) \cdot \big( f(\Bz')-f(\Bw') \big) \right|^p}{|\Bz-\Bw|^{d+(s+1)p}}} d{\Bw}d{\Bz}\\ \nonumber
    &\leq \frac{2^{p-1}k^p }{\lambda^2} \|\nabla \Phi_{\lambda}\|^{n+(s+1)p}_{L^{\infty}(D_-)}\,\left([{\Bu}]^p_{\mathcal{X}^{s,p}(D)} + (1+ \lambda)^p \|\nabla f\|^{p}_{L^{\infty}}|u_n|^p_{W^{s,p}(D)}\right).
\end{align*}
A similar estimate as above also holds for $[E_\mu(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}$ where $\lambda, k$ are replaced by $\mu$ and $\ell$. It is easy to see that combining the two estimates and keeping in mind (\ref{NormPhi}) and the explicit formulae (\ref{klmq}) and (\ref{Def.LambdaMu}), we obtain after some calculations the bound
\begin{equation*}
 [E(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})} \leq C[{\Bu}]^p_{\mathcal{X}^{s,p}(D)}+c(1+M)^{\frac{n}{2}+2p}
 M^{p}|\Bu|^p_{W^{s,p}(D)}.
\end{equation*}
It remains is to estimate the third term $\int_{D_{-}}\int_{D} \frac{\left|(E(\Bu)(\Bx) - E(\Bu)(\By)) \cdot(\Bx-\By)\right|^p}{|\Bx-\By|^{n + (s+1)p}}d\By d \Bx$ in \eqref{Int-split}. To that end, we denote the integral by $I_{mix}$ and for $\Bx\in D_{-}$ write 
\[E({\Bu})(\Bx) = k\Bu(\Phi_\lambda(\Bx)) + \ell\Bu(\Phi_\mu(\Bx)) + (m-k)(\boldsymbol{0}', u_n(\Phi_\lambda(\Bx))) + (q-\ell)(\boldsymbol{0}', u_n(\Phi_\lambda(\Bx))), \]
It then follows by algebraic calculations and using the relation between $k, \ell, m$, and $q$ that for $\Bx\in D_{-}$ and $\By\in D$: 
\begin{equation*}
\begin{split}
&(E(\Bu)(\Bx) - E(\Bu)(\By)) \cdot(\Bx-\By)\\
&=k \big( \Bu(\Phi_{\lambda}(\Bx)) - \Bu(\By) \big) \cdot \big( \Phi_{\lambda}(\Bx) - \By \big) + \ell \big( \Bu(\Phi_{\mu}(\Bx)) - \Bu(\By) \big) \cdot \big( \Phi_{\mu}(\Bx) - \By \big) \\
&\quad + (k-m) \big( y_n - f(\Bx') \big) \cdot \big( u_n(\Phi_{\lambda}(\Bx)) - u_n(\Phi_{\mu}(\Bx)) \big)\,.
\end{split}
\end{equation*}
We then have the  estimate that 
\[
\begin{split}
    I_{mix}&\leq C k^{p} \int_{D_{-}}\int_{D} \frac{\left|\big( \Bu(\Phi_{\lambda}(\Bx)) - \Bu(\By) \big) \cdot \big( \Phi_{\lambda}(\Bx) - \By \big)\right|^p}{|\Bx-\By|^{n + (s+1)p}}d\By d \Bx\\
    &+ C \ell^{p} \int_{D_{-}}\int_{D} \frac{\left|\big( \Bu(\Phi_{\mu}(\Bx)) - \Bu(\By) \big) \cdot \big( \Phi_{\mu}(\Bx) - \By \big)\right|^p}{|\Bx-\By|^{n + (s+1)p}}d\By d \Bx\\
    &+C|k-m|^{p} \int_{D_{-}}\int_{D} \frac{\left|\big( y_n - f(\Bx') \big) \cdot \big( u_n(\Phi_{\lambda}(\Bx)) - u_n(\Phi_{\mu}(\Bx)) \big)\right|^p}{|\Bx-\By|^{n + (s+1)p}}d\By d \Bx\\
    &= I^{1}_{mix}+ I_{mix}^2 +I_{mix}^3 
\end{split}
\]
The first two terms $I_{mix}^1$ and $I_{mix}^2$ can be estimated in similar ways. To demonstrate, making  the change of variables $\Bz= \Phi_\lambda (\Bx)$, we obtain that 
\[
I_{mix}^1 = \frac{Ck^p}{\lambda}\int_{D}\int_{D} \frac{\left|\big( \Bu(\Bz) - \Bu(\By) \big) \cdot \big( \Bz - \By \big)\right|^p}{|\left(\Phi_\lambda\right)^{-1}(\Bz)-\By|^{n + (s+1)p}}d\By d \Bz.
\]
We now use \eqref{flat-vs-curved} to estimate the latter by $C [\Bu]_{\mathcal{X}^{s,p}(D)}$. We finish the proof by estimating $I^{3}_{mix}$. Making the variable change $\Bz= \Phi_{1}(\Bx)$ to work solely in $D,$ we have that 
\begin{align*}
I^{3}_{mix} &\leq C \int_{D}\int_{D}\frac{|y_n- f(\Bz')|^p |u_{n}(\Phi_{\lambda}((\Phi_1)^{-1}(\Bz)))-u_{n}(\Phi_{\mu}((\Phi_1)^{-1}(\Bz)))|^p}{|(\Phi_1)^{-1}(\Bz)-\By|^{n + (s+1)p}} d\By d\Bz\\
&= C \int_{D} J(\Bz)| u_{n}(\Bz', f(\Bz')+\lambda(z_n-f(\Bz')))-u_{n}(\Bz', f(\Bz')+\mu(z_n-f(\Bz')))|^{p} d\Bz
\end{align*}
where for each $\Bz\in D$ 
\[
J(\Bz) = \int_{D} \frac{|y_n-f(\Bz')|^p}{(|\Bz'-\By'|^2 + |(y_n-f(\Bz')) + (z_n - f(\Bz'))|^2)^{\frac{d+(s+1)p}{2}}} d\By \leq \frac{C}{|z_n - f(\Bz')|^{sp}}
\]
as shown in Lemma~\ref{lem:A.1}. As a consequence, we have that 
\begin{equation}
\label{I_mix^3}
I^{3}_{mix}\leq C  \int_{D} \frac{| u_{n}(\Bz', f(\Bz')+\lambda(z_n-f(\Bz')))-u_{n}(\Bz', f(\Bz')+\mu(z_n-f(\Bz')))|^{p}}{|z_n - f(\Bz')|^{sp}} d\Bz.
\end{equation}
Finally, an application of Lemma~\ref{lem:2.4} together with (\ref{I_mix^3}) completes the proof of the proposition.
\end{proof}



We are now ready to prove Theorem~\ref{thm:Korn.Epigraph}.

\begin{proof}[Proof of Theorem~\ref{thm:Korn.Epigraph}]
Theorem~\ref{thm:Korn.Epigraph} follows by an application of the above proposition and Korn's second inequality for $\mathbb R^n,$ \cite{MScott2018Korn}. 
Indeed, by the fractional Korn's second inequality for vector fields defined on $\mathbb{R}^n$ proved in \cite[Theorem~1.1]{MScott2018Korn}
we have, on the one hand, for a constant $c_0=c_0(n,p,s),$ that 
\begin{align*}
|\Bu|_{W^{s,p}(D)}\leq  |E(\Bu)|_{W^{s,p}(\mathbb{R}^n)}
\leq  c_0\|E(\Bu)\|_{\mathcal{X}^{s,p}(\mathbb{R}^n)}.
\end{align*}
On the other hand, Proposition \ref{prop:extension} yields for constants $C,c>0$ that

\begin{equation*}
\|E(\Bu)\|_{\mathcal{X}^{s,p}(\mathbb{R}^n)}\leq C\|\Bu\|_{\mathcal{X}^{s,p}(D)} + c(1+\|\nabla f\|_{L^{\infty}})^{2+\frac{n}{2p}}\|\nabla f\|_{L^{\infty}} |\Bu|_{W^{s,p}(D)}).
\end{equation*}
Consequently, we obtain 
\begin{equation*}
|\Bu|_{W^{s,p}(D)}\leq  c_0C\|\Bu\|_{\mathcal{X}^{s,p}(D)} + cc_0(1+\|\nabla f\|_{L^{\infty}})^{2+\frac{n}{2p}}\|\nabla f\|_{L^{\infty}} |\Bu|_{W^{s,p}(D)},
\end{equation*}
which would yield (\ref{Korn.sec.epi}) provided 
$M_0$ fulfills $(1+M_0)^{2+\frac{n}{2p}}M_0<\frac{1}{cc_0}.$ This completes the proof of the theorem.  

\end{proof}


\subsection{Fractional Korn's second inequality in bounded domains}
In this section we provide a proof of inequality (\ref{thm:korn2nd}) in Theorem~\ref{Intro:mainthm}. As already mentioned, we will adopt a partition of unity argument employed in \cite{MS2022}. For the convenience of the reader we repeat the arguments here. Before we present the proof, we make  the following observations related to estimates involving the product $\psi \Bu$ of $\Bu\in \mathcal{X}^{s,p}(\Omega)$ and $\psi\in W^{1, \infty}(\Omega)$. First, such a product $\psi \Bu$ belongs to $ \mathcal{X}^{s,p}(\Omega)$ with the estimate 
\begin{align*}
    [\psi \Bu]_{\mathcal{X}^{s,p}(\Omega)} \leq c\,\|\psi\|_{W^{1, \infty}} \| \Bu\|_{\mathcal{X}^{s,p}(\Omega)}, 
\end{align*}
where $c$ depends only on $n, p, s$, and $ \text{diam}(\Omega)$. This is precisely \cite[Lemma 3.1]{MS2022}. 
Second, due to \cite[Lemma 3.2]{MS2022}, if $\Omega\subset \tilde{\Omega}$, and there exists $\beta>0$ such that for all $\By\in \tilde{\Omega}\setminus \Omega$
\[
\text{dist}(\By, \text{supp($\psi$)}) \geq \beta >0,
\]
then after extending the product by $\boldsymbol{0}$ on $\tilde{\Omega}\setminus \Omega$, it will belong to $\mathcal{X}^{s,p}(\tilde{\Omega})$ with the similar estimate 
\begin{align}\label{prod-zero-ext}
[\psi \Bu]_{\mathcal{X}^{s,p}(\tilde{\Omega})} 
\leq c_1(\beta)|\psi\|_{W^{1,\infty}}\|\Bu\|_{\mathcal{X}^{s,p}(\Omega)},
\end{align}
where $c_1(\beta)$ depends only on $n, p, s, \text{diam}(\Omega)$, and  $\beta.$ Both statements can be proven by a direct evaluation of the $\mathcal{X}$ seminorm of the product $\psi \Bu,$ see 
\cite{MS2022} for details. We are now ready to present the proof of the the theorem. 

\begin{proof}[Proof of inequality \eqref{thm:korn2nd} of Theorem~  \ref{Intro:mainthm}] 
Let $M_0$ be the constant found in Theorem \ref{thm:Korn.Epigraph}. Suppose that $\Omega$ is a bounded Lipschitz domain with local Lipschitz constant $\leq M_0$. By definition, we may choose an open set $\Omega_0\Subset \Omega$ and open balls $B_{r_j}(\By_j)$, for $j=1, \dots, N$ with centers $\By_j\in \partial \Omega$  such that 
\begin{enumerate}
\item $\Omega=\cup_{j=0}^{N}\Omega_j$ where $\Omega_j = \Omega\cap B_{r_j}(\By_j)$  for $j=1, \dots, N$.

\item For every $1\leq j\leq N$, define $T_j: B_{r_{j}}(\By_j) \to \mathbb{R}^{n}$ to be the operator consisting of the translation $\By_j \to 0$ and a rotation such that $T_j(\partial \Omega\cap B_{r_j}(\By_j) )$ coincides with part of the graph of a Lipschitz function $f_j:\mathbb{R}^{n-1} \to \mathbb{R}$ with $\|\nabla f_j\|_{L^{\infty}(\mathbb R^{n-1})} \leq M_0$. Note that the function $f_j$ is initially only defined on the open unit cube in $\mathbb R^{n-1},$ but we extend it into all of $\mathbb R^{n-1}$ by Kirzbraun's theorem \cite{Kirzbraun1934}, preserving the Lipschitz constant. This is necessary for the reduction of the situation to epigraphs in $\mathbb R^{n-1}.$
\end{enumerate}
Set $Q_j = T_j (B_{r_j}(\By_j)),$ and also define
\[\begin{split}
Q_j^+ := \{ \Bx \in Q_j \, : \, x_n > f_j(\Bx') \}\,, &\qquad
Q_j^- := \{ \Bx \in Q_j \, : \, x_n < f_j(\Bx') \}\,, \\
K_j^+ := \{ \Bx \in \mathbb{R}^n \, : \, x_n > f_j(\Bx') \}\,, &\qquad
K_j^- := \{ \Bx \in \mathbb{R}^n \, : \, x_n < f_j(\Bx') \}\,.
\end{split}\]
We may choose the map $T_j$ so that  $T_j(\Omega_j) = Q_{j}^{+}$. Note that $T_j$ is a bi-Lipschitz map with Lipschitz constant depending only on $n$ and $\Omega$.  Let $\{ \varphi_j \}_{j=0}^N \subset C^{\infty}_c(\mathbb{R}^n;\mathbb{R}^n)$ be a $C^{\infty}$ partition of unity subordinate to the collection $\{\Omega_0\}\cup \{ B_{r_j}(\By_j) \}_{j=1}^N$. Then for every $1\leq j\leq N,$ we have $\text{supp}(\varphi_j) \subset B_{r_j}(\By_j),$  $\text{dist}( \By, \text{supp}(\varphi_j)) \geq \beta_j > 0$ for every $\By \in \Omega \setminus \overline{\Omega_j} $. We also have $\text{supp} (\varphi_0)\subset \Omega_0$, $\text{dist}( \By, \text{supp}(\varphi_0)) \geq \beta_0 > 0$ for every 
$\By \in \Omega \setminus \overline{\Omega_0},$ and $\textstyle \sum_{j=0}^N \varphi_j \equiv 1$ on $\Omega$. 

Suppose now $\Bu\in C^{1}(\Omega;\mathbb{R}^n)$. Define $\Bu_j := \varphi_j \Bu$, for $j=0, 1, \dots, N.$ We consider $\Bu_0$ first. After extending it by $\boldsymbol{0}$ to $\mathbb{R}^n$, we have that $\Bu_0\in \mathcal{X}^{s,p}(\mathbb{R}^n)$ and that by the fractional Korn's inequality on $\mathbb{R}^n$,  \cite{MScott2018Korn} and \eqref{prod-zero-ext}
\begin{align}
\label{est-for-u0}
|\Bu_0|_{W^{s,p}(\Omega)} &\leq c[\Bu_0]_{\mathcal{X}^{s,p}(\mathbb{R}^n)}\\ \nonumber
&\leq cc^0_{1}(\beta)\|\varphi_0\|_{W^{1,\infty}} \|\Bu\|_{\mathcal{X}^{s,p}(\Omega)}.
\end{align}
For $j=1,\dots,N$ applying again \eqref{prod-zero-ext} using the semi norm $|\cdot|_{W^{s,p}(\Omega)}$ instead of $[\cdot]_{\mathcal{X}^{s,p}(\Omega)}$ we have  
\begin{equation}\label{est-for-ujs}
|\Bu_j|_{W^{s,p}(\Omega)} \leq  c_1^j(\beta_j)\|\varphi_j\|_{W^{1,\infty}}
\|\Bu_j\|_{W^{s,p}(\Omega_j)}
\end{equation}
where $c_1^j(\beta)$ depends only on $s,p,n,$ $\text{diam} (\Omega)$ and $\beta_j.$ Now since $T_j$ consists of a rotation and a translation, $\nabla T_j$ is a constant rotation, with $T_j(\Bx) - T_j(\By) = (\nabla T_j) (\Bx-\By)$. Therefore, writing $R_j := \nabla T_j$, define $\Bv_j(\Bx) := R_j \Bu_j(T_j^{-1}(\Bx))$. Then we have $\Bv_j \in W^{s,p}(Q_j^+)$ and that for each $\By\in K^{+}_j\setminus Q_j^{+},$ $\text{dist}(\By, Supp(\Bv_j))\geq \tilde{\beta}_j>0$ for some positive constant $\tilde \beta_j$. Moreover, 
\begin{equation*}
\|\Bv_j\|_{L^{p}(Q_j^{+})} = \|\Bu_j\|_{L^{p}(\Omega_j)},\,\|\Bv_j\|_{W^{s,p}(Q_j^{+})} = \|\Bu_j\|_{W^{s,p}(\Omega_j)},\,\text{and } [\Bv_j]_{\mathcal{X}^{s,p}(Q_j^+)}=[\Bu_j]_{\mathcal{X}^{s,p}(\Omega_j)}.
\end{equation*}
We will demonstrate the last equality as the others can be established similarly. By changing of coordinates,
\begin{equation*}
\begin{split}
[\Bv_j]_{\mathcal{X}^{s,p}(Q_j^+)}^p &= \int_{\Omega_j}\int_{\Omega_j}{ \frac{\left| \big( R_j \Bu_j(\Bx) - R_j \Bu_j(\By) \big) \cdot \big( T_j(\Bx) - T_j(\By) \big)  \right|^p}{|T_j(\Bx) - T_j(\By)|^{d+sp+p}} }d{\By}d{\Bx} \\
	&=  \int_{\Omega_j}\int_{\Omega_j}{ \frac{\left| \big( R_j \Bu_j(\Bx) - R_j \Bu_j(\By) \big) \cdot \big( R_j \Bx - R_j \By \big)  \right|^p}{|R_j \Bx - R_j \By|^{d+sp+p}} }d{\By}d{\Bx} \\
	&=\int_{\Omega_j}\int_{\Omega_j}{ \frac{\left|R_j^{\intercal} R_j \big( \Bu_j(\Bx)-\Bu_j(\By) \big) \cdot \big( \Bx-\By \big)  \right|^p}{|\Bx - \By|^{d+sp+p}} }d{\By}d{\Bx}\\
	&= [\Bu_j]_{\mathcal{X}^{s,p}(\Omega_j)}^p.
\end{split}
\end{equation*}
Extending $\Bv_j$ by $\boldsymbol{0}$ on $K^{+}_j\setminus Q_j^{+}$, we have that $\Bv_j\in C^{1}(K_j^{+};\mathbb{R}^n)$. Applying fractional Korn's inequality for epigraphs, Theorem \ref{thm:Korn.Epigraph}, we have 
\begin{align} 
\label{Q-to-K}
|\Bv_j|_{W^{s,p}(Q_j^{+})} &\leq |\Bv_j|_{W^{s,p}(K_j^{+})}\\ \nonumber
&\leq C\|\Bv_j\|_{\mathcal{X}^{s,p}(K^{+}_j)},
\end{align}
where $C$ only depends on $s, p, n$ and $M_0$. We may also apply \eqref{prod-zero-ext} to estimate further as 
\begin{equation}
\label{est-rotated-u} 
\|\Bv_j\|_{\mathcal{X}^{s,p}(K^{+}_j)} \leq c_2^j(\tilde{\beta_j})
\|\Bv_j\|_{\mathcal{X}^{s,p}(Q^{+}_j)} 
\end{equation}
We combine now \eqref{est-for-ujs}, \eqref{Q-to-K}, and \eqref{est-rotated-u} to obtain 
\begin{equation}\label{est-for-uj-by-uj}
|\Bu_j|_{W^{s,p}(\Omega)} \leq 
C \|\Bu\|_{\mathcal{X}^{s,p}(\Omega)}, \qquad j=1, 2, \dots, N,
\end{equation}
where $C$ is a positive constant that depends on $s,p,n, \text{diam}(\Omega)$, $M_0$, and the partition of unity. Therefore by \eqref{est-for-u0} and \eqref{est-for-uj-by-uj}, we have 
\[
[\Bu]_{W^{s,p}(\Omega)} = \Big[\sum_{j=0}^{N} \Bu_j\Big]_{W^{s,p}(\Omega)} \leq \sum_{j=0}^{n}[\Bu_j]_{W^{s,p}(\Omega)} \leq C\|\Bu\|_{\mathcal{X}^{s,p}(\Omega)}.
\]
The estimate for vector fields  $\Bu$ in $W^{s,p}(\Omega;\mathbb{R}^n)$ follows by density. This completes the proof. 
\end{proof}















\subsection{Fractional Korn's first inequality in bounded domains}
In this section we provide a proof of inequality (\ref{thm:korn1st}) in Theorem~\ref{Intro:mainthm}].   
\begin{proof}[Proof of inequality \eqref{thm:korn1st} of Theorem~\ref{Intro:mainthm}]
Assume in contradiction (\ref{thm:korn1st}) fails to hold. It then follows that there exist a sequence $\Bu_k\in W^{s,p}(\Omega,\mathbb R^n)$ and a sequence of skew-symmetric matrices $\BA_k\in \mathrm{skew}(\mathbb R^n)$ such that 
\begin{equation}
\label{3.38}
|\Bu_k-\BA_k\cdot \Bx|_{W^{s,p}(\Omega)}=1\quad \text{and} \quad [\Bu_k]_{\mathcal{X}^{s,p}(\Omega)}\leq 1/k,\quad k=1,2,\dots
\end{equation}
We may also assume that for each $k$ the average of ${\Bu}_k-\BA_k\cdot \Bx$ over $\Omega$ is $\boldsymbol{0}$ by shifting it by a vector 
$\Bb_k\in \mathbb R^n$ if necessary. Upon passing to the fields $\Bv_k=\Bu_k-\BA_k\cdot \Bx-\Bb_k,$ we can further assume without loss of generality that $\BA_k=0$ and $\Bb_k=0$.  Thus  the minimality conditions 
\begin{equation}
\label{3.39}
|\Bv_k|_{W^{s,p}(\Omega)}\leq |\Bv_k-\BA\cdot \Bx|_{W^{s,p}(\Omega)}\quad \text{for any}\quad \BA\in \mathrm{skew}(\mathbb R^n), \ k=1,2,\dots
\end{equation}
hold and by Poincar\'e's inequality, the sequence $\Bv_k$ is bounded in $W^{s,p}(\Omega;\mathbb{R}^n)$. From the compactness theorem,  
\cite[Theorem~7.1]{NPV},  
the sequence $\{\Bv_k\}$ is pre-compact in $L^p(\Omega),$ thus we can assume without loss of generality that 
\begin{equation}
\label{3.40}
\Bv_k\to \Bv \quad \text{in}\quad L^p(\Omega),
\end{equation}
for some field $\Bv\in L^p(\Omega).$ We then have by Korn's second inequality (\ref{thm:korn2nd}) and \eqref{3.39} that
\begin{align*}
 \|\Bv_k-\Bv_m\|_{W^{s,p}(\Omega)}&\leq C([\Bv_k-\Bv_m]_{\mathcal{X}^{s,p}(\Omega)}+\|\Bv_k-\Bv_m\|_{L^p(\Omega)})\\ \nonumber
 & \leq  C([\Bv_k]_{\mathcal{X}^{s,p}(\Omega)}+[\Bv_m]_{\mathcal{X}^{s,p}(\Omega)}+\|\Bv_k-\Bv_m\|_{L^p(\Omega)})\\ \nonumber
 &\leq C(1/k+1/m+\|\Bv_k-\Bv_m\|_{L^p(\Omega)}),
 \end{align*}
thus the conditions \eqref{3.40} implies that the sequence $\{\Bv_k\}$ is Cauchy and thus is convergent in $W^{s,p}(\Omega).$ This gives 
\begin{equation}
\label{3.42}
\Bv_k\to \Bv \quad \text{in}\quad W^{s,p}(\Omega).
\end{equation}
From \eqref{3.38}) and \eqref{3.42} we have
\begin{align*}
[\Bv]_{\mathcal{X}^{s,p}(\Omega)}&\leq [\Bv_k]_{\mathcal{X}^{s,p}(\Omega)}+[\Bv-\Bv_k]_{\mathcal{X}^{s,p}(\Omega)}\\
&\leq 1/k+[\Bv-\Bv_k]_{W^{s,p}(\Omega)}\to0
\end{align*}
as $k\to\infty,$ thus  $[\Bv]_{\mathcal{X}^{s,p}(\Omega)}=0,$ which gives 
\begin{equation}
\label{3.43}
\Bv(\Bx)=\BA\cdot \Bx+\Bb,\quad\text{for a.e.}\quad\Bx\in\Omega,
\end{equation}
for some constant skew-symmetric matrix $\BA\in \mathbb R^{n\times n}$ and some vector $\Bb\in\mathbb R^n$ \cite[Proposition 1.2]{Temam-Miran}. 
Note that then we have by \eqref{3.38}, \eqref{3.39},  \eqref{3.42}, and \eqref{3.43}:
\begin{align*}
1&=[\Bv_k]_{W^{s,p}(\Omega)}
 \leq [\Bv_k-\BA\cdot \Bx]_{W^{s,p}(\Omega)}
 = [\Bv_k-\Bv]_{W^{s,p}(\Omega)}\to 0 
\end{align*}
as $k\to\infty,$ which is a contradiction. 

\end{proof}



\section{Fractional Korn's inequality for planar convex domains}\label{sec:convex}
As we discussed in the introduction, we conjecture that the smallness of the Lipschitz constant of the boundary of the domain is not necessary for the validity of the Fractional Korn's inequalities. In this section, we will support this hypothesis by demonstrating the validity of the inequality 
in the case of planar convex domains. We first obtain the result for planar convex polygons. The argument of the proof mimics the strategy we used for smooth domains. We begin by proving the inequality for angular domains.  We then cover the boundary of the convex polygonal domain by balls centered on the boundary. The resulting intersecting sets are either wedges (bounded angular domains) or half balls over which  we will have the appropriate estimates. Finally, we use a partition of unity argument to obtain the estimates over the convex polygon. 
In this section, vectors defined on the planar domains are represented as $\Bu = (u_1, u_2)$. 


\setcounter{equation}{0}
\subsection{The case of angular domains}


Consider an angular planar domain $D$ with an angle of span in the interval $(0,\pi)$. Upon an affine variable change we may  assume without loss of generality that 
$D$ is given by 
\begin{equation}
\label{4.1}
D=\{\Bx\in\mathbb R^2 \ : \ 0<x_1,\ \ \alpha x_1<x_2\}, \quad \alpha\in\mathbb R.
\end{equation}
Note that $D$ is exactly half of the epigraph supported by the function $f(x_1) = \alpha x_1$ defined over $(0,\infty)$.   
In that case, we set 
\begin{equation*}
D_{-}=\{\Bx\in\mathbb R^n \ : \ 0<x_1,\ \ x_2<\alpha x_1\}.
\end{equation*}
Notice that $\overline{D}\cup D_{-} = \mathbb{R}^{2}_{x_{1} \geq 0 } = \{(x_1, x_2)\in \mathbb{R}^{2}: x_1\geq 0\}.$ We begin by demonstrating the existence of an extension operator to prove that vector fields defined over $D$ can be extended to $\mathbb{R}^{2}_{x_1 > 0}$ accompanied with an appropriate control of their nonlocal norm.  We use the extension operator defined in \cite{nitsche1981korn} for planar angular domains where it is shown to map $W^{1,2}(D;\mathbb{R}^2)$ to $W^{1,2}(\mathbb{R}^{2}, \mathbb{R}^{2})$. 
\begin{proposition}
Let $s\in (0, 1)$, $1<p<\infty$, and let $D$ be given by (\ref{4.1}). Then, there exists a bounded extension operator $E: W^{s,p}(D,\mathbb{R}^2)\to W^{s, p}( \mathbb{R}^2_{x_1 > 0},\mathbb{R}^2)$ such that  $E(\Bu)(\Bx)= \Bu(\Bx)$ for $\Bx\in D$. Moreover, there exists a constant $C>0$ depending only on $p, s,$ and $\alpha,$ such that   
\begin{equation}
\label{4.3}
[E(\Bu)]_{\mathcal{X}^{s,p}(\mathbb{R}^{2}_{x_1 > 0})}\leq C ([\Bu]_{\mathcal{X}^{s,p}(D)}+\|\Bu\|_{L^p(D)}), 
  \end{equation}
  for all $\Bu\in W^{p,s}(D,\mathbb R^2).$ 
\end{proposition}
  

\begin{proof}
Following \cite{nitsche1981korn}, we set   $E(\Bu)(\Bx) = \tilde{E}_{\lambda} (\Bu)(\Bx) + \tilde{E}_{\mu}(\Bu)(\Bx)$, for $\Bx\in D_{-}$, where 
\begin{align}\label{3.4}
\tilde{E}_{\lambda} (\Bu)(\Bx) &= (k u_1^\lambda(\Bx) + \alpha k (1+\lambda) u_2^\lambda(\Bx), m u_2^\lambda(\Bx))\nonumber\\ 
\tilde{E}_{\mu} (\Bu)(\Bx) &= (\ell u_1^\mu(\Bx) - \alpha k (1+\lambda) u_2^\mu(\Bx), q u_2^\mu(\Bx)),
\end{align} and $E(\Bu)(\Bx) = \Bu(\Bx)$, if $\Bx\in D$. 
The constants $\lambda$, $\mu$, $k$, $\ell$, $m$, $q$, satisfy the constraints \eqref{eq:ReflectionConditions}-(\ref{Def.LambdaMu}), and the functions $u_i^{\lambda}$ and $u_i^{\mu}$ are defined as before in \eqref{ulambdamu}. Note that this is the extension for epigraphs with the additional summand $\alpha k(1+\lambda)(u_2^\lambda(\Bx)-u_2^\mu(\Bx))$ in the 1st component of 
$E(\Bu)(\Bx)$ for $\Bx\in D_{-}$.  The proof of the estimate in \eqref{4.3} follows the calculations done for the case of the epigraphs. Below we sketch the proof only including those calculations that are new. As before, we begin by decomposing the integral as  
\begin{align}
\label{angle-decomp}
[E(\Bu)]^p_{\mathcal{X}^{s,p}(\mathbb{R}^{2}_{x_1 > 0})}= [E(\Bu)]^p_{\mathcal{X}^{s,p}(D)} &+  [E(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}\nonumber\\
&+ 2\int_{D_{-}}\int_{D} \frac{\left|(E(\Bu)(\Bx) - E(\Bu)(\By)) \cdot(\Bx-\By)\right|^p}{|\Bx-\By|^{n + (s+1)p}}d\By d \Bx.
\end{align}
We nned to estimate the last two terms. Clearly, 
\[
[E(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})} \leq 2^{p-1}\left([\tilde{E}_{\lambda}(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}  +[\tilde{E}_{\mu}(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}\right).
\]
A simple calculation reveals that the additional summands $\alpha k(1+\lambda)u_2^\lambda(\Bx)$ and $\alpha k(1+\lambda)u_2^\mu(\Bx)$ make it possible to simplify further. Indeed, after change of variables $\Bz= \Phi_{\lambda}(\Bx)=(x_1, \alpha x_1 + \lambda(\alpha x_1 - x_2))$  and $\Bw= (y_1, \alpha y_1 + \lambda(\alpha y_1 - y_2))$, we have that 
\[
[\tilde{E}_{\lambda}(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})} = C(\alpha, \lambda, k, p, s) [\Bu]^{p}_{\mathcal{X}^{s,p}(D)}
\]
Similar estimates also holds for $[\tilde{E}_{\mu}(\Bu)]^p_{\mathcal{X}^{s,p}(D_{-})}$, after noting that the relations between the parameters in \eqref{eq:ReflectionConditions}, implies that $\alpha k(1 + \lambda) = -\alpha \ell(1 + \mu)$. The point here is that 
the additional summand $\alpha \,k(1+\lambda)(u_2^\lambda(\Bx)-u_2^\mu(\Bx))$ in the first component of the extension facilitates a cancellation of an extra term that would appear if we otherwise use the extension operator \eqref{ext-epigraph} treating the domain as a Lipschitz domain. What is left now is estimating the mixed integral $\int_{D_{-}}\int_{D}\dots d\By d\Bx$ appearing in \eqref{angle-decomp}. This can be estimated as in the proof of Proposition \ref{prop:extension}. The only difference is that there will be an additional term due to the new term 
$\alpha k(1+\lambda)(u_2^\lambda(\Bx)-u_2^\mu(\Bx))$. This amounts to estimating the expression 
\begin{equation}
\label{4.6} 
\mathrm{II_{new}}=\int_{D_{-}}\int_D\frac{|(u_2^\lambda(\Bx)-u_2^\mu(\Bx))\cdot(\Bx'-\By')|^p}{|\Bx-\By|^{2+p+ps}}d\By d\Bx
\end{equation} 
in terms of the norm of ${\Bu}$ in $\mathcal{X}^{s,p}(D)$. 
To prove \eqref{4.6}, by the change of variable  $\Bz=\Phi_1(\Bx),$ we have 
\begin{align*}
\mathrm{II_{new}}&=C\int_{D}\int_D\frac{|(u_2^\lambda(\Phi_1^{-1}(\Bz))-u_2^\mu(\Phi_1^{-1}(\Bz)))\cdot(\Bz'-\By')|^p}{|\Phi_1^{-1}(\Bz)-\By|^{2+p+ps}}d\By d\Bz\\ \nonumber
&\leq C\int_D I(\Bz)|u_2^\lambda(\Phi_1^{-1}(\Bz))-u_2^\mu(\Phi_1^{-1}(\Bz))|^p d\Bz,
\end{align*} 
where for any fixed $\Bz\in D$ we have set
\[
I(\Bz)=\int_D \frac{|\Bz'-\By'|^p}{|\Phi_1^{-1}(\Bz)-\By|^{2+p+ps}} d\By.
\] 
Using Lemma \ref{lem:A.2} from the appendix we have that for each $\Bz\in D,$  
\begin{equation*}
I(\Bz) \leq \frac{C}{|z_2-f(\Bz')|^{ps}}, \end{equation*} 
with a constant $C$ that depends only on $p,s,$ and the Lipschitz constant of $f$, which is $\alpha$ in this case. 
As a consequence, we have 
\begin{align}
\label{4.9} 
\mathrm{II_{new}} &\leq C \int_D \frac{|u_2^\lambda(\Phi_1^{-1}(\Bz))-u_2^\mu(\Phi_1^{-1}(\Bz))|^p}{|z_2-f(\Bz')|^{ps}}d\Bz\\ \nonumber
 &= C \int_D \frac{|u_2(\Bz',f(\Bz')+\lambda(z_2-f(\Bz')))-u_2(\Bz',f(\Bz')+\mu(z_2-f(\Bz')))|^p}{|z_2-f(\Bz')|^{ps}}d\Bz. 
 \end{align} 
We now apply Lemma \ref{lem:2.4} to estimate the last integral in (\ref{4.9}) by the seminorm  $[\Bu]_{\mathcal{X}^{s,p}(D)}.$ 


\end{proof}


\begin{remark}
The above proposition in particular implies the fractional Korn's second inequality for planar angular domains. 
Indeed, for any $\Bu\in W^{s,p}(D,\mathbb R^2)$ we have $E(\Bu)\in W^{s,p}(\mathbb{R}^{2}_{x_1 > 0},\mathbb R^2)$, thus by Proposition~\ref{ext-epigraph} (the epigraph is supported by a hyperplane), we can further extend the extension $E(\Bu)\colon \mathbb{R}^{2}_{x_1 > 0}\to \mathbb R^2$ into $\mathbb R^2$ with the new extension still denoted by $E(\Bu)$ and with the bound 
\begin{equation}
\label{4.10}
[E(\Bu)]_{\mathcal{X}^{s,p}(\mathbb{R}^{2})}\leq 
C\|E(\Bu)\|_{\mathcal{X}^{s,p}(\mathbb{R}^{2}_{x_1 > 0})}.
\end{equation}
Next, the fractional Korn's inequality for the entire space \cite{MScott2018Korn}  gives
\begin{equation}
\label{4.11}
|\Bu|_{W^{s,p}(\mathbb{R}^2)} \leq C\|E(\Bu)\|_{W^{s,p}(\mathbb{R}^{2})}.
\end{equation}
Consequently, combining (\ref{4.10}), (\ref{4.11}) and (\ref{4.3}) we arrive at 
\begin{align*}
|\Bu|_{W^{s,p}(D)}\leq |\Bu|_{W^{s,p}(\mathbb{R}^2)}
\leq C\|E(\Bu)\|_{W^{s,p}(\mathbb{R}^{2})}
\leq C \|E(\Bu)\|_{\mathcal{X}^{s,p}(\mathbb{R}^{2}_{x_1 > 0})} 
\leq C ([\Bu]_{\mathcal{X}^{s,p}(D)}+\|\Bu\|_{L^p(D)}).
\end{align*}
\end{remark}


\subsection{The case of general planar convex domains}

In this subsection, we only show that extension of vector fields defined in planar convex domains $D$ to some neighborhood of $D,$ with controlled $\|\cdot\|_{X^{s,p}}$ norm is possible. We prove the following extension result.  
\begin{theorem}
Let $\Omega\subset\mathbb R^2$ be a bounded convex domain, and let $s\in(0,1)$ and $p\in (1,\infty)$ with $ps\neq 1.$ For any $\epsilon>0,$ there exists a neighborhood $U(\Omega,\epsilon)$ of $\Omega$ and a constant $\tilde C=\tilde C(\Omega,n,p,s,\epsilon)>0,$ such that for any $\Bu\in W^{s,p}(\Omega,\mathbb R^n),$ there exists an extension $E(\Bu)$ in $U,$ such that 
\begin{equation}
\label{4.12}
[E(\Bu)]_{\mathcal{X}^{s,p}(U)}\leq \tilde C\|\Bu\|_{\mathcal{X}^{s,p}(\Omega)}+\epsilon [\Bu]_{W^{s,p}(\Omega)}.
\end{equation}
\end{theorem}


\begin{proof}

Let us parametrize the boundary of $\Omega$ (which is a Lipschitz curve) by the arc length: $\partial\Omega=\BGg(t)\colon[0,1]\to\mathbb R^2,$ where we assume without loss of generality that the length of $\BGg$ equals 1. Let $\Bn(t)$ denote the external unit normal to $\BGg(t),$ which exists for a.e. $t\in [0,1]$. We can assume, without loss of generality that $\Bn(0)$ and $\Bn(1)$ exist. Let $\theta(t)$ denote the angle between $\Bn(t)$ and the positive $x_1$ axis of a fixed coordinate system. It is clear that $\theta(t)\colon[0,1]\to \mathbb R$ is a non-decreasing function, and that when the parameter $t$ runs over the interval $[0,1],$ the unit normal $\Bn(t)$ rotates in the positive direction, making a whole round. Thus we have 
\begin{equation}
\label{4.13} 
\bigvee_{0}^1(\theta)=\theta(1)-\theta(0)\leq 2\pi.
\end{equation}
Fix $\epsilon>0.$ As $\theta$ is non-decreasing, the set of discontinuities of $\theta$ is countable. Let $T=\{t_i\}_{i=1}^\infty\subset [0,1]$ be the set of discontinuities of 
$\theta.$ As $\BGg$ is convex, the left and right limits $\theta(t_i-)$ and $\theta(t_i+)$ exists for every $i\in\mathbb N,$ and (\ref{4.13}) implies that 
\begin{equation*}
\sum_{i=1}^\infty(\theta(t_i+)-\theta(t_i-))\leq 2\pi. 
\end{equation*}
Therefore we can choose an index $N\in\mathbb N$ such that 
\begin{equation}
\label{4.15} 
\sum_{i=N+1}^\infty(\theta(t_i+)-\theta(t_i-))<\epsilon,
\end{equation}
i.e., the total rotational jumps of the unit normal $\Bn$ at the points $t_{N+1},t_{N+2},\dots$ is at most $\epsilon.$ This allows one to reduce the situation to the case of finitely many points of discontinuity as will be seen below. We will demonstrate below how one should extend $\Bu$ outside $D$ near the points $\Bp_k=\BGg(t_k),$ $k=1,2,\dots,N$ and apart from them. 
To that end we choose $\eta>0$ small enough and cover the interval $[0,1]$ by open intervals $\{I_l\}_{l=1}^L$ of length $\leq \eta$ with the following properties. 
\begin{itemize}
\item[(C1)]  Each interval $I_l$ can contain at most one of the points $t_k,$ $k=1,2,\dots,N$. Moreover, the collection of intervals $I_l$ that contain the points $t_k,$. $k=1,2,\dots,N,$ is a disjoint collection. Thus we will denote those intervals by $I_k^\ast,$ $k=1,2,\dots,N$. One further has 
\begin{equation*} 
\bigvee_{I_k^\ast}(\theta)\leq \theta(t_k+)-\theta(t_k-)+\epsilon.
\end{equation*}
\item[(C2)]  If the interval $I_l$ does not contain any of the points $t_k,$ $k=1,2,\dots,N$, then 
\begin{equation*}
\bigvee_{I_l}(\theta)\leq \epsilon.
\end{equation*}
\item[(C3)] For each $k=1,\dots,N$, up to a rotation, $\BGg(I_k)$ is the graph of a Lipschitz function.  
\end{itemize}
This can be achieved by covering every point of the compact interval $[0,1]$ by small open intervals (then choosing a finite sub-cover), utilizing (\ref{4.15}) and the definition of $\theta(t+)$ and $\theta(t-)$ for every point $t\in [0,1].$ Next we will extend the field $\Bu$ beyond $\Omega$ at the boundary in each of the intervals $I_l,$ $l=1,2,\dots,L.$  The cases of $I_k^\ast$ intervals and the ones that do no contain any of the points $t_k$ require different types of extensions and treatment. Therefore, we consider the two cases separately.\\
\textbf{The case of jump points $t_k.$}\\
Let the index $k\in \{1,\dots,N\}$ be fixed. The curve $\BGg$ has supporting lines at $t_k$ to the left and right that form an angle $\theta_k=\pi-(\theta(t_k+)-\theta(t_k-)),$ where $0<\theta_k<\pi.$ We choose the local coordinate system so that the point $\BGg(t_k)$ is the origin, the supporting line on the left of $t_k$ is the $x_2$ axis, and the supporting line on the right of $t_k$ is the graph of the function $x_2=\alpha x_1$ for $x_1>0$ as shown in Figure 1 below. Suppose that the curve supported at $t_k$ by $x_2=\alpha x_1$ is represented by the function $x_2= f(x_1)$. 
\begin{figure}[htbp] 
\centering
\begin{tikzpicture}[scale=1]
\draw[->] (-1,0)--(5,0) node[xshift=.3cm] {$x_1$};
\draw [->](0,-4)--(0,3) node[yshift=.3cm] {$x_2$};
       \draw (0,0) circle(3pt) [fill] ;
       \node at (-0.5, 0.3) {$\BGg(t_{k})$};
\draw [fill=white!80!gray](0,0) to[out=90,in=180] (3,3) to (3,0) to (0,0);
\node at (2,3.3){$\BGg_1$};
\draw [fill=white!80!gray](0,0) to[bend right] (3,-1.2) to (3, 0);
\node at (3.25,-1.2){$\BGg_2$};
\node at (2,-0.8){$x_2= f(x_1)$};
\draw [blue,line width=1pt](0,0) to (0,3);
\draw [red, dashed,line width=1pt,fill=white!80!purple](0,0) to[out=270] (3,-4.5) to (3,-1.2) to[bend left] (0.2,-0.2);
\draw [red, dashed,line width=2pt](0,0) to[out=270] (1.5,-4);
\draw [green,line width=1pt](0,0) to (3,-4);
\node at (3,-3){$x_2=\alpha x_1$};
\node at (1,-4) {$\BGg^{*}_{1,\mu}$}; 
\node at (0.9,-2) {$\BGg^{*}_{1,\lambda}$}; 
\node at (2,1) {$\Omega^k$};
\node at (2.2,-2) {$\Omega^k_{-}$};
\end{tikzpicture}
\caption{The curves $\BGg_1$ and $\BGg_2$ are the parts of $\BGg$ on the top and the bottom of the domain. $\BGg_2$ represents the graph of $x_2=f(x_1)$.  The green line is the graph of $x_2=\alpha x_1$ for $x_1>0,$ that is tanget to $\BGg_2$ at $(0,0)$. Here the picture corresponds to the case $t\in I_k^\ast.$ }
\end{figure}   
It then follows that
 $f(x_1)=\alpha x_1+c(x_1)x_1,$ where $c(x_1)\geq 0$ for $x_1\geq 0,$ and $\lim_{x_1\to 0+}c(x_1)=0,$ while on the curve 
$\BGg_1$ we have $\lim_{x_1\to 0+}\frac{x_1}{x_2}=0.$ In this case one needs to extend $\Bu$ by the formula (\ref{3.4}), treating the case similar to the angular domain case, i.e., 
\begin{equation}
\label{4.18}
\begin{cases}
[E(\Bu)(\Bx)]_1=ku_1^\lambda(\Bx)+lu_1^\mu(\Bx)+\alpha k(1+\lambda)(u_2^\lambda(\Bx)-u_2^\mu(\Bx)), & \Bx\in \Omega^k_{-},\\
[E(\Bu)(\Bx)]_2=mu_2^\lambda(\Bx)+nu_2^\mu(\Bx), & \Bx\in \Omega^k_{-},
\end{cases}
\end{equation}
where in this particular case the function $f(x_1)$ is not $\alpha x_1$ (in general) and $\Phi_\lambda(\Bx)=\Bx^\lambda=(x_1, f(x_1)+\lambda(f(x_1)-x_2)),$ where 
$\Bx=(x_1,x_2).$ We denote the image of  the curve $\BGg_1$ under the mappings $\Phi_\lambda^{-1}(\Bx)$ and $\Phi_\mu^{-1}(\Bx)$ by $\BGg_{1,\lambda}^\ast$ and $\BGg_{1,\mu}^\ast$ respectively. These curves will be between the negative part of the $x_2$ axis and the curve $\BGg_2$ and both will touch the $x_2$ axis. Also, because of the condition $\lambda<\mu,$ the curve $\BGg_{1,\mu}^\ast$ will go below $\BGg_{1,\lambda}^\ast$ as shown in Figure 1. This will allow one to extend $\Bu$ in the region between the curves $\BGg_2$ and $\BGg_{1,\mu}^\ast,$ thus the extended field will be defined in the region between the curves $\BGg_1$ and $\BGg_{1,\mu}^\ast,$ the union of which has no exterior normal rotational jump at $t_k,$ and thus can be regarded as the new boundary and the next extension can be done about it as will be shown in the case of \textit{small rotational jump of the normal,} done consequently. It remains to prove that \eqref{4.18} works. Indeed, it is easy to see that in the $\Omega_{-}^k\Omega_{-}^k$ integral, every summand will be estimated as before in the proof of Proposition~3.2, except there will occur a term (and a similar $\mu$ term)
\begin{equation}
\label{4.19} 
C\int_{\Omega^k}\int_{\Omega^k}\frac{|(u_2^\lambda(\Bz)-u_2^\lambda(\Bw))\cdot(f(z_1)-f(w_1)-\alpha(z_1-w_1))|^p}{|\Phi_\lambda^{-1}(\Bz)-\Phi_\lambda^{-1}(\Bw)|^{n+p+ps}}d\Bz d\Bw,
\end{equation}   
which can be estimated as follows. As $f$ is Lipschitz, we can write 
$$f(z_1)-f(w_1)-\alpha(z_1-w_1))=(z_1-w_1)\int_{0}^1[f'(w_1+t(z_1-w_1))-\alpha]dt.$$
The rotation of the unit normal to $\BGg_2$ is of order $\epsilon,$ thus we have 
$$|f'(w_1+t(z_1-w_1))-\alpha|\leq C\epsilon,\quad t\in [0,1],$$
hence we can estimate \eqref{4.19} as 
\begin{equation*}
\int_{\Omega^k}\int_{\Omega^k}\frac{|(u_2^\lambda(\Bz)-u_2^\lambda(\Bw))\cdot(f(z_1)-f(w_1)-\alpha(z_1-w_1))|^p}{|\Phi_\lambda^{-1}(\Bz)-\Phi_\lambda^{-1}(\Bw)|^{n+p+ps}}d\Bz d\Bw\leq C\epsilon^p[\Bu]_{W^{s,p}(\Omega^k)}.
\end{equation*}   
The next step is to extend the obtained field in $\Omega^k_{-}\cup\Omega^k$ about the curve $\BGg_1\cup\BGg_{1,\lambda}^\ast$ to the left, which apparently has small 
(order $\epsilon$) rotation of the unit normal. This kind of situation is treated below.\\ 
\textbf{The case when the normal has a small total rotational jump.}\\
This is the case of any of the intervals $I_l$ that do not contain the points $t_k,$ $k=1,2,\dots,N,$ as well as the curves $\BGg_1\cup\BGg_{1,\mu}^\ast$ for $k=1,2,\dots,N.$ This case corresponds to the locally small Lipschitz constant case of the boundary function $f,$ as in the previous sections, where clearly the standard Nitsche's extension \eqref{ext-epigraph} (or (\ref{4.18})) without the additional term 
$\alpha k(1+\lambda)(u_2^\lambda(\Bx)-u_2^\mu(\Bx))$ will work. Hence we get a partition of the boundary of $\Omega$ into small curves, such that the vector field $\Bu$ has been extended about each portion of the boundary to the complement $\Omega^c$ with good control on the $\mathcal{X}$ seminorm. Apparently, we can assume without loss of generality, that initially, we have constructed these intervals $I_l$ by first covering the boundary of $\Omega$ by small balls and the remaining part in $\Omega$ by $\Omega_0$ exactly in the same way as in Section~3.2. Finally, the analysis of Section~3.2 will go through proving (\ref{4.12}). 





\end{proof}





\section*{Acknowledgements}
Davit Harutyunyan's research is supported by NSF 
DMS-2206239. Tadele Mengesha's research is supported by NSF DMS-1910180 and DMS-2206252. James Scott acknowledges support from NSF DMS-1937254 and NSF DMS-2012562.


