---
abstract: 'We prove that for bounded Lipschitz domains in $\rN$ Korn’s first inequality holds for vector fields satisfying homogeneous mixed tangential and normal boundary conditions.' 
author:
- 
- 
bibliography:
- 'paule.bib'
- 'Litbank.bib'
---

Introduction
============

Recently, motivated by [@desvillettesvillanikornnormal; @desvillettesvillanitrendglequiboltzmann] and inspired by the ideas and techniques presented in [@paulymaxconst0; @paulymaxconst1; @paulymaxconst2] for estimating the Maxwell constants, we have shown in [@bauerpaulytangnormkorn] that Korn’s first inequality, i.e., $$\begin{aligned}
\mylabel{firstkornsqrttwo}
\normltom{\na v}
&\leq\ck\,\normltom{\sym\na v},\end{aligned}$$ holds with $\ck=\sqrt2$ for all vector fields $v\in\hoom$ satisfying (possibly mixed) homogeneous normal or homogenous tangential boundary conditions and for all piecewise $\coo$-domains $\om\subset\rN$, $N\geq2$, with concave boundary parts.  In this contribution, we extend to any bounded (strong) Lipschitz domain $\om\subset\rN$, $N\geq2$. As pointed out in [@desvillettesvillanitrendglequiboltzmann], this Korn inequality has an important application in statistical physics, more precisely in the study of relaxation to equilibrium of rarefied gases modeled by Boltzmann’s equation. 

Preliminaries
=============

We will utilize the notations from [@bauerpaulytangnormkorn].  Throughout this paper and unless otherwise explicitly stated, let $\om\subset\rN$, $N\geq2$, be a bounded domain with strong Lipschitz boundary $\ga:=\p\om$, i.e., locally $\ga$ can be represented as a graph of a Lipschitz function.  As in [@bauerpaulytangnormkorn], we introduce the standard scalar valued Lebesgue and Sobolev spaces by $\ltom$ and $\hoom$ as well as $$\hocom:=\ovl{\cicom}^{\hoom},$$ respectively, where $\cicom$ denotes the test functions yielding the usual Sobolev space $\hocom$ with zero boundary traces.  These definitions extend component-wise to vector or matrix, or more general tensor fields and we will use the same notations for these spaces.  Moreover, we will consistently denote functions by $u$ and vector fields by $v$. We define the vector valued $\ho$-Sobolev space $\hoctom$ resp.  $\hocnom$ as closure in $\hoom$ of the set of test vector fields $$\begin{aligned}
\mylabel{CtCn_def}
\cictom
:=\setb{v|_{\om}}{v\in\cicrN,\,\tcomp{v}=0},\qquad
\cicnom
:=\setb{v|_{\om}}{v\in\cicrN,\,\ncomp{v}=0},\end{aligned}$$ respectively, generalizing homogeneous tangential resp.  normal boundary conditions.  Here, $\nu$ denotes the a.e.  defined outer unit normal at $\ga$ giving a.e.  the normal resp.  tangential component $$\ncomp{v}:=\nu\cdot v|_{\ga},\quad
\tcomp{v}:=v|_{\ga}-\ncomp{v}\nu$$ of $v$ on $\ga$. We assume additionally that $\ga$ is decomposed into two relatively open subsets $\gat$ and $\gan:=\ga\setminus\ovl{\gat}$ and introduce the vector valued $\ho$-Sobolev space of mixed boundary conditions $\hoctnom$ as closure in $\hoom$ of the set of test vector fields $$\begin{aligned}
\mylabel{Ctn_def}
\cictnom
:=\setb{v|_{\om}}{v\in\cicrN,\,\tcomp{v}|_{\gat}=0,\,\ncomp{v}|_{\gan}=0}.\end{aligned}$$

Korn’s Second Inequality
------------------------

It is well known that Korn’s second inequality can easily be proved by a simple $\hmo$-argument using Necas inequality.  Let us illustrate a simple and short proof: In the sense of distributions we have e.g. for all vector fields $v\in\ltom$ that the components of $\na\na v$[^1] consist only of components of $\na\sym\na v$, i.e., $$\begin{aligned}
\mylabel{nasymna}
\forall\,i,j,k=1,\dots,N\qquad
\p_{i}\p_{j}v_{k}
&=\p_{i}\sym_{j,k}\na v
+\p_{j}\sym_{i,k}\na v
-\p_{k}\sym_{i,j}\na v,\end{aligned}$$ where $\sym_{j,k}T:=(\sym T)_{j,k}$. By e.g. [@sohrbook 1.1.3 Lemma] we have (for scalar functions) the Necas estimate $$\begin{aligned}
\mylabel{necas}
\exists\,c&>0&
\forall\,u&\in\ltom&
c\,\normltom{u}
&\leq\normhmoom{\na u}+\normhmoom{u}
\leq(\sqrt{N}+1)\normltom{u},\end{aligned}$$ where $\hmoom:=\big(\hocom\big)'$ and e.g. by using the full $\hoom$-norm $$\normhmoom{u}
:=\sup_{0\neq\varphi\in\hocom}
\frac{\scpltom{u}{\varphi}}{\norm{\varphi}_{\hoom}},\qquad
\normhmoom{\na u}
:=\sup_{0\neq\phi\in\hocom}
\frac{\scpltom{u}{\div\phi}}{\norm{\phi}_{\hoom}}.$$ For the original results of see the works of Necas, e.g. [@necasbook1967; @necaspaper1967a], from the 1960s. 

Necas’ estimate can be refined to $$\begin{aligned}
\mylabel{necasproj}
\exists\,c&>0&
\forall\,u&\in\ltzom:=\set{u\in\ltom}{\scpltom{u}{1}=0}&
c\,\normltom{u}
&\leq\normhmoom{\na u}
\leq\sqrt{N}\normltom{u}.\end{aligned}$$ The best constant $c>0$ in is also called inf-sup- or LBB-constant as by using the $\hoom$-half norm $$\begin{aligned}
c=\inf_{0\neq u\in\ltzom}
\frac{\normhmoom{\na u}}{\normltom{u}}
=\inf_{0\neq u\in\ltzom}\sup_{v\in\hocom}
\frac{\scpltom{u}{\div v}}{\normltom{u}\normltom{\na v}}
=c_{\mathsf{LBB}}.\end{aligned}$$ We note that the LBB-constant can be bounded from below by the inverse of the continuity constant $c_{A}$ of the $\ho$-potential operator (often called Bogovskii operator) $A:\ltzom\to\hocom$ with $\div Au=u$, i.e., $$\begin{aligned}
\forall\,u&\in\ltzom&
\normltom{\na Au}
&\leq c_{A}\normltom{u}.\end{aligned}$$ This follows directly by setting $v:=Au$ (note that $\na Au\neq0$ for $0\neq u\in\ltzom$) and $$c_{\mathsf{LBB}}
\geq\inf_{0\neq u\in\ltzom}
\frac{\normltom{u}^2}{\normltom{u}\normltom{\na Au}}
\geq\frac{1}{c_{A}}.$$

We immediately get:

\[Korn’s second inequality\] There exists $c>0$ such that for all $v\in\hoom$ $$\begin{aligned}
\normltom{\na v}
&\leq c\big(\normltom{\sym\na v}
+\normltom{v}\big).\end{aligned}$$

Let $v\in\hoom$. Combining and we estimate $$\begin{aligned}
\normltom{\na v}
&\leq c\,\big(\normhmoom{\na\na v}+\normhmoom{\na v}\big)\\
&\leq c\,\big(\normhmoom{\na\sym\na v}+\normhmoom{\na v}\big)
\leq c\,\big(\normltom{\sym\na v}+\normltom{v}\big),\end{aligned}$$ showing the stated result. 

By standard mollification we see that the restrictions of $\cicrN$-vector fields to $\om$ are dense in $${{\mathsf{S}}(\om)}:=\set{v\in\ltom}{\sym\na v\in\ltom},$$ even if $\om$ just has the segment property.  Especially $\hoom$ is dense in ${{\mathsf{S}}(\om)}$. This shows immediately:

\[$\ho$-regularity\] It holds ${{\mathsf{S}}(\om)}=\hoom$.

Let $v\in{{\mathsf{S}}(\om)}$. By density, there exists a sequence $(v_{n})\subset\hoom$ converging to $v$ in ${{\mathsf{S}}(\om)}$. By Theorem \[seckorn\] $(v_{n})$ is a Cauchy sequence in $\hoom$ converging to $v$, yielding $v\in\hoom$.

The latter arguments show, that for any domain allowing for Necas’ estimate Korn’s second inequality Theorem \[seckorn\] holds.  In these domains we have also the $\ho$-regularity Theorem \[horeg\], provided that the segment property holds. 

\[lqremone\] is well known to hold also in the $\mathsf{L}^q/\,\mathsf{W}^{-1,q}$-setting for $1<q<\infty$. As and the mollification techniques are available for general $q$, it follows that Theorem \[seckorn\] and Theorem \[horeg\] immediately extend to the $\mathsf{L}^q/\,\mathsf{W}^{1,q}/\,\mathsf{S}^{q}$-setting for all $1<q<\infty$.

Poincaré Inequality for Elasticity
----------------------------------

To apply standard solution theories for linear elasticity, such as Fredholm’s alternative for bounded domains or Eidus’ limiting absorption principle [@eiduslabp] for exterior domains, it is most important to ensure for bounded domains the compact embedding $$\begin{aligned}
\mylabel{ecp}
{{\mathsf{S}}(\om)}\hookrightarrow\ltom.\end{aligned}$$ As long as Korn’s second inequality, i.e., the continuous embedding ${{\mathsf{S}}(\om)}\hookrightarrow\hoom$, holds true, the compact embedding follows immediately by Rellich’s selection theorem, i.e., the compact embedding $\hoom\hookrightarrow\ltom$. As shown in [@weckelacomp], there are bounded irregular domains, more precisely bounded domains with the $p$-cusp property (Hölder boundaries), see [@witschremmax Definition 3] or [@weckelacomp Definition 2], with $1<p<2$, for which Korn’s second inequality fails and so the embedding ${{\mathsf{S}}(\om)}\subset\hoom$ by the closed graph theorem[^2], but the important compact embedding remains valid.  More precisely, by [@weckelacomp Theorem 2] the compact embedding holds for bounded domains having the $p$-cusp property with $1\leq p<2$[^3], and implies immediately a Poincaré type inequality for elasticity by a standard indirect argument.  For this we define $${{{\mathsf{S}}_{0}}(\om)}:=\set{v\in{{\mathsf{S}}(\om)}}{\sym\na v=0}
=\set{v\in\ltom}{\sym\na v=0}.$$ It is well known that even for any domain $\om$ $${{{\mathsf{S}}_{0}}(\om)}={\mathcal{R}}$$ holds, where ${\mathcal{R}}:=\set{Sx+a}{S\in{\mathfrak{so}}\,\wedge\,a\in\rN}$ is the space rigid motions and ${\mathfrak{so}}={{\mathfrak{so}}(N)}$ the vector space of constant skew-symmetric matrices.  This follows easily for $v\in{{{\mathsf{S}}_{0}}(\om)}$ by approximating $\om$ by smooth domains $\om_{n}$, in each of which $v_{n}:=v|_{\om_{n}}$ equals the same rigid motion $r\in{\mathcal{R}}$.

\[Poincaré inequality for elasticity\] Let $\om$ be bounded and possess the $p$-cusp property with some $1\leq p<2$. Then there exists $c>0$ such that for all $v\in{{\mathsf{S}}(\om)}\cap{\mathcal{R}}^{\bot}$ $$\normltom{v}
\leq c\,\normltom{\sym\na v}.$$ Equivalently, for all $v\in{{\mathsf{S}}(\om)}$ $$\normltom{v-r_{v}}
\leq c\,\normltom{\sym\na v},\qquad
r_{v}:=\pi_{{\mathcal{R}}}v.$$

Here and throughout the paper, we denote orthogonality in $\ltom$ by $\bot$. Moreover, $\pi_{{\mathcal{R}}}$ denotes the $\ltom$-orthogonal projector onto the rigid motions ${\mathcal{R}}$.

If the assertion was wrong, there exists a sequence $(v_{n})\subset{{\mathsf{S}}(\om)}\cap{\mathcal{R}}^{\bot}$ with $\normltom{v_{n}}=1$ and $\normltom{\sym\na v_{n}}\to0$. By we can assume without loss of generality that $(v_{n})$ converges in $\ltom$ to some $v\in\ltom$. But then $v\in{{{\mathsf{S}}_{0}}(\om)}\cap{\mathcal{R}}^{\bot}=\{0\}$, in contradiction to $1=\normltom{v_{n}}\to\normltom{v}=0$.

Under the assumptions of Theorem \[poincareela\], the variational static linear elasticity problem, for $f\in\ltom$ find $v\in{{\mathsf{S}}(\om)}\cap{\mathcal{R}}^{\bot}$ such that $$\forall\,\phi\in{{\mathsf{S}}(\om)}\cap{\mathcal{R}}^{\bot}\qquad
\scpltom{\sym\na v}{\sym\na\phi}
=\scpltom{f}{\phi},$$ is uniquely solvable with continuous resp.  compact inverse $\ltom\to{{\mathsf{S}}(\om)}$ resp.  $\ltom\to\ltom$, which shows that Fredholm’s alternative holds for the corresponding reduced operators. 

Korn’s First Inequality
=======================

By Rellich’s selection theorem, Theorem \[seckorn\] and an indirect argument we can easily prove:

\[Korn’s first inequality without boundary conditions\] There exists $c>0$ such that for all $v\in\hoom$ with $\na v\bot\,{\mathfrak{so}}$ $$\begin{aligned}
\mylabel{firstkornbot}
\normltom{\na v}
&\leq c\,\normltom{\sym\na v}. 
\intertext{Equivalently for all $v\in\hoom$}
\nonumber
\normltom{\na v-S_{v}}
&\leq c\,\normltom{\sym\na v},\qquad
S_{v}:=\frac{1}{|\om|}\skw\int_{\om}\na v.\end{aligned}$$

Here, $S_{v}=\pi_{{\mathfrak{so}}}\na v$ is the $\ltom$-orthogonal projection of $\na v$ onto ${\mathfrak{so}}$.

The equivalence is clear by the orthogonal projection.[^4] If was wrong, there exists a sequence $(v_{n})\subset\hoom$ with $\na v_{n}\bot\,{\mathfrak{so}}$ and $\normltom{\na v_{n}}=1$ and $\normltom{\sym\na v_{n}}\to0$. Without loss of generality we can assume $v_{n}\bot\,\rN$. By Poincare’s inequality $(v_{n})$ is bounded in $\hoom$. Thus, by Rellich’s selection theorem we can assume without loss of generality that $(v_{n})$ converges in $\ltom$ to some $v\in\ltom$. By Theorem \[seckorn\] $(v_{n})$ is a Cauchy sequence in $\hoom$. Therefore $(v_{n})$ converges in $\hoom$ to $v\in\hoom\cap(\rN)^{\bot}$ with $\sym\na v=0$ and $\na v\bot\,{\mathfrak{so}}$. But then $\na v$ is even constant and belongs to ${\mathfrak{so}}$. Hence $\na v=0$[^5] in contradiction to $1=\normltom{\na v_{n}}\to\normltom{\na v}=0$.

Using Poincare’s inequality we immediately obtain:

\[Korn’s first inequality without boundary conditions\] There exists $c>0$ such that for all $v\in\hoom\cap(\rN)^{\bot}$ with $\na v\bot\,{\mathfrak{so}}$ $$\begin{aligned}
\norm{v}_{\hoom}
&\leq c\,\normltom{\sym\na v}.\end{aligned}$$

In order to prove Korn’s first inequality in $\hoctnom$ we need a Poincaré type estimate on this space.  It should be noted that in general mixed boundary conditions are not sufficient to rule out a kernel of the gradient operator.  For example, consider the cube $\om:=(0, \,1)^3\subset \reals^3$ with $\gat$ being the union of the top and bottom together with the constant vector field $r(x):=(0,\,0,\,1)^t$. Then $r\in \hoctnom$. On this account, such constant vector fields have to be excluded separately. 

\[Poincaré inequality with tangential or normal boundary conditions\] There exists $c>0$ such that for all $v\in\hoctnom\cap \big(\hoctnom\cap\reals^N\big)^\perp$ $$\normltom{v}
\leq c\,\normltom{\na v}.$$

If the assertion was wrong, there exists some sequence $(v_{n})\subset\hoctnom\cap \big(\hoctnom\cap\reals^N\big)^\perp$ with $\normltom{v_{n}}=1$ and $\normltom{\na v_{n}}\to0$. Thus, by Rellich’s selection theorem we can assume without loss of generality that $(v_{n})$ converges in $\ltom$ to some $v\in\ltom$. Hence, $(v_{n})$ is a Cauchy sequence in $\hoom$ and converges in $\hoom$ to $v\in\hoctnom\cap \big(\hoctnom\cap\reals^N\big)^\perp$ with $\na v=0$. Therefore, $v$ is a constant in $\hoctnom\cap\rN$ and must vanish in contradiction to $1=\normltom{v_{n}}\to\normltom{v}=0$.

As an easy consequence we get

$\na\hoctnom$ is a closed subspace of $\ltom$.

Let $(v_{n})\subset\hoctnom$ such that $\na v_{n}\to G\in\ltom$ in $\ltom$. Without loss of generality we can assume $(v_{n})\subset\hoctnom\cap \big(\hoctnom\cap\reals^N\big)^\perp$, otherwise we replace $v_n$ by $$\tilde{v}_n:=v_n-\pi_{\hoctnom\cap\rN}v_n\in\hoctnom\cap \big(\hoctnom\cap\reals^N\big)^\perp,$$ where $\pi_{\hoctnom\cap\rN}$ is the orthogonal projector onto $\hoctnom\cap\rN$. Because of Lemma \[poincaretannor\] $(v_{n})$ is a Cauchy sequence in $\hoctnom$, which converges in $\hoom$ to $v\in\hoctnom$. Hence, $G\ot\na v_{n}\to\na v\in\na\hoctnom$.

To exclude the kernel of the $\sym\na$-operator on $\hoctnom$, we define $${\mathcal{K}}:=\set{\na v}{v\in\hoctnom,\,\sym\na v=0}
=\na\big({\mathcal{R}}\cap\hoctnom\big)
={\mathfrak{so}}\cap\na\hoctnom.$$

\[Korn’s first inequality with tangential or normal boundary conditions\] There exists $c>0$ such that for all $v\in\hoctnom$ with $\na v\bot\,{\mathcal{K}}$ $$\begin{aligned}
\mylabel{firstkorntannorineq}
\normltom{\na v}
\leq c\,\normltom{\sym\na v}.\end{aligned}$$ Equivalently, for all $v\in\hoctnom$ $$\begin{aligned}
\normltom{\na v-\pi_{{\mathcal{K}}}\na v}
&\leq c\,\normltom{\sym\na v}.\end{aligned}$$

Here, $\pi_{{\mathcal{K}}}$ denotes the $\ltom$-orthogonal projector onto ${\mathcal{K}}$.

Equivalence is again clear by the orthogonal projection.  If was wrong, there exists a sequence $(v_{n})\subset\hoctnom$ with $\na v_{n}\bot\,{\mathcal{K}}$ and $\normltom{\na v_{n}}=1$ and $\normltom{\sym\na v_{n}}\to0$. Without loss of generality we can assume $(v_{n})\subset\hoctnom\cap \big(\hoctnom\cap\reals^N\big)^\perp$. By Lemma \[poincaretannor\] $(v_{n})$ is bounded in $\hoom$, and thus, using Rellich’s selection theorem, we can assume without loss of generality that $(v_{n})$ converges in $\ltom$ to some $v\in\ltom$. By Theorem \[seckorn\] $(v_{n})$ is a Cauchy sequence in $\hoom$. Therefore, $(v_{n})$ converges in $\hoom$ to $v\in\hoctnom$ with $\sym\na v=0$ and $\na v\bot\,{\mathcal{K}}$. But then, $\na v$ is even a constant in ${\mathfrak{so}}$, i.e., $\na v\in{\mathcal{K}}$, in contradiction to $1=\normltom{\na v_{n}}\to\normltom{\na v}=0$.

\[lqremtwo\] Similar to Remark \[lqremone\], all the results from Theorem \[firstkorn\] to Theorem \[firstkorntannor\] extend to the $\mathsf{L}^q/\,\mathsf{W}^{1,q}$-setting for all $1<q<\infty$ with the obvious modifications.  The same holds true for all results presented in the subsequent sections. 

Discussing the Set ${\mathcal{K}}$
----------------------------------

In this section we shall discuss which combinations of domains $\om$ and boundary parts $\gat$ allow for a non-constant rigid motion $r\in\hoctnom\cap{\mathcal{R}}$, i.e., ${\mathcal{K}}\neq\{0\}$. We start with the case $\gat=\ga$, i.e, with the full tangential boundary condition. 

If $\gat=\ga$, then ${\mathcal{K}}=\{0\}$ and there exists a constant $c>0$ such that for all $v\in\hoctom$ $$\begin{aligned}
\normltom{\na v}
&\leq c\,\normlt{\sym\na v}.\end{aligned}$$

We give a proof by contradiction.  Assume $r\in{\mathcal{R}}\cap\hoctom$ and $r\not=0$. Let us define the null space $\calN_r:=\setlr{x\in\rN}{r(x)=0}$. Then $\calN_r$ is an empty set or an affine plane in $\rN$ with dimension $d_{\calN_r}\leq N-2$. We recall that $\nu$ is the outer unit normal at $\ga$ defined a.e.  on $\ga$ w.r.t.  the $(N-1)$-dimensional Lebesgue measure.  Since $r$ is normal on $\ga$, we conclude for almost all $x\in\ga\setminus\calN_r$ $$\begin{aligned}
\label{rnu}
\nu(x)=\pm\frac{r(x)}{\norm{r(x)}}.\end{aligned}$$ Because $\om$ is locally on one side of the boundary $\ga$, the unit normal $\nu$ cannot change sign in in any connected component of $\ga\setminus\calN_r$. But since $d_{\calN_r}\leq N-2$, it follows that $\ga\setminus\calN_r$ is connected, and w.l.o.g.  $$\begin{aligned}
\label{sign}
\nu(x)=\frac{r(x)}{\norm{r(x)}}\qquad\text{for almost all }x\in\ga\setminus\calN_r.\end{aligned}$$ As $\ga\cap\calN_r$ has measure zero, we can replace $\ga\setminus\calN_r$ by $\ga$ in .  With Gau[ß]{}’ theorem we conclude $$\begin{aligned}
0=\int_\om\div r
=\int_\ga\nu\cdot r
=\int_{\ga}\norm{r}>0,\end{aligned}$$ a contradiction. 

Next we turn to the full normal boundary condition, i.e. $\gat=\emptyset$. In [@desvillettesvillanikornnormal] it is proved that for smooth bounded domains $\om\subset\rN$ Korn’s first inequality holds for all $v\in \hocnom$, i.e. ${\mathcal{K}}=\left\{0\right\}$, if and only if $\om$ is not axisymmetric.  Furthermore an explicit upper bound on the constant is given.[^6] In that contribution and here axisymmetry is defined as follows. 

\[axisymmetry-def\] $\om$ is called axisymmetric if there is a non-trivial rigid motion $r\in\calR$ tangential to the boundary $\ga$ of $\om$, i.e. $0\neq r\in\hocnom$.

In a more elementary and canonical approach in $\rt$ a domain is called axisymmetric w.r.t.  an axis $a$ if it is a body of rotation around this axis.  In order to show that in $\rt$ both concepts coincide for bounded Lipschitz domains, we make use of the invariance of a Lipschitz boundary under the flow of a tangential vector field. 

\[tangentialflow-lem\] Let $\om\subset\rN$ be a (not necessarily bounded) domain with a (strong) Lipschitz boundary $\ga$ and $r:\rN\to\rN$ a locally Lipschitz continuous vector field that is tangential on $\ga$ a.e.  w.r.t.  the $(N-1)$-dimensional Lebesgue measure on $\ga$. Let $p\in\ga$ and let $t\mapsto\gamma(t)$ the maximal solution of the ordinary differential equation $$\label{ODE}
\dot{\gamma}=r(\gamma),\quad 
\gamma(0)=p$$ existing on the interval $I_p$. Then for all $t\in I_p$ $$\gamma(t)\in\ga.$$

This proposition is a variant of Nagumo’s invariance theorem, see [@aubincellina:84 Theorem 2, p. 180], c.f.  also [@nagumo:42], where the tangential condition on $r$ is defined in terms of a so called ’Bouligand contingent cone’.  As we need this statement for a Lipschitz boundary we give a self-contained proof in the Appendix. 

The next lemma states that for bounded domains in $\rt$ both definitions of axisymmetry coincide.  An elementary proof is provided in the appendix. 

\[equivalence\] Let $\om\subset\rt$ be a bounded Lipschitz domain. 

-   Assume $\sigma,b\in\rt$, $\norm{\sigma}=1$ and let $g=\setlr{\lambda\sigma+b}{\lambda\in\reals}$. Assume that $\om$ is axisymmetric w.r.t.  the axis $g$. Then the vector field $r$ with $r(x):=\sigma\wedge(x-b)$ is a rigid motion, which is tangential at $\ga$, i.e. $r\in\calR\cap\hocnom$.

-   Let $r\in\calR\cap\hocnom$, $r(x)=\omega\,\sigma\wedge x+b$ for all $x\in\rt$ with $\sigma,b\in\rt$, $\norm{\sigma}=1$ and $\omega\in\reals$. Then $\omega\neq 0$, $\scp{b}{\sigma}=0$, and $\om$ is axisymmetric w.r.t.  the axis $g=\setlr{\lambda \sigma+\frac{1}{w}\sigma\wedge b}{\lambda\in\reals}$.

\[unbounded\] There are rigid motions tangential to the boundary of some domains in $\rt$, which do not exhibit any axis of symmetry.  Consider, for example, a domain $\om$ built from a plane square which simultaneously is lifted along and rotated around the axis perpendicular to it, e.g. $$\om:=\setb{\big(x_1\cos(t)-x_2\sin(t),x_1\sin(t)+x_2\cos(t),t\big)^t}
{\norm{x_1}+\norm{x_2}<1,\,t\in\reals}.$$ Then $r(x):=(-x_2, x_1, 1)^t$ is tangential to $\ga$.

Using Definition \[axisymmetry-def\], Korn’s first inequality for normal boundary conditions is more or less obvious. 

Let $\gat=\emptyset$. Then Korn’s first inequality holds for all $v\in\hocnom$, if and only if ${\mathcal{K}}=\{0\}$, if and only if $\om$ is not axisymmetric. 

The first ’if and only if’ is just the assertion of Theorem \[firstkorntannor\].  For the second ’if and only if’ according to the definition of axisymmetry the only remaining issue is to prove that there is no constant vector field tangential to a bounded Lipschitz domain (in that case we would have a non-trivial rigid motion, which gives no contribution to ${\mathcal{K}}$).  Assume that a constant vector $0\neq a\in\rN$ tangential to $\ga$ exists, i.e. $a\in\hocnom$, and let $\hat{x}\in\ga$. Then according to Proposition \[tangentialflow-lem\] the unbounded curve $t\mapsto\hat{x}+ta$ would remain in $\ga$, which contradicts the boundedness of $\om$.

The latter proof shows that a bounded domain is axisymmetric if and only if there is a non-constant rigid motion tangential to the boundary. 

For mixed boundary conditions there are domains of rather special type with ${\mathcal{K}}\neq\{0\}$. Consider, for example, a half cylinder $$\om:=\setlr{x\in\rt}{x_1>0,\,x_1^2+x_2^2<1,\,0<x_3<1},$$ or more generally, the domain $$\om:=\setlr{(r\cos\phi,r\sin\phi,x_3)^t}{\phi_1<\phi<\phi_2,\,0<x_3<1,\,0< r<h(x_3)}$$ with $\gat:=\ga\cap\setlr{(r\cos \phi_{1/2},r\sin\phi_{1/2},x_3)^t}{0\leq r,\,0<x_3<1}$ and for some positive Lipschitz function $ h:\reals\to\reals$ and some $-\pi<\phi_1<\phi_2<\pi$. Define $r(x):=(-x_2, x_1, 0)^t$. Then $r$ is a rigid motion and $r\in\hoctnom$. In the next theorem we will show that in $\rt$ all bounded domains $\om$ with ${\mathcal{K}}\neq \left\{0\right\}$ are compositions of subdomains of this kind. 

\[gemischteRandbedingungen-lem\] Let $\om\subset\rt$ be a bounded Lipschitz domain and let $\emptyset\neq\gat\neq\ga$. Assume that there is a non-constant rigid motion $r\in\calR\cap\hoctnom$, $r(x)=\omega\,\sigma\wedge x+b$ for all $x\in\rt$ with $\omega\in\reals$ and $\norm{\sigma}=1$. Define $g_r\subset\rt$ by $g_r:=\setlr{\lambda\,\sigma+\frac{1}{\omega}\,\sigma\wedge b}{\lambda\in\reals}$. Then $\scp{\sigma}{b}=0$, $\gat$ is a subset of a union of affine planes, where each of these planes contains $g_r$. Every connected component of $\gan$ is a subset of a surface which is axisymmetric w.r.t.  $g_r$.

By this theorem the aforementioned cube, i.e. $\om=(0,1)^3\subset\rt$ with $\gat$ being the union of the top and bottom faces, has a trivial kernel ${\mathcal{K}}=\{0\}$, which means Korn’s first inequality Theorem \[firstkorntannor\] holds on $\hoctnom$, while Poincaré’s inequality Lemma \[poincaretannor\] only holds on $\hoctnom\cap\big((0,0,1)^t\big)^\perp$.

First we note that the scalar-product $\scp{\sigma}{b}$ is independent of the chosen Cartesian coordinates, i.e. if we choose another positively oriented Euclidian coordinate system $(y_1, y_2,y_3)$ and represent the vector field $r$ by means of the $y$-coordinates, then there exist vectors $\sigma_y,b_y\in\rt$ with $\norm{\sigma_y}=1$ and $r(y)=\omega\,\sigma_y\wedge y+b_y$ for all $y\in\rt$. Furthermore $\scp{\sigma_y}{b_y}=\scp{\sigma}{b}$. In the same way the representation of the axis $g_r$ associated to $r$ is independent of the Cartesian coordinates chosen; in $y$-coordinates we have $g_r=\setlr{\lambda\,\sigma_y+\frac{1}{\omega}\sigma_y\wedge b_y}{\lambda\in\reals}$.

Suppose $r\in\calR\cap\hoctnom$ and that $r$ is not constant.  We fix some $p\in\gat$ together with a neighborhood $U\subset\rt$ of $p$, an open subset $V\subset\reals^2$, Euclidian coordinates $(x_1,x_2,x_3)=(x',x_3)$ and a Lipschitz map $h:V\subset\reals^2\to\reals$, such that for all $x\in U$ we have $x=(x',x^3)\in\gat$ if and only if $x^3=h(x')$. Since $r$ is normal and by Rademacher’s theorem, we have $$\begin{aligned}
\label{normal-equation}
r(x',h(x'))
&=f(x')\big(\na_{x'}h(x'),-1\big)^t\end{aligned}$$ with some function $f:V\subset\reals^2\to\reals$ a.e.  in $V$.

In $x$-coordinates $r$ can be represented by $r(x)=\omega\,\sigma\wedge x+b$ with some $b,\sigma\in\rt$, $\norm{\sigma}=1$ and $0\neq\omega\in\reals$. From we conclude $$\begin{aligned}
\label{K1}
b_1+\omega\,\sigma_2h(x')-\omega\,\sigma_3x_2 
&=f(x')\p_1h(x'),\\
\label{K2}
b_2+\omega\,\sigma_3x_1-\omega\,\sigma_1h(x')
&=f(x')\p_2h(x'),\\
\label{K3}
b_3+\omega\,\sigma_1x_2-\omega\,\sigma_2x_1
&=-f(x').\end{aligned}$$ We differentiate (in the sense of distributions) w.r.t.  $x_2$ and w.r.t.  $x_1$, compute the difference as well as the sum of the resulting equations, and conclude using $$\begin{aligned}
\label{sigma3}
\sigma_3
&=\sigma_1\p_1h+\sigma_2\p_2h,\\ 
\label{gemischt}
0
&= f\p_1\p_2h.\end{aligned}$$ Differentiating w.r.t.  $x_1$ and w.r.t.  $x_2$ yields $$\label{doppelt}
f\p_1^2h=f\p_2^2h=0.$$ Now we multiply by $\sigma_1$, by $\sigma_2$, equate the resulting equations for $\sigma_1 \sigma_2 h$, use , , and obtain $$\label{skp}
0=\scp{b}{\sigma}.$$ From , we conclude that $\na_{x'}h$ is constant on connected components of $V\cap\{f\neq 0\}$. Therefore, $h$ is an affine function on each part and continuous on the whole of $V$. Note that $\{f=0\}$ is a subset of the line $\mathcal N_{\sigma,b}:=\setlr{x'\in\reals^2}{b_3+\omega\,\sigma_1 x_2-\omega\,\sigma_2 x_1=0}$. Now we extend the affine function from one connected component of $V\cap\{f\neq 0\}$ to $\reals^2$ and call the resulting affine function $\tilde{h}$. Because of the plane $\mathcal E_{\tilde{h}}:=\setb{(x',\tilde{h}(x'))}{x'\in\reals^2}$ is collinear to $g_r$. Recalling $\scp{\sigma}{b}=0$, it is straightforward to check that $g_r$ is the affine kernel of $r$. Now we use this fact together with the collinearity of $\mathcal E_{\tilde{h}}$ and $g_r$ in order to prove $g_r\subset \mathcal E_{\tilde{h}}$. It is sufficient to show that $\mathcal E_{\tilde{h}}\cap\{r=0\}$ is not void.  But in view of and this is obvious. 

Now let $p\in\gan$. Since $\scp{\sigma}{b}=0,$ the solutions $\gamma$ of $\dot{\gamma}=r(\gamma)$ are circles, contained in planes perpendicular to $g_r$ and with centers on $g_r$ (See also the computations in the proof of Lemma \[equivalence\].).  Hence, applying Proposition \[tangentialflow-lem\], every connected component is a subset of some hyper surface being axisymmetric w.r.t.  $g_r$.

Appendix
========

Clearly, it is sufficient to prove the invariance locally.  Since $\ga$ is Lipschitz, after rotation there is a neighborhood $U=V\times I$ of $p$ with $V\subset\reals^{N-1}$, $I\subset\reals$, orthonormal coordinates $(x^1,\ldots,x^N)=(x',x^N)\in V\times I$, a point $x'_0\in V$ and a Lipschitz continuous function $h:V\to I$ such that $p=(x'_0,h(x'_0))$, and for all $x\in U$ we have $x\in\ga$ iff $x^N= h(x')$. By Rademacher’s theorem $h$ is differentiable a.e.  with respect to the $(N-1)$-dimensional Lebesgue measure on $V$, and $\na_{x'}h\in\li(V)$. Furthermore, the set of the $N-1$ vectors $$t_1(x'):=\big(1,0,\ldots,0,\p_1h(x')\big)^t,\,\ldots\,,
t_{N-1}(x'):=\big(0,\ldots,0,1,\p_{N-1}h(x')\big)^t$$ gives a basis of the tangential space of $\ga$ in the point $(x',h(x'))$ for almost all $x'\in V$. Therefore, on $\ga\cap\,U$ we have two representations of the vector field $r$, one representation in the coordinate vectors of $x^1,\ldots,x^N$ holding on the whole of $U$, $$r(x)=r_U(x)=\big(r_U^1(x),\ldots,r_U^N(x)\big)^t,$$ and the functions $r_U^i$, $i=1,\ldots,N$, are Lipschitz continuous functions on $U$. On the other hand, for almost all $x'\in V$ $$r(x',h(x'))=r_V^1(x')t_1 (x')+\cdots+r_V^{N-1}(x')t_{N-1}(x').$$ We define $r_V:=(r_V^1,\ldots,r_V^{N-1})^t$. Comparison yields a.e.  on $V$ and for all $i=1,\ldots,N-1$ $$\label{Koeffizientenfunktionen}
r_U^i(x',h(x'))=r_V^i(x').$$ Hence, $r_V$ is Lipschitz continuous on $V$. Furthermore, $$\label{N-Funktion}
r_U^N (x',h(x'))
=r_V^1(x')\p_1h(x')+\cdots+r_V^{N-1}(x')\p_{N-1}h(x')
=r_V(x')\cdot\na_{x'}h(x')$$ holds for almost all $x'\in V$. Since $h$ is Lipschitz on $V$ and $r_U^N$ is Lipschitz on $U$, $r_V\cdot\na_{x'}h$ is also Lipschitz on $V$. Now we define the flow of $r_V$: For $x'\in V$ we set $\psi(\,\cdot,x')$ as the solution of the ordinary differential equation $$\label{Def-ODE-V}
\dot{\psi}(t,x')
=r_V\big(\psi(t,x')\big),\quad\psi(0,x')=x'.$$ Since $r_V$ is Lipschitz on $V$, we can restrict the flow such that for some $\eps>0$ and some neighborhood $\bar{V}\subset V$ of $x_0'$ the solution $\psi$ is Lipschitz continuous on $(-\eps,\eps)\times\bar{V}$. Next we lift up this flow to $\ga$ and define $$\gamma_V(t):=\big(\psi(t,x'_0),h(\psi(t, x'_0))\big)^t.$$ By definition $\gamma_V(0)=p$ and $\gamma_V(t)\in\ga$ for all $t\in(-\eps,\eps)$.

In the next step we have to prove that $\gamma_V$ is also a solution of on $(-\eps,\eps)$. With regard to it only remains to prove that the mapping $t\mapsto h(\psi(t,x'_0))$ is classically differentiable with derivative $\p_t\big(h(\psi(t,x'_0))\big)=r^N_U\big(\psi(t,x'_0),h(\psi(t,x'_0))\big)$. We denote the $l$-dimensional Lebesgue measure by ${\mathcal{L}}^l$. For all $t\in(-\eps,\eps)$ it holds that $\psi(t,\,\cdot\,)$ is a bi-Lipschitz homeomorphism with inverse Lipschitz transformation $\psi(t,\,\cdot\,)^{-1}=\psi(-t,\,\cdot\,)$. Therefore, if ${\mathcal{L}}^{N-1}(\psi(t,\,\cdot\,)(\tilde{V}))=0$ for some set $\tilde{V}\subset\bar{V}$, then also ${\mathcal{L}}^{N-1}(\tilde{V})=0$, because $\tilde{V}=\psi(-t,\,\cdot\,)\big(\psi(t,\,\cdot\,)(\tilde{V})\big)$. Fix a measurable set $V_0\subset V$ such that ${\mathcal{L}}^{N-1}(V_0)=0$ and $h$ is classically differentiable for every $x'\in V\setminus V_0$. Let us define $$W_0:=\setlr{(t,x)\in(-\eps,\eps)\times\bar{V}}{\psi(t,x)\in V_0}.$$ Then $W_0$ is measurable and using Tonelli’s and Fubini’s theorems and the change of variable formula we obtain $$\begin{aligned}
{\mathcal{L}}^N(W_0) 
&=\int_{(-\eps,\eps)\times\bar{V}}{\mathbf 1}_{W_0}
\leq c\int_{(-\eps,\eps)}\int_{V_0}1=0.\end{aligned}$$ Therefore, and since $\psi$ is differentiable w.r.t.  $t$ everywhere, we have by using $$\label{derivative-formula}
\p_th(\psi(t,x'))
=\na h(\psi(t,x'))\cdot\p_t\psi(t,x')
=r^N_U\big(\psi(t, x'),h(\psi(t, x'))\big)$$ for almost all $(t,x')\in(-\eps,\eps)\times\bar{V}$. Consequently this formula holds in the distributional sense.  Because $h\circ\psi$ is continuous and its distributional derivative w.r.t.  $t$ is also continuous, it is also differentiable w.r.t.  $t$ in the classical sense.  This can be seen as follows: We define $$v(t,x')
:=h(\psi(0,x'))
+\int_0^tr^N\big(\psi(\tau,x'),h(\psi(\tau,x'))\big)\,\mathrm{d}\tau.$$ The vector field $v$ is classically differentiable w.r.t.  $t$ and $\p_tv(t, x')=r^N_U\big(\psi(t,x'),h(\psi(t,x'))\big)$ holds for all $(t,x')\in(-\eps,\eps)\times\bar{V}$. Furthermore, for all $\phi\in\cic\left((-\eps,\eps)\times\bar{V}\right)$ $$\int_{(-\eps,\eps)\times\bar{V}}
(v-h\circ\psi)\p_t\phi=0.$$ This yields $h\circ\psi(t, x')=v(t, x')+w(x')$. Since for all $x'\in\bar{V}$ we have $h\circ\psi(0,x')=v(0,x')$, we finally conclude $w= 0$ on $\bar{V}$ and hence $v=h\circ\psi$.

For (i) we choose $\sigma_1,\sigma_2\in\rt$ such that the set $\{\sigma_1,\sigma_2,\sigma\}$ gives a positively oriented orthonormal basis of $\rt$. Let $x\in\ga$ and define $d:=\dist(g,x)$. Since $\om$ is axisymmetric w.r.t.  $g$, for all $t\in\reals$ $$\begin{aligned}
\gamma(t)
&:=\scp{x}{\sigma}\sigma
+\big(\scp{b}{\sigma_1}+d\cos(t)\big)\sigma_1
+\big(\scp{b}{\sigma_2}+d\sin(t)\big)\sigma_2\in\ga.\end{aligned}$$ Therefore, $\dot{\gamma}(t)$ is a tangential vector at $\ga$ located in $x$. On the other hand $$\begin{aligned}
r(x)
&=\sigma\wedge(x-b)
=\sigma_2\scp{x-b}{\sigma_1}-\sigma_1\scp{x-b}{\sigma_2}\\
&=\sigma_2\bscp{\big(\scp{b}{\sigma_1}+d\cos(t)\big)\sigma_1-b}{\sigma_1}
-\sigma_1\bscp{\big(\scp{b}{\sigma_2}+d\sin(t)\big)\sigma_2-b}{\sigma_2}\\  
&=\sigma_2d\cos \left(t\right)-\sigma_1d\sin \left(t\right)
=\dot{\gamma}\left(t\right),\end{aligned}$$ which yields $r\in\hocnom\cap\calR$.

No we turn to the proof of (ii).  If $\omega=0$ then $x(t)=x_0+tb$ remains in $\ga$ for all $t$ if $x_0\in\ga$ (Proposition \[tangentialflow-lem\]) and $\om$ would be unbounded.  Therefore, we have $\omega\neq0$. We choose again $\sigma_1,\sigma_2\in\rt$ such that the set $\{\sigma_1,\sigma_2,\sigma\} $ gives an orthonormal basis of $\rt$ with positive orientation.  The solution of the ordinary differential equation system $$\begin{aligned}
\dot{s}_1
&=-\omega s_2+\scp{b}{\sigma_1},
&
\dot{s}_2
&=\omega s_1+\scp{b}{\sigma_2},
&
\dot{s}_3
&=\scp{b}{\sigma},\\
s_1(0)
&=\scp{\hat{x}}{\sigma_1},
&
s_2(0)
&=\scp{\hat{x}}{\sigma_2},
&
s_3(0)
&=\scp{\hat{x}}{\sigma}\end{aligned}$$ is given by $$\begin{aligned}
s_1(t)
&=c_1\cos(\omega t)-c_2\sin(\omega t)-\frac{1}{\omega}\scp{b}{\sigma_2},\\
s_2(t)
&=c_1\sin(\omega t)+c_2\cos(\omega t)+\frac{1}{\omega}\scp{b}{\sigma_1},\\
s_3(t)
&=\scp{\hat{x}}{\sigma}+t\scp{b}{\sigma},\end{aligned}$$ where $c_1$ and $c_2$ are uniquely defined by the initial conditions on $s_1$ and $s_2$. Then $$x(t):=s_1(t)\sigma_1+s_2(t)\sigma_2+s_3(t)\sigma$$ is the unique solution of $$\dot{x}=r(x),\quad x(0)=\hat{x}.$$ Due to Proposition \[tangentialflow-lem\] and since $r\in\hocnom$, we have $x(t)\in\ga$ for all $t\in\reals$. Because $\om$ is bounded, we conclude $\scp{b}{\sigma}=0$. Therefore, the trajectory $t\mapsto x(t)$ is a circle lying in a plane perpendicular to $\sigma$ with center $$-\frac{1}{\omega}\scp{b}{\sigma_2}\sigma_1
+\frac{1}{\omega}\scp{b}{\sigma_1}\sigma_2
+\scp{\hat{x}}{\sigma}\sigma.$$ Consequently, $\om$ is axisymmetric w.r.t.  to $g$.

[^1]: We denote by $\na v$ the transpose of the Jacobian of $v$ and by $\na\na v$ the tensor of second derivatives of $v$.

[^2]: The identity mapping $\id_{\mathsf{S}}:{{\mathsf{S}}(\om)}\to\hoom$ is continuous, if and only if $\id_{\mathsf{S}}$ is closed, if and only if ${{\mathsf{S}}(\om)}\subset\hoom$.

[^3]: For $p=1$ the $1$-cusp property equals the strict cone property, which itself holds for strong Lipschitz domains. 

[^4]: We can also compute it by hand: For $v\in\hoom$ with $\na v\bot\,{\mathfrak{so}}$ we see $$|S_{v}|^2
    =\frac{1}{|\om|}\scp{\skw\int_{\om}\na v}{S_{v}}
    =\frac{1}{|\om|}\scpltom{\na v}{S_{v}}
    =0$$ since $S_{v}\in{\mathfrak{so}}$. For $v\in\hoom$ and $T\in{\mathfrak{so}}$ we have $$\begin{aligned}
    \scpltom{\na v-S_{v}}{T}
    &=\int_{\om}\scp{\skw\na v}{T}-\scpltom{S_{v}}{T}
    =\scp{\int_{\om}\skw\na v}{T}-|\om|\scp{S_{v}}{T}
    =0,\end{aligned}$$ implying $v+s_{v}\in\hoom$ with $\na(v+s_{v})=(\na v-S_{v})\bot\,{\mathfrak{so}}$ and $\sym\na(v+s_{v})=\sym(\na v-S_{v})=\sym\na v$, where $s_{v}(x):=S_{v}x$.

[^5]: We note that even $v\in\rN$ holds and thus $v=0$.

[^6]: In [@desvillettesvillanikornnormal] a $\co$-boundary is assumed, but it seems that for the proof of [@desvillettesvillanikornnormal Lemma 4] actually a $\ct$-boundary is needed in order to guaranty $\ho$-regularity of $\na\phi$, where $\phi$ is the solution of [@desvillettesvillanikornnormal (14)].