text:
\section{Introduction}\label{Introduction}

 Let $ \Omega \subset {\bf R} ^2$ be a
domain and suppose that we have a decomposition of the boundary $
\partial \Omega = D\cup N$ with $ D  \cap N = \emptyset$.
We consider the mixed  boundary value problem for the linear Stokes system
\begin{equation}\label{MP}
\left\{ \begin{aligned}
&-\Delta u + \nabla p = f , \qquad &&\mbox{in } \Omega \\
& - \div u = g , \qquad            &&\mbox{in } \Omega \\
&2 \nu \epsilon(u)  - p \nu = f _N,\qquad  &&\mbox{on } N \\
&u = f _D , \qquad                 &&\mbox{on }  D .
\end{aligned}
\right.
\end{equation}
In the boundary value problem (\ref{MP}),  the functions $f$, $f_D$,
$f_N$ and $g$ are given
and we look for a vector-valued function $ u :  \Omega \rightarrow
{\bf R}^2$ and scalar function $p : \Omega \rightarrow {\bf R}$ which
satisfy the above conditions.   We use $ \epsilon(u) $ to denote the
symmetric part of the gradient of $u$ and $ \nu$ is the outer unit
normal to $ \partial \Omega$.

The goal of this note is to  give conditions on the decomposition
$\partial \Omega = D\cup N$
that allow us to  construct the Green function for the
boundary value problem (\ref{MP})
in a Lipschitz domain in two dimensions. Our argument
begins with an idea of D.~Mitrea and I.~Mitrea \cite{MR2763343} who
construct Green functions for second order elliptic equations with
Dirichlet boundary conditions in two dimensions by extrapolating from
the standard theory of weak solutions with gradient in $L^2$.  We
expect the Green function to have a gradient in the Lorentz space $L^
{ 2, \infty }$, or weak $L^2$, and an argument that involves
perturbing the function space allows us to
extend the existence theory from $L^2$ to weak $L^2$.

Once we have the existence of a solution with gradient in $L^ { 2,
\infty}( \Omega)$, we use local regularity estimates for solutions
to establish pointwise estimates for the Green function. These
estimates are  mainly of  interest near the boundary since
 the interior regularity of solutions is
well-understood.

The approach we use is  limited to two dimensions. To study the Green
function for a
boundary value problem for a second order operator
in an $n$-dimensional domain, one would need to
study solutions that have their gradient in $L^ { n / ( n -1 ) ,
\infty}$. When $ n =2$, $ n /(n-1) =2$ and it is not
difficult to pass from the standard theory of weak solutions to results for functions that have a gradient  in $L^{2, \infty}$.
When $ n>2$, the gap from $L^2$ to $L^ { n/(n-1) , \infty}$ is too large for this approach to be fruitful.
  There is a well-known path  to construct the
Green function for the Stokes operator or elliptic systems with
Neumann or Dirichlet boundary conditions in Lipschitz domains in three
dimensions.  This begins with work of Pipher and Verchota
\cite{MR94j:35030} and Dahlberg and Kenig \cite{DK:1990} and continues
in many papers. The recent of work of D.~Mitrea and I.~Mitrea
\cite{MR2763343} includes a construction of the Green function with
Dirichlet boundary conditions for elliptic systems in three
dimensions.  However, the range of exponents for which we can study
the mixed boundary value problem in the non-tangential sense is
smaller than the range for other boundary value problems  and thus we are not able to adapt these arguments to construct the Green function for the mixed problem in three dimensions.
Furthermore, we are interested in understanding the Green function as a
step towards studying the  mixed problem in the non-tangential sense.

We recall only a few high points in the study of Green functions for
elliptic operators.
A classic result of Littmann, Stampachhia and Weinberger
\cite{LSW:1963} gives pointwise estimates for the fundamental solution
(or Green function)
of an elliptic equation in the plane with bounded and measurable
coefficients.
 Dolzmann and M\"uller
\cite{MR1354111} construct the Green function in a $C^1$-domain for an
elliptic operator with continuous coefficients.
Auscher and collaborators \cite{MR1600066} consider elliptic
equations with complex coefficients. Such operators may also be viewed
as elliptic systems.
Dong and Kim
\cite{MR2485428} construct the Green function for elliptic systems for
operators with bounded and measurable coefficients and with
Dirichlet boundary conditions.

There is a large body of work related to the study of the Stokes
equations in Lipschitz  domains.  Fabes, Kenig, and Verchota
\cite{FKV:1988} treat the Neumann and Dirichlet problems. They
establish the existence of  solutions with
the non-tangential maximal function of the
gradient in $L^2( \partial \Omega)$. The results for the
Dirichlet problem were extended to an optimal range of $L^p$-spaces by
Z.~Shen \cite{MR1223521}.  The work of D.~Mitrea and I.~Mitrea \cite{MR2763343}
mentioned above gives a construction of the Green function for the
Stokes operator with Dirichlet boundary conditions in   two and three
dimensions.  The
mixed problem for the Laplacian
in Lipschitz domains was the subject of a problem posed
by Kenig in his CBMS lectures \cite{CK:1994}. Recent progress may be
found in the article of Taylor, Ott, and Brown \cite{MR3034453}. This
work and related results on the Lam\'e system \cite{MR3040944} in two
dimensions rely on estimates for the Green function with mixed
boundary conditions. One motivation  for the present note  is
an interest in
developing the properties of the Green function that are needed to attack
the mixed problem for the linear Stokes system in two dimensions.

There is also substantial interest in studying the mixed problem for
the linearized Stokes equations in
polyhedral domains and obtaining optimal regularity results. Many
polyhedral domains are also Lipschitz domains, however the class of
polyhedral domains includes domains that are not Lipschitz in the
sense defined below, at least in dimension three and higher.  Lipschitz
domains, of course, include many domains that are not polyhedral and
are of interest because the class of Lipschitz domains includes domains
with interesting features at all length scales.  Our treatment of the
mixed problem includes conditions on the decomposition of the boundary
that are scale invariant as well.  We refer the reader to Maz'ya and
Rossmann \cite{MR:2007,MR2563641} and the references cited therein for
additional background on the mixed problem for the Stokes system  in polyhedral
domains.

Finally, we note that there has been recent work on the mixed problem
in domains that are more general than Lipschitz. See work of Auscher,
Badr, Haller-Dintelmann, and Rehberg \cite{ABHR:2012},
Haller-Dintelmann, Jonsson, Knees, and Rehberg \cite{HJKR:2012}, and
Brewster, D.~Mitrea, I.~Mitrea, and M.~Mitrea \cite{BMMM:2012}.
Roughly speaking, one needs the set where Neumann data is specified to be
Lipschitz, while the Dirichlet set is allowed to be more general. The
work of Brewster {\em et. al. }has a weaker condition near $N$. It
would be interesting to construct Green functions for the mixed problem
in a similar setting.  One impediment to carrying out the work
reported here in a more general setting is the difficulty of treating
the equation $ \div u = f$ as in Proposition \ref{Bogo}.

In section \ref{WeakSolutions} we give a  standard weak formulation of
the mixed problem in (\ref{WMP}).    Using this notion of a  weak
solution
we
 define the {\em Green function }for the boundary value problem
(\ref{MP}). The Green function is a pair $ (G(x,y), \Pi(x,y))$ where $
G^ { \alpha \beta }: \Omega \times \Omega \rightarrow {\bf R} $ and $
\Pi ^ \alpha : \Omega \times \Omega \rightarrow {\bf R}$ with $ \alpha,
\beta =1,2$. If $(u,p)$ is a weak solution (as defined below)  of the mixed problem with data
$f$ and $g$ taken from $ C^ \infty _ c ( \Omega)$ and the boundary data $
f_D$ and $f_N$ are zero, then the solution $u$ is given by
\[
u ^ \alpha (x) = \int _ \Omega G^{ \alpha \beta } ( x,y ) f^ \beta (y)
 + \Pi ^ \alpha (x,y ) g(y ) \, dy .
\]
Since we have uniqueness of weak solutions, it is immediate that the
Green function is unique.
\note{ Give up on representation formula for the pressure. Leave this
  as an open problem for now. }

Our main result is the following theorem.  The reader will need to
refer to section \ref{Defs} for a  detailed statement of our conditions
on the domain $ \Omega$ and the decomposition of the boundary $
\partial \Omega = D \cup N$.   In the theorem below, we use $d$ to
denote the diameter of $ \Omega$.

\begin{theorem}
\label{Green}
Let $\Omega $ be a Lipschitz domain and suppose that $ D$ and $ N$
satisfy the conditions (\ref{AhlDav}), (\ref{DOpen}), and
(\ref{NOpen}).
 There exists a Green function $(G, \Pi)$ for the
boundary value problem (\ref{MP}) and the Green function satisfies the
following:

\begin{align}
& |G(x,y) | \leq C ( 1 + \log ( d / |x-y |)   )
\label{GreenUp} \\
& \| \nabla G(x, \cdot ) \|_ { L^ { 2, \infty } ( \Omega ) } + \|
\Pi (x, \cdot ) \|_ { L^ { 2, \infty } ( \Omega)} \leq C   \label{GreenLorentz} \\
& \nabla_y G(x, \cdot ) , \Pi(x, \cdot ) \in  L^ q ( \Omega
\setminus \Omega _ \rho( x))  \text{ for  } \rho > 0, \frac 1 q> \frac
1 2 -\kappa \label{GreenGrad} \\
& |G(x,y ) - G(x, z ) | \leq  C \left ( \frac { |y -z | } { |x-y | }
\right) ^ \gamma , \qquad |x-y| > 2 |  y-z | \label{GreenHolder} \\
& G^ { \alpha \beta } (x,y) = G^ { \beta \alpha } (y,x) . \label{GreenSymm}
\end{align}
The parameter $ \kappa>0 $ depends on the Lipschitz character of
$\Omega$ and  the H\"older exponent $ \gamma$ depends only on $M$.
\note{define $d$}

Furthermore  if $u$ is the weak solution of (\ref{WMP}) with  $ f_N \in L^ t ( N)$, $f \in L^ t (
\Omega )$, and $g \in L^ {t} _{1, \partial \Omega}( \Omega)$ for some $t>1$, then we have
\begin{multline}
\label{GreenRep}
u^ \alpha (x) =
 \int _ \Omega G^ { \alpha \beta } (x,y) ( f^ \beta (y)
- \frac { \partial g } {\partial y _ \beta } (y) )
+ \Pi ^ \alpha(x,y)  g(y)  \, dy
\\
 + \int _N G^ { \alpha \beta }(x,y)  f_N ^ \beta
(y) \, d\sigma.
\end{multline}
\end{theorem}

The paper will proceed in the following manner.
In section \ref{Defs} we introduce the function spaces and machinery
needed to construct the Green function.
Section \ref{WeakSolutions} establishes the existence of weak
solutions. Section \ref{Local} gives the local regularity needed to
establish the pointwise estimates for the Green function and section
\ref{TheEnd} provides the details of the proof of Theorem \ref{Green}.

\section{Notations and definitions}
\label{Defs}
\subsection{Domains}

We assume that $ \Omega \subset {\bf R}^2 $ is a {\em Lipschitz domain}.
Thus $ \Omega$ is a bounded, connected, open set and if $ x\in
\partial\Omega$, then the boundary near $x$ is given by
the graph of a Lipschitz function. More precisely, this means that we
have  constants $M >0$ and $ R_0>0$ so that for each $ x \in \partial
\Omega$, we may find a Lipschitz function $ \phi : {\bf R} \rightarrow
{\bf R}$  such that
\begin{align}
\label{LipDef1}
\Omega \cap Z_{200 R_0 } (x) & = \{ y : y _ 2 > \phi (y_1) \}  \cap Z _
       {200 R _ 0} (x) \\
\label{LipDef2}
\partial \Omega \cap Z_{200 R_0 } (x) & = \{ y : y _ 2 = \phi (y_1) \}  \cap Z _
       {200 R _ 0} (x),
\end{align}
where for $x=(x_1,x_2) \in {\bf R}^2$,  $ Z _ \rho ( x) = \{ y : |x_1 - y _1 | < \rho,
|x_2- y _ 2 | < (4M+2) \rho\} $ is a {\em coordinate cylinder } centered at $ x\in \partial\Omega $.
The coordinate system in (\ref{LipDef1}-\ref{LipDef2}) is assumed to
be a rotation of the standard coordinate system.
Since the domain is bounded, we may  fix a covering of $ \partial \Omega$
by a finite collection of coordinate cylinders $ \{ Z_{R_0}(x_1),
\dots,  Z_{R_0}(x_N)\}$ and we will use these cylinders in the
constructions  below.

Next we define {\em boundary intervals }$\sball x \rho \subset  \partial \Omega$.
If $ 0 < \rho < 100R_0$ and $ x \in
\partial \Omega$, then we set $ \sball x \rho  = Z_ \rho (x) \cap
\partial \Omega$. We also define {\em local domains } $ \Omega_\rho (x) \subset \Omega $.
 These sets will be disks in the interior of
$ \Omega$ and will provide a convenient family of sets for studying
the local regularity of solutions of a boundary value problem near the
boundary. For $ 0 < \rho < 100R_0$, if the distance from $ x $ to $
\partial \Omega$, $ \mathop{\rm dist}\nolimits(x, \partial \Omega) > \rho$, then we let $
\locdom x \rho = \{ y : |x-y | < \rho \}$ be the disk centered at $x$
and of radius $ \rho$. If $ \mathop{\rm dist}\nolimits(x, \partial \Omega) \leq
 \rho$, then $x$ lies in some coordinate cylinder $ Z$  and we write
$ x = \hat x + s e_2$ with $ \hat x \in \partial \Omega\cap Z$
in the coordinate system of $Z$ and $s>0$.
We define the local domain  $ \locdom x  \rho = \Omega \cap  Z _
\rho (\hat  x)$. Since the definitions of $ \sball x \rho$ and $\locdom x
\rho$ depend on a coordinate cylinder, there may be several choices
for these sets.
\note{ as defined above, the boundary intervals do not depend on choice of
coordinate cyliner}
Our estimates will hold for any such choice with the
convention that when two of these objects appear in an estimate we
use the same coordinate cylinder to define both of them.
The local domains $ \locdom x \rho$ are star-shaped Lipschitz domains
with Lipschitz constant depending only on $M$, the Lipschitz constant
for $ \Omega$. This will be helpful below as we will need to know that
various estimates hold uniformly over all local domains $\locdom x
\rho$.   See Ott and Brown \cite[p.~4376]{MR3040944} for a definition
of  star-shaped Lipschitz domains and for more details.

We assume that the set  $D \subset \partial\Omega $ satisfies the {\em
  Ahlfors-David regularity condition}. Thus with $M$ the constant that
controls the local behavior of our domain,
\begin{equation}\label{AhlDav}
M^{-1}\rho \leq \sigma (\Delta _ \rho (x) ) \leq M  \rho,\  \qquad 0 < \rho  < R_0,\,
x\in  D.
\end{equation}
Above, $ \sigma $ stands for surface measure.
We also require the sets $D$ and  $N$  to have nonempty interior
in the following quantitative sense.
We assume that there exists $M> 0$ such that
\begin{gather}
 \label{DOpen}
\text{There exists  $ x\in D$ so that
 $\sball x  { M^ { -1} R_0}  \subset D$, }
\\
\label{NOpen}
\text{There exists  $ x\in N$ so that
 $\sball  x  { M^ { -1} R_0}  \subset N$. }
\end{gather}
The assumption on $N$ is used in Proposition \ref{Bogo} to solve the
equation $ \div u =f$ even
when $f$ does not have mean value zero.
The assumption on $D$ is used  in Appendix \ref{KornApp} to obtain
coercivity of the
quadratic form.   Note
that this condition is only at the scale of the domain and is not
assumed to hold at every scale. In our proof of local regularity
below, we will consider mixed problems on local domains $ \locdom x \rho$ with $ \rho $ arbitrarily small.
 We will  have freedom to specify boundary conditions on $ \locdom x \rho \setminus \partial \Omega$ and
 will be able to guarantee that the conditions  (\ref{NOpen}) and (\ref{DOpen}) hold on all
 local domains.

The estimates in this paper are of two types. We will prove local
estimates for solutions that hold at scales $ \rho$  with $ 0 < \rho
< R_0$ and with a constant that depends only on $M$ and the indices
of any Lorentz spaces that appear in the estimate. In estimates over
the entire domain, the constants will also depend on the
 collection of coordinate cylinders that
cover the boundary and such constants will be said to depend on the
Lipschitz character of $ \Omega$.

\subsection{Function spaces}

For $ 1 < q < \infty $, $ 1 \leq r \leq \infty$,
we let $L^ { q,r}  $  denote the standard Lorentz space as defined in
\cite{BL:1976}, for example.
For $ k = 1, 2, \dots$, we will use
$ L^ { q,r}_{k}( \Omega)$ to denote   the  Lorentz-Sobolev space of functions
with $k$ derivatives in $L^ { q, r}(\Omega)$,
\[
L^ { q,r } _k ( \Omega ) = \{ u : D^ \alpha u \in L^ { q, r} ( \Omega
) \text{ for all } |\alpha | \leq k \}
\]
with the scale-invariant norm defined by
\[
\| u \|^q_ {L^ { q, r } _k ( \Omega ) }
=  \sum _ { |\alpha | \leq k } R_0 ^ { q(|\alpha | - k) }  \|D^ \alpha u
\| ^ q_ { L ^ { q, r } (     \Omega ) }.
\]
In the special case when $ q = r$, we will drop the second index and
note that this gives the standard Sobolev space $ L^ q _k(\Omega)$. We
will need to know that the spaces $ L^{q,r}_k ( \Omega ) $ form
a real interpolation scale and that the spaces $
L^q_k(\Omega)$ are a
complex interpolation scale (in $q$). See
 \cite[Proposition 2.4]{JK:1995} for  this result in
the setting of complex interpolation and
\cite[Proposition 2.1]{MR2763343} for real interpolation.

If $D \subset \partial   \Omega$ is a closed subset, define $ C^
\infty _D ( \bar \Omega)$ to be the collection of
 functions that have derivatives of all orders in $ \bar \Omega$,
 whose derivatives extend continuously to $ \bar \Omega$ and
that vanish on a neighborhood of $D$.  For $ 1 < q< \infty $, $ 1 \leq r
\leq \infty$, we define
$L^ { q,r } _ {1,D} ( \Omega)$ to be the closure of $ C^ \infty _D(
\bar \Omega)$ in $L^ { q, r} _{1}( \Omega)$.
Since we do not have a nice dense class in the space $ L^ { q, \infty
} _1( \Omega)$, the above definition does not serve to define the
spaces $L^ { q, \infty}_{1, D}( \Omega)$.   An alternate definition is
suggested by the work of Haller-Dintelmann, Jonsson, Knees, and
Rehberg \cite{HJKR:2012}
  who  use the
machinery developed by Jonson and Wallin \cite{MR820626} to construct a projection $
{\cal P } : L^ q _ { 1} ( \Omega) \rightarrow L^ q _{ 1, D} (
\Omega)$.  By interpolation, we have that the map $ {\cal P} : L^ { q, r
  } _1 ( \Omega) \rightarrow L^ { q,r } _1 ( \Omega) $.
Thus  we may  define  $ L^ { q, r } _{1, D}(
\Omega) = {\cal P } ( L^ { q, r } _ 1 ( \Omega))$ and since $ {\cal
  P}^2= {\cal P}$  the space $ L^{q,
  r}_{1,D}( \Omega)$ will be a closed subspace of  $L^ {
  q,r}_1(\Omega)$.   It is straightforward to show that the family of
spaces $L^ { q,r } _ {1, D}( \Omega)$ is an interpolation scale and
thus  with   $[\cdot, \cdot ] _{ \theta }$ denoting the operation of
complex interpolation, the
spaces $ L^q_{1,D}( \Omega)$ satisfy
\begin{equation}
\label{Complex}
[ L^ { q_0}_{1,D} ( \Omega), L ^ { q_ 1 }_ { 1, D} ( \Omega) ]_ \theta
= L^ { q_ \theta } _ { 1, D} ( \Omega), \quad \frac 1 { q_ \theta } =
\frac { 1- \theta } { q _0 } +  \frac \theta { q_ 1 }.
\end{equation}
We will use  $( \cdot, \cdot ) _ {  \theta ,r } $
to denote  the real interpolation operation and  we have
\begin{equation}  \label{Real}
( L^ { q_0, r_0} _{1,D}( \Omega) , L^ { q_1, r_1}_{1, D} ( \Omega ) ) _ {
    \theta , r} =  L^ { q_ \theta, r}_{1,D} ( \Omega ) , \qquad
  \frac 1 { q_\theta } = \frac { 1-\theta } { q_0 } + \frac \theta { q_1} .
\end{equation}
These results hold in our setting where the domain is Lipschitz and
the set $ D$ satisfies the Ahlfors-David regularity condition
(\ref{AhlDav}).  In fact these results hold more generally and we
refer to
 the work of Haller-Dintelmann {\em et.~al.} \cite{HJKR:2012},
Auscher {\em et. al. } \cite{ABHR:2012}
and
 Brewster  {\em et.~al.}
\cite[section 6]{BMMM:2012} who consider these issues on a   larger
family of domains.

\note{ Proof of the real interpolation property.
Let $X_{q,r} = {\cal P } ( L^ { q,r}_1( \Omega))$.

We claim that $ (X_{q_0} , X_ {q_1} ) _{ \theta , r } = X_{q_ \theta,
  r }$.

We first observe that $ {\cal P }  : L^ {q_i}_1(\Omega) \rightarrow X _ {
  q_i}$ and thus it follows that
\[
{\cal P }  (
( L^ { q_0}_1(\Omega), L^ {  q_1}_1 (\Omega) ) _ { \theta, r}
) \subset (X_{q_0}, X_{q_1} ) _ { \theta, r}.
\]
But we have $ ( L^ { q_0}_1(\Omega), L^ {
  q_1}_1(\Omega) ) _ { \theta, r} = L^ { q_\theta , r} _1(\Omega) $,
thus we conclude
\[
X_ { q_ \theta, r} ={\cal P } (  L^ { q_ \theta, r } _ 1 (\Omega)  )
\subset (X_{q_0}, X_{q_1} ) _ { \theta, r}.
\]

To obtain the reverse inclusion, we observe that $ {\cal P }^2 = {\cal
  P}$  on each  $ X_{q}$ and thus on  $
( X_ { q_ 0} ,
X_{q_1})_{\theta, r} \subset X_ {q_0} + X_{q_1}$.
Since we have $i: X_q \rightarrow L^ { q}_1(\Omega) $, interpolation
implies   that the natural inclusion
$i:(X_{q_0}, X_{q_1} ) _ { \theta, r}  \subset L^ { q_\theta,
  r}_1$. Applying $ \cal P$ gives
${\cal P } ( (X_{q_0}, X_{q_1} ) _ { \theta, r} ) =
(X_{q_0}, X_{q_1} ) _ { \theta, r}  \subset {\cal P}( L^ { q_ \theta, r} _1 (
\Omega)) = X_ {q_\theta, r} $.

The above uses the identification of the real interpolation spaces for
$L^q_1(\Omega)$.  Reference?

The same argument also establishes that the spaces $L^q_{1, D} (\Omega)$
behave as expected under complex interpolation.

Remark: More simply with $ \cal P$  and the natural inclusion, we have
that $ L^ {q,r}_{1,D}$ is a retract of $L^ {q,r }_1$.

Can we prove density of $ C^\infty _D ( \bar \Omega)$ in
$ L^ { q, r }_{1,D}(\Omega)$ provided $r,q< \infty$.

We will need that when $ q=r=2$, then $ L^ {2,2}_{1,D}( \Omega)$ is the
standard space used in the treatment of the mixed problem as, for
example, in the work Auscher {\em et. al.}\cite{ABHR:2012}.
 }

 We recall that a version of H\"older's
inequality holds in the Lorentz spaces (see \cite[Theorem 3.5]{MR0146673}). For
a set $ {\cal O }$,
\begin{equation}
\label{LorentzHolder}
\int _ {\cal O }| f g |\,d y \leq C\| f \|_ {L^ { q_0, r _ 0 } ( \cal
  O)}
\| g\| _ { L ^ { q _ 1, r _1}( \cal O ) } , \qquad  \frac 1  { q _
  0 } + \frac 1  { q _ 1 } =1 , \ \frac 1 { r _0 } + \frac 1 { r _1} \geq
1.
\end{equation}
Furthermore, if $1<q< \infty$ and $ 1\leq r < \infty$, then
with $ X^*$ denoting the dual space of a Banach space $X$, we have
\begin{equation}
\label{DualLorentz}
 L^ { q, r } ( \Omega) ^* = L^ {q', r'}( \Omega), \qquad \frac 1 q +
 \frac 1 { q' }= \frac 1 r + \frac 1 { r'} = 1.
\end{equation}
The result (\ref{DualLorentz})  follows from a general result about duality and
interpolation, see Theorem 3.7.1 in the monograph of Bergh and
L\"ofstrom \cite{BL:1976}.

\subsection{Inequalities}

The weak formulation for the Stokes operator introduced below will use
the quadratic form
$a : L^ { q,r } _ { 1, D} ( \Omega)
\times L^ { q', r' } _ { 1,D} ( \Omega ) \rightarrow {\bf R} $ given by
\begin{equation}
\label{FormDef}
a(u,v) = 2 \int _ { \Omega } \epsilon _i ^\alpha (u) \epsilon _ i ^
  \alpha (v) \, dy .
\end{equation}
Here $ \epsilon ^ \alpha _ i (u) = \frac 1 2 ( \frac { \partial u ^
  \alpha } { \partial x _ i } +\frac  { \partial u ^ i } {\partial x _
  \alpha } ) $ denotes the symmetric part of the gradient of $u$ and
we use the convention that repeated indices are summed.
Our conditions imply that we may find a constant $c$ depending only on
the global Lipschitz character  of $ \Omega$ such that
\begin{equation}
\label{Korn}
a(u,u) \geq c \| \nabla u \| _ { L^ 2 ( \Omega)}, \qquad u \in L^ 2 _{
  1, D} ( \Omega).
\end{equation}
See Proposition \ref{Coercive} in Appendix \ref{KornApp}.

Next we give a small extension of a result of Bogovskii
\cite{MR631691}
which will
allow us to solve $\div u =f$ with $ u \in L^2_{1,D}( \Omega)$ when $f
$ is in $L^2 ( \Omega)$.
In the proof below, we will use $L^ 2 _0 ( E)$ to denote  $\{ f \in L^2 (E)
: \int _E f =0\}$.
The main fact from Bogovskii we will need is that there is
a linear map  $B: L^2 _0( \locdom x r ) \rightarrow L^{2}_{1, \partial \locdom x r }
( \Omega)$  which satisfies $ \div B f =f$ and the norm of this
operator is bounded by $ C/r$ where the constant $C$ depends only on
the constant $M$ which controls the local  Lipschitz character of $
\Omega$.
The argument below extends this result to general domains in a form
that is useful for the study of the mixed problem.

\note{
An $L^p$-result holds as well.
 }

\begin{proposition}
\label{Bogo}
 Let $ \Omega $ be a connected Lipschitz domain with a
  decomposition of the boundary $ \partial \Omega = D \cup N$ and
  assume that $N$ satisfies (\ref{NOpen}).    There exists
  a linear  map $ B : L^ 2 ( \Omega ) \rightarrow L^ 2 _ { 1, D} (
  \Omega)$ that satisfies
\[
\div B f =f
\]
and $ \|  Bf \| _ { L^ 2 _{ 1,D}( \Omega) } \leq C \|f \|_ { L^
  2 ( \Omega) }$, where $C$ depends on the global character of $
\Omega$.
\end{proposition}

\begin{proof}
We begin by covering $\Omega$ by a collection of local domains,
$ \locdom {x_j}{R_0/2}$, $ j =0,1,\dots,N$, and set $ \omega _j
= \locdom {x_j } {R_0}$ and $ \Omega _k = \cup _ {  j=0 } ^ k \omega
_j $, $ k=0, \dots, N$. Using the assumption (\ref{NOpen}) and
reindexing the domains $\{\omega_j \}$, we may
assume that there is a surface interval $ \sball x
s \subset \partial \omega _0 \cap  N$ with $ s $ comparable to $R_0$.
Furthermore, since $\Omega$ is
connected, we may order the domains so that $ \locdom {x_k
}{R_0/2} \cap ( \cup _ { j = 0 } ^ { k-1} \locdom { x_j } { R_0
/2}  ) \neq \emptyset$.

 We will inductively define a sequence of maps $
P_k : L^ 2 ( \Omega) \rightarrow L^2 ( \omega _0)\times  L^2_0( \omega
_1)\times
\cdots \times  L^2 _0 ( \omega _k ) $, $k=0, \dots, N$ so that $ P_k f = ( f_0, \dots,
f_k)$  satisfies
\begin{equation}
\label{Hype}
\sum _ { j = 0 } ^ k f_ j = f \cdot \chi_{ \Omega_k } , \qquad \sum _
{ j = 0 } ^ k \| f_j \| _{L^ 2 ( \omega_j ) } \leq C \|f\|_{ L^
2( \Omega_k )}.
\end{equation}
In the first sum of (\ref{Hype}) we set $f_j$ to be zero outside
$ \omega_j$. We define $ P_0$ by $P_0 f =f \cdot  \chi _ { \omega_0 } $ and
clearly (\ref{Hype}) holds for $k=0$.

Before we define $ P_k$, $k =1, \dots,N$, we observe that since
$ \Omega _ { R_0 /2 } ( x _k ) \cap ( \cup _ { j = 0 } ^ { k-1} \Omega
_{R_0/2} ( x_j )) \neq \emptyset$, we have that the measure of $\omega
_k \cap \Omega_{k-1}$ is comparable to $R_0^2$. If $ f \in
L^2 ( \omega _k )$, we write $ f\cdot \chi _ { \Omega _k }  = g  + f _k$
with
\[
f_k = f \cdot \chi _{ \omega _k \setminus \Omega _{ k-1}} - \frac
{\chi_{ \omega _k  \cap \Omega _ { k
-1 }}}
 {
|\omega _k \cap \Omega _{ k -1} | }
\int _ { \omega _k \setminus \Omega _{k-1} } f \, dy
\]
and $ g =f - f_k$. The inequality of Cauchy-Schwarz and our
observation that the measure of  $ \omega _k \cap \Omega_{k-1}$  is
comparable to $R_0^2$ implies that
\[
\| f_k \|_{ L^2 ( \omega_k )} + \|g \|_{ L^2 ( \Omega _{ k -1})}
\leq \| f\| _ { L^2 ( \Omega_k)}.
\]
We use our induction hypothesis that (\ref{Hype}) holds for $ k
-1$ and let $(f_0, \dots, f_{ k-1}) = P_{k-1} g$. It is easy to
see that $ P_k f = ( f_0, \dots, f_k )$ is the desired decomposition.

Given the decomposition of $f \in L^2 ( \Omega)$, $f=(f_0, \dots,
f_N)$, we may use the result of Bogovskii  in each $ \omega _j$, $j
=1, \dots N$,
to find $u_j$ which satisfies $ \div u _j =
f _j$ and $ u _ j \in L^ 2 _ { 1, \partial \omega _j }( \omega _j
)$. Recall
that $L^ 2 _ { 1, \partial \omega _j }( \omega _j ) $ is
the Sobolev space of functions with one derivative in $L^2$ and which
vanish on the boundary $ \partial \omega_j$. To define $u _0$, we use the
existence of the surface interval $\sball x s$ noted above to find a
vector-valued
function $ \eta \in C^\infty _D(\bar \Omega)$,
$\mathop{\rm supp}\nolimits  \eta \subset \omega _0 \cup \sball x s $ and so that
$ 1 = \int _ {\sball x s} \eta \cdot \nu\, d\sigma = \int _ { \omega
_0 } \div \eta \, dy. $  We let $ u _ 0 = v _0 + \eta \int _ {\omega
_0 } f_0 \, dy $ where $ v _0 $ is the solution of $ \div v _0 = f_0
- \div \eta \int _{ \omega _0 } f_0\, dy $ guaranteed by Bogovskii's
result. Finally, we let $ u = \sum _ { j =0 } ^ N u _j$
where we have defined each $u _j $ to be zero outside
$ \omega_j$.
It is immediate
that $ u \in L^2 _ { 1, D} ( \Omega )$ and satisfies $ \div u
=f$.
\end{proof}

\section{Weak solutions}
\label{WeakSolutions}
The goal of this section is to show that we can solve the mixed
problem when the right-hand side of \eqref{WMP} lies $L^ { 2,1} _{ 1,D}
( \Omega ) ^*$, the dual of the Lorentz-Sobolev space $ L^ { 2,1 } _ { 1,D}
(\Omega) $.   Since functions in  $L^ { 2,1 } _ 1 ( \Omega)$ are
continuous, the Dirac delta measure lies in the dual of $L^ { 2, 1} _1
( \Omega) $ and thus we are able to construct the Green function as
the solution of (\ref{MP}) when $f$ is the Dirac delta measure,
$f_N=0$ and $g =0$.

We give a weak formulation of the mixed problem. We will
need to consider this formulation, not only on the Sobolev space $L^ 2
_ { 1, D} ( \Omega) $ but also on Lorentz-Sobolev spaces with indices
$q$ near 2.
To simplify the notation below, we introduce spaces $ S_ {q,r} = L ^ {
  q, r } _ { 1,D} ( \Omega) \times L^ { q,r } ( \Omega)$ and  then  $ S^
* _{ q, r} $ will  denote the dual of $S_ { q, r}$. As with the Lorentz and
Lebesgue spaces, we will drop the second index when both indices take
the same value and use $ S_q = S_ {q,q}$.

\note{ The particular form is not important, provided we have coercivity
  and boundedness.
}

We give a weak formulation of (\ref{MP}) in the space $S_{
  q,r}$. For this problem, we will require
$ f\in L^ { q, r} _{ 1, D}(\Omega) ^*$,
$g \in L^ { q,r}( \Omega) $ and $f_N$ to lie in the dual
of the image of $L^ { q, r} _{1, D} ( \Omega)$ under the trace
operator.  We give a weak formulation of (\ref{MP})  in the special
case that $f_D=0$.
We say that $(u, p) $ is a weak solution of (\ref{MP}) with $f_D=0$ if we have
\begin{equation}
\label{WMP}
\left\{
\begin{array}{ll}
 a(u,\phi) - \int _ { \Omega } p \div \phi  \, dy
\\ \qquad \qquad = \langle f, \phi\rangle
+ \langle f_N, \phi \rangle _{ \partial \Omega}
-
 \int _ \Omega \nabla g \cdot  \phi \, dy,
\qquad
& \phi
\in L^ { q' , r'}_{1,D} ( \Omega)  \\
 - \div u = g, \\
( u,p) \in S_ {q,r}.
\end{array}
\right.
\end{equation}
\note{
It is not clear if there are simple conditions on $g$  that guarantee the
right-hand side is in the dual of $L^{ q',r'}_{1,D} ( \Omega)$.
This weak formulation is not quite right--we weren't allowed to
  integrate by parts to obtain $ \int g \div \phi$. When $g$ is zero,
  this is not a problem, but for general $g$, there is something more
  to be said.

This might affect the Caccioppoli inequality, too.
}
The three terms on the right of the first equation in (\ref{WMP}) have different roles in the
 Stokes system. However,  for the purposes of
establishing existence of solutions, it
makes sense to simplify the problem by treating
the three terms as  one.  We will consider the case when the
right-hand side of the first equation in (\ref{WMP}) gives an element
in  the dual of $L^ { q', r'} _{ 1,
  D} ( \Omega)$.   It is not obvious how to give general conditions on
$ f_N$, $f$, and $g$ which guarantee that this will happen. However,
our construction of the Green function will not require us to consider
this question.

\note{  Can we find an example of a function $g$ in $L^2$ for which the right-hand side is not in $L^2_{-1}$? }

We introduce a map $ T: S_ {q,r} \rightarrow S_ {q',r'}^*$ given
by $ T(u,p ) = ( \lambda , \mu)$ where  $ \lambda $ is in the dual of
$ L^ { q', r' } _ { 1, D} ( \Omega)$ and $ \mu$ is in the dual of $ L^ {
  q', r'} ( \Omega)$ and are given by
\begin{align}
\label{Tdef1}
\lambda ( \phi ) &= a(u,\phi) -\int_\Omega p \div \phi \, dy  \\
\label{Tdef2}
\mu(h)  & = - \int _ \Omega h \div u \, dy .
\end{align}
 Thus, establishing the existence of a solution to
(\ref{WMP}) is equivalent to showing that the map
\begin{equation}
\label{UpsideDown}
T : S_ {q, r}  \rightarrow S_ {q',r' } ^* \text{ is invertible.}
\end{equation}
The next theorem outlines the main steps in establishing
(\ref{UpsideDown}).
\begin{theorem}
\label{Exist}
Let $ \Omega $ be a Lipschitz domain and suppose that the
decomposition
of the boundary $ \partial \Omega = D\cup N$ satisfies
(\ref{AhlDav}), (\ref{DOpen}), and
(\ref{NOpen}). Then $ T$ is a bounded operator satisfying
\begin{align}
& T : S_ {q,r } \rightarrow S_ {q',r'}^*, \qquad 1 < q <  \infty , 1 \leq
  r \leq \infty \label{Bounded} \\
& T : S_ 2\rightarrow S_2 ^* \text{ is invertible}. \label{Energy}
\end{align}
Moreover, there exists $\kappa > 0$ such that
\begin{align}
& T: S_q \rightarrow S_{q'} ^ * \text{ is invertible if } |\frac 1 2 -
  \frac 1 q | < \kappa \label{Jiggle} \\
& T ^ { -1 } : S _ { q' ,r' } ^ * \rightarrow S_ { q, r } , \ |\frac
  1 2 - \frac 1 q | < \kappa, \ 1 < r \leq \infty. \label{Useful}
\end{align}
The constant $ \kappa$ depends only on the Lipschitz character of $
\Omega$. The norms of the operators depend on the Lipschitz character
of $ \Omega$ and the exponents $q$ and $r$ for the Lorentz spaces.
\end{theorem}

\begin{proof} The first claim (\ref{Bounded}) is an easy consequnce of
the extension of H\"older's inequality to the Lorentz spaces,
(\ref{LorentzHolder}). The second claim (\ref{Energy})
 is a standard  existence
theorem for weak solutions of the Stokes system, but  we sketch the details due to
its fundamental importance. We follow the argument given by
Maz'ya and Rossman for polyhedral domains \cite{MR2563641}.

In the proof of the second claim (\ref{Energy}), we
will work
 in  the subspace of functions that
are divergence free.
Thus we let
\[
H = \{ u \in L^ { 2} _{ 1, D} ( \Omega) : \div u =0\}
\]
and  we define $ H^ \perp$ to be the orthogonal complement of $ H$
in $L^ 2 _ { 1, D}( \Omega) $.
\comment{ where we use
\[
\int _ \Omega \frac { \partial u ^ \alpha} {\partial x _i } \frac {
  \partial v ^ \alpha }{\partial x _ i }\, dx
\]
as the inner product
on $L^2 _ {1, D} (
\Omega)$. }
  According to Proposition \ref{Bogo}, we have a map $ B' :
L^ 2 ( \Omega ) \rightarrow L^ 2_ { 1, D} ( \Omega)$ with $\div B'f =
f$. We let $ P $ be the orthogonal projection from $L^ 2 _{1, D}(
\Omega)$
onto $H ^ \perp$ and
then define $ B = PB'$. It is easy to see that $ B : L^ 2 ( \Omega )
\rightarrow H ^ \perp$ is an isomorphism and $ \div Bf =f$.

Given $ (\lambda, \mu )$ in $S_2^*$, we need to find
$(u,p) $ in $ S_2 $ which satisfies $T(u,p) =
(\lambda , \mu )$.
Since  $ \mu $ lies in $L^2 ( \Omega)^*$,
 there exists $ g\in L^2 ( \Omega)$ such that
$ \mu ( h) = \int_ { \Omega } gh\,d y $.
From  Proposition \ref{Bogo}, we may find
$ w = Bg$ in $L^ 2 _{1, D} (\Omega)$ which satisfies $ \div w =g$.
The form $a$ is coercive  on $ L^ { 2} _{1, D}(
\Omega)$ (see (\ref{Korn})),
hence  it is also coercive on the subspace $H$ and by
Lax-Milgram we may find
$ v \in L^ { 2 } _{1, D}( \Omega)$ which satisfies
\begin{equation}
\label{ExA}
a(v,\phi) = \lambda ( \phi) - a( w, \phi), \qquad  \phi \in H.
\end{equation}
We let $u = v + w$ and turn to constructing the pressure $p$. As a step in this
direction, we let $ \tau (h ) = \lambda ( Bh) - a ( u, Bh) $ for $ h
\in L^ 2( \Omega)$. As $ B : L^ 2 ( \Omega ) \rightarrow L^ 2 _ { 1,
  D} ( \Omega)$ is continuous, it follows that $ \tau \in L^ 2 (
\Omega) ^*$ and hence we may find $ p \in L^ 2 ( \Omega)$ so that
\[
\tau(h) = - \int_\Omega ph\,d y.
\]
Since $B ( L^2 ( \Omega)) = H ^ \perp$, we have
\begin{equation}
\label{ExB}
a(u, \phi) - \int_\Omega p \div \phi \, dy = \lambda ( \phi), \qquad \phi \in
H^ \perp.
\end{equation}
Combining (\ref{ExA}) and (\ref{ExB}) and observing that $ \div u =
g$, we have found a pair $(u,p)$ which satisfies $ T(u,p) = (\lambda ,
\mu)$.

To establish that $ T $ is injective, we suppose that $T(u,p)
=(0,0)$ with $(u,p) \in S_2$. If $ \mu$ as defined in (\ref{Tdef2}) is
zero, we have $ \div u =0$. Recalling  (\ref{Tdef1})  it follows that $
a(u,u) =0$ and then the
coercivity of $a$ implies that $ u =0$. Once we have that $u=0$,
we  let $ \phi = B p$ in the first line of (\ref{WMP}) and conclude
that $ \int_ \Omega p^2  \, dy =0$ and hence $p =0$.

We turn to the third statement in Theorem \ref{Exist}, (\ref{Jiggle}).  The
family $S_q$ is a complex interpolation scale and by Corollary 4.5.2 in
Bergh and L\"ofstrom \cite{BL:1976} it follows that $S_q^*$ is also a
complex interpolation scale. Using (\ref{Bounded}) and (\ref{Energy}),
(\ref{Jiggle}) follows from a general result of Sneiberg
\cite{MR0634681} (see also Tabacco-Vignati and Vignati
\cite{TV:1988}).

If we choose $ q$ with $ \frac 1 2 < \frac 1 q < \frac 1 2 + \kappa
$ with $ \kappa $ as (\ref{Jiggle}), we obtain
\[
T^ { -1} : ( S^*_q , S^ * _ {q'}) _ { 1/2, r} \rightarrow  (S_{q'} , S_
q )_{ 1/2, r}.
\]
At least for $ 1<r \leq \infty$, we have $( S^*_q , S^ * _ {q'}) _ {
\theta, r} = ((S_ {q'}, S_q)_{\theta ,r'})^*$ (see \cite[Theorem
3.7.1]{BL:1976}).  Thus using our characterization of the real
interpolation spaces for $L^q_{1, D}( \Omega)$ in (\ref{Real}) we obtain
\eqref{Useful}.

\note{
Is the projection needed in the definition of $B$?

Check that $ B: L^2 (\Omega) \rightarrow H^ \perp$ really is onto.

}
\end{proof}

\section{Local regularity}
\label{Local}
In this section, we will need the following version of the Poincar\'e
inequality. A proof may be found in  our previous work
\cite[Appendix A]{MR3040944}.
  Below,  we let $-\!\!\!\!\!\!\int _ { \locdom x r } u \, dy =
|\locdom x  r|^{-1} \int_{\locdom x r} u\, dy$ denote the average of
the function $u$ on $ \locdom x r $.

Let $ u $ be in $L^ q _ {1, D} ( \Omega)$ and let $ \bar u _ {x, r}$
be defined by
\[
\bar u _ {x, r } = \left \{ \begin{aligned}  0 , \qquad & \mathop{\rm dist}\nolimits (
  \locdom x r , D ) = 0\\
-\!\!\!\!\!\!\int _ { \locdom x r } u \, dy , \qquad & \mathop{\rm dist}\nolimits ( \locdom x {  r }
, D) > 0 .
\end{aligned}
\right.
\]
Then for $ 1 \leq q < 2 $ we have
\begin{equation}
\label{PoSo}
\left (\int  _ { \locdom x { r } } |u- \bar u _ { x, r}| ^ { 2q / ( 2-q)} \, dy \right )
^ { \frac  1 q - \frac 1 2 } \leq C \left ( \int _ { \locdom x { 2r} }
|\nabla u | ^ q \, dy \right ) ^ { 1/q}.
\end{equation}
This assumes that the set $D$ satisfies condition (\ref{AhlDav}).
Note that the  integral on the right-hand side of (\ref{PoSo}) is
over a larger set.  In the case where $ \mathop{\rm dist}\nolimits(\locdom x r , D) =0$,
the Ahlfors-David condition (\ref{AhlDav})  guarantees that $u$
vanishes on a subset of
$\partial \locdom x {2r}$ with measure proportional to $r$ and this
allows us to obtain an estimate for $u$ on $\locdom x {2r}$.
 The expansion of the  domain of integration is
not needed  in the case  where $ \bar u _ {x, r } \neq 0$.

The existence of the Green function is an immediate consequence of
Theorem \ref{Exist} since we may define $ G(x,\cdot) ,\Pi (x, \cdot) $
by
$(G^ { \alpha \cdot } (x, \cdot), \Pi^ \alpha (x, \cdot) ) =
T^ { -1} ( e_ \alpha\delta _x , 0 )$. However, more work is needed to obtain
the estimates of our main result, Theorem \ref{Green}. Thus we
consider the local regularity of solutions of the boundary value
problem (\ref{WMP}) before giving the argument for existence.

 \begin{proposition}
\label{Reverse}
Let $ \Omega$ be a Lipschitz domain with $\partial\Omega = D \cup N$ and
$D$ satisfying
(\ref{AhlDav}).  Suppose that $(u, p ) \in S_{q_1}$ is a solution of
the weak mixed problem with $ f_N =0$ and $g =0$.  Let $ \locdom x
\rho$ be a local domain and suppose that $ \eta $ is a smooth cutoff function
which is one on $\locdom x \rho$ and zero outside $\locdom x {
2\rho}$. There exists a positive number $ \kappa$ so that if $
\frac 1 { q_1} < \frac 1 2 + \kappa$, then for each $q$ with $
\frac 1 2 > \frac 1 q > \frac 1 2 - \kappa$, we have
\begin{multline*}
\| p \| _ {L^ q ( \locdom x \rho ) }+
\| \nabla u \| _ {L^q ( \locdom x \rho )}
\\
\leq \| \eta f \| _ { L^ {q'} _{1,D} ( \locdom x {2 \rho} ) ^*} +
\frac 1 \rho  (  \| \nabla u \| _ { L^ {\tilde q }  (
  \locdom x { 4 \rho} ) }  +\|p\|_ {L^ {\tilde q} ( \locdom x { 2\rho})}) .
\end{multline*}
Here, we define $ \tilde q $  by $ \frac 1 {\tilde q} =  \frac 1 2 + \frac 1
  q $.
 \end{proposition}
\note{
 It might make more sense to apply the norm in $L^ {q'} _{ 1, D'} (
\locdom x { 2 \rho} )^* $ to $\eta f$, however in most of our
applications,  $ \eta f $ is zero
and thus there is no point in going to the extra trouble.
}

\begin{proof} We fix a local domain $ \locdom x \rho$ and cutoff
  function $ \eta $ as in the statement of the Proposition. We will
  show that $(\eta (u - \bar u _ {x, 2 \rho}), \eta p)$ is a solution of a mixed problem in $
  \locdom x { 2\rho}$.  We note that our definitions of $ \bar u _ {x,
    2\rho}$ and of $ \eta$ guarantee that $ \eta ( u - \bar u _ { x,
    2\rho})$
will vanish on $D$.  To define the Dirichlet set $ D' \subset
\partial  \locdom x { 2 \rho}$, we will consider two cases. Case 1: $\bar
  \Omega _ { 2 \rho }(x)  \subset \Omega$. In this case,  $\locdom x{2\rho}$
  is a disk and  we define $ D'$
  to be an arc of length $ \pi \rho$ in  the boundary. Case 2: $\partial
  \locdom x { 2\rho } \cap \partial \Omega \neq \emptyset$. In this
  case, we let  $ {\cal S}  = \partial \locdom x {2 \rho}\cap \Omega \cap \{ y : |x_1 -
  y _1 | = 2\rho\}$ denote the sides of $ \locdom x { 2\rho}$ and then
  we put $ D' = (D \cap  \partial \locdom x {2\rho}) \cup { \cal S }$.
Finally, we set $ N'  =
  \partial  \locdom x { 2\rho}\setminus D'$ and
 leave it as an
  exercise to check that the decomposition $ \partial \locdom x
  {2\rho} = D'\cup N' $ satisfies the Ahlfors-David
  regularity condition (\ref{AhlDav}) and the conditions  (\ref{DOpen}), and
  (\ref{NOpen}) with constants that are independent of
$x$ and $ \rho$.
Thus, we may find $ \kappa$ that is independent of $x$ and $\rho$
 so  that (\ref{Jiggle}) of  Theorem \ref{Exist} holds
for $q$ with $ | \frac 1 2 - \frac 1 q | < \kappa$.

We will show that, with $T$ the map defined in
(\ref{Tdef1}-\ref{Tdef2}) for $ \locdom x {2\rho}$, we have
\begin{multline}
\label{ReverseEstimate}
\| T(\eta (u - \bar u _ { x, 2\rho}), \eta p )\| _ { S_{q'} ^ *
  (\locdom x { 2\rho} ) } \\
\leq \frac C \rho ( \| \nabla u  \| _ { L^ { \tilde   q }  ( \locdom x
   {4 \rho}) } +
\|p  \| _ { L^ { \tilde   q }  ( \locdom x {2 \rho}) } )
+ \| \eta f \| _ { L^ { q' } _ { 1, D' } ( \locdom x { 2\rho} ) ^*}.
\end{multline}
With this, the estimate of the theorem will follow from the definition
of $ \eta$ and Theorem \ref{Exist}. Our assumption that $ (u,p) \in S_
{q_ 1} (\Omega)$ with $ \frac 1 { q _ 1} < \frac 1 2 + \kappa$
guarantees that $ (\eta( u- \bar u _{x, 2\rho}) , \eta p) $ is the unique solution (\ref{WMP})
in $S_{q_1}( \locdom x { 2 \rho})$.

Thus, we turn to the proof of (\ref{ReverseEstimate}).  We fix $ q$
with
$\frac 1 2 > \frac 1 q > \frac 1  2  - \kappa$ and choose
$\phi \in L^ { q' } _ { 1, D' } ( \locdom x { 2 \rho } ) $.  We use $
\eta \phi$ in the weak formulation of the mixed problem in $\Omega$ satisfied by $
u$  to obtain
\[
a( u- \bar u _ { x, 2\rho} , \eta \phi ) + \int_ \locdom x { 2\rho} p \div( \eta \phi )\, dy =
\langle f,  \eta \phi \rangle .
\]
Using the product rule, we may rewrite this as
\begin{multline}
\label{Happy}
a ( \eta ( u - \bar u _ { x, 2\rho} ), \phi ) + \int_ { \locdom x { 2 \rho}}
\eta p \div \phi\,d y
= \langle \eta f , \phi \rangle    \\
+ \int_ { \locdom x { 2\rho} } - p \nabla \eta \cdot  \phi
 + \epsilon^ \alpha _ i ( u - \bar u
_ {x , 2\rho } ) \frac {\partial \eta }{ \partial y _ i  } \phi ^ \alpha
- \frac { \partial \eta }{ \partial y _ i } ( u - \bar u _ { x , 2\rho })
^ \alpha \epsilon _ i ^ \alpha ( \phi ) \, dy \\
= \langle \eta f , \phi \rangle  + I + II + III.
\end{multline}
We claim that
\begin{equation}
\label{Grumpy}
|I+II+III| \leq \frac
 C  \rho (  \| p \| _ { L^ {\tilde q}  ( \locdom x { 2 \rho}
  ) }
+ \| \nabla u \|_{ L^ {\tilde q }  ( \locdom x {
    2 \rho} )})
\|\nabla \phi \| _      { L^ {q'} ( \locdom x { 2\rho} ) }.
\end{equation}
Furthermore,  it is clear from the product rule and the
Sobolev-Poincar\'e inequality (\ref{PoSo}) that $ - \div ( \eta ( u
- \avg u x { 2\rho}) )$  satisfies
\begin{equation}
\label{Sneezy}
\|
 - \div ( \eta ( u
- \avg u x { 2\rho} ) )\| _ { L^q ( \locdom x { 2 \rho }) }
 \leq
\frac C \rho \| \nabla u \| _ { L^ { \tilde q } ( \locdom x { 4 \rho}
  )}.
\end{equation}
Combining (\ref{Grumpy}) and  (\ref{Sneezy}) gives
(\ref{ReverseEstimate}) and hence the Proposition. Note that we may
replace $D'$ in (\ref{ReverseEstimate}) by $D$ in the  Proposition
since the norm of $ \eta f$ in  $L^{p,1}_D(\Omega)^*$ is an increasing function
in the set $D$.

Thus it remains to prove the claim (\ref{Grumpy}).
To estimate  $I$ and $II$,  we use  H\"older's inequality and the
Sobolev-Poincar\'e inequality (\ref{PoSo})
\begin{align*}
| I | +|II|
 & \leq \frac C  \rho (  \| p \| _ { L^ {\tilde q}  ( \locdom x { 2 \rho}
  ) }
+ \| \nabla u \|_{ L^ {\tilde q }  ( \locdom x {
     2 \rho} )})
 \| \phi \| _      { L^ {\tilde q '} ( \locdom x { 2 \rho} ) }
\\
&\leq
\frac C  \rho (  \| p \| _ { L^ {\tilde q}  ( \locdom x { 2 \rho}
  ) }
+ \| \nabla u \|_{ L^ {\tilde q }  ( \locdom x {
    2 \rho} )})
\|\nabla \phi \| _      { L^ {q'} ( \locdom x { 2\rho} ) }.
\end{align*}
To estimate $III$, we
use H\"older's inequality and then apply  the Sobolev-Poincar\'e
inequality \eqref{PoSo}
to $ u - \bar u _ {x , 2 \rho }$ to obtain
\begin{align*}
|III| &\leq \frac  C \rho
\| u - \bar u _ { x, 2\rho }\| _ {L ^ {q}( \locdom x { 2 \rho} ) }
\| \nabla \phi \| _ {L^ {q'} ( \locdom x { 2\rho } )}
\\
&\leq \frac C \rho   \| \nabla u \| _ { L^ { \tilde q } ( \locdom x { 4\rho} ) }
 \| \nabla \phi \| _ { L^ {q'} ( \locdom x { 2\rho} )} .\\
\end{align*}
Combining the last two displayed  estimates gives the claim
(\ref{Grumpy}) and hence the Proposition.
\end{proof}

We will need the following version of the Caccioppoli inequality  from
 Giaquinta (\cite[Theorem 1.1]{MR641818}).  Let $ u$ be a
solution of (\ref{WMP}) with $ f = 0$ and $ g =0 $  on $\locdom x { 2
  \rho}$
and $ f_N = 0 $ on $\sball  x { 2\rho} \cap N$, then we have
\begin{equation}
\label{Caccioppoli}
(-\!\!\!\!\!\!\int _ { \locdom x \rho } |\nabla u |^2 \, dy ) ^ { 1/2}  \leq
C -\!\!\!\!\!\!\int _ { \locdom x { 2\rho }} |u|\, dy .
\end{equation}
Giaquinta and Modica prove this estimate with an $L^2$ norm on the
right and in the interior. The argument may be adapted to work near
the boundary. It is well-known that we may vary the exponent on the
right-hand side, see \cite{MR1239172} for example.

\begin{corollary}
\label{Continuous}
Suppose that $ \Omega$ is Lipschitz domain  and $ \partial \Omega = D
\cup N$ with $D$ satisfying
(\ref{AhlDav}).  Let $ \kappa $ be as in Proposition \ref{Reverse}.
Suppose that $ (u,p) \in S_{q}$, $ 1/q < 1/2 + \kappa$  is a solution of (\ref{WMP}) in $\Omega$ with $ f_N
=0$,    $ f_ D=0$, and $g=0$  and $ f $ is zero on $\locdom x {2\rho}$. Then we have
the estimates
\begin{gather}
\label{LocalHolder}
|u(z) - u (y) | \leq C \left ( \frac { |z-y|} \rho \right) ^ \gamma
 -\!\!\!\!\!\!\int _ { \locdom x {2\rho}} |u(y)- \avg u x {2\rho} |\, dy , \qquad y, z \in \locdom x
\rho
\\
\label{MVT}
|u(x) | \leq C  -\!\!\!\!\!\!\int _ { \locdom x \rho} |u(y) |\, dy .
\end{gather}
\end{corollary}

\note{ Need to check hypotheses needed for caccioppoli. }

\note{ A comment about exponent $\gamma$. }

\begin{proof} The first estimate  (\ref{LocalHolder}) follows from
Proposition \ref{Reverse},    Morrey's inequality, and the Caccioppoli
inequality (\ref{Caccioppoli}). This gives the H\"older estimate with
$0< \gamma < 2 \kappa$ with $ \kappa$ as in  Proposition \ref{Reverse}.
To
obtain the second inequality, we write
$
|u(x) | \leq |u(x) - u (z) | + |u(z) |.
$
We may average this inequality with respect to $z$ in $ \locdom x {\rho/2}$
and then use   (\ref{LocalHolder}).
\end{proof}

\section{Proof of Theorem \ref{Green}}
\label{TheEnd}

We are ready to give the proof of our main result,
Theorem \ref{Green}, from the introduction.
\begin{proof}[Proof of Theorem \ref{Green}]
We begin by noting that functions in the Lorentz-Sobolev space
$L^ { 2,1 } _{1,D} ( \Omega)$   are continuous. To see this, we
observe that since $ \Omega$ is Lipschitz,  we have an extension
operator
 $E: L^{ 2,1} _1( \Omega) \rightarrow
L^ { 2,1 } _1( {\bf R} ^2)$. If $f$ is
 in $L^{ 2,1}_1( {\bf R}^2)$, we may represent $f$ as a first order
 Riesz potential
and use H\"older's inequality  in Lorentz spaces (\ref{LorentzHolder})
to conclude  that  $f$ is bounded and continuous
(see  \cite{MR607898}, for example).
Thus the  Dirac delta measure concentrated at a
point $x$ in $ \Omega $ lies in $ L^ { 2, 1 } _ { 1, D} ( \Omega)
^*$.  For each $ x \in \Omega$, $ \alpha =1$ or 2, we put
\[
( G^{ \alpha \cdot} ( x, \cdot  ) , \Pi^ \alpha (x, \cdot) )
= T^ { -1 } (e_\alpha \delta _x , 0),
\]
where  $e_\alpha$ is a unit vector and $ T $ is the map defined
in (\ref{Tdef1}-\ref{Tdef2}).  From  (\ref{Useful}), we have that  $
T:S_{2,\infty} \rightarrow S_{2,1}^*$ is invertible and hence
\[
\| G(x, \cdot ) \| _ {L^ { 2, \infty }_{1,D} (\Omega)} + \| \Pi(x,
\cdot)  \| _ { L ^ { 2, \infty } ( \Omega) }  \leq C.
\]
Thus, we have established (\ref{GreenLorentz}) of Theorem
\ref{Green}.
Since we have  $ \nabla G(x,\cdot)$ in
$L^{2, \infty} (\Omega) $ it follows that  for each local domain $
\locdom x \rho$
\[
\int _ { \locdom x \rho } |\nabla _y G(x,y) | \, dy \leq C\rho.
\]
Using the Sobolev-Poincar\'e inequality (\ref{PoSo}), we obtain
that $ G (x, \cdot )$ is in the space $BMO_D(\Omega)$ as defined by
Taylor, Kim, and Brown \cite[p.~1579]{TKB:2012}. Following the arguments
from Taylor, Kim, and Brown,   the local
regularity estimates in Corollary \ref{Continuous} imply  the upper bound
(\ref{GreenUp}) and the H\"older estimate (\ref{GreenHolder}) for $
G(x, \cdot)$.
The estimate for the gradient of $G $ in (\ref{GreenGrad}) follows
immediately from Proposition \ref{Reverse}.

To establish the representation formula, (\ref{GreenRep}), we observe
that if $u$ is a solution of (\ref{WMP}) with $f_N \in L^t(N)$, $f \in
L^ t ( \Omega)$,  and  $ \nabla g \in L^ { t } ( \Omega)$ for some $t>1$,
then we have that the solution $(u,p)$ lies in $S_q$ for some $q>2$.  From
the weak formulation (\ref{WMP}) for the equation for $G$, we obtain
\[
a( G^ { \alpha \cdot} (x, \cdot) , u ) - \int _ \Omega \Pi ^ \alpha (
x, y) g(y) \,dy   = u^ \alpha (x) - \int _ \Omega \nabla g \cdot G^ {
  \alpha \cdot }(x,y) \, dy  .
\]
On the other hand, using $ G^ { \alpha \cdot} (x, \cdot) $ as the test function in the
weak formulation of the boundary value problem satisfied by $u$ gives
\[
a(u, G^ { \alpha \cdot } (x, \cdot ) ) = \int _ \Omega G ^ {\alpha
\beta} f^ \beta (y) \, dy + \int _ { N } G^ { \alpha \beta } (x, y)
f_N^ \beta (y)  \, d\sigma .
\]
We recall that the form $a(\cdot, \cdot)$ is symmetric and
equate  the two expressions for $a(u, G^ { \alpha \cdot} (x, \cdot)
)$ which  gives the representation formula (\ref{GreenRep}).   If $ g $ is
smooth and compactly supported in $ \Omega$, then integration by parts
shows that the
term in (\ref{GreenRep}) involving
$G$ and derivatives of $g$ is zero. This
allows one to see that $G$ is a Green function as defined
before Theorem \ref{Green}.

Next we prove the symmetry property for the Green function. On a formal
level, the proof follows quickly from the weak formulations of the
problems for $ G ^{ \alpha \cdot } (x, \cdot)$ and $ G^{ \beta
  \cdot}(y, \cdot) $ and the symmetry of the form $a(\cdot, \cdot)$.
These combined give
\[
a( G^ { \alpha \cdot } (x, \cdot) , G^ { \beta \cdot}(y,\cdot)  )
= G^ { \alpha\beta} (y,x) = G^{ \beta \alpha } (y,x).
\]
However, this formal argument fails since we do not have that $ G(x,
\cdot)$ lies in $L^{ 2,1 } _{1, D}( \Omega)$.  The argument can be
made rigorous by approximating the $ \delta $-functions  by
normalized characteristic functions of balls centered at $x$ and $y$
and letting the radii tend to zero. See the work of Taylor, Kim, and
Brown \cite{TKB:2012} for a similar argument.
\end{proof}

We close with a few related questions for further investigation. Many
of the
methods used in the work reported here apply also to variable
coefficient equations.
It would be interesting
to extend the construction of the Green function to operators with
non-constant coefficients that are
 modelled on  the Stokes system and to domains that are more
general than Lipschitz.

One point that remains open is finding a representation for the
pressure. One may define another row of the Green matrix to be a
solution of the system
\[
- \Delta G^{3\cdot}(x, \cdot ) + \nabla \Pi^3(x, \cdot)  = 0 , \qquad
-\div G^ { 3\cdot}(x, \cdot) = \delta _x
\]
and use $G^ { 3 \cdot}$ to give a representation of the
pressure. See, for example, the work of  Maz'ya and Rossmann
\cite{MR:2005}.  One expects that $ G^ {3\cdot} (x, \cdot ) $ lies in
$L^{2,
  \infty}( \Omega)$ uniformly for $ x\in \Omega$. However, we are
unable to obtain such estimates.

\note{
Need to prove symmetry. This may require a limiting argument.

Representation for the pressure. Do we have the right bvp?

}

redpajama_set_name: RedPajamaArXiv


text:
Year : 2020 | Volume : 68 | Issue : 5 | Page : 1224--1225
Approach to a Patient with Stroke After Flying
Nazim Ata1, Erdem Cevık2,
1 Aeromedical Research and Training Center, Eskisehir, Turkey
2 Van Military Hospital, Van, Turkey
Dr. Nazim Ata
USAEM Baskanligi, Eski Hava Hastanesi, Odunpazari, Eskisehir
Arriving at a definitive diagnosis in case of a patient with neurological symptoms after a low altitude flight is very critical, while at the same time presenting a dilemma in aeromedical decision making. Symptoms of stroke can closely mimic those of neurologic manifestations of decompression sickness. This paper discusses the approach to a case of a 28-year-old male helicopter passenger who developed left-sided numbness of the body after a 1.5 h flight at 13,000 feet altitude and was hospitalized with the diagnosis of transient ischemic attack.
Ata N, Cevık E. Approach to a Patient with Stroke After Flying.Neurol India 2020;68:1224-1225
Ata N, Cevık E. Approach to a Patient with Stroke After Flying. Neurol India [serial online] 2020 [cited 2021 Jan 16 ];68:1224-1225
Available from: https://www.neurologyindia.com/text.asp?2020/68/5/1224/299138
Arterial gas embolism is rare in altitude exposure. Cerebral symptoms occurring after altitude exposure are usually attributed to decompression sickness (DCS).[1] DCS is a relatively rare entity in both military and civil aviation. It is caused by the formation of gas bubbles in blood or body tissues during or after rapid reduction of the ambient pressure. The symptoms of DCS range from mild, such as pain, to serious ones like neurological manifestations.[2] Altitude-induced DCS is usually seen at altitudes in excess of 18,000 feet as a consequence of loss of cabin pressure. DCS may also be encountered in some nonpressurized aircraft that are approved for flight above 20,000 feet.[3] Rarely, some individuals can develop this problem at lower altitudes. Usually, there are one or more risk factors for DCS in these rare cases.[4] Vascular obstruction causing stroke-like signs and symptoms can also mimic the neurological manifestations of DCS.[5] Hence, diagnosis of altitude-induced neurological DCS can pose a challenge, since it can be easily confused with other stroke etiologies. Signs of deep vein thrombosis (DVT) on Doppler ultrasonography or a right to left shunt on echocardiography support the transient ischemic attack (TIA) diagnosis. History of exposure to altitude supports the DCS diagnosis. The purpose of this paper is to underline the importance of a methodical approach and definitive diagnosis in a patient who develops symptoms similar to those of stroke after a low-altitude flight.
A 28-year-old male (height, 175 cm; weight, 98.4 kg; body mass index; 32.13 kg/m2) was admitted to the emergency room with a numbness on the left side of body after a 90 min flight in a helicopter at 13,000 feet altitude. Emergency medicine specialist and neurologist reviews revealed a blood pressure of 150/90 mmHg, heart rate of 70 beats/min, and respiratory rate of 15 breaths/min with saturation of 98% on room air. Left hemi-hypoesthesia was observed during neurological examination. His mental status, cranial nerves, motor system, reflexes, and coordination were normal. The patient claimed to have been sitting throughout the flight and did not give any history of hard exercise before or during the flight. There were no complaints of chest pain, joint pain, shortness of breath, or skin rash. Complete blood count and routine blood tests and electrocardiography were also normal. The patient was a heavy smoker (three packs per day) and had a history of hypertension for the last 3 years. His initial presentation was consistent with a diagnosis of stroke, and hence, he was hospitalized with the initial diagnosis of TIA. He was started on antiplatelet drugs, calcium channel blockers, and peripheral vasodilators. The physicians who treated the patient overlooked DCS at first because they were not flight surgeons. They performed tests to determine the etiology of stroke. Bilateral lower extremity arterial-venous and carotid-vertebral artery color Doppler ultrasonography and echocardiography were performed as part of the search for an etiologic factor. Patent foramen ovale, DVT, and other pathologies were ruled out. The patient's symptoms resolved completely within the first 24 h. There was no evidence of acute ischemia or presence of gas in the cerebral arteries on diffusion MRI scanning. Aerospace medicine specialist who is in charge in hyperbaric oxygen treatment facility in another city was asked for consultation. Hyperbaric oxygen therapy (HBOT) was not recommended in this case, since the patient's symptoms had resolved completely. The patient's recovery was uneventful and he was discharged after 5 days with a final diagnosis of TIA.
Symptoms and signs of stroke can closely mimic those of neurologic DCS. Hence, neurological DCS should also have been considered in the differential diagnosis in this case; especially since the patient had a history of exposure to altitude before developing the symptoms.
Altitude-induced DCS can be easily confused with more common stroke etiologies. A neurological examination is generally inadequate to distinguish between different stroke etiologies. A definitive diagnosis is made primarily by the patient's anamnesis.[5] However, in many patients, even this proves to be inadequate.
The presence of risk factors such as heavy smoking and hypertension may suggest a thromboembolic disease. Although the patient's symptoms and his subsequent recovery were suggestive of TIA, the diagnosis of DCS should also have been considered in this case. The patient's age, absence of any signs of DVT on Doppler ultrasonography or a right to left shunt on echocardiography go against this diagnosis. A review of existing literature suggests that DVT can occur after 10 or more flight hours, which could have led to the TIA. Moreover, only 6% of patients with DVT have stroke.[6] Altitude-induced DCS should have been considered as a diagnosis for this patient, keeping in view the history of exposure to altitude. However, a diagnosis of altitude-induced DCS could also be argued against, since the exposure to altitude was below 18,000 ft and any specific DCS risk factors like a history of diving or dehydration were absent.
Typically, 18,000 feet is generally considered the threshold for occurrence of altitude-induced DCS symptoms. However clinically insignificant gas bubbles have been documented at lower altitudes as well.[7] A study by Butler on 111 cases of DCS below 18,000 feet altitude revealed that 35% of these cases were seen in the real flight, whereas the rest were observed in altitude chamber flights. In total, 75% of these cases had a history of physical exercise and prior exposure to altitude.[8] The most common symptom of DCS was pain only; neurological manifestations were present in only 18% of patients.[7]
The most common predisposing factors for DCS are high altitude, prolonged exposure to altitude, low temperature, physical exercise, injury, dehydration, diving before flying, alcohol consumption, hypoxia, obesity, smoking, and advanced age.[7],[9] The patient was a heavy smoker and obese; however, any other predisposing factors for low-altitude were absent.
In patients suffering from stroke after flight or altitude exposure, arriving at a definitive diagnosis (TIA or DCS) might be difficult. Under these conditions, in addition to standard medical treatment, HBOT should also be considered as a supportive treatment modality. Prompt treatment in accordance with US Navy Treatment can be beneficial in such cases. Diagnostic tests for ischemic cerebrovascular diseases should be performed concurrently. Resolution of symptoms after HBOT would confirm the diagnosis of DCS. Hence, HBOT would act as both a therapeutic and diagnostic measure.
In the absence of predisposing factors and lack of training in HBOT for physicians, the diagnosis of DCS can remain elusive. An awareness and strong suspicion are essential for the diagnosis and management of DCS. An early diagnosis and prompt referral for treatment would go a long way in decreasing the morbidity and mortality.[10]
The authors certify that they have obtained all appropriate patient consent forms. In the form, the patient(s) has/have given his/her/their consent for his/her/their images and other clinical information to be reported in the journal. The patients understand that their names and initials will not be published and due efforts will be made to conceal their identity, but anonymity cannot be guaranteed.
1 Vann RD, Butler FK, Mitchell SJ, Moon RE. Decompression illness. Lancet 2011;377:153-64.
2 Rudge FW. Decompression sickness presenting as a viral syndrome. Aviat Space Environ Med 1991;62:60-1.
3 Black WR, DeHart RL. Decompression sickness: An increasing risk for the private pilot. Aviat Space Environ Med 1992;63:200-2.
4 Rudge FW. A case of decompression sickness at 2,437 meters (8,000 feet). Aviat Space Environ Med 1990;61:1026-7.
5 Weenink RP, Hollmann MW, van Hulst RA. Acute neurological symptoms during hypobaric exposure: Consider cerebral air embolism. Aviat Space Environ Med 2012;83:1084-91.
6 Belvís R, Masjuan J, García-Barragán N, Cocho D, Martí-Fàbregas J, Santamaría A, et al. Stroke and pulmonary thromboembolism after a long flight. Eur J Neurol 2005;12:732-4.
7 Fabian BG. Case report: Inflight decompression sickness affecting the temporomandibular joint. Aviat Space Environ Med 1998;69:517-8.
8 Butler W, Webb J. Decompression sickness below 18.000 feet: A large case series. Aviat Space Environ Med 2013;84:293.
9 Voge VM. Probable bends at 14,000 feet: A case report. Aviat Space Environ Med 1989;60:1102-3.
10 Auten JD, Kuhne MA, Walker HM, Porter HO. Neurologic decompression sickness following cabin pressure fluctuations at high altitude. Aviat Space Environ Med 2010;81:427-30.

redpajama_set_name: RedPajamaCommonCrawl


text:
"What is this thing?" Pentagon officially releases video of three UFO sightings
7 months ago admin1
Are we really alone in the universe?
According to three newly released videos from the Pentagon, the answer may be a resounding "no".
As recently covered by CNN.com, the Department of Defense has officially acknowledged that UFOs or "Unidentified Aerial Phenomena" have been seen in recent years by Navy pilots and have released several videos that were recorded (scroll down to see all three in full). ).
The videos were released to the public after it was determined that they do not reveal "any sensitive capabilities or systems," according to Pentagon spokeswoman Sue Gough, and also that the video's release "does not affect any subsequent investigation of incursions." military in airspace". by unidentified aerial phenomena".
Now comes the tricky part, as millions of armchair ufologists across the country are scrambling to figure out exactly what was shown in the videos. Surprising to service members who reacted in awe at how quickly objects were moving, a voice in the videos speculates that it could be a drone.
The videos begin to be released "to clear up any misconceptions by the public about whether or not the images that were circulating were real and whether or not there are more videos," Gough said.
Their veracity was confirmed by the Navy in September of last year, and it is possible that they may have inspired "formal guidelines" in the Navy on how their pilots can report when they believe they have seen possible UFOs.
Navy pilot describes intense sighting: "It was extremely abrupt"
One of the surefire ways to tell a legitimate sighting of something alien in nature among the sighting of an everyday aircraft or celestial phenomenon is whether or not the craft appears to have a certain next-level "intelligence" behind it.
The suspected or even confirmed alien ships will move in several different directions, almost as if the pilot is playing some kind of futuristic video game. Ducks, dips, dips, 360 degree turns at the drop of a hat and everything in between characterize sightings that are alien in nature.
According to retired US Navy pilot David Fravor, a 2017 sighting involved a craft that moved in ways he couldn't explain. This may be the first clue as to its authenticity.
"As I approached him… he quickly accelerated south and disappeared in less than two seconds," Fravor said. "That was extremely abrupt, like a ping pong ball bouncing off the wall. He would crash and go the other way."
The Navy videos were first released between December 2017 and March 2018 by To The Stars Academy of Arts & Sciences, a company co-founded by former Blink-182 frontman Tom DeLonge, who says he has studied information about UFOs and other mysterious phenomena in the night sky. .
The Pentagon has already studied records of UFO encounters as part of a secret program launched at the request of Senator Harry Reid of Nevada.
The program ran for five years, from 2007 to 2012, according to the Pentagon, after it was assessed that there were more important priorities that needed funding.
"These aircraft – let's call them aircraft – are exhibiting characteristics that are not currently in the US inventory or any foreign inventory that we are aware of," former program chief Luis Elizondo said of the objects surveyed.
Elizondo says he resigned from the Defense Department in 2017 to protest secrecy surrounding the program as well as funding issues.
According to its original backer Reid, the release of the video "only scratches the surface of the research and materials available. The US needs to take a serious, scientific look at this and any potential national security implications."
While no official proclamation has been made as to what the sightings constitute yet, some members of Congress are still interested in the issue, with senators receiving a confidential briefing from Navy officials on UFOs this past summer.
"If pilots at Oceana or elsewhere report flight hazards that interfere with training or put them at risk, Senator Warner wants answers. It doesn't matter if it's weather balloons, little green men or something else entirely – we can't ask our pilots to put their lives on the line unnecessarily," Rachel Cohen, spokeswoman for Democrat Virginia Senator Mark Warner, told CNN. at the time.
Pilots who have seen the UFOs are far from convinced they are from this Earth. "Look at that thing – it's spinning!" Pilots are heard expressing shock and confusion.
Another later exclaims, "What the fuck is this thing?" Two of the videos were shot in 2015 and one was shot in 2004. More information on this development as it becomes available.
For now, check out all three videos that have been released below. Are these videos conclusive evidence of an alien presence in our skies?
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text:
Mattress-in-a-box, Reinvigorating a sleepy sector
Harsha Wickremasinghe Head of Business Intelligence
Five years ago most people would have rubbished the idea that you would buy a mattress online without ever setting foot in a store. After all, choosing a mattress has traditionally been a very complicated process characterised by a mind-boggling range of options such as pocket springs, coils, damask covers and hand-tufted edges. The only way to make real sense of these was to visit a (typically uninspiring) store. Once there, you realise that all the brands offer some or all the same features as each other, but at wildly differing price points. Having tried the mattress out for a few minutes (under the glare of an eager salesperson), you find yourself walking out of the store more confused than when you first entered.
Disruptors Enter the Fray
The bed and mattress market has a very traditional retail structure, dominated by a handful of large retailers, which in turn sell branded products from a relatively concentrated global manufacturer base, alongside their own label offerings. Their bulky nature makes mattresses a logistical nightmare, online penetration was extremely low and the multiple layers in the value-chain offered scope for considerable mark-ups, which are ultimately paid for by customers via inflated prices.
The sector was ripe for exploitation by a disruptor that would revolutionise both the product range and the buying process. In early 2014 pureplay Casper Sleep launched in the US, followed later that year by Eve Sleep in the UK. Their business models are predicated on offering a curated range (standardised products takes the hard work out of choosing the best mattress), slick branding and marketing, a strong focus on customer service (100 day home trial and free delivery & returns), all of which is executed online and comes packaged in a box that can be delivered with ease by a one-man courier to the top floor of the tightest of urban apartments.
Casper has been phenomenally successful, raising $55m at a pre-money valuation of $500m just 18 months after launch, and generating more than $200m revenue in 2016. It has expanded into Germany (Europe's largest bed market) and the UK. Eve Sleep has also turbocharged its business, increasing revenues from £2.6m in 2015 to £12m in 2016. Eve is raising £35m from a partial IPO, valuing the Business at £140m, and is staking its success on rapid expansion across Europe. As of 31 December 2016 Eve's European sales amounted to 26.3% of overall revenue, compared to 64.6% in the UK.
Their success has spawned myriad competitors in the US, the UK and Europe , all vying for a share of the market. Some are clearly adopting a 'Me-Too' approach from both a product and marketing perspective, whilst others are making subtle differences to product composition (e.g. using micro springs as opposed to purely foam / latex-bases), or pitching at the more premium end of the price spectrum (e.g. Sapira). Even traditional retailers and manufacturers have launched their own mattress-in-a-box brands, such as bed specialist Dreams with Hyde & Sleep and Sealy with Cocoon.
All the hype and noise generated by this new breed of disruptors poses a number of challenges that will stress-test their businesses as they scale up. The proliferation of competitors, many of which are broadly similar in their propositions, may start to cause confusion amongst the very customers they tried to entice with simplicity in the first place. Furthermore, with the typical replacement cycle for a mattress being at least eight years, there is a requirement to stretch the product offering beyond merely mattresses, bed linen and pillows.
A key to differentiating is to spend lavishly on marketing (after all they are creating lifestyle brands), which requires deep pockets to succeed. Eve's FY16 Cost Per Acquisition (CPA) was £245, its Average Order Value (AOV) £423 and UK conversion rate between 0.66% and 1.30%. The former is clearly unsustainable (and Eve's medium term blended CPA target is £100), but it is a clear indication that over the next few years we will witness a shakeout of weaker brands that simply do not have the financial firepower to spend on marketing to drive brand awareness amongst consumers.
Bricks-and-Mortar Still Dominates
Despite their impressive growth, the online mattress-in-a-box companies account for just a tiny sliver of the overall bed and broader furniture markets, and a number are also currently loss-making (Eve posted an £11.3m EBITDA loss in 2016) Traditional bricks-and-mortar retailers remain the sector's dominant channel, and will continue to be for the foreseeable future.
These young pureplay brands have also begun to realise that in order to really drive volumes and build critical mass, they simply cannot ignore physical retail. Simba has teamed up with John Lewis (which has a two-year exclusive agreement to sell the brand), whilst Eve Sleep has concessions in Debenhams, Next and Fenwick. Casper has also announced that it will be offering its pillows, sheets and other accessories (though not mattresses for the moment) via 1,200 Target stores in the US from next month.
Our view is that the long-term success of mattress-in-a-box players will be based on having a strong omni-channel proposition. Truly successful lifestyle brands will note that in order to develop a well-entrenched relationship with their customers, they need to anticipate the more complex purchasing journey they undertake. These retailers will need to gain an in-depth, holistic view of their customers through data collected across both on-and-offline channels. This in turn should help to create stronger loyalty and a more meaningful brand engagement with customers.
M&A Prospects
The entrance of the pureplay mattress-in-a-box companies has certainly injected new life into a sector which has been one of the last bastions of traditional retail. Their savvy marketing, curated ranges and a focus on a customer-centric experience are clearly winning over shoppers.
Traditional retailers in the sector are, for the time being, more than happy to sit back and let these young upstarts fight it out amongst themselves as they grapple for market share across multiple geographies. Once the shakeout of weaker competitors is complete, and clear winners begin to emerge, this is when we expect the likes of Steinhoff and other leading furniture retailers to seriously turn their attention to M&A opportunities in the mattress-in-a-box sector.
Harsha Wickremasinghe
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When I woke up on the surgery table, I thought it was all a dream. The white plaster walls. The IV drip in my arm. The periwinkle gown. None of it was real.
The moment I realized that I was awake, that this was really happening, I burst into tears. Couldn't stop.
The nurse comforted me with a glass of ice water while the anesthesiologist injected a nerve block into my knee.
Except he was wrong. I felt… many things.
Fear. Shame. Regret. Anxiety. Mostly fear. So much fear.
Fear of the wall of pain that I knew would descend once the block wore off. Fear that even with a metal plate affixed to my bones they would never fully heal. Fear that I wasn't strong enough to handle all of this. Fear that I would disappoint everyone who perceives me as "the brave one." Fear that I would disappoint myself.
As I write these words, it's been 32 hours since I woke up and wished I was dreaming.
The block has worn off. The pain is excruciating.
I trust that this experience is a form of mental & spiritual training — a PhD course in courage. Re-shaping me in ways I can't understand yet. Shaping me into a better, more capable version of myself.
This is my training ground.
While I didn't voluntarily step onto the field to receive this particular training, while I would love to avoid it, or postpone it, while I would love to flee this temporary discomfort, here I am, and here we go.
Sometimes you enroll in an educational program willingly — and sometimes God hits the "registration" button for you.
Either way: you've got lessons to learn.


Completed in 2009, Frank Gehry's Fabrikstrasse 15 is an icon on the growing Novartis Basel campus. In the evening its brilliant sculptural form is underscored by layers of light — all on the interior — that gently wash the facade, illuminate the workstations, and glow from within its core.
Basel, Switzerland Breaking the bounds of of Vittorio Magnago Lampugnani's master plan, Fabrikstrasse 15 by Frank Gehry stands in a surprising juxtaposition to the serene array of rectilinear buildings that dominate the Novartis campus. It is located at the geographic heart of the campus, in full view of the company's renovated 1939 Forum 1 International Headquarters building, and across the street from a refined stretch of porticoed offices and labs by Adolf Krischanitz, Rafael Moneo, Lampugnani, and Yoshio Taniguchi. The highly visible, independent site gave the architect freedom to exploit his expansive, free-spirited style.


Located on the west side of the city, West Middle School serves approximately 600 students each year in grades 6-8.
We are proud to be a neighborhood school in a community with a strong commitment to education and collaboration. Just take a look at a number of items we have accomplished by clicking the report card on the right. Each and everyday WMS staff members work to build strong relationships with our students and parents. Our focus continues to be what is best for our students!
If you have any questions, or thoughts to share, please do not ever hesitate to contact us directly. Our door is always open. Thank you for your continued support of West Middle School, Where Everyone Succeeds Together!


text:
Priti Patel reveals country list for quarantine hotels
The shelves in the Brazilian supermarket collapse in a domino effect and kill a worker
Shocking moment Towering shelves in a Brazilian supermarket fall on top of each other in a domino effect, crushing one worker to death and injuring eight others
Eight people, including employees and customers, were injured in the accident on Friday
The shelves collapsed in the Mateus Atacarejo Mix supermarket in São Luís
The dead victim was identified as 21-year-old shop worker Elane de Oliveira Rodrigues
She had worked in the supermarket for three months before she was killed
By Janet Tappin Coelho for Mailonline
Released: 4:53 AM EDT, Oct 4, 2020 | Updated: 5:09 AM EDT, Oct 4, 2020
This is the shocking moment when the shelves of a Brazilian supermarket collapse in a domino effect, pushing a worker to death.
Eight people, including employees and customers, were injured in the accident late Friday evening at the Mateus Atacarejo Mix supermarket in São Luís, northern Brazil.
The dead victim was identified as 21-year-old Elane de Oliveira Rodrigues. She had worked in the supermarket for three months and had been crushed under falling products in the aisle where she had stacked shelves.
Six other people in the same location were wounded and rescued by emergency services. They were taken to the hospital and have since been released. Two other victims, rescued by people at the scene, suffered abrasions and minor injuries and were allowed to go home.
Eight people, including employees and customers, were injured in the accident late on Friday evening in the Mateus Atacarejo Mix supermarket in São Luís, northern Brazil (Image: shelves collapse in the supermarket)
Rescue workers searching stacks of supermarket goods after the accident
The retail store was at its busiest and full of shoppers around 8:15 p.m. at the time of the disaster.
CCTV captured the moment when the metal structures to the right of the screen began to fall, creating a domino effect on other huge shelves in front of them, piled high with goods.
Salespeople and customers at the checkouts stopped when they heard a thunderous noise. Then everyone panicked and fled when they saw the shelves fall on top of each other.
As people stormed back into the slaughter to search for survivors, screams of the injured rang out and the footage showed huge piles of food and commercial products mixed with metal shops and rubble.
A man was seen carrying a small child out of the rubble.
And one woman cried in despair and disbelief as she filmed the horror scene.
Eyewitnesses formed chain gangs outside the store to remove the piles of damaged supplies. A video shows lines of people throwing the products from one to the other to clear the area and locate survivors buried underneath.
The words Neto Verdelho posted in the video are: "Never lose faith in humanity! People who help take things out to find the people below. & # 39;
Teams of firefighters, police and emergency services (SAMU) reacted immediately to the scene.
The search for victims continued throughout the night with rescuers in reflective vests calling for silence as they called to the victims buried under the storage shelves to signal where they were by ringing their cell phones or making a noise with them they could be found.
The rescue operation lasted more than 11 hours and involved over 270 firefighters. The search officially ended on Saturday morning
Ruan Alysson, the victim's cousin, told the local press: & # 39; (Elane) was so happy with her new job because she had just started working as a stacker for the company three months ago.
"The last picture she took of herself on her cell phone in her blue work t-shirt was in the workshop."
In a statement, the Mateus Group, which operates one of the city's largest supermarket chains, regretted the accident and stressed that everything was done to find, rescue and support the victims. They later confirmed the death of the worker.
"It is with great regret that the Mateus Group joins the family of Elane de Oliveira Rodrigues, who unfortunately died in the accident."
There is still no information on what caused the collapse, but there are suspicions that an accident with a forklift may have caused one of the gondolas to fall, leading to the tragedy.
A full investigation has been launched by health and safety officials and the police.
(tagsToTranslate) Dailymail (t) Messages (t) Brazil
Boris Johnson says Donald Trump will be fine.
Jacinda Ardern and her archenemy & # 39; The Crusher & # 39; gave their votes early to avoid crowds on election day


Apple displaced Google to become the top brand in the world in 2018 while Facebook, mired in data breach controversies, fell to ninth place in the top 100 brands globally, an annual report said on Thursday. According to global brand consultancy Interbrand's "Best 100 Global Brands 2018" report, Amazon achieved a 56 per cent growth to become the third top brand globally. According to the ranking, Apple's brand value grew 16 per cent (year-on-year) to $214.5 billion.
It went on to become the first company in the US to hit the $1 trillion market cap. "In the second place, Google was up 10 per cent to $155.5 billion while Amazon is valued at $100.8 billion," according to report findings. Microsoft (valued at $92.7 billion) was fourth while Coca-Cola (valued at $66.3 billion) came fifth followed by Samsung on the sixth spot. Facebook's brand value has declined 6 per cent this year in the wake of the Cambridge Analytica data scandal.


Shawanda Lorraine Everett, 41, 401 E. North St., Kinston, Oct. 9, misdemeanor larceny. Bond: $500. Arresting officer: K. Jones.
Arkeem Jivor Simmons, 19, 1512 Lawrence Lane, Kinston, Oct. 10, felony second degree burglary, felony larceny after breaking and entering. Bond: None listed. Arresting officer: Information not available.
Novatni Lagurdie Nicholson, 32, 1002 Desmond St., Kinston, Oct. 10, felony second degree burglary, felony larceny after breaking and entering. Bond: $30,000. Arresting officer: J. Poole.
Lakeyish Croom, 25, 316 S. Davis St., Kinston, Oct. 9, misdemeanor failure to appear. Bond: $600. Arresting officer: B. Wells.
Deforrorra Locust, 66, 1410 Parker St., Kinston, Oct. 10, misdemeanor failure to appear. Bond: $600. Arresting officer: P. Bielby.
Deforrarra Locust, 71, 1410 Parkers Ave., Kinston, Oct. 10, misdemeanor second degree trespassing, misdemeanor breaking and entering. Bond: $600. Arresting officer: LoPresti.
Joshua NMN Smith, 29, 308 Glennwood Ave., Kinston, Oct. 10, two counts misdemeanor order for arrest/failure to appear. Bond: $6,000. Arresting officer: LoPresti.
Tashawn Marquez Burney, 16, 307 Sherwood Ave., Kinston, Oct. 10, misdemeanor second degree trespassing. Bond: $500. Arresting officer: LoPresti.
Montrell Ray Armwood, 31, 606-A Mitchell St., Kinston, Oct. 10, misdemeanor domestic criminal trespass. Bond: None. Arresting officer: K. Jones.
Demournay Jamar Pugh, 25, 803-B Roseanne Drive, Kinston, Oct. 10, felony breaking and entering, felony larceny after breaking and entering. Bond: $35,000. Arresting officer: R. Brock.
Daimion Sutton, 21, 2103 Carlton Drive, Kinston, Oct. 10, felony breaking and entering, felony larceny after breaking and entering, misdemeanor larceny. Bond: $35,000. Arresting officer: R. Brock.
Daniel Demel Mitchell, 29, 2112 Brandy Station Ct., Kinston, Oct. 10, two counts misdemeanor failure to appear. Bond: $500. Arresting officer: S. Archino.
David Arthur Deters, 46, 1500 W. Washington Ave., Kinston, Oct. 10, misdemeanor failure to appear/speeding. Bond: $500. Arresting officer: J. Eubanks.
Deallo Laroque Hines, 20, 3261 Middle St., Kinston, Oct. 8, misdemeanor possess marijuana up to 1/2 ounce, misdemeanor maintain place for controled substance. Bond: $2,000. Arresting officer: E. Mills.
Mandrell Eugene Swinson, 40, 221 W. Boundary St., La Grange, Oct. 9, felony failure to appear. Bond: $500. Arresting officer: E. Mills.
John W. Suggs, 48, 521 Lincoln St., Kinston, Oct. 5, misdemeanor shoplifting concealment of goods. Bond: $2,000. Arresting officer: P. Bielby.
Ernest Claude Rhodes, 75, 1108 Candlewood Drive, Kinston, Oct. 5, misdemeanor failure to appear. Bond: $1,000. Arresting officer: S. Kivett.
Johnny Gentry Phillips, 37, 2545 Salina Drive, Kinston, Oct. 6, misdemeanor second degree trespassing. Bond: $1,000. Arresting officer: B. Biggins.
Mark Anthony Jones II, 25, 905 Waters St., Kinston, Oct. 7, felony possession with intent to sell/deliver marijuana, misdemeanor carrying concealed gun, misdemeanor possess drug paraphernalia. Bond: $10,000. Arresting officer: T. Moore.
Corey Maurice Bryant, 17, 107 E. Grainger Ave., Kinston, Oct. 5, misdemeanor larceny. Bond: $500. Arresting officer: P. Bielby.
Jerome NMN Brown, 47, 17 Carver Court, Kinston, Oct. 5, misdemeanor possess drug paraphernalia. Bond: $500. Arresting officer: T. Moore.
Peshan Katrail Sutton, 24, 4524 Ash Lane, Grifton, Oct. 8, misdemeanor assault on a female. Bond: None. Arresting officer: J. Hewitt.
Joyce Maxine Moore, 64, 1509 N. Queen St., Kinston, Oct. 8, misdemeanor second degree trespassing. Bond: $300. Arresting officer: J. Hewitt.
Jospeh D. Braxton, 36, 902 W. Lenoir Ave., Kinston, Oct. 8, misdemeanor driving while impaired. Bond: Other. Arresting officer: D. Hardy.
Samuel NMN Hines, 37, 804 Tulip Lane, Kinston, Oct. 11, misdemeanor failure to appear. Bond: $500. Arresting officer: P. Bielby.
Dorothy Ann Pitts, 42, 308 Glenwood Ave., Kinston, Oct. 11, misdemeanor contempt of court. Bond: $1,171. Arresting officer: J. Eubanks.
Sonequa Johnson, 26, 1914 W. Washington, Oct. 11, misdemeanor contempt of court. Bond: $1,000. Arresting officer: S. Kivett.


text:
Электронная книга: Scott Pratt L. «Logic. Inquiry, Argument, and Order»
An enlightening introduction to the study of logic: its history, philosophical foundations, and formal structures Logic: Inquiry, Argument, and Order is the first book of its kind to frame the study of introductory logic in terms of problems connected to wider issues of knowledge and judgment that arise in the context of racial, cultural, and religious diversity. With its accessible style and integration of philosophical inquiry and real-life concerns, this book offers a novel approach to the theory of logic and its relevance to questions of meaning and value that arise in the world around us. The book poses four problems for logic: Is logic separate from experience? Does logic require dualisms? Can logic reconcile opposed ways of understanding the world? And when things are divided, does the boundary have a logic? The author begins the exploration of these questions with a discussion of the process of analyzing and constructing arguments. Using the logical theories of C. S. Peirce, John Dewey, and Josiah Royce to frame the investigation, subsequent chapters outline the process of inquiry, the concept of communicative action, the nature of validity, categorical reasoning through the theory of the syllogism, and inductive reasoning and probability. The book concludes with a presentation of modal logic, propositional logic, and quantification. Logic is presented as emerging from the activities of inquiry and communication, allowing readers to understand even the most difficult aspects of formal logic as straightforward developments of the process of anticipating and taking action. Numerous practice problems use arguments related to issues of diversity and social theory, and the book introduces methods of proving validity that include Venn diagrams, natural deduction, and the method of tableaux. Logic: Inquiry, Argument, and Order is an ideal book for courses on philosophical methods and critical reasoning at the upper-undergraduate and graduate levels. It is also an insightful reference for anyone who would like to explore a cross-cultural approach to the topic of logic.
Издательство: "John Wiley&Sons Limited"
Купить за 9990.07 руб и скачать на Litres
Logic — • A historical survey from Indian and Pre Aristotelian philosophy to the Logic of John Stuart Mill Catholic Encyclopedia. Kevin Knight. 2006. Logic Logic … Catholic encyclopedia
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logic, history of — Introduction the history of the discipline from its origins among the ancient Greeks to the present time. Origins of logic in the West Precursors of ancient logic There was a medieval tradition according to which the Greek philosopher … Universalium
Logic — For other uses, see Logic (disambiguation). Philosophy … Wikipedia
Argument — This article is about the subject as it is studied in logic and philosophy. For other uses, see Argument (disambiguation). In philosophy and logic, an argument is an attempt to persuade someone of something, by giving reasons or evidence for… … Wikipedia
logic — logicless, adj. /loj ik/, n. 1. the science that investigates the principles governing correct or reliable inference. 2. a particular method of reasoning or argumentation: We were unable to follow his logic. 3. the system or principles of… … Universalium
Informal logic — (or, occasionally, non formal logic) is the study of arguments as presented in ordinary language, as contrasted with the presentations of arguments in an artificial, formal, or technical language (see formal logic ). Informal logic emerged in… … Wikipedia
Aristotle's logic and metaphysics — Alan Code PART 1: LOGICAL WORKS OVERVIEW OF ARISTOTLE'S LOGIC The Aristotelian logical works are referred to collectively using the Greek term 'Organon'. This is a reflection of the idea that logic is a tool or instrument of, though not… … History of philosophy
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Stephill Generators Ltd were proud to be invited to attend the opening of Pickerings Plants new depot in Tiverton, Devon on Friday 16th June 2017. We can report that the new facility is very impressive and the opening was well supported by customers and suppliers alike, with Mark Chamberlain – Pickerings MD - reporting that several orders were taken on the day. Visitors were treated to a variety of refreshments and a delicious hog roast lunch.


text:
Orličky (deutsch: Adlerdörfel, 1304: Reinprechtsdorf, 1588: Worlitschka) ist ein Dorf mit 305 Einwohnern im Kreis Ústí nad Orlicí (Wildenschwert) im Pardubický kraj, Tschechien. Es liegt im südlichen Teil des Adlergebirges zwischen Letohrad (Geiersberg) und Králíky (Grulich).

Die Gründung des Ortes geht vermutlich auf die Kolonisation des böhmischen Königs Přemysl Ottokar zurück. Dieser siedelte in der Gegend Deutsche an, die größtenteils aus Franken kamen. Im höchsten Punkt war früher eine Glasbläserei angesiedelt. Durch den hohen Holzverbrauch der Glaserei wurden die Wälder gerodet und neues Land für weitere Ansiedlungen gewonnen. Ein zweiter Gewerbezweig war die Bürsterei. Durch die raue Natur gab es trotzdem nicht viel Arbeit und viele Einwohner gingen bis nach Prag zur Arbeit. Nach Hause kamen sie meist nur an Feiertagen. Ab 1890 zogen die meisten Familien in die Industriestädte Deutschlands.

Weblinks
 Adlerdörfel, Historie und Daten
 Homepage Orličky, tschechisch

Einzelnachweise

redpajama_set_name: RedPajamaWikipedia


The staff and contractors engaged by Northern Fencing Specialists (NFS) are extensively trained to work in the rail environment. NFS provides all compliant documentation and training required to work safely. Safety is our number one priority and is non-negotiable at NFS.
Northern Fencing is the choice for many Rail organisations throughout Australia as it has the flexibility to analyse your specific requirements and tailor a solution that satisfies those requirements.


text:
Q: Error: Class org.opencv.core.Mat could not be found When i generating header file with JNI using 'javah' it shows
Error: Class org.opencv.core.Mat could not be found.

System: Mac OSX
Using: Android Studio and opencv 3.1 for android
Commandline:
javah -classpath /Users/lumeng/Library/Android/sdk/platforms/android-23/android.jar:. com.lumenghz.test.OpencvHelper

Who can help me?
Edit-1
I also tried
javah -d ~/Documents/androidworkspace/RumenTest/test/src/main/jni -classpath /Users/lumeng/Library/Android/sdk/platforms/android-23/android.jar:. com.lumenghz.test.OpencvHelper

But still got the same error:
Error: Class org.opencv.core.Mat could not be found.

A: You are giving wrong class path here:

com.lumenghz.test.OpencvHelper

Give full class path like this:

~/Documents/androidworkspace/RumenTest/openCVLibrary310/build/intermediates/classes/debug:. com.lumenghz.test.OpencvHelper

  javah -d (JNI Folder Location) -classpath (JAR Locaion);(class Path)

example as per your paths-
 javah -d ~/Documents/androidworkspace/RumenTest/test/src/main/jni -classpath /Users/lumeng/Library/Android/sdk/platforms/android-23/android.jar;~/Documents/androidworkspace/RumenTest/openCVLibrary310/build/intermediates/classes/debug: com.lumenghz.test.OpencvHelper

A: Solved
I should declare the classpath of opencv-sdk-android which been builded.
So the commandLine should be like this:
javah -classpath ~/Documents/androidworkspace/RumenTest/openCVLibrary310/build/intermediates/classes/debug:. com.lumenghz.test.OpencvHelper

redpajama_set_name: RedPajamaStackExchange


Barristers in England and Wales are specialist advocates and advisers who provide expert authorized providers when representing their lay and professional shoppers in courts and other legal contexts. Demonstrates an understanding of economic disclosure in family legislation issues. Barristers are regulated by the Bar for the jurisdiction where they practise, and in some nations, by the Inn of Courtroom to which they belong. A barrister, who could be thought-about as a jurist, is a lawyer who represents a litigant as advocate before a courtroom of acceptable jurisdiction.
Demonstrates an understanding of capability and events diminished capacity within the family regulation context. Because of this solicitors are restricted from doing so. In these two courts, barristers dress within the traditional English manner, as do the judges and other legal professionals.
Demonstrates data of felony regulation and felony process together with the next statutes and related case legislation: b. Managed Medication and Substances Act and Regulations. In sure areas (however not crime or conveyancing ), barristers may now settle for directions from a shopper instantly ("Direct Access").


Everyone needs a bottle opener, and our newest bottle opener is also a keychain! Constructed of once-fired military grade brass, the 7.62mm bullet bottle opener is a must-have for all. This keychain also features an aluminum dog tag keychain laser-engraved with the 3rd ID patch.
* CUSTOMIZE ITwith your name, rank, service dates, or phone number on the reverse side.
* 100% Made and Sourced in the USA.