text:
\section{Introduction}
In the Naghdi shell model, the strain energy is a sum of bending, membrane, and transverse shear strain energies.
The three strains are expressed in terms of the model primary variables comprising  displacement of the shell mid-surface and rotation of normal fibers,
in which derivatives and function values are combined
together in a complicated manner by the curvature tensors and Christoffel symbols.
Korn's  inequality for shells \cite{BCM}  establishes an equivalence between the strain energy norm of the primary variables
and the their  usual Sobolev norms, which ensures  the wellposedness of the
the Naghdi shell model in Sobolev spaces.
The situation for the Koiter shell model is similar, which excludes the transverse shear effect in shell deformation and only uses
mid-surface displacement as the primary variables, for which  a Korn's inequality is also available \cite{BCM}.
When analyzing conforming finite element methods for shells \cite{Arnold-Brezzi, Bernadou, Bathe-book},
such continuous version of Korn's inequalities play a fundamental role
in that
they attribute most of the numerical analysis and estimates to that in Sobolev spaces.
To deal with discontinuous Galerkin methods for shells \cite{Koiter-shell, Naghdi-shell},
it is desirable to have appropriate generalizations of the Korn's inequalities that  can be applied to piecewise defined discontinuous functions.
It seems that such discrete
Korn's inequalities do not trivially follow from the Korn's inequalities for shells.
Methods of proving discrete Korn's inequality for plane elasticity \cite{Brenner-2} can not be easily adapted to shell problems either.
It is the purpose of
this paper to establish such discrete Korn's inequalities for both the Naghdi and Koiter shell models, for the most general geometry of shell mid-surfaces,
without making any additional assumption on the regularity of shell model solutions.

Let $\t\Omega\subset\R^3$
be the middle surface of a shell of thickness $2\eps$.
It is  the image of a two-dimensional domain
$\Omega\subset\R^2$ through a mapping $\!Phi$ that has third order continuous derivatives.
The Naghdi shell model is a two-dimensional model defined on $\Omega$.
In the following, we use super scripts to indicate contravariant components of tensors, and
use subscripts to indicate covariant components. Greek super and subscripts take their values in $\{1, 2\}$,
and Latin scripts take their values
in $\{1, 2, 3\}$. Summation rules with respect to repeated and super and subscripts will also be used.
We use $H^\alpha(\tau)$ to denote the $L^2$ based Sobolev space of order $\alpha$ of functions defined on $\tau$, in which  the
norm and semi-norm are denoted by $\|\cdot\|_{k, \tau}$ and  $|\cdot|_{k, \tau}$, respectively. We also use $H^0(\tau)$ to denote $L^2(\tau)$.
 When $\tau=\Omega$, we simply write the space as $H^\alpha$.

The coordinates $x_\alpha\in\Omega$
furnish the curvilinear coordinates on $\t\Omega$ through the mapping $\!Phi$.
We assume that at any point on the surface,
along the coordinate lines,
the two tangential vectors
$\!a_{\alpha}={\partial\!Phi}/{\partial x_{\alpha}}$
are linearly independent.
The unit vector
$\!a_3=(\!a_1\x\!a_2)/|\!a_1\x\!a_2|$ is normal to $\t\Omega$.
The triple $\!a_i$ furnishes the covariant basis on $\t\Omega$.
The contravariant basis
$\!a^i$ is defined by the relations
$\!a^{\alpha}\cdot\!a_{\beta}=\delta^{\alpha}_{\beta}$ and $\!a^3=\!a_3$,
in which $\delta^{\alpha}_{\beta}$ is the Kronecker delta.
It is obvious that $\!a^{\alpha}$ are also tangent to the surface.
The metric tensor has the covariant components
$a_{\alpha\beta}=\!a_{\alpha}\cdot\!a_{\beta}$,  the determinant of
which is denoted by $a$. The contravariant components
are given by
$a^{\alpha\beta}=\!a^{\alpha}\cdot\!a^{\beta}$.
The curvature tensor
has covariant components
$b_{\alpha\beta}=\!a_3\cdot\partial_{\beta}\!a_{\alpha}$, whose
mixed components are $b^{\alpha}_{\beta}=a^{\alpha\gamma}b_{\gamma\beta}$.
The symmetric tensor $c_{\alpha\beta}=b^\gamma_\alpha b_{\gamma\beta}$ is called the third
fundamental form of the surface.
The Christoffel symbols
are defined by
$\Gamma^{\gamma}_{\alpha\beta}
=\!a^{\gamma}\cdot\partial_{\beta}\!a_{\alpha}$,
which are symmetric with respect to the subscripts.

The Naghdi shell model \cite{Naghdi} uses the covariant components $u_\alpha$ of the
tangential displacement vector $\!u=u_\alpha\!a^\alpha$, coefficient $w$ of the normal displacement $w\!a^3$ of the shell mid-surface,  and
the covariant components $\theta_\alpha$ of the  normal fiber rotation vector $\!theta=\theta_\alpha\!a^\alpha$ as the primary variables.
The bending strain tensor, membrane strain tensor, and transverse shear strain vector associated with a deformation
represented by
such a set of primary variables are
\begin{equation}\label{N-bending}
\rho_{\alpha\beta}(\!theta, \!u, w)=
\frac12(\theta_{\alpha|\beta}+\theta_{\beta|\alpha})-\frac12(b^\gamma_\alpha u_{\gamma|\beta}+b^\gamma_\beta u_{\gamma|\alpha})+c_{\alpha\beta}w,
\end{equation}
\begin{equation}\label{N-metric}
\gamma_{\alpha\beta}(\!u,w)=
\frac12(u_{\alpha|\beta}+u_{\beta|\alpha})
-b_{\alpha\beta}w,
\end{equation}
\begin{equation}\label{N-shear}
\tau_\alpha(\!theta, \!u, w)=\partial_\alpha w+b^\gamma_\alpha u_\gamma+\theta_\alpha.
\end{equation}
Here, the covariant derivative of a vector $u_\alpha$ is defined by
\begin{equation}\label{covariant-derivative}
u_{\alpha|\beta}=\partial_{\beta}u_{\alpha}-\Gamma^{\gamma}_{\alpha\beta}
u_{\gamma}.
\end{equation}
The loading forces on the shell body and upper and lower surfaces
enter the shell model as resultant loading forces per unit area  on the shell middle surface,
of which the tangential force density is
$p^{\alpha}\!a_{\alpha}$ and transverse force density $p^3\!a_3$.
Let the boundary $\partial\t\Omega$ be divided to $\partial^D\t\Omega\cup\partial^S\t\Omega\cup\partial^F\t\Omega$.
On $\partial^D\t\Omega$ the shell is clamped, on $\partial^S\t\Omega$ the shell is soft-simply supported, and
on $\partial^F\t\Omega$ the shell is free of displacement constraint and subject to force or moment  only.
(There are $32$ different ways to specify boundary conditions at any point on the shell boundary, of which we consider the three most typical.)
Let $\!H^1=H^1\x H^1$. The shell model is
defined in the Hilbert space
\begin{multline}\label{N-space}
H=\{(\!phi, \!v, z)\in \!H^1\x\!H^1\x H^1;\  v_\alpha \text{ and } z \text{ are }0\  \text{on}\ \partial^D\Omega\cup\partial^S\Omega, \\
\text{ and  }\theta_\alpha \text{ is }0\
\text{on}\ \partial^D\Omega\}.
\end{multline}
The model determines $(\!theta, \!u, w)\in H$
such that
\begin{multline}\label{N-model}
\frac13\int_{\Omega}
a^{\alpha\beta\lambda\gamma}\rho_{\lambda\gamma}(\!theta, \!u, w)
\rho_{\alpha\beta}
(\!phi, \!v, z)\sqrt a\d x_1 \d x_2\\
+\eps^{-2}\left[\int_{\Omega}
a^{\alpha\beta\lambda\gamma}\gamma_{\lambda\gamma}(\!u,w)
\gamma_{\alpha\beta}(\!v,z)+\kappa\mu\int_{\Omega}a^{\alpha\beta}\tau_\alpha(\!theta, \!u, w)\tau_\beta(\!phi, \!v, z)\right]\sqrt a\d x_1\d x_2
\\
=
\int_{\Omega}
(p^{\alpha}v_{\alpha}+
p^3z)\sqrt a\d x_1\d x_2
+\int_{\partial^S\t\Omega}r^\alpha\phi_\alpha\d\t s
+\int_{\partial^F\t\Omega}\left(q^\alpha v_\alpha+q^3z+r^\alpha\phi_\alpha\right)\d\t s
\\ \forall\
(\!phi, \!v,z) \in H.
\end{multline}
Here, $q^i$ and $r^\alpha$ are the force resultant and
moment resultant on the shell edge \cite{Naghdi}.
The factor $\kappa$ is a shear correction factor.
The last two integrals on the shell edge is taken with respect to the arc length of the boundary of $\t\Omega$.
The fourth order contravariant tensor
$a^{\alpha\beta\gamma\delta}$ is the elastic tensor of the shell,
defined by
\begin{equation*}
a^{\alpha\beta\gamma\delta}=\mu (a^{\alpha\gamma}a^{\beta\delta}+a^{\beta\gamma}a^{\alpha\delta})+
\frac{2\mu\lambda}{2\mu+\lambda}
a^{\alpha\beta}a^{\gamma\delta}.
\end{equation*}
Here, $\lambda$ and $\mu$ are the Lam\'e coefficients of the elastic material. It satisfies the condition that
there are constants $C_1$ and $C_2$
that only depend on the shell mid-surface and the Lam\'e coefficients of the shell material such that for any
tensor $\varsigma_{\alpha\beta}$
\begin{equation*}
\sum_{\alpha,\beta=1}^2|\varsigma_{\alpha\beta}|^2\le C_1a^{\alpha\beta\lambda\gamma}\varsigma_{\alpha\beta}\varsigma_{\lambda\gamma},\quad
a^{\alpha\beta\lambda\gamma}\varsigma_{\alpha\beta}\varsigma_{\lambda\gamma}\le C_2\sum_{\alpha,\beta=1}^2|\varsigma_{\alpha\beta}|^2.
\end{equation*}
Also, there are constants $C_1$ and $C_2$ that only depend on the shell mid-surface such that for any vector $\varsigma_\alpha$,
\begin{equation*}
\sum_{\alpha=1}^2|\varsigma_\alpha|^2\le C_1 a^{\alpha\beta}\varsigma_\alpha \varsigma_\beta,\quad
a^{\alpha\beta}\varsigma_\alpha \varsigma_\beta\le C_2\sum_{\alpha=1}^2|\varsigma_\alpha|^2.
\end{equation*}
These inequalities together with a Korn's inequality \cite{BCM} assures that
the Naghdi shell model  \eqref{N-model} has a unique solution in the space $H$. The Korn's inequality states that
there is a $C$ that could be dependent on the shell mid-surface such that
\begin{multline}\label{N-Korn-continuous}
\|\!theta\|_{\!H^1}+\|\!u\|_{\!H^1}+\|w\|_{H^1}\\
\le C\left[
\sum_{\alpha,\beta=1}^2\|\rho_{\alpha\beta}(\!theta, \!u, w)\|^2_{L^2}+
\sum_{\alpha,\beta=1}^2\|\gamma_{\alpha\beta}(\!u, w)\|^2_{L^2}+
\sum_{\alpha=1}^2\|\tau_{\alpha}(\!theta, \!u, w)\|^2_{L^2}+f^2(\!theta, \!u, w)\right]^{1/2}\\
\forall\ (\!theta,\!u, w)\in \!H^1\x\!H^1\x H^1.
\end{multline}
Here $f$ is a continuous  seminorm satisfying the rigid body motion condition that if  $(\!theta, \!u, w)$ defines a rigid body motion and
$f(\!theta, \!u, w)=0$ then $(\!theta, \!u, w)=0$.
The displacement functions  $(\!theta, \!u, w)\in \!H^1\x\!H^1\x H^1$ defines a rigid body if and only if
$\rho_{\alpha\beta}(\!theta, \!u, w)=0$,  $\gamma_{\alpha\beta}(\!u, w)=0$,  and $\tau_{\alpha}(\!theta, \!u, w)=0$ \cite{BCM}.

We assume that $\Omega$ is a bounded polygon.
Let $\T_h$ be a shape regular, but not necessarily quasi-uniform,
triangulation on $\Omega$.
Shape regularity of triangulations
is a crucial notion in this paper. It is worthwhile to recall its definition
here. Considering a triangle, we let $r$ and $R$ be the radii of its
inscribed circle and circumcircle, respectively.
Then the ratio $R/r$ is called its shape regularity constant, or simply shape regularity.
For a triangulation, the maximum
of shape regularities of all its triangles
is called the shape regularity of the triangulation \cite{Ciarlet-FEM-book}, denoted by $\K$.
We will need to consider a (infinite) class of triangulations. For a class, the
{\em shape regularity} $\K$ is the supreme of all the shape regularities
of its triangulations.
For the triangulation $\T_h$, we use $\T_h$ to denote the set of all (open) triangular elements
of the partition,
and use $\Omega_h$ to denote the union of
all the open triangular elements.
We use $\E^0_h$ to denote the set of all interior (open) edges and $\E^\partial_h$ all
boundary edges, and let $\E_h=\E^0_h\cup\E^\partial_h$. We use $h_\tau$ to denote the diameter of an element
$\tau\in\T_h$ and use
$h_e$ to denote the length of an edge $e\in \E_h$.

Let $H^1_h$ be the space of piecewise $H^1$ functions subordinated to the triangulation
$\T_h$.
A function in $H^1_h$ is independently defined on
every element $\tau\in\T_h$ on which it belongs to $H^1(\tau)$.
A function $u$ in $H^1_h$
is certainly
in $L^2(\Omega_h)$.
On an edge $e\in\E^0_h$, a function $u$ may have two different traces from the
two elements sharing $e$. We use $\lbra u\rbra$ to denoted the absolute value of difference
of the two traces, which is the jump of $u$ over $e$.
In the space
$H^1_h$, we define a norm
\begin{equation}\label{H1hnorm}
\|u\|_{H^1_h}:=\left[\sum_{\tau\in\T_h}\|u\|^2_{1,\tau}
+\sum_{e\in\E^0_h}\frac{1}{h_e}\int_e\lbra u\rbra^2\d s\right]^{1/2}.
\end{equation}
Let $\!H^1_h=H^1_h\x H^1_h$.
We prove the discrete Korn's inequality for Naghdi shell that
for all $(\!theta,\!u, w)\in \!H^1_h\x\!H^1_h\x H^1_h$
\begin{multline}\label{N-Korn-discrete}
\|\!theta\|^2_{\!H^1_h}+\|\!u\|^2_{\!H^1_h}+\|w\|^2_{H^1_h}\\
\le C\left[
\sum_{\alpha,\beta=1}^2\|\rho_{\alpha\beta}(\!theta, \!u, w)\|^2_{0, \Omega_h}+
\sum_{\alpha,\beta=1}^2\|\gamma_{\alpha\beta}(\!u, w)\|^2_{0, \Omega_h}+
\sum_{\alpha=1}^2\|\tau_{\alpha}(\!theta, \!u, w)\|^2_{0, \Omega_h}+f^2(\!theta, \!u, w)\right.\\
\left. +\sum_{e\in\E^0_h}
\frac{1}{h_e}\int_e
\sum_{\alpha=1}^2\left(\lbra \theta_\alpha\rbra^2+\lbra u_\alpha\rbra^2\right)\d s
+\sum_{e\in\E^0_h}\frac{1}{h_e}\int_e\lbra w\rbra^2\d s\right].
\end{multline}
Here $f$ is a seminorm such that $f(\!theta, \!u, w)\le C(\|\!theta\|_{\!H^1_h}+\|\!u\|_{\!H^1_h}+\|w\|_{H^1_h})$
and it satisfies the rigid body motion condition that
if  $(\!theta, \!u, w)$ defines a rigid body motion and $f(\!theta, \!u, w)=0$ then $(\!theta, \!u, w)=0$.
The constant $C$ could be dependent on the shell mid-surface and the shape regularity $\K$ of the triangulation
$\T_h$, but otherwise is independent of the triangulation. We shall simply say that such a constant is independent of $\T_h$.

The Koiter shell model \cite{Koiter} uses the covariant components $u_\alpha$ of the tangential displacement $u_\alpha\!a^\alpha$
and the coefficient $w$ of the normal  displacement $w\!a^3$
of the shell mid-surface as the primary variable. Such a displacement
deforms the surface $\t\Omega$ and changes its curvature and metric tensors. The linearized change in curvature tensor is the bending
strain tensor. It is expressed in terms of the displacement components as
\begin{equation}\label{K-curvature}
\rho^K_{\alpha\beta}(\!u, w)=\partial^2_{\alpha\beta}w-\Gamma^{\gamma}_{\alpha\beta}
\partial_{\gamma}w+b^{\gamma}_{\alpha|\beta}u_{\gamma}+
b^{\gamma}_{\alpha}u_{\gamma|\beta}+b^{\gamma}_{\beta}u_{\gamma|\alpha}-c_{\alpha\beta}w.
\end{equation}
The linearized change of metric tensor is the membrane strain tensor $\gamma_{\alpha\beta}(\!u,w)$ that has the same expression as in Naghdi model
\eqref{N-metric}.
There is no transverse shear in the Koiter model. Indeed the bending strain tensor \eqref{K-curvature} can be obtained from the bending strain
tensor of the Naghdi model \eqref{N-bending} by eliminating the rotation vector $\!theta$ using the zero-shear condition
that $\tau_\alpha(\!theta, \!u, w)=0$, and multiplying the result by $-1$. I.e.,
\begin{equation*}
\rho^K_{\alpha\beta}(\!u, w)=-\rho_{\alpha\beta}(\!theta, \!u, w) \text{ with } \theta_\alpha=-\partial_\alpha w-b^\beta_\alpha u_\beta.
\end{equation*}

As in the Naghdi model,
we let the boundary $\partial\t\Omega$ be divided to $\partial^D\t\Omega\cup\partial^S\t\Omega\cup\partial^F\t\Omega$.
On $\partial^D\t\Omega$ the shell is clamped, on $\partial^S\t\Omega$ the shell is simply supported, and
on $\partial^F\t\Omega$ the shell is free of displacement constraint and subject to force or moment  only.
(There are $16$ different ways to specify boundary conditions at any point on the shell boundary, of which we consider the three most
typical.)
The shell model is
defined in the Hilbert space
\begin{multline}\label{K-space}
H^K=\{(\!v, z)\in \!H^1\x H^2\ | \ v_\alpha \text{ and } z \text{ are }0\  \text{on}\ \partial^D\Omega\cup\partial^S\Omega, \\
\text{ and the normal derivative of  }z\text{ is }0\
\text{on}\ \partial^D\Omega\}.
\end{multline}
The model determines $(\!u, w)\in H^K$
such that
\begin{multline}\label{K-model}
\frac13\int_{\Omega}
a^{\alpha\beta\lambda\gamma}\rho^K_{\lambda\gamma}(\!u, w)
\rho^K_{\alpha\beta}
(\!v, z)\sqrt a\d x_1\d x_2
+\eps^{-2}\int_{\Omega}
a^{\alpha\beta\lambda\gamma}\gamma_{\lambda\gamma}(\!u,w)
\gamma_{\alpha\beta}(\!v,z)\sqrt a\d x_1\d x_2\\
=
\int_{\Omega}
(p^{\alpha}v_{\alpha}+
p^3z)\sqrt a\d x_1\d x_2
+\int_{\partial^S\t\Omega}mD_{\t\!n}z\d\t s
+\int_{\partial^F\t\Omega}\left(q^\alpha v_\alpha+q^3z+mD_{\t\!n}z\right)\d\t s
\\
\forall\ (\!v,z) \in H^K.
\end{multline}
Here, $q^i$ and $m$ are resultant loading functions on the shell boundary, which can be calculated from force resultants and
moment resultants on the shell edge \cite{Koiter}. The scalar $z$ can be viewed as
defined on $\t\Omega$. We let $\t\!n=\t n^\alpha\!a_\alpha$ be the unit outward normal to $\partial\t\Omega$ that is tangent to
$\t\Omega$. The derivative $D_{\t \!n}z:=\t n^\alpha\partial_\alpha z$ is the directional derivative  in the direction of $\t\!n$ with respect to arc length.
The elastic tensor
$a^{\alpha\beta\gamma\delta}$ is the same as in Naghdi model.
The wellposedness of the Koiter model \eqref{K-model}
hinges on a Korn's inequality for Koiter shell \cite{BCM} that  there is a constant $C$ such that
\begin{multline}\label{K-Korn-continuous}
\|\!u\|_{\!H^1}+\|w\|_{H^2}\le C
\left[\sum_{\alpha,\beta=1,2}\|\rho^K_{\alpha\beta}(\!u, w)\|^2_{L^2}+\sum_{\alpha,\beta=1,2}\|\gamma_{\alpha\beta}(\!u, w)\|^2_{L^2}+f^2(\!u, w)\right]^{1/2}
\\
\ \ \forall\
\!u\in \!H^1,\ w\in H^2.
\end{multline}
Here $f(\!u, w)$ is a semi-norm on $\!H^1\x H^2$ that satisfies the rigid body motion condition that
if $(\!u, w)$ defines a rigid body motion and $f(\!u, w)=0$ then $\!u=0$ and $w=0$.
A displacement $(\!u, w)\in \!H^1\x H^2$ defines  a rigid body motion of the shell mid-surface
if there are constant vectors $\!c$ and $\!d$ such that  $u_\alpha\!a^\alpha+w\!a^3=\!c+\!d\x\!Phi(x_1, x_2)$.
This is equivalent to $\rho^K_{\alpha\beta}(\!u, w)=0$ and  $\gamma_{\alpha\beta}(\!u, w)=0$ \cite{BCM}.

For the Koiter model, on the triangulation $\T_h$, we also need to consider piecewise $H^2$ functions that are independently defined on each element, with
the norm defined by
\begin{multline}\label{H2hnorm}
\|w\|_{H^2_h}:=
\left[\sum_{\tau\in\T_h}\|w\|^2_{2,\tau}
+\sum_{e\in \E^0_h}\left(\sum_{\alpha=1, 2}
h^{-1}_e\int_{e}\lbra \partial_\alpha w\rbra^2\d s
+
h^{-1}_e\int_{e}\lbra w\rbra^2\d s\right)\right]^{1/2}.\hfill
\end{multline}
Let $f(\!u, w)$ be a semi-norm that is continuous with respect to this norm such that there is a $C$ only dependent on $\K$ of $\T_h$
and
\begin{equation*}
|f(\!u, w)|\le C(\|\!u\|_{\!H^1_h}+\|w\|_{H^2_h})\ \forall\ (\!u, w)\in \!H^1_h\x H^2_h.
\end{equation*}
We  assume that $f$ satisfies the rigid body motion condition that
if $(\!u, w)\in \!H^1\x H^2$ defines a rigid body motion and $f(\!u, w)=0$ then $\!u=0$ and $w=0$.
We have the discrete Korn's inequality for Koiter shell that for all $\!u\in \!H^1_h$ and $w\in  H^2_h$
\begin{multline}\label{K-Korn-discrete}
\|\!u\|_{\!H^1_h}+\|w\|_{H^2_h}\le C
\left[
\sum_{\alpha,\beta=1,2}\|\rho^K_{\alpha\beta}(\!u, w)\|^2_{0,\Omega_h}+\sum_{\alpha,\beta=1,2}\|\gamma_{\alpha\beta}(\!u, w)\|^2_{0,\Omega_h}\right.
\\
\left.+\sum_{e\in \E^0_h}\left(
\sum_{\alpha=1,2}h^{-1}_e\int_{e}\lbra u_{\alpha}\rbra^2\d s
+\sum_{\alpha=1, 2}
h^{-1}_e\int_{e}\lbra \partial_\alpha w\rbra^2\d s
+
h^{-1}_e\int_{e}\lbra w\rbra^2\d s\right)
+f^2(\!v, z)
\right]^{1/2}.
\end{multline}

With this inequality established, one may add the term $\sum_{e\in \E^0_h}h^{-3}_e\int_{e}\lbra w\rbra^2\d s$ to both sides to obtain a new inequality.
Let $\lbra \partial_s w\rbra$ and $\lbra \partial_n w\rbra$ be the jumps in the tangential and normal derivatives of $w$ over and edge $e$, respectively. Then
we have the identity
\begin{equation*}
\sum_{\alpha=1}^2\lbra \partial_\alpha w\rbra^2=\lbra \partial_s w\rbra^2+\lbra \partial_n w\rbra^2.
\end{equation*}
If $w$ is a piecewise polynomial, we have the inverse inequality $\int_e\lbra \partial_s w\rbra^2\le Ch^{-2}_e\int_e\lbra w\rbra^2$.
Thus for piecewise polynomials $u_\alpha$ and
$w$, we have the following variant of discrete Korn's inequality for Koiter shell.
\begin{multline}\label{K-Korn-discrete-h-3}
\|\!u\|_{\!H^1_h}+\|w\|_{\bar H^2_h}\le C
\left[
\sum_{\alpha,\beta=1,2}\|\rho^K_{\alpha\beta}(\!u, w)\|^2_{0,\Omega_h}+\sum_{\alpha,\beta=1,2}\|\gamma_{\alpha\beta}(\!u, w)\|^2_{0,\Omega_h}\right.
\\
\left.+\sum_{e\in \E^0_h}\left(
\sum_{\alpha=1,2}h^{-1}_e\int_{e}\lbra u_{\alpha}\rbra^2\d s
+
h^{-1}_e\int_{e}\lbra \partial_nw\rbra^2\d s
+
h^{-3}_e\int_{e}\lbra w\rbra^2\d s\right)
+f^2(\!u, w)
\right]^{1/2}.
\end{multline}
Here
\begin{multline}\label{H2hnorm-h-3}
\|w\|_{\bar H^2_h}:=
\left[\sum_{\tau\in\T_h}\|w\|^2_{2,\tau}
+\sum_{e\in \E^0_h}\left(
h^{-1}_e\int_{e}\lbra \partial_n w\rbra^2\d s
+
h^{-3}_e\int_{e}\lbra w\rbra^2\d s\right)\right]^{1/2}.\hfill
\end{multline}
This inequality is useful for analysis of discontinuous Galerkin methods for Koiter shell.

In proving these discrete Korn's inequalities, a important tool is a compact embedding theorem
in the space $H^1_h$,
which is proved in Section~\ref{sec-compact}.
In Section~\ref{Korn-Naghdi} we prove the discrete Korn's inequalities \eqref{N-Korn-discrete}
for the Naghdi shell, and
Section~\ref{Korn-Koiter} is devoted Koiter shell model.
Throughout the paper, $C$ will be a generic constant that may depend on the domain $\Omega$, the mapping $\!Phi$ that defines the shell mid-surface, and shape
regularity $\K$ of a triangle, of a triangulation, or of a class of triangulations. But otherwise,
the constant is independent of triangulations. An integral $\int_\tau u(x_1, x_2)\d x_1 \d x_2$ or $\int_{\partial\Omega}v(s)\d s$
will be simply written as $\int_\tau u$ or $\int_{\partial\Omega}v$, in which
the integration variable and measure should be clear from the context.

\section{Compact embedding in the space of piecewise $H^1$ functions}
\label{sec-compact}
We need a discrete analogue of the  Rellich–-Kondrachov compact embedding theorem, which will play a fundamental role
in proving the discrete Korn's inequalities for shells.
There are several papers relevant to compact embedding in piecewise function spaces.
In \cite{Feng}, such a result is stated under  the assumption that the triangulation $\T_h$ is quasi-uniform.
In \cite{Ern}, a theorem was proved for piecewise polynomials under the assumption that the maximum mesh size tends to zero.
In \cite{Buffa}, there is a sub-mesh condition to be verified.
Although these theories are developed in more general settings, their results do not readily meet our needs.
We only assume that $\T_h$ is a shape regular triangulation of the polygon $\Omega$. Our proof also clearly
shows that the statement could break down when an interior angle of $\Omega$ tends to zero or $2\pi$.

\subsection{A trace theorem}\label{subsec-trace}
We first prove a trace theorem for functions in $H^1_h$.
This result itself is a generalization of a trace theorem of Sobolev space theory.
It will be used in proving the discrete compact embedding theorem, and in verifying the continuity
of the seminorm denoted by $f$ in the right hand side of \eqref{N-Korn-discrete}, \eqref{K-Korn-discrete}, and
 \eqref{K-Korn-discrete-h-3}.
We will need the following trace theorem on an element \cite{A-DG}.
\begin{lem}
Let $\tau$ be a triangle, and $e$ one of its edges. Then there is a
constant $C$ depending on the shape regularity of $\tau$ such that
\begin{equation}\label{trace}
\int_eu^2\le C\left[h_e^{-1}\int_{\tau}u^2+h_e\sum_{\alpha=1}^2\int_{\tau}|\partial_\alpha u|^2\right]
\ \ \forall\ u\in H^1(\tau).
\end{equation}
\end{lem}
\begin{thm}\label{tracetheorem}
Let $\T_h$ be a shape regular, but not necessarily quasi-uniform triangulation of $\Omega$. There
exists a constant $C$ such that
\begin{equation}\label{Omega-trace}
\|u\|_{L^2(\partial\Omega)}\le C \|u\|_{H^1_h}\ \ \forall\ u\in H^1_h.
\end{equation}
\end{thm}
\begin{proof}
Let $\!phi$ be a piecewise smooth
vector field on $\Omega$ whose normal component is continuous across any straight line segment, and
such that $\!phi\cdot\!n=1$ on $\partial\Omega$.
On each element
$\tau\in\T_h$, we have
\begin{equation*}
\int_{\partial\tau}u^2\!phi\cdot\!n=\int_\tau \div(u^2\!phi)=
\int_\tau(2u\nabla u\cdot\!phi+u^2\div\!phi).
\end{equation*}
Here $\nabla u$ is the gradient of $u$. Summing up the above equations over all elements of $\T_h$, we get
\begin{equation*}
\int_{\partial\Omega}u^2=-\sum_{e\in\E^0_h}\int_e\lbra u^2\!phi\rbra
+\int_{\Omega_h}(2u\nabla u\cdot\!phi+u^2\div\!phi).
\end{equation*}
If $e$ is the border between the elements
$\tau_1$ and
$\tau_2$ with outward normals $\!n_1$ and $\!n_2$, then
$\lbra u^2\!phi\rbra=u^2_1\!phi_1\cdot\!n_1+u^2_2\!phi_2\cdot\!n_2$,
where $u_1$ and $u_2$ are restrictions of $u$
on $\tau_1$ and $\tau_2$, respectively.  It is noted that although $\!phi$ may be discontinuous
across $e$, it normal component is continuous, i.e., $\!phi_1\cdot\!n_1+\!phi_2\cdot\!n_2=0$.
On the edge $e$, we have
$|\lbra u^2\!phi\rbra|\le|\lbra u^2\rbra|\|\!phi\|_{0,\infty,\Omega}$.
Here, $|\lbra u^2\rbra|=|u_1^2-u_2^2|=2|\lbra u\rbra\lbrac u\rbrac|$,
with $\lbrac u\rbrac=(u_1+u_2)/2$ being the average. We have
\begin{multline}\label{jump-on-e}
\int_e|\lbra u^2\!phi\rbra|\le
2|\!phi|_{0,\infty,\Omega}
\left[h_e^{-1}\int_e\lbra u\rbra^2\right]^{1/2}\left[h_e\int_e\lbrac u\rbrac^2\right]^{1/2}\\
\le C
|\!phi|_{0,\infty,\Omega}\left[h_e^{-1}\int_e\lbra u\rbra^2\right]^{1/2}
\left[\int_{\delta e}u^2+h_e^2\int_{\delta e}|\nabla u|^2\right]^{1/2}.
\end{multline}
Here, $|\!phi|_{0,\infty,\Omega}$ is the Sobolev norm in the space $[W^{0,\infty}(\Omega)]^2$ and $C$ only depends on the shape regularity of $\tau_1$ and $\tau_2$.
We used $\delta e=\tau_1\cup\tau_2$ to denote the ``co-boundary''
of edge $e$,
and we used the trace
estimate \eqref{trace}.
It then follows from the Cauchy--Schwarz inequality that
\begin{equation*}
\|u\|^2_{L^2(\partial\Omega)}\le C(|\!phi|_{0,\infty,\Omega}+|\div\!phi|_{0,\infty,\Omega})
\left[\|u\|^2_{L^2}+\int_{\Omega_h}|\nabla u|^2
+\sum_{e\in\E^0_h}\frac{1}{|e|}\int_e\lbra u\rbra^2\right].
\end{equation*}
Here the constant $C$ only depends on the shape regularity of $\T_h$.
The dependence on $\Omega$ of the $C$ in \eqref{Omega-trace}
is hidden in the $\!phi$ in the above inequality.
\end{proof}

\subsection{Compact embedding in $H^1_h$}
\label{subsec-compact}
For $\delta>0$, we define a boundary strip $\Omega_\delta$ of width $\mathcal{O}(\delta)$ for the domain $\Omega$.
Let $\Omega_\delta^0=\Omega\setminus \overline{\Omega_\delta}$ be the interior domain.
The interior domain $\Omega^0_\delta$ has the property that if a point is in $\Omega^0_\delta$, then the
disk centered at the point with radius $\delta$ entirely lies in $\Omega$.
We first show that when the strip is thin, the $L^2(\Omega_\delta)$
norm of the restriction on $\Omega_\delta$ of a function in $H^1_h$ must be small.
\begin{lem}\label{bd-shift-theorem}
There is a constant $C$ such that
\begin{equation}\label{stripe}
\int_{\Omega_{\delta}}u^2\le C\delta\|u\|^2_{H^1_h}\ \ \forall\ u\in H^1_h.
\end{equation}
Here $\Omega_\delta$ is a boundary strip of width $\delta$ attached to $\partial\Omega$.
\end{lem}
\begin{proof}
We choose a piecewise smooth a vector field $\!phi$ whose normal component is continuous
across any curve such that $\!phi=0$ on the inner boundary of $\Omega_{\delta}$, and
$\div\!phi=1$ and $|\!phi|\le C\delta$ on $\Omega_{\delta}$.
(A construction of such $\!phi$ is given below.)
We then extend $\!phi$ by zero onto the entire domain $\Omega$. The extended, still denoted by $\!phi$,
is a piecewise smooth vector field whose normal components
is continuous over any curve in $\Omega$.
We thus have
\begin{equation*}
\int_{\Omega_{\delta}}u^2=
\int_{\Omega}u^2\div\!phi
=\sum_{\tau\in\T_h}\int_{\tau}u^2\div\!phi=
-\sum_{\tau\in\T_h}\int_{\tau}2u\nabla u\cdot\!phi+
\sum_{\tau\in\T_h}\int_{\partial\tau}u^2\!phi\cdot\!n.
\end{equation*}
The last term can be written as
\begin{equation*}
\sum_{\tau\in\T_h}\int_{\partial\tau}u^2\!phi\cdot\!n=
\int_{\partial\Omega}u^2\!phi\cdot\!n+\sum_{e\in\E^0_h}\int_e\lbra u^2\!phi\rbra.
\end{equation*}
Since the normal components of $\!phi$ is continuous
on edges in $\E^0_h$, we use the same argument as in the proof of Theorem~\ref{tracetheorem}, cf., \eqref{jump-on-e},
to get
\begin{equation*}
\sum_{e\in\E^0_h}\int_e|\lbra u^2\!phi\rbra|\le C
|\!phi|_{0,\infty,\Omega}\left[\|u\|^2_{L^2}+\sum_{\tau\in\T_h}\int_{\tau}h^2_\tau|\nabla u|^2
+\sum_{e\in\E^0_h}\frac{1}{h_e}\int_e\lbra u\rbra^2\right].
\end{equation*}
It then follows from Theorem~\ref{tracetheorem} that
\begin{equation} \label{bd-shift}
\int_{\Omega_{\delta}}u^2\le C
|\!phi|_{0,\infty,\Omega}\left[\|u\|^2_{L^2}+\int_{\Omega_h}|\nabla u|^2
+\sum_{e\in\E^0_h}\frac{1}{h_e}\int_e\lbra u\rbra^2\right].
\end{equation}
The proof is complete since $|\!phi|_{0,\infty, \Omega}\le C\delta$.
The constant $C$ depends on $\Omega$ in terms of its interior angles and exterior angles at convex and
concave vertexes, respectively.
\end{proof}

We describe a way to choose the boundary strip and construct the vector field $\!phi$ that was
used in the proof.
This field can be constructed by piecing together several special vector fields.
We need some vector fields on rectangles, wedges, and circular disks.
On the $xy$-plane, we consider the vertical rectangular strip $R=(0,\delta)\x (0, l)$.
On this strip, we consider $\!psi_R=\langle x, 0\rangle$. This vector
field satisfies the condition that $\div\!psi_R=1$, $\!psi_R=0$ on the left side,
$\!psi_R\cdot\!n=0$ on the top and bottom sides and the maximum of $|\!psi_R|$ is $\delta$ that is attained
on the right side.
On a wedge $W$ with its vertex at the origin, we consider the vector field
$\!psi_W=\langle x, y\rangle/2$.
This $\!psi_W$ satisfies the conditions that
$\div\!psi_W=1$, $\!psi_W\cdot\!n=0$ on the two sides of $W$, and $|\!psi_W|=\rho/2$
at a point in $W$ whose distant from the origin is $\rho$.
On a circular disk $C$ centered at the origin and of radius $\rho$, we consider the vector field
$\!psi_C=(1-\rho^2/r^2)\langle x, y\rangle/2$. Here $r=(x^2+y^2)^{1/2}$.
This vector field satisfies the condition that $\div\!psi_C=1$ on the disk except at the center
where it is singular. It points toward the center  every where. And it is zero on the boundary.
We use this field on a sector of the circle $C$, on the two radial sides of which
we have $\!psi_C\cdot\!n=0$.
\begin{figure}[!ht]
\centerline{\input{boundarystrip.pspdftex}}
\caption{\label{boundarystrip}Boundary strip $\Omega_\delta$ near a convex vertex (left)
and a concave vertex (right).}
\end{figure}
With these special vector fields, we can then assemble the $\!phi$ on a boundary strip
$\Omega_\delta$. Along the interior side
of each straight segment of
$\partial\Omega$ we choose a uniform strip of thickness $\delta$. These strips overlap near
vertexes. If $\Omega$ is convex at a vertex, we introduce a wedge whose vertex is
at the intersection of the interior boundary of the uniform strips, and whose sides are orthogonal
to the meeting straight segments, see the left figure of Figure~\ref{boundarystrip}.
If $\Omega$ is concave at a vertex, we resolve it by using a circular sector, centered near the vertex
and outside of $\Omega$. The radius of the circle is slightly bigger than $\delta$
such that the arc is continuously connected to the
interior edges of the meeting strips, and the two radial sides are orthogonal to the meeting boundary
segments, see the right figure in Figure~\ref{boundarystrip}.
With such treatment of the vertices, the boundary strip $\Omega_\delta$
is composed of
rectangular strips attaching to major portions of straight segments of
$\partial\Omega$, portion of wedges at convex vertexes, and portion of circular sectors
at concave vertexes, see the shaded region in Figure~\ref{boundarystrip}.
We then transform $\!psi_R$, $\!psi_W$, and $\!psi_C$ to various parts of $\Omega_\delta$
and assemble a $\!phi$
on $\Omega_\delta$. The vector field $\!phi$ thus constructed is zero on the interior boundary
of $\Omega_\delta$. Its normal components are continuous across any curve, and
$\div\!phi=1$ on $\Omega_\delta$.
The thickness of $\Omega_\delta$ is the constant $\delta$ for the rectangular part.
It is maximized to $\delta/\sin\frac{\theta}2$ at a convex vertex. It is minimized to $\alpha\delta$
at the concave vertex, with $0<\alpha<1$, a value can be chosen as, for example, $1/2$. The norm $|\!phi|$
has a maximum $\delta/\sin\frac{\theta}2$ at a convex vertex with $\theta$ being the interior angle.
Thus when $\theta$ is small, $|\!phi|$ is big, and the estimate
\eqref{bd-shift} deteriorates.
The norm $|\!phi|$ also has a local maximum near a concave vertex.
It is bounded as
\begin{equation*}
|\!phi|\le\delta\frac{1-\alpha\sin\frac{\theta}{2}}{(1-\alpha)\sin\frac{\theta}{2}}.
\end{equation*}
When the exterior angle is small this maximum would be  big.
Also, one needs to choose $\alpha$ away from $1$ and $0$,
to maintain a moderate thickness of the strip and a reasonable bound for $|\!phi|$
which affect the estimate \eqref{bd-shift}.
We remark that the constant $C$ in the estimate \eqref{stripe} could tend to infinity if
an interior angle of the polygonal domain $\Omega$ tends to zero or $2\pi$.

We then prove that functions in $H^1_h$ are ``shift-continuous'' in $L^2$, as
stated in the following lemma. We extend a function $u\in H^1_h$
to a function $\tilde u$ on the whole $\RR^2$ by zero.
\begin{lem}\label{shift-continuity}
There is a constant $C$  such that
\begin{equation}\label{whole-shift}
\int_{\RR^2}[\tilde u(\!x+\!rho)-\tilde u(\!x)]^2\mathrm{d}\!x\le C|\!rho|\|u\|^2_{H^1_h}\ \ \forall\ u\in H^1_h.
\end{equation}
\end{lem}
\begin{proof}
Because for an element $\tau\in\T_h$, smooth functions are dense in $H^1(\tau)$, we only need
to prove \eqref{whole-shift}
for functions that are smooth on each element of $\T_h$. Let $u$ be such a piecewise smooth function.
Let $\!rho$ be an arbitrary short vector. We take $\delta=|\!rho|$ and choose a boundary strip $\Omega_\delta$.
The interior part $\Omega^0_{\delta}$ of the domain has the property that
if $\!x\in \Omega^0_{\delta}$ then the line segment $[\!x, \!x+\!rho]\subset \Omega$. We have
\begin{equation*}
\int_{\RR^2}[\tilde u(\!x+\!rho)-\tilde u(\!x)]^2\mathrm{d}\!x
=\int_{\Omega_\delta^0}[u(\!x+\!rho)-u(\!x)]^2\mathrm{d}\!x
+\int_{\RR^2\setminus\Omega_\delta^0}[\tilde u(\!x+\!rho)-\tilde u(\!x)]^2\mathrm{d}\!x.
\end{equation*}
Using Lemma~\ref{bd-shift-theorem}, we bound the second term as
\begin{equation}\label{bd-strip-est}
\int_{\RR^2\setminus\Omega_\delta^0}[\tilde u(\!x+\!rho)-\tilde u(\!x)]^2\mathrm{d}\!x\le
\int_{\Omega_{2\delta}}u^2(\!x)\mathrm{d}\!x\le C|\rho|\|u\|^2_{H^1_h}.
\end{equation}
We then
focus on the first term.
This integral can be taken on an equal measure subset of
$\Omega^0_\delta$. This subset is obtained by removing a zero measure subset that
is composed of such point $\!x$:
$\!x$ or $\!x+\!rho$ is on an open edge $e\in\E^0_h$,
or the closed straight line segment $[\!x, \!x+\!rho]$ connecting $\!x$ and $\!x+\!rho$ contains any vertex of $\T_h$,
or $[\!x, \!x+\!rho]$ overlaps some edges of $\E^0_h$.
By such exclusion, for any $\!x$ in the remaining set, both the ends of $[\!x, \!x+\!rho]$
are in the interior of some open triangular elements, and $[\!x, \!x+\!rho]$ contains no
vertex. The restriction of $u$ on $[\!x, \!x+\!rho]$ is a piecewise smooth one dimensional function,
which may have
a finite number of jumping points in the open straight line segment $(\!x, \!x+\!rho)$.
By the fundamental theorem
of calculus, we have
\begin{equation*}
u(\!x+\!rho)-u(\!x)=\int_0^1\nabla u(\!x+t\!rho)\cdot\!rho \mathrm{d}t+\sum_{p\in
[\!x, \!x+\!rho]\cap\E^0_h}\lbra u\rbra_p
\end{equation*}
Note that the integrand in the integral may make no sense at $t$, if
$\!x+t\!rho\in\E^0_h$, where $u$ may jump. These points are excluded from the integration, where the
jumping effect is resolved in the second term.
On the segment $[\!x, \!x+\!rho]$, $u$ may have a jump at $p\in [\!x, \!x+\!rho]\cap\E^0_h$, which is denoted by
$\lbra u\rbra_p$ that is the value of $u$ from the side of $\!x$ minus that from the side of $\!x+\!rho$.
We thus have
\begin{equation*}
[u(\!x+\!rho)-u(\!x)]^2\le
|\!rho|^2\int_0^1|\nabla u(\!x+t\!rho)|^2\mathrm{d}t+\left[\sum_{p\in[\!x, \!x+\!rho]\cap\E^0_h}\lbra u\rbra_p\right]^2.
\end{equation*}
When we take integral on $\Omega^0_\delta$ (minus the aforementioned zero-measure subset),
the first term is bounded as follows.
\begin{equation*}
\int_{\Omega^0_{\delta}}
|\!rho|^2\int_0^1|\nabla u(\!x+t\!rho)|^2\mathrm{d}t\mathrm{d}\!x=
|\!rho|^2\int_0^1\int_{\Omega^0_{\delta}}|\nabla u(\!x+t\!rho)|^2\mathrm{d}\!x\mathrm{d}t\\
\le
|\!rho|^2\int_{\Omega_h}|\nabla u|^2\mathrm{d}\!x.
\end{equation*}

To estimate the jumping related second term, we write $\lbra u\rbra_p=h_e^{1/2}h_e^{-1/2}\lbra u\rbra_p$
if $p\in [\!x, \!x+\!rho]\cap \E^0_h$ is on the edge $e$, and
use the Cauchy-Schwarz inequality to obtain the following estimate.
\begin{equation*}
\left[\sum_{p\in[\!x, \!x+\!rho]\cap\E^0_h}\lbra u\rbra_p\right]^2\le
\left[\sum_{e\cap[\!x, \!x+\!rho]\ne\emptyset}h_e^{-1}\lbra u\rbra^2_{e\cap[\!x, \!x+\!rho]}\right]
\left[\sum_{e\cap[\!x, \!x+\!rho]\ne\emptyset}h_e\right].
\end{equation*}
We show below that
there is a $C$, depending on the domain $\Omega$ and the
shape regularity $\K$ of $\T_h$ , such that
\begin{equation}\label{zigzag}
\sum_{e\cap[\!x, \!x+\!rho]\ne\emptyset}h_e\le C.
\end{equation}
We then have
\begin{equation*}
\int_{\Omega^0_{\delta}}\left[\sum_{p\in
[\!x, \!x+\!rho]\cap\E^0_h}\lbra u\rbra_p\right]^2\le
C\int_{\Omega^0_{\delta}}
\sum_{e\cap[\!x, \!x+\!rho]\ne\emptyset}h_e^{-1}\lbra u\rbra^2_{e\cap[\!x, \!x+\!rho]}\mathrm{d}\!x.
\end{equation*}
Every term in the right hand side is associated with a particular edge $e\in \E^0_h$. Each edge
$e\in\E^0_h$ is relevant to at most the points
in the parallelogram $\Omega_e$ in the Figure~\ref{shift}.
Thus, by changing the order of sum and integral, we have
\begin{multline*}
\int_{\Omega^0_{\delta}}
\sum_{e\cap[\!x, \!x+\!rho]\ne\emptyset}h_e^{-1}\lbra u\rbra^2_{e\cap[\!x, \!x+\!rho]}\mathrm{d}\!x
\le
\sum_{e\in\E^0_h}h_e^{-1}\int_{\Omega_e}\lbra u\rbra^2_{e\cap[\!x, \!x+\!rho]}\mathrm{d}\!x\\
\le
\sum_{e\in\E^0_h}h_e^{-1}\sin\langle\!rho, e\rangle|\!rho|\int_{e}\lbra u\rbra^2
\le
|\rho|\sum_{e\in\E^0_h}h_e^{-1}\int_{e}\lbra u\rbra^2.
\end{multline*}
Here, $\langle\!rho, e\rangle$ is the angle between the vector $\!rho$ and the edge $e$.
\begin{figure}[!ht]
\centerline{\input{edge-shift.pspdftex}}
\caption{\label{shift}
A $\!rho$ shift and $\Omega_e$ for an edge $e$.}
\end{figure}
Therefore, we have
\begin{equation*}
\int_{\Omega^0_{\delta}}[u(\!x+\!rho)-u(\!x)]^2d\!x\le
|\!rho|^2|\nabla u|^2_{0, \Omega_h}+|\!rho|\sum_{e\in\E^0_h}h_e^{-1}\int_e\lbra u\rbra^2.
\end{equation*}
Note that the second term may carry the smaller coefficient $|\!rho|\max\{h,|\!rho|\}$
such that the two terms
are closer in the order. But we do not need such refined estimates.
We thus proved
\begin{equation}\label{interior-shift}
\int_{\Omega^0_{\delta}}[u(\!x+\!rho)-u(\!x)]^2\mathrm{d}\!x\le C|\!rho|\|u\|^2_{H^1_h}\ \ \forall\ u\in H^1_h.
\end{equation}
The shift continuity \eqref{whole-shift} then follows from \eqref{interior-shift} and  \eqref{bd-strip-est}.
We have shown that the set of zero extended functions is shift continuous
in $L^2(\RR^2)$.
\end{proof}
We give a proof for the estimate \eqref{zigzag}.
Let $l$ be a straight line cutting through $\Omega$. Let $\T_h$ be a shape regular triangulation
with regularity constant $\K$. Then the sum of lengths of mesh line segments
intersecting $l$ is bounded independent of the triangulation. More specifically,
we prove that there is a constant $C$, depending on the shape regularity $\K$,
but otherwise independent of the triangulation $\T_h$
such that
\begin{equation}\label{zigzag1}
\sum_{e\in\E_h\text{ and }e\cap l\ne\emptyset}h_e\le C.
\end{equation}
We shall use some facts that follow from the shape regularity assumption.
There is a minimum angle $\theta_\K$ among all angles of triangles of $\T_h$. The number of edges
sharing a vertex is bounded by a constant $C$ that only depends on $\K$.
Let $e_1$ and $e_2$ be two edges
sharing a
vertex. There are constants $C_1$ and $C_2$ depending on $\K$ such that
$h_{e_1}\le C_1 h_{e_2}$ and $h_{e_2}\le C_2 h_{e_1}$.
Without loss of generality, we assume $l$ is horizontal. We also
assume that $l$ does not pass any vertex. (This assumption was already made in the context of \eqref{zigzag}.
It can be removed by a slight modification
of the following argument.)
\begin{figure}[!ht]
\centerline{\input{cut-triangle.pspdftex}}
\caption{\label{cut-tri}A line cutting through the triangulation.}
\end{figure}
We first trim the set of intersecting edges  $\{e\in\E_h\text{ and }e\cap l\ne\emptyset\}$ to simplify the set without
significantly reducing its sum of lengths of all edges.
Consider the left most edge intersecting $l$,  of which only one end vertex
is shared by some other edges intersecting $l$. Let $A$ be this end vertex, and assume it
is above $l$. We examine all the edges intersecting $l$ and sharing $A$ in the counterclockwise order.
We discard all such edges but the last one that is $AB$ in Figure~\ref{cut-tri}.
(The next edge sharing $A$, as $AC$, does not intersect $l$.) The edge $BC$ must intersect $l$.
There could be other edges intersecting $l$ and sharing the vertex $B$.
We examine all the edges sharing $B$ and intersecting $l$ in the clockwise order. We discard
all but the last one. (It is $BC$ in the figure.) Now the vertex $C$ is in the same situation as $A$, and
we can determine the edge $CD$ using the same rule as for $AB$. Then we determine $DE$, $EF$, and so forth.
We do the trimming all the way
to the right end of $l$. This procedure touches every edge
intersecting $l$, by either trimming an edge off or keeping it. Every edge deleted has at least one vertex-sharing
edge retained.
The remaining edges constitute a continuous piecewise straight path
as represented by the thick line in the figure. We denote this set by $\E^l_h$.
It follows from the aforementioned facts about the shape regular triangulation that there is
a constant $C$, depending on $\K$ only, such that
\begin{equation*}
\sum_{e\in\E_h\text{ and }e\cap l\ne\emptyset}h_e\le C\sum_{e\in\E^l_h}h_e.
\end{equation*}
We consider a typical triangle bounded by $l$ and $\E^l_h$, as the shaded one
in the figure, whose sides are $a$, $b$, and $c$. Let the angle $\angle DEF$ be denoted by $\theta$.
Then $\theta\ge\theta_\K$. Note that
$c^2=a^2+b^2-2ab\cos\theta$. If $\theta$ is obtuse, then $a+b\le \sqrt 2 c$.
Otherwise, we have $c^2=(a^2+b^2)(1-\cos\theta)+(a-b)^2\cos\theta\le (a^2+b^2)(1-\cos\theta)$. Thus
$a+b\le\sqrt{\frac2{1-\cos\theta}}c$. In any case, we have $a+b\le\sqrt{\frac2{1-\cos\theta_\K}}c$.
We thus proved
\begin{equation*}
\sum_{e\in\E^l_h\cap\E^0_h}h_e\le\sqrt{\frac2{1-\cos\theta_\K}}|l\cap\Omega|.
\end{equation*}
From this, \eqref{zigzag1} follows. Here $|l\cap\Omega|$ is the length of the line segment $l\cap\Omega$ which does exceed the diameter of $\Omega$.
We can now prove the following compact embedding theorem.
\begin{thm}\label{uniformcompactembedding}
Let $\T_{h_i}$ be a (infinite) class of shape regular but not necessarily quasi-uniform
triangulations of the polygonal domain $\Omega$, with a shape regularity constant $\K$.
For each $i$, let $H^1_{h_i}$ be the space of piecewise $H^1$ functions, subordinated to the
triangulation $T_{h_i}$,
equipped with the norm \eqref{H1hnorm}.
Let $\{u_i\}$ be a sequence such that $u_i\in H^1_{h_i}$ for each $i$ and
there is a constant $C$, such that $\|u_i\|_{H^1_{h_i}}\le C$ for all $i$.
Then, the sequence $\{u_i\}$ has a convergent subsequence in $L^2$.
\end{thm}
\begin{proof}
It follows from \eqref{whole-shift} that the sequence $\{u_i\}$ is a shift-continuous
subset of $L^2$. This, together with the estimate \eqref{tracetheorem}, verifies the
condition for a subset of $L^2$ to be compact, see Theorem~2.12 in \cite{Adams}.
\end{proof}

\section{Discrete Korn's inequality for Naghdi shell}
\label{Korn-Naghdi}
In this section, we prove the discrete Korn's inequality \eqref{N-Korn-discrete} for the Naghdi shell model.
We let $H_h=\!H^1_h\x \!H^1_h\x H^1_h$, and define a norm in this space by
\begin{equation}\label{h-norm}
\|(\!theta, \!u, w)\|_{H_h}:=\left[\sum_{\alpha=1,2}\left(\|\theta_\alpha\|^2_{H^1_h}+\|u_\alpha\|^2_{H^1_h}\right)+\|w\|^2_{H^1_h}\right]^{1/2}.
\end{equation}
Recall that the norm in $H^1_h$ is defined by \eqref{H1hnorm}.
Let $f(\!theta, \!u, w)$ be a semi-norm that is continuous with respect to the $H_h$  norm such that there is a $C$ and
\begin{equation}\label{N-f-continuous}
|f(\!theta, \!u, w)|\le C\|(\!theta, \!u, w)\|_{H_h}\ \forall\ (\!theta, \!u, w)\in H_h.
\end{equation}
We also assume that $f$ satisfies the condition that
if $(\!theta, \!u, w)\in  \!H^1\x \!H^1\x H^1$ defines a rigid body motion, as explained in the introduction,
and $f(\!theta, \!u, w)=0$ then $\!theta=0$, $\!u=0$, and $w=0$.

We define a discrete energy norm
on the space $H_h$ by
\begin{multline}\label{triple-norm}
\|(\!theta, \!u, w)\|_{E_h}:=
\left[\sum_{\alpha,\beta=1,2}\left(\|\rho_{\alpha\beta}(\!theta, \!u, w)\|^2_{0, \Omega_h}+\|\gamma_{\alpha\beta}(\!u, w)\|^2_{0,\Omega_h}\right)+
\sum_{\alpha=1,2}\|\tau_\alpha(\!theta, \!u, w)\|^2_{0,\Omega_h}\right.
\\
\left.+\sum_{e\in \E^0_h}h^{-1}_e\left(
\sum_{\alpha=1,2}\int_{e}\left(\lbra \theta_{\alpha}\rbra^2+\lbra u_{\alpha}\rbra^2\right)
+\int_{e}\lbra w\rbra^2\right)
+f^2(\!theta,\!u, w)\right]^{1/2}.
\end{multline}
The main result of this section is the  following theorem.
\begin{thm}\label{N-Korn-thm}
We assume that $\Omega\subset\RR^2$ is a polygon, on which $\T_h$ is a shape regular but not necessarily quasi-uniform
triangulation with a regularity constant $\K$. Let $f(\!theta, \!u, w)$ be a seminorm on $H_h$
satisfies the condition \eqref{N-f-continuous}, and the rigid body motion condition.
There exists a constant $C$ that could be dependent on the shell mid-surface
and shape regularity of the triangulation, but otherwise independent of the triangulation,
such that
\begin{equation}\label{N-Korn-inequality}
\|(\!theta, \!u, w)\|_{H_h}\le C\|(\!theta, \!u, w)\|_{E_h}\ \forall\ (\!theta, \!u, w)\in H_h.
\end{equation}
\end{thm}
To prove this theorem, we need a discrete Korn's inequality for plane elasticity for piecewise functions in $H^1_h$, see (1.21) of \cite{Brenner-2}.
It says
that there is a constant $C$ that might be dependent on the domain $\Omega$ and the shape regularity $\K$ of the triangulation $\T_h$, but
otherwise independent of $\T_h$ such that
\begin{equation}\label{Korn-Brenner}
\sum_{\alpha=1,2}\|u_\alpha\|^2_{H^1_h}\le C\left[\sum_{\alpha=1,2}\|u_\alpha\|^2_{0,\Omega_h}
+\sum_{\alpha, \beta=1,2}\|e_{\alpha\beta}(\!u)\|^2_{0,\Omega_h}+\sum_{e\in\E^0_h}
h^{-1}_e\int_{e}\sum_{\alpha=1,2}\lbra u_{\alpha}\rbra^2\right].
\end{equation}
Here $e_{\alpha\beta}(\!u)=(\partial_\beta u_\alpha+\partial_\alpha u_\beta)/2$ is the plane elasticity strain which is the symmetric part of the gradient of $\!u$.

\begin{proof}[Proof of Theorem~\ref{N-Korn-thm}]
From \eqref{Korn-Brenner},  the definitions \eqref{N-bending}, \eqref{N-metric}, and \eqref{N-shear}
of $\rho_{\alpha\beta}$, $\gamma_{\alpha\beta}$, and $\tau_\alpha$,
and the
definition of the discrete energy norm \eqref{triple-norm}, we see that
there is a constant $C$ such that
\begin{multline}\label{Korn-thm-proof1}
\|(\!theta, \!u, w)\|_{H_h}^2\le C\left[\|(\!theta,\!u, w)\|_{E_h}^2+\sum_{\alpha=1,2}\left(\|\theta_\alpha\|^2_{0,\Omega_h}+\|u_\alpha\|^2_{0, \Omega_h}\right)
+\|w\|^2_{0,\Omega_h}\right]\\
\forall\ (\!theta, \!u, w)\in H_h.
\end{multline}
On a fixed triangulation $\T_h$, it then follows from the Rellich--Kondrachov
compact embedding theorem and Peetre's lemma (Theorem 2.1, page 18 in \cite{Raviart}) that
there is a constant $C_{\T_h}$ such that
\begin{equation*}
\|(\!theta, \!u, w)\|_{H_h}\le C_{\T_h}\|(\!theta,\!u, w)\|_{E_h}\ \ \forall\ (\!theta, \!u, w)\in H_h.
\end{equation*}
We need to show that for a class  of shape regular triangulations, such $C_{\T_h}$ has an upper bound that only
depends on the shape regularity $\K$ of the whole class.
If this is not true, there would exist a sequence
of triangulations $\{\T_{h_n}\}$ and an associated sequence of functions $(\!theta^n, \!u^n, w^n)$ in
$H_{h_n}$
such that
\begin{equation}\label{N-energy-to-0}
\|(\!theta^n, \!u^n, w^n)\|_{H_{h_n}}=1 \text{ and  }\|(\!theta^n,\!u^n, w^n)\|_{E_{h_n}}\le 1/n.
\end{equation}
It follows from Theorem~\ref{uniformcompactembedding}
that there is a subsequence,
still denoted by $(\!theta^n, \!u^n, w^n)$, such that
\begin{equation}\label{converge-in-L2}
\lim_{n\to\infty}(\!theta^n, \!u^n, w^n)=(\!theta^0, \!u^0, w^0)\text{ in }\!L^2\x\!L^2\x L^2.
\end{equation}
We show that the limiting functions $\theta^0_\alpha$, $u^0_\alpha$, and $w^0$ are all in $H^1$,
this limit defines a rigid body motion,  and it is zero, which will lead to a contradiction.

First, we show that $w^0$ is actually in $H^1$ and we have that
$\partial_\alpha w^0+\theta^0_\alpha+b^\beta_\alpha u^0_\beta=0$.
Since $\lim_{n\to\infty}(\!theta^n, \!u^n, w^n)=(\!theta^0, \!u^0, w^0)$ in $\!L^2\x\!L^2\x L^2$,
in view of the definition \eqref{N-shear}, we have
\begin{equation*}
\lim_{n\to\infty}\tau_\alpha(\!theta^n, \!u^n, w^n)=\tau_\alpha(\!theta^0, \!u^0, w^0)\ \text{ in } H^{-1}.
\end{equation*}
In the above expressions, the derivatives $\partial_\alpha w^n$ and $\partial_\alpha w^0$ are understood in distributional sense.

For any compactly supported smooth functions $\phi^\alpha\in\D(\Omega)$, for each $n$, we have
\begin{multline*}
\langle\tau_\alpha(\!theta^n, \!u^n, w^n),\phi^\alpha\rangle
=
-\int_{\Omega_{h_n}}w^n\partial_\alpha\phi^\alpha
+\int_{\Omega_{h_n}}(\theta^n_\alpha+b^\beta_\alpha u^n_\beta)\phi^\alpha
\\=
\int_{\Omega_{h_n}}(\partial_\alpha w^n+\theta^n_\alpha+b^\beta_\alpha u^n_\beta)\phi^\alpha-
\sum_{e\in\E^0_{h_n}}\int_e\lbra w^n\rbra_{n_\alpha}\phi^\alpha.
\end{multline*}
Here, $n_\alpha$ are the components of the unit normal $\!n$ to the edge $e$.
If $e$ is shared by $\tau_1$ and $\tau_2$ with unit outward normals being $\!n_1$ and $\!n_2$, on which the restrictions of
$w$ are $w_1$ and $w_2$,
then $\lbra w\rbra_{n_\alpha}=w_1n_{1\alpha}+w_2 n_{2\alpha}$ is the jump of $w$ with respect to $\!n$.
Summation convention is also used in $\lbra w^n\rbra_{n_\alpha}\phi^\alpha$ with $\alpha$ being viewed as a repeated
sub and super scripts.
Using H\"older inequality and the trace inequality \eqref{trace}, we get
\begin{multline*}
|\langle\tau_\alpha(\!theta^n, \!u^n, w^n),\phi^\alpha\rangle|\le
\|\tau_\alpha(\!theta^n,\!u^n,w^n)\|_{0,\Omega_{h_n}}\|\phi^\alpha\|_{0,\Omega}\\
+
C\left[\sum_{e\in\E^0_{h_n}}h^{-1}_e\int_e\lbra w^n\rbra^2\right]^{1/2}\sum_{\alpha=1}^2\left[|\phi^\alpha|^2_{0,\Omega}+
\sum_{\tau\in\T_{h_n}}h^2_\tau|\phi^\alpha|^2_{1,\tau}\right]^{1/2}.
\end{multline*}
Since $\|(\!theta^n,\!u^n, w^n)\|_{E_{h_n}}\to 0$,
we have
\begin{equation*}
\lim_{n\to\infty}\tau_\alpha(\!theta^n, \!u^n, w^n)=0\ \text{ in } H^{-1}.
\end{equation*}
Therefore, in $H^{-1}$, we have
$\tau_\alpha(\!theta^0, \!u^0, w^0)=\partial_\alpha w^0+\theta^0_\alpha+b^\beta_\alpha u^0_\beta=0$.
Since $\theta^0_\alpha$ and $u^0_\alpha$ are in $L^2$, this equation shows that the weak derivatives of $w^0$ are in $L^2$. Thus we have $w^0\in H^1$ and
\begin{equation}\label{shear=0}
\tau_\alpha(\!theta^0, \!u^0, w^0)=\partial_\alpha w^0+\theta^0_\alpha+b^\beta_\alpha u^0_\beta=0.
\end{equation}

Next we show $u^0_\alpha\in H^1$ and $\gamma_{\alpha\beta}(\!u^0, w^0)=0$.
We have $\lim_{n\to\infty}\gamma_{\alpha\beta}(\!u^n, w^n)=\gamma_{\alpha\beta}(\!u^0, w^0)$ in $H^{-1}$, in which the derivatives
$\partial_\alpha u^n_\beta$ and $\partial_\alpha u^0_\beta$ are understood in the distributional sense.
Let $\phi^{\alpha\beta}$ be an arbitrary symmetric
tensor valued function  with components in $\D(\Omega)$. For each $n$ we have
\begin{multline*}
\langle \gamma_{\alpha\beta}(\!u^n, w^n),\phi^{\alpha\beta}\rangle=-\int_{\Omega}u^n_\alpha\partial_\beta\phi^{\alpha\beta}
-\int_{\Omega}(\Gamma^\lambda_{\alpha\beta}u^n_\lambda+b_{\alpha\beta}w^n)\phi^{\alpha\beta}\\
=\int_{\Omega_{h_n}}\gamma_{\alpha\beta}(\!u^n, w^n)\phi^{\alpha\beta}-\sum_{e\in\E^0_{h_n}}\int_e\lbra u^n_{\alpha}\rbra_{n_\beta}\phi^{\alpha\beta}.
\end{multline*}
It follows from this equation and the assumption
$\lim_{n\to\infty}\|(\!theta^n,\!u^n, w^n)\|_{E_{h_n}}=0$ that we have $\lim_{n\to\infty}\langle \gamma_{\alpha\beta}(\!u^n, w^n),\phi^{\alpha\beta}\rangle=0$.
Thus $\gamma_{\alpha\beta}(\!u^0, w^0)=0$, in which the derivatives are understood in the distributional sense.
In view of the definition \eqref{N-metric}, we have
\begin{equation*}
e_{\alpha\beta}(\!u^0)=\Gamma^\lambda_{\alpha\beta}u^0_\lambda+b_{\alpha\beta}w^0.
\end{equation*}
Since $u^0_\alpha$ and $w^0$ are in $L^2$, so $e_{\alpha\beta}(\!u^0)\in L^2$.
In the sense of distribution, there is the identity that \cite{BCM, Lions}
\begin{equation*}
\partial_{\alpha\beta}u^0_\lambda=\partial_\beta e_{\alpha\lambda}(\!u^0)+\partial_\alpha e_{\lambda\beta}(\!u^0)-\partial_\lambda e_{\alpha\beta}(\!u^0).
\end{equation*}
From this we see that $\partial_{\alpha\beta}u^0_\lambda\in H^{-1}$. It follows from a Lemma of J.L. Lions (whose assumption on the domain
is met by our polygon, see page 110 of \cite{Lions} and page
124 of \cite{BCM}) and
the fact $\partial_{\beta}u^0_\lambda\in H^{-1}$ that
$\partial_{\beta}u^0_\lambda\in L^2$. Therefore, we proved that $u^0_\alpha\in H^1$, and we have
\begin{equation}\label{membrane=0}
\gamma_{\alpha\beta}(\!u^0, w^0)=0.
\end{equation}

We then show that $\theta^0_\alpha\in H^1$ and $\rho_{\alpha\beta}(\!theta^0, \!u^0, w^0)=0$.
As in the above, in the space $H^{-1}$, we have $\lim_{n\to\infty}\rho_{\alpha\beta}(\!theta^n, \!u^n, w^n)=\rho_{\alpha\beta}(\!theta^0, \!u^0, w^0)$,
in which the derivatives are understood in the distributional sense.
For arbitrary $\phi^{\alpha\beta}\in\D(\Omega)$, and for any $n$, we have the identity
\begin{multline*}
\langle \rho_{\alpha\beta}(\!theta^n, \!u^n, w^n) , \phi^{\alpha\beta}\rangle=
\int_{\Omega_{h_n}}\rho_{\alpha\beta}(\!theta^n, \!u^n, w^n)\phi^{\alpha\beta}
-\sum_{e\in\E^0_{h_n}}\int_e\lbra \theta^n_{\alpha}\rbra_{n_\beta}\phi^{\alpha\beta}
+\sum_{e\in\E^0_{h_n}}\int_eb^\gamma_\alpha\lbra u^n_{\gamma}\rbra_{n_\beta}\phi^{\alpha\beta}.
\end{multline*}
Using the assumption $\lim_{n\to\infty}\|(\!theta^n,\!u^n, w^n)\|_{E_{h_n}}\to 0$ and this
equation, we get
\begin{equation*}
\lim_{n\to\infty}\langle \rho_{\alpha\beta}(\!theta^n, \!u^n, w^n),\phi^{\alpha\beta}\rangle=0.
\end{equation*}
Therefore $\rho_{\alpha\beta}(\!theta^0,\!u^0, w^0)=0$, in which the derivatives $\partial_\alpha\theta^0_\beta$ are  understood in
in the distributional sense.
In view of the definition \eqref{N-bending} of $\rho_{\alpha\beta}(\!theta^0,\!u^0, w^0)$,
and the proved regularity that $u^0_\alpha\in H^1$, we see that $e_{\alpha\beta}(\!theta^0)$
are in $L^2$. From this and the argument used above, we see $\partial^2_{\alpha\beta}\theta_\gamma\in H^{-1}$. Using the Lemma of J. L. Lions again,
we get the regularity $\theta^0_\alpha\in H^1$, and the equation
\begin{equation}\label{bending=0}
\rho_{\alpha\beta}(\!theta^0, \!u^0, w^0)=0.
\end{equation}

According to Lemma~3.4 of \cite{BCM},
using the regularities that $\theta^0_\alpha$, $u^0_\alpha$, and $w^0$ are all in $H^1$, and the equations \eqref{shear=0}, \eqref{membrane=0}, and \eqref{bending=0},
we conclude that  the displacement functions
$(\!theta^0, \!u^0, w^0)$ defines a rigid body motion.

Finally, we show that $(\!theta^0, \!u^0, w^0)=0$.
It follows from the bound \eqref{Korn-thm-proof1}, the assumption \eqref{N-energy-to-0}, and the convergence \eqref{converge-in-L2}
that
$\lim_{n\to\infty}\|(\!theta^n-\!theta^0, \!u^n-\!u^0, w^n-w^0)\|_{H_{h_n}}=0$. Since $f$ is uniformly continuous
with respect to the norm of $H_{{h_n}}$ and since $f(\!theta^n, \!u^n, w^n)\to 0$ ($f$ is a part
of the energy norm \eqref{triple-norm}), we see
$f(\!theta^0, \!u^0, w^0)=0$. Thus $(\!theta^0, \!u^0, w^0)=0$. Therefore,
$\lim_{n\to\infty}\|(\!theta^n, \!u^n, w^n)\|_{H_{h_n}}=0$,
which is contradict to the assumption that
$\|(\!theta^n, \!u^n, w^n)\|_{H_{h_n}}=1$.
\end{proof}

As an example, we take
\begin{equation*}
f(\!theta, \!u, w)=
\left[\sum_{e\in\E^D_h}\int_{e}\sum_{\alpha=1,2}\theta^2_\alpha+\sum_{e\in\E^{S}_h\cup\E^D_h}
\left(\int_{e}\sum_{\alpha=1,2}u^2_{\alpha}+\int_{e}w^2\right)
\right]^{1/2}.
\end{equation*}
It follows from Theorem~\ref{tracetheorem} that there is a $C$ such that
the continuity condition \eqref{N-f-continuous} is satisfied by this $f$.
Under the assumption that  the measure of $\partial^D\Omega$ is positive,
it is verified in \cite{BCM} that if $(\!theta, \!u, w)$ defines a rigid body motion and  $f(\!theta, \!u, w)=0$ then $\!theta=0$, $\!u=0$, and $w=0$.
With this $f$ in the Korn's inequality \eqref{N-Korn-inequality},
we could add boundary penalty term
\begin{equation*}
\sum_{e\in\E^D_h}\int_{e}h^{-1}_e\sum_{\alpha=1,2}\theta^2_\alpha+\sum_{e\in\E^{S}_h\cup\E^D_h}
\left(h^{-1}_e\int_{e}\sum_{\alpha=1,2}u^2_{\alpha}+h^{-1}_e\int_{e}w^2\right)
\end{equation*}
to the squares of both sides of \eqref{N-Korn-inequality} to obtain an inequality that is  useful in  analysis of discontinuous Galerkin
methods for Naghdi shell in which the essential boundary conditions are enforced by Nitsche's consistent boundary penalty method.

\section{Discrete Korn's inequality for Koiter shell}
\label{Korn-Koiter}
The wellposedness of the Koiter model \eqref{K-model}
is based on the Korn's inequality for Koiter shell that
there is a constant $C$ such that
\begin{multline}\label{Korn}
\sum_{\alpha=1,2}\|u_\alpha\|^2_{H^1}+\|w\|^2_{H^2}\le C
\left[\sum_{\alpha,\beta=1,2}\|\rho^K_{\alpha\beta}(\!u, w)\|^2_{L^2}+\sum_{\alpha,\beta=1,2}\|\gamma_{\alpha\beta}(\!u, w)\|^2_{L^2}+f^2(\!u, w)\right]
\\
\ \ \forall\
\!u\in \!H^1,\ w\in H^2.
\end{multline}
Here $f(\!u, w)$ is a semi-norm on $\!H^1\x H^2$ that satisfies the condition that
if $\!u, w$ defines a rigid body motion, as explained in the introduction, and $f(\!u, w)=0$ then $\!u=0$ and $w=0$.
We  generalize this inequality to piecewise functions on $\Omega_h$.
For Koiter shell problems.
we also need to use the discrete space $H^2_h$
that is composed of piecewise
$H^2$ functions with the norm defined by \eqref{H2hnorm}.

For $(\!u, w)\in \!H^1_h\x H^2_h$, we define a norm
\begin{equation}\label{K-h-norm}
\|(\!u, w)\|_{H^K_h}=\left(\sum_{\alpha=1,2}\|u_\alpha\|^2_{H^1_h}+\|w\|^2_{H^2_h}\right)^{1/2}.
\end{equation}
Let $f(\!u, w)$ be a semi-norm that is continuous with respect to this norm such that there is a $C$ that only depends on the shell mid-surface
and the regularity $\K$ of the triangulation $\T_h$
\begin{equation}\label{K-f-continuous}
|f(\!u, w)|\le C\|(\!u, w)\|_{H^K_h}\ \forall\ (\!u, w)\in \!H^1_h\x H^2_h.
\end{equation}
We assume that $f$ satisfies the rigid body motion condition that
if $(\!u, w)\in \!H^1\x H^2$ defines a rigid body motion and $f(\!u, w)=0$ then $\!u=0$ and $w=0$.
We define the discrete energy norm on the space $H^K_h$.
\begin{multline}\label{K-triple-norm}
\|(\!u, w)\|_{E^K_h}=
\left[
\sum_{\alpha,\beta=1,2}\|\rho^K_{\alpha\beta}(\!u, w)\|^2_{0,\Omega_h}+\sum_{\alpha,\beta=1,2}\|\gamma_{\alpha\beta}(\!u, w)\|^2_{0,\Omega_h}\right.
\\
\left.+\sum_{e\in \E^0_h}\left(
\sum_{\alpha=1,2}h^{-1}_e\int_{e}\lbra u_{\alpha}\rbra^2
+\sum_{\alpha=1, 2}
h^{-1}_e\int_{e}\lbra \partial_\alpha w\rbra^2
+
h^{-1}_e\int_{e}\lbra w\rbra^2\right)
+f^2(\!u, w)
\right]^{1/2}.
\end{multline}
We then have the following generalization of the Korn's inequality for Koiter shells to piecewise functions.
\begin{thm}\label{K-Korn-thm}
There exists a constant $C$ that could be dependent on the shell mid-surface,
and the shape regularity $\K$ of $\T_h$, but otherwise independent of the triangulation,
such that
\begin{equation}\label{K-Korn-inequality}
\|(\!u, w)\|_{H^K_h}\le C\|(\!u, w)\|_{E^K_h}\ \forall\ w\in H^2_h,\ u_{\alpha}\in H^1_h.
\end{equation}
\end{thm}
\begin{proof}
Koiter's shell model is a restriction of the Naghdi's model on the subspace of zero-shear
deformations. The discrete Korn's inequality \eqref{K-Korn-inequality} for Koiter shell can be derived from
the discrete Korn's inequality for Naghdi shell \eqref{N-Korn-inequality}.

For $u_\alpha\in H^1_h$ and $w\in H^2_h$, we define piecewise function $\theta_\alpha\in H^1_h$ by
\begin{equation}
\theta_\alpha=-\partial_\alpha w-b^\gamma_\alpha u_\gamma.
\end{equation}
We then have $\rho_{\alpha\beta}(\!theta, \!u, w)=-\rho^K_{\alpha\beta}(\!u, w)$ and $\tau_\alpha(\!theta, \!u, w)=0$.
There are constants $C_1$  and $C_2$ only depending on components of the mixed curvature tensor $b^\gamma_\alpha$
such that for all $e\in E^0_h$ and for all $\tau\in\T_h$
\begin{multline*}
C_1\left(
\sum_{\alpha=1,2}\int_{e}\lbra u_{\alpha}\rbra^2
+\sum_{\alpha=1, 2}
\int_{e}\lbra \partial_\alpha w\rbra^2\right)
\le
\sum_{\alpha=1,2}\int_{e}\lbra u_{\alpha}\rbra^2
+\sum_{\alpha=1, 2}
\int_{e}\lbra \theta_\alpha\rbra^2
\\
\le C_2 \left(
\sum_{\alpha=1,2}\int_{e}\lbra u_{\alpha}\rbra^2
+\sum_{\alpha=1, 2}
\int_{e}\lbra \partial_\alpha w\rbra^2
\right),
\end{multline*}
\begin{multline*}
C_1\left(\sum_{\alpha=1}^2\|u_\alpha\|_{1,\tau}+\|w\|_{2,\tau}\right)\le
\sum_{\alpha=1}^2\|\theta_\alpha\|_{1,\tau}+\sum_{\alpha=1}^2\|u_\alpha\|_{1,\tau}+\|w\|_{1,\tau}\\
\le C_2
\left(\sum_{\alpha=1}^2\|u_\alpha\|_{1,\tau}+\|w\|_{2,\tau}\right).
\end{multline*}
From these, we get  the equivalences
\begin{equation}
C_1\|(\!u, w)\|_{H^K_h}\le \|(\!theta, \!u, w)\|_{H_h}\le C_2\|(\!u, w)\|_{H^K_h}
\end{equation}
and
\begin{equation}
C_1\|(\!u, w)\|_{E^K_h}\le \|(\!theta, \!u, w)\|_{E_h}\le C_2\|(\!u, w)\|_{E^K_h}.
\end{equation}
The continuity \eqref{K-f-continuous} implies the condition \eqref{N-f-continuous}.
The inequality \eqref{K-Korn-inequality} then follows from the Korn's inequality for Naghdi shell \eqref{N-Korn-inequality}.
\end{proof}

As an example for the semi-norm satisfying the continuity condition \eqref{K-f-continuous}, we take
\begin{equation*}
f(\!u, w)=
\left[\sum_{e\in\E^{S}_h\cup\E^D_h}
\left(\int_{e}\sum_{\alpha=1,2}u^2_{\alpha}+\int_{e}w^2\right)
+\sum_{e\in\E^D_h}\int_{e}(D_{\!n}w)^2\right]^{1/2}.
\end{equation*}
It follows from Theorem~\ref{tracetheorem} that there is a $C$ only dependent on $\K$ such that
the continuity condition \eqref{K-f-continuous} is satisfied by this $f$.
Under the assumption that  the measure of $\partial^D\Omega$ is positive,
it is verified in \cite{BCM} that if $(\!u, w)\in \!H^1\x H^2$ defines a rigid body motion and $f(\!u, w)=0$ then $\!u=0$ and $w=0$.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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The term "shrub rose" is somewhat confusing because a rose, any rose, is a shrub. When it comes to roses, the term "shrub" refers to the appearance of the rose plant, which grows like a bush. Probably the most popular shrub rose in landscaping today is the Knock Out rose. Shrub roses are winter hardy with excellent disease tolerance. These are not shy roses; they can easily reach six feet tall, but are fairly simple to tame to a manageable height, even trained into a hedge. Although the flowers from shrub roses carry little fragrance, they are available in a wide assortment of colors. Red, pink, yellow and white are the most commonly found colors.
Some roses are sold by the name rather than the type of rose. Like David Austin roses. These are referred to as English Roses, and can be hybrid teas or floribundas. These sturdy, disease-resistant roses can grow to five feet high and wide and are available in many colors. "Graham Thomas" is a well-known choice with its yellow blossoms, "Darcey Russell" blooms a deep dark pink, and both "Heritage" and "Alnwick Rose" are pale pink, outstanding for their old rose scent.
Hardy enough to withstand salt water spray on a sea-shore, rugosas are the workhorses of the rose family. These thorny shrubs form a thicket. Covered in single petal blooms, they are often called "wild roses." Their huge red rose hips that persist into winter make these toughies a stand out.
Then there are climbing roses. They do just that – wind and climb their way up arbors and along fences. Some varieties can climb up trees, reaching 15 to 20 feet at maturity.
All roses appreciate a sunny location with good airflow and well-drained soil.
With so many varieties to choose from, roses have a place in just about any landscape. Add one, or more, to your garden and take the time to…stop and smell the roses!


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pdfposter: Scale and tile PDF images/pages to print on multiple pages.
pdfposter can be used to create a large poster by building it from multiple pages or printing it on large media. It expects as input a PDF file, normally printing on a single page. The output is again a PDF file, maybe containing multiple pages together building the poster. The input page will be scaled to obtain the desired size.
To get an impression, please visit the example section in the manual.
pdfjoin: Join several PDF documents into a single one.
pdfjoin is a command line tool, a droplet and a Python library to join several PDF documents into a single one. It will join all files passed as argument into a new file. As a special feature it can operate as a "droplet".
pdfbook: Rearrange pages from a PDF file for printing books or booklets, creating a new PFD file.
python-ghostscript : Ghostscript, is a well known interpreter for the PostScript language and for PDF. This package implements a interface to the Ghostscript C-API using ctypes. Both a low-level and a pythonic, high-level interface are provided.


text:
West Virginia man sentenced for stealing firearms from ATF
By Jeff Pegues
August 27, 2019 / 6:21 PM / CBS News
Search for stolen firearms
Search for stolen firearms 01:31
A West Virginia man has been sentenced to 14 years in prison for stealing and reselling thousands of firearms from the Bureau of Alcohol, Tobacco, Firearms and Explosives (ATF). As CBS News first reported in March, it was an inside job.
Christopher Lee Yates, a former contract security guard at an ATF facility, over the course of three years, between 2016 to 2019, stole thousands of firearms, gun parts and ammunition and then sold the weapons across the country, according to a press release by the West Virginia U.S. attorney's office.
This major breach of security was an embarrassing episode for the ATF, which launched a search across 49 of the 50 states for the firearms. As CBS News reported, some of the firearms parts were found on the streets of Philadelphia — fashioned into a gun. Hundreds of stolen firearms parts came from agents' guns that were supposed to be disposed of by the ATF.
The ATF has recovered 4,625 of stolen firearms and firearms parts, according to the U.S. attorney's press release, including 80 handguns, "including ATF duty weapons, 10 to 15 AR-style lower receivers, and four machine guns."
Yates, who is 52 years old, pleaded guilty to a single count of possession of stolen firearms and one count of theft of government property. In addition to the 14-year sentence, the judge in the case issued a $300,000 financial judgment against him.
When Yates was arrested, officials found "a firearm concealed in a book bag on the front passenger seat floorboard" in his vehicle. Agents later determined the pistol had a serial number on it. The ATF computer system confirmed the firearm had been listed as "disposed" by the ATF's National Disposal Branch on August 2017.
First published on August 27, 2019 / 6:21 PM
Biden signs executive actions on first day as president

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Express, Passport-Google, React - Maintaining correct session information I've been pouring over SO for a few days now trying to understand my problem, but haven't had much luck.
Using passport-google in my node/express application, I would like to authenticate a user and successfully redirect to my React app (bootstraped using create-react-app) running on a different port with the proper express session set for that port running React.
This is my flow, the problem, and what I've gathered so far:

*

*User clicks Login with Google on my React application.

*Passport redirects and presents the user with a login screen and ultimately sends the user back to auth/google/callback on my express application

*auth/google/callback sends its successRedirects to my React app on port 3000.
At this point it seems looks like the express session I want tied to port 3000 is connected to port 8000. To be clear, calling a custom /session endpoint on port 8000 which sends back the express session shows me what I want, while calling it from 3000 shows me nothing. I'm guessing this happens because it was on port 8000 where the initially. Based off this post I think I may be right.
Also, what really confuses me is that when I am redirected back to my React app on port 3000, I see the connect.sid cookie with the correct sessionId I see on port 8000. It feels like it did and didn't work at the same time. Why would the connect.sid be set on port 3000, yet any call to req.session delivers no passport information? When/how does that get set? To my understanding passport sets it in the serialization/deserialization process
Also, in general, is what I'm trying to accomplish feasible/worth the effort? I would have thought I would have an easier time figuring this out but it seems like passport might be a good authentication strategy when you are only working with the server itself and a templating language like ejs or even a framework like Next.js.
Ultimately what I want to do is have the user data retrieved from Google handy so I can use it to make calls against my db. I guess if I have the connect.sid as a cookie on the React app I could use that to look up the session information on my MongoStore which looks to have what I need, but would it be better just to have that info registered on the server in my session? Also I'm not sure if it's good practice to make my cookies accessible with the httpOnly flag on express-sessions and parsing the cookie using document.cookie. I've also used WebSockets on a past project to send information from a server to a client (to negate this same problem), but am not sure if that's a bandaid to my lack of understanding the passport flow or a viable alternative to a problem passport wasn't built to alleviate.
Here is some of the auth flow in no particular order. Any ideas, comments, suggestions, or corrections would be greatly appreciated!
router.get("/google", passport.authenticate("google", { scope: ["profile"] }))
router.get(
    "/google/callback",
    passport.authenticate("google",
    {
        successRedirect: "http://localhost:3000",
        failureRedirect: "http://localhost:3000/auth-failed",
    }, (req, res) => {
        // empty
    })
)

app.use(
  session({
    secret: process.env.APP_SECRET || "keyboardcat",
    store: new MongoStore({ mongooseConnection: dbConnection }),
    resave: false,
    saveUninitialized: false,
    cookie: {
      httpOnly: false
    }
  })
)

passport.serializeUser((user, done) => {
    done(null, { _id: user._id })
})

passport.deserializeUser((id, done) => {
    User.findOne(
        { _id: id },
        (err, user) => {
            if (err) {
                console.log(err)
            }
            done(null, user)
        }
    )
})

Sidenote: I actually got this working using Next.js because it eliminated the problem of dealing with different ports. If I continue to find it as a viable solution, I could be persuaded to ditch my create-react-app.

redpajama_set_name: RedPajamaStackExchange


text:
2016/17 Archive Page
2016/2017 SEASON - APRIL 22
'It's coming': Lessons from Vancouver in Canada's opioid crisis
Canada's growing fentanyl crisis is at the centre of Saturday's W5 investigation in Vancouver -- ground zero for overdoses with more than 1,000 deaths last year alone.
Reporter witnesses possible overdose
Reporter Jon Woodward explains what happened when he witnessed a possible overdose while on assignment in Vancouver's Downtown Eastside.
'Worse than AIDS': Frontline workers race to save lives in opioid crisis
How frontline responders are coping in Vancouver, where the epidemic's death toll is rising.
Heroin as drug treatment saves lives, millions of dollars: clinic
Why a clinic says it's providing heroin to drug users for free, and how it's saving lives and millions of dollars.
25 per cent of visits at downtown Vancouver hospital are for drug overdoses
B.C. is on a trajectory to hit over 1,300 overdose deaths in 2017, with 400 of those in Vancouver alone.
Serena Ryder on her rise from child prodigy to rockstar, and her battle with crippling depression
W5 on Saturday: Serena Ryder talks about her rise to international stardom and her struggle with depression
From Vimy Ridge to the NHL, Red Dutton's remarkable life
Plight of thalidomide 'sample babies' who don't qualify for gov't compensation
W5's Kevin Newman profiles the plight of the thalidomide "sample babies" -- Canadians who claim their mothers were given free samples of the morning sickness drug in the late 1950s and early 60s, but don't qualify for compensation from government.
The dangerous sport of rooftopping
W5 investigates the dangerous sport of rooftopping - where modern urban adventurers seek thrills at dizzying heights, walking where even professionals fear to tread, trying to get that one great picture or rush.
Town divided over plan to store nuclear waste near Lake Huron
An Ontario community is divided over plans to bury dangerous nuclear waste deep underground, not far from the shores of Lake Huron.
Door-to-door furnace sales schemes that victimize Canadians
This 51st season of W5 launches with an investigation into the surging number of Canadians agreeing to pay thousands of dollars more to be comfortable in their homes than they need to.
Families sue nursing home giant for neglected care they claim loved ones suffered
W5 this week: Sandie Rinaldo tells the story of dozens of families who are going to court against one of the largest owners of nursing homes, seeking damages for the neglected care they claim their loved ones suffered.
W5 returns to 'Trumpland' two months later, and finds it restless
On W5 this week: Kevin Newman retraces a journey he took just before the November U.S. election, to see how American voters in the swing state of Pennsylvania feel now about their choice to elect Donald Trump as president.
Hidden camera investigation finds most Calgary car dealerships breach consumer rules
The Automobile Protection Association goes undercover and finds Calgary car consumers are still likely exposed to questionable sales tactics and advertising practices -- even after a government push to clean up the industry.
At age 18, Lance Stroll is the first Canadian to race in Formula 1 in 11 years
On Saturday, W5 profiles 18-year-old Canadian race car driver Lance Stroll, who finds himself going into his rookie season in the prestigious Formula 1 Racing Circuit.
Undercover investigation reveals what goes on inside Montreal animal research lab
A national council responsible for monitoring animal welfare in research laboratories is investigating hidden camera footage obtained by CTV's W5 that appears to show mistreatment of dogs, pigs and monkeys used for testing at a Montreal-based facility.
Investigation reveals decline in fighting is changing hockey
Fighting has been a part of hockey since its earliest days. But an analysis of hockey statistics by TSN and W5 shows in recent years, the incidence of hockey fighting has dropped dramatically at all levels.
Inside an illegal call centre scamming millions from Canadians
Saturday on W5: Kevin Newman's investigation of an illegal call centre in Mumbai reveals a Canadian connection that led to arrests in Ontario.
How a former NHL star lost millions to a fraud involving an unscrupulous financial advisor
On W5 Saturday: TSN's Rick Westhead probes an investment scheme that targeted professional athletes, including former NHL defenceman Bryan Berard.
Are pedophile hunters snaring the innocent?
W5 investigates the practices and backgrounds of "Creep Catchers" -- a group of online vigilantes who hunt for possible pedophiles and expose them.
'A Tribe Called Red': Juno nominations propel DJs into mainstream
W5 profiles 'A Tribe Called Red': an indigenous electronica group hitting its creative stride, just as a real conversation is starting to take hold in Canada about how we do better by indigenous people.
Terror plotters claim they were groomed by Mounties and were relieved bombs were fake
For the first time since they were convicted of terrorism related offences in 2015, then set free last July, John Nuttall and Amanda Korody are speaking out in an exclusive interview with W5's Victor Malarek.
After a year of care in Canada, triple amputee returns to Ukraine
W5 follows Canadian surgeons on a medical mission to Kiev. There, they help a 12-year-old triple amputee who lost his limbs when a grenade detonated outside his home in Ukraine.
Cross-border custody battles: W5 finds more parental child abductions going international
This Saturday's W5 investigates cross-border custody battles, as more parental child abductions go international.
Music producer Bob Ezrin opens up about drugs, depression & loss
One of Canada's most notable music producers is opening up about his struggle with depression and his son's death, in hopes of shattering perceptions of shame often associated with mental health.
An exclusive look at Canada's elite Special Forces
CTV's Chief News Anchor and Senior Editor Lisa LaFlamme delivers a never-before-seen look inside Canada's ultra-elite team of soldiers training foreign troops to fight ISIS.
Freely available online porn exposes children to violent, abusive sex
Joseph Deschambault's fits of rage began when he was just nine years old. Looking for answers, his parents realized that Joseph had lost his innocence when he found hard core Internet pornography.
How a dog's sense of smell could soon be a legitimate diagnostic tool
W5's Sandie Rinaldo investigates how a dog's superior sense of smell could soon be a legitimate, and much less invasive, diagnostic tool.
Bungled arms case results in multi-million dollar secret settlement by federal government
A Vancouver businessman has been paid millions of dollars in a secret settlement with the Canadian government after the Canada Border Services Agency wrongly accused him of exporting military technology to China in violation of export controls, a W5 investigation reveals.
W5 investigates cases of sexual assault in Ontario nursing homes
On this week's W5, an investigation into troubling cases of sexual assault in Ontario nursing homes.
A reunion with conjoined twins, successfully separated a decade ago
On W5: a remarkable journey of rediscovery which began a decade ago at Toronto's Hospital for Sick Children when two tiny twins, identical brothers, were born conjoined. Watch the reunion with the boys who are now living separately in a remote African village.
The rising value and volatility of our fresh water
When a bottled water giant takes billions of litres of water out of the ground each year, where do local communities fit in? And why do governments charge so little when they are nearly giving away a precious resource? Victor Malarek investigates.
Ebola-ravaged Liberia today: what's possible when people unite and politics is swept aside
The Ebola virus ravaged families and left a broken economy of survivors. CTV's Kayla Hounsell travelled to Liberia, a country which now stands as an example to the world of what's possible, when politics is swept aside and people unite to help.
Teens take emotional pilgrimage to Nazi death camps in Poland
This week, W5's Sandie Rinaldo joins a group of Canadian high school students, chaperones, and Holocaust survivors on a journey from Toronto to Poland as they embark on an annual pilgrimage known as the March of the Living, an emotionally grueling tour of Nazi concentration camps.
Investigation uncovers how turkeys are treated in slaughterhouses
W5 investigates how turkeys get to your plate. Undercover footage from the animal rights group Mercy for Animals reveals the what goes on in the slaughterhouse. It may look cruel -- but as W5 discovered under Canada's laws -- it is entirely legal.
Former students allege psychological, physical and sexual abuse at Ontario Christian school
This week, W5 investigates troubling claims of decades of abuse -- psychological, physical, and sexual -- from former students attending an Ontario Christian school from the 1970s through to the 1990s.
Who were the victims of alleged nursing home killer Elizabeth Wettlaufer?
W5's Sandie Rinaldo sat down with family members and friends of four of the alleged victims of former nurse Elizabeth Wettlaufer, to hear their stories of reliving grief and dealing with the prospect that their loved ones' lives were cut short
Photo gallery: Remembering the victims
In late October, police charged former Ontario nurse Elizabeth Tracy Mae Wettlaufer with eight counts of first-degree murder. The alleged victims died between 2007 and 2014: James Silcox, Maurice Granat, Gladys Millard, Helen Matheson, Mary Zurawinski, Helen Young, Maureen Pickering and Arpad Horvath.
Deadly epidemic follows illicit trade in pain killer fentanyl
W5 examines the illegal use of fentanyl, a pain-killing drug reported to be 80 to 100 times more powerful than morphine.
Catching up with Hani al Moulia, a young Syrian refugee who is thriving
W5 catches up with Hani al Moulia, a young Syrian refugee we first profiled six months ago. He was in a hurry to make his mark and contribute to his new home. As W5's Kevin Newman discovered, Hani hasn't wasted a moment.
Toronto hopes to reverse bad-weather curse for 104th Grey Cup
This year's Grey Cup host city, Toronto, has never been known for the energy it brought to the championship. But as TSN's Brian Williams reports, this year should be different.
What are the secrets of people who live to 100? Labs are starting to find out
W5's Kevin Newman discover labs across Canada are part of an international effort among scientists to get to the bottom of how a few people live to be a hundred.
What will America be after Nov. 8? Canadians brace for U.S. election impact
W5 is anchored in Philadelphia as the most tumultuous U.S. presidential election in memory comes down to voting day.
Family searching for answers after hospital tragedy runs into secrecy laws
A W5-Toronto Star investigation into secrecy laws that protect hospitals while keeping families in the dark after tragic events.
Undercover video provides rare glimpse into possible auto insurance fraud
Saturday at 7 p.m.: W5 takes you inside a hidden camera investigation into bogus accident claims, giving you rare insight into a problem that results in higher auto insurance costs for all motorists.
Exclusive: Billionaire Charles Bronfman on his fortune, philanthropy, and some of his regrets
W5 exclusive: an interview with billionaire Charles Bronfman, who reflects on his family's fortune, philanthropy and shares some regrets.
Calling 911 may not get you the medical help that will save your life
October 8 on W5: In Canada, calling 911 doesn't always guarantee the paramedic who arrives can do everything in their power to save you. Kevin Newman asks why.
W5 responds: Ontario Paramedics' critique of report on 911 in Canada
An exclusive look at Canada's Special Forces and the war against ISIS
This Saturday on W5: CTV's Chief News Anchor and Senior Editor Lisa LaFlamme delivers a never-before-seen look inside Canada's ultra-elite team of soldiers training foreign troops to fight ISIS.
OCT 1 SEASON PREMIERE
Alarming hoaxes by hackers who send tactical teams to fake crises
On this weeks' W5: an investigation into alarming incidents that seem to be happening more and more in big cities -- tactical teams responding to a crisis like a gunman at large or a hostage taking. But in some cases, the calls are sophisticated fakes that are happening so often it has a name: swatting.


Brown rebukes the Members of the R.A.
He had now lived beyond the full term of threescore years and ten, and his days had become sad. His only son Oliver, a youth of great promise both as painter and writer, had died, and this bitterness was followed soon after by the death of his wife. While still struggling manfully with ever-continuing money difficulties, he could not repress complaint to a friend at his evil fortune. His health was fast failing, and he was seven hundred pounds in debt, which was indeed a disgrace to his country. The profession of an artist is an expensive one; a writer needs but his pens, ink and paper with a little space to write in, but an artist must have large rooms with many appliances, as necessary to him as scaffolding to a builder in his operations. He must have money at all stages of figure subjects to pay models and to buy materials; during the probationary period of his career he may bear many discomforts with patience, but when he has the full responsibilities of life; he must have the opportunity of repose away from sight of his work, that he may not miss refreshment of mind and body. Brown had always been most studious, industrious, and frugal, and had produced many noble works of which the country must eventually be proud; yet here he was in his last years suffering in mind as though he had been a profitless ne'er-do-well.


text:
Le mont Carihuairazo (également orthographié Carihuayrazo) est un volcan coiffé par une caldeira voisin de la plus haute montagne de l'Équateur, le Chimborazo. Son sommet s'élève à une altitude de .

Volcan des Andes équatoriennes
Sommet en Équateur
Sommet des Andes
Province de Pichincha

redpajama_set_name: RedPajamaWikipedia


In May, David Marks was named head of Wells Fargo Capital Finance, which joined the newly-formed Wells Fargo Commercial Capital group in January. In his new position at WFCF, Marks completes a full circle after a 30-year career at the bank, which began by doing workouts in the oil and gas industry. Marks reminisces about his career and shares his vision for the future with ABF Journal editor Nadine Bonner.
Deep in the heart of Texas, the ABL world is heating up. Despite reports that the landscape is flat and banks have become reluctant bedfellows, Chris Capriotti and two members from the JPMorgan ABL team moved to Texas Capital Bank recently to ramp up its ABL division and transform a regional bank into a national player.
When Blue Hills Bank, a 140-year old community bank, decided to start an ABL division, it offered veteran ABL lenders Keith Broyles and Yvonne Kizner the opportunity to build the new division from scratch. The pair jumped at the chance. They spoke with ABF Journal Editor Nadine Bonner and shared their optimism and excitement about this new venture.
Last January CIT Commercial Services named industry veteran Marc Heller president of the division, succeeding Jon Lucas. Heller responds to ABF Journal's questions, provides insights into his personal experiences in factoring and offers some history of the industry and a look ahead.
In this Q&A with ABL industry veteran John DePledge, the new business head and national sales manager for Citi Commercial Banks' ABL unit discusses his plans for the team, his build-out strategy and why the ABL market is going to get more competitive in the near future.
In October, Hitachi Capital America established Hitachi Capital Canada. Before 2012 came to a close, the company acquired Hennessey Capital. In early 2013, Hitachi announced that Hennessey would offer equipment financing. ABFJ catches up with William H. Besgen, president and COO of Hitachi Capital America, Thomas Cross, general manager of Hitachi Capital's Trade Finance Division, and Michael Semanco, president and COO of the Hennessey Capital Division, to learn more about these strategic moves.
Last April, the commercial finance industry learned that the venerable turnaround firm of NHB Advisors had been acquired by a newly formed corporate consulting firm bearing the name Gavin/Solmonese. While acquisitions of turnaround firms either in whole or in part have made industry headlines regularly this year — indeed, Deloitte acquired CRG Partners and CVC Capital Partners took a majority stake in AlixPartners — this announcement was a bit unique.
ABL industry veteran Andrew Moser follows both his head and his heart. A self-described "24/7 kind of guy" has assembled a team under the Harbinger Group Inc. umbrella to bring innovative approaches to small- and middle-market borrowers, investors and lending partners.
With a seasoned team on board nimbly assembled in 90 days last summer, Mike Maiorino finds himself at home at People's United Bank where he was tapped to launch People's United Business Capital. While the model maybe "right down the middle" in terms of the broad spectrum of asset-finance companies, when it comes to customer's and their financing needs, Maiorino shuns a "one size fits all" approach.


Story #1 relates to my new novel On the Edge of the Loch: A Psychological Novel set in Ireland.
I received an email from a reader, a Canadian man, confessing he had 'fallen head over heels in love with Cilla', and asking if he were to visit Ireland, what were his chances of meeting a girl just like her. He sounded entirely serious and wrote well.
But . . . Cilla is a character from the above novel.
I explained that Cilla is exceptional, like most fictional characters, but that some people say Ireland in real life is full of exceptional characters (it is, like no place else on earth!).
Nonetheless, I did not want to mislead him or cause him to spend money he might not have. So I added a caveat: I told him that in my experience the right person is often just around the corner, and that if he adopts that view, and acts on it, Cilla might be much closer than he thinks – and that I'd start there. Then one day, who knows, I said, maybe he and 'Cilla' would visit Ireland together (Tourism Ireland take note).
He hasn't responded – and might not. So, as we say here, that might be that.
On the Edge of the Loch: A Psychological Novel set in Ireland can be ordered from Amazon as an eBook or print book – http://amzn.to/28RRRdt – at reduced launch prices.
Next Next post: Imagination and Inspiration: If You Were a Crow or Crocodile or Gorilla . . .


Samarthia joined the Department of Environment and Geography in December 2007 from her previous post from the Centre for Business Relationships, Accountability Sustainability and Society (BRASS) at Cardiff University, Cardiff. Samarthia had worked at the School of Management and Economics, Queen's University of Belfast prior to joining Cardiff University. Samarthia has held various research positions in India and has extensive research experience in the field of Sustainable development & technological policy issues; Social & economic aspects of sustainability. Samarthia's research interests include resilience in food systems, such as in the Greater Mekong Subregion.
My research expertise is in the area of sustainability, policy and governance, providing coherent frameworks for highly innovative research into mechanisms to de-couple economic development from environmental deterioration to operate within ecosystem limits.
My research focuses on three key areas: (i) Sustainability and the agri-food sector, tackling a wide range of issues from climate change adaptation and mitigation to understanding the ways in which current behaviours or changed behaviours may affect the sustainability of consumption and production systems in the future; (ii) Sustainability and the automotive sector exploring environmental, economic and social costs of enhanced transport systems, especially the impacts of bio-based material substitution in the automotive sector; (iii) Environment, society & governance, examining the complexity & the impact of climate change policy on businesses in the supply chain and mapping social and knowledge flows in ecosystems.
This project uses new technology platforms developed at Cornell University, USA, to access valuable genetic variation from ancestral wild species of rice, introduce this to modern day elite cultivars that are widely grown in India and identify new varieties that have increased yields under drought conditions due to the presence of small segments of DNA from the wild species. In addition however, the adoption of new rice varieties by local communities is dependent on multiple factors in addition to yield performance with grain quality being one of the most important. Cultural and sociological issues such as the role of gender also play a significant part in determining the success of a new cultivar and in order to address this project will perform a socioeconomic study to assess the likely uptake patterns of any new varieties and their impact on farmers' incomes. The impact of new rice cultivars on yield, per unit cost of production, and total production per specific input (land, labour, fertilizers, seeds) will be evaluated and on-farm testing and participatory evaluation will be an integral component of cultivar field trials. This project will also model the impact of new varieties to quantify how yields are enhanced in drought-prone landscapes currently and in the future as climate changes. This information along with the outputs from our socio-economic studies will inform policy makers in developing countries about the socio-economic benefits of new varieties and thus improve adoption of the most effective strategies to address food insecurity and poverty.