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\begin{document}
\title[Green's matrix in two dimensions]
{Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains}

\author[H. Dong]{Hongjie Dong}
\address[H. Dong]{Division of Applied Mathematics, Brown University,
182 George Street, Providence, RI 02912, United States of America}
\email{hdong@brown.edu}
\thanks{Hongjie Dong was partially supported by the
National Science Foundation under agreement No. DMS-0111298
and a start-up funding from the Division of Applied Mathematics of
Brown University.}

\author[S. Kim]{Seick Kim}
\address[S. Kim]{Centre for Mathematics and its Applications, The Australian National University, ACT 0200, Australia}
\email{seick.kim@maths.anu.edu.au}
\thanks{Seick Kim is supported by the Australian Research Council.}

\subjclass[2000]{Primary 35A08, 35B65; Secondary 35J45}

\keywords{Green function, Green's matrix, fundamental solution, fundamental matrix, second order elliptic system, measurable coefficients.}

\begin{abstract}
We study Green's matrices for divergence form, second order strongly elliptic
systems with bounded measurable coefficients in two dimensional domains.
We establish existence, uniqueness, and pointwise estimates of
the Green's matrices.
\end{abstract}

\maketitle

\mysection{Introduction} \label{intro}
In this article, we study Green's matrices for divergence form,
second order strongly elliptic systems with bounded measurable coefficients
in two dimensional domains.
More precisely, we are concerned with the Green's matrix for elliptic systems
\begin{equation*}
\sum_{j=1}^N L_{ij} u^j\coloneqq \sum_{i=1}^N\sum_{\alpha,\beta=1}^2
D_\alpha(A^{\alpha\beta}_{ij}(x) D_\beta u^j), \quad i=1,\ldots,N
\end{equation*}
in an open connected set $\Omega\subset \mathbb R^2$. Here, $A^{\alpha\beta}_{ij}(x)$
are bounded measurable functions on $\Omega$ satisfying the strong ellipticity
condition.
By a Green's matrix we mean an $N\times N$ matrix valued function
$\vec G(x,y)=(G_{ij}(x,y))_{i,j=1}^N$ defined on
$\{(x,y)\in\Omega\times\Omega: x\neq y\}$
satisfying the following properties
(see Theorem~\ref{thm1} below for more precise statement):
\begin{align*}
&\sum_{j=1}^N L_{ij} G_{jk}(\cdot,y)=-\delta_{ik}\delta_y (\cdot)
\quad \forall y\in\Omega,\\
&G_{ij}(\cdot,y)=0\quad\text{on}\quad
\partial\Omega \quad \forall y\in\Omega,
\end{align*}
where $\delta_{ik}$ is the Kronecker delta symbol and
$\delta_{y}(\cdot)$ is the Dirac delta function with a unit mass at $y$.
In the scalar case (i.e., when $N=1$), the Green's matrix becomes a real
valued function and is usually called the Green's function.

We prove that if $\Omega$ has either finite volume or finite width, then
there exists a unique Green's matrix in $\Omega$; see Theorem~\ref{thm1}.
The same is true when $\Omega$ is a domain above a Lipschitz
graph (e.g., $\Omega=\mathbb R^2_+$); see Theorem~\ref{thm2}.
We also establish growth properties of the Green's matrices
including logarithmic pointwise bounds.
We emphasize that we do not require $\Omega$ to be bounded nor to have
a regular boundary in Theorem~\ref{thm1}. Compared to the result of
Dolzmann and M\"uller \cite{DM}, where $\Omega$ is assumed
to be a bounded Lipschitz domain, our result is an improvement
in this respect.
Although there is no Green's matrix for $\Omega=\mathbb R^2$,
there is a possible definition of a fundamental matrix in the entire plane.
Such a construction was carried out by Kenig and Ni \cite{KN} in the scalar
case and by Auscher, McIntosh, and Tchamitchian \cite{AMcT}
in the systems setting
(In fact, Auscher et al. considered complex coefficients elliptic equations
in \cite{AMcT}, but with appropriate changes their strategy carries over
to more general elliptic systems).
For the completeness of presentation, we include the result of Auscher et al.
\cite{AMcT} in Section~\ref{sec5}.

Let us briefly review the history of works in this area.
In the scalar case, the basic facts about Green's functions of
symmetric elliptic operators in bounded domains were proved by
Littman, Stampacchia, and Weinberger \cite{LSU}.
The study of the Green's functions for nonsymmetric elliptic operators
in bounded domains $\Omega\subset \mathbb R^n$ ($n\geq 3$) was carried out
by Gr\"{u}ter and Widman \cite{GW}.
As it is mentioned earlier, there is no Green's function for $\Omega=\mathbb R^2$;
the fundamental solution $-(1/2\pi)\ln |x-y|$ of Laplace equation
changes sign and is not considered as a Green's function from a point
of view of the classical potential theory (see e.g., \cite{Doob}).
Nevertheless, it is still possible to define a fundamental solution in $\mathbb R^2$.
By using the maximum principle, Kenig and Ni \cite{KN} constructed one
for symmetric elliptic operators.
In \cite{CL}, Chanillo and Li derived that the fundamental solution constructed
by Kenig and Ni is a function of bounded mean oscillation in $\mathbb R^2$.
Also, we would like to bring attention to a paper by Escauriaza
\cite{Escauriaza} on the fundamental solutions of elliptic and parabolic
equations in nondivergence form.
In the systems setting, the Green's matrices of the elliptic systems
with continuous coefficients in bounded $C^1$ domains have been discussed by
Fuchs \cite{Fuchs}  and Dolzmann and M\"uller \cite{DM}.
In fact, Dolzmann and M\"uller improved the strategy of Fuchs and showed
the existence and pointwise estimate for Green's matrix in bounded Lipschitz
domains $\Omega\subset\mathbb R^2$ without imposing any regularity on the
coefficients.
Recently, Hofmann and Kim \cite{HofKim1} gave a unified approach in
studying Green's functions/matrices in arbitrary domains $\Omega\subset \mathbb R^n$
($n\geq 3$) valid for both scalar equations and systems of elliptic type by
considering a class of operators $L$ such that weak solutions of
$L\vec u=0$ satisfy an interior H\"older estimate.
However, like the method used in Gr\"uter and Widman \cite{GW}, the method of
Hofmann and Kim heavily relied on the assumption
that $n\geq 3$ and could not be applied to the two dimensional case.
An parabolic extension of the result by Hofmann and Kim
was carried out in a very recent paper by Cho, Dong, and Kim \cite{CDK}.
In particular, Cho et al. proved that so called
``Dirichlet heat kernel'' of a strongly elliptic system exists
in any domain $\Omega\subset \mathbb R^2$ (see Corollary~2.9 in \cite{CDK}).
In fact, our basic strategy is to make use of their result and
construct the Green's matrix out of the ``Dirichlet heat kernel'' by
integrating in $t$-variable.

The organization of this paper is as follows.
In Section~\ref{main}, we introduce some notations and then state our main
results, Theorem~\ref{thm1} and Theorem~\ref{thm2}.
We give the proof of Theorem~\ref{thm1} in Section~\ref{sec3} and
that of Theorem~\ref{thm2} in Section~\ref{sec4}.
Finally, in Section~\ref{sec5} we introduce the result of Auscher et al.
\cite{AMcT} regarding construction of a fundamental matrix for an elliptic
system in the entire plane.

\mysection{Preliminaries and main results} \label{main}

\subsection{Strongly elliptic systems in $\mathbb R^2$} \label{sec2.1}
Throughout this article, the summation convention over repeated indices
shall be assumed.
Let $L$ be a second order elliptic operator of divergence type acting on
vector valued functions
$\vec{u}=(u^1,\ldots,u^N)^T$ ($N\geq 1$) defined on an open set
$\Omega\subset \mathbb R^2$ in the following way:
\begin{equation}
\label{eqP-01}
L\vec{u} = D_\alpha (\vec{A}^{\alpha\beta}\, D_\beta \vec{u})
\quad\left(\coloneqq \sum_{\alpha=1}^2 \sum_{\beta=1}^2
D_\alpha (\vec{A}^{\alpha\beta}\, D_\beta \vec{u})\right),
\end{equation}
where $\vec{A}^{\alpha\beta}=\vec{A}^{\alpha\beta}(x)$
($\alpha,\beta=1,2$)
are $N$ by $N$ matrices satisfying the strong ellipticity condition, i.e.,
there is a number $\lambda>0$ such that
\begin{equation}
\label{eqP-02}
A^{\alpha\beta}_{ij}(x)\xi^j_\beta\xi^i_\alpha
\ge \lambda \abs{\vec{\xi}}^2
\coloneqq\lambda\sum_{i=1}^N\sum_{\alpha=1}^2|\xi^i_\alpha|^2
\quad\forall x\in\Omega.
\end{equation}
We also assume that $A^{\alpha\beta}_{ij}$ are bounded, i.e.,
there is a number $\Lambda>0$ such that
\begin{equation}
\label{eqP-03}
\sum_{i,j=1}^N\sum_{\alpha,\beta=1}^2
|A^{\alpha\beta}_{ij}(x)|^2\le \Lambda^2\quad\forall x\in\Omega.
\end{equation}
We do not impose any further condition other than \eqref{eqP-02}
and \eqref{eqP-03} on the coefficients.
Especially, we do not assume the symmetry of the coefficients.
The transpose operator ${}^t\!L$ of $L$ is defined by
\begin{equation}
\label{eqP-05}
{}^t\!L\vec{u} = D_\alpha ({}^t\!\vec{A}^{\alpha\beta} D_\beta \vec{u}),
\end{equation}
where ${}^t\!\vec{A}^{\alpha\beta}=(\vec{A}^{\beta\alpha})^T$
(i.e., ${}^t\!A^{\alpha\beta}_{ij}=A^{\beta\alpha}_{ji}$).
Note that the coefficients ${}^t\!A^{\alpha\beta}_{ij}$ satisfy
the conditions \eqref{eqP-02}, \eqref{eqP-03} with the same constants $\lambda, \Lambda$.
\subsection{The function space $Y^{1,2}_0(\Omega)$}
The function space $Y^{1,2}_0(\Omega)$ is defined as the set of all weakly
differentiable functions on $\Omega$ such that $D u\in L^2(\Omega)$ and
$u \eta \in W^{1,2}_0(\Omega)$ for any $\eta\in C^\infty_c(\mathbb R^2)$.
An open set $\Omega\subset \mathbb R^2$ is said to be a Green domain if
$\{u 1_\Omega: u\in C^\infty_c(\mathbb R^2)\}\nsubset W^{1,2}_0(\Omega)$.
We ask the readers to refer \cite[\S 1.3.4]{MZ}
for the proofs of lemmas stated below.
\begin{lemma}
\label{lem:P02}
Let $\Omega\subset \mathbb R^2$ be a Green domain and $B\subset\mathbb R^2$ be a ball.
Then, there is a constant $C=C(\Omega,B)$ such that
\begin{equation}
\label{eq:P178}
\norm{u}_{L^2(\Omega\cap B)}\leq C\norm{D u}_{L^2(\Omega)}\quad
\forall u\in Y^{1,2}_0(\Omega).
\end{equation}
\end{lemma}

\begin{lemma}
\label{lem:P03}
Let $\Omega\subset \mathbb R^2$ be a Green domain.
Then $Y^{1,2}_0(\Omega)$ is a Hilbert space when endowed with the inner product
\begin{equation}
\label{eq:P179}
\ip{u,v}\coloneqq\int_\Omega D_i u D_i v.
\end{equation}
\end{lemma}

\begin{lemma}
\label{lem:P04}
Let $\Omega\subset \mathbb R^2$ be a Green domain.
Then $C^\infty_c(\Omega)$ is a dense subset of the Hilbert
space $Y^{1,2}_0(\Omega)$ equipped with the inner product \eqref{eq:P179}.
\end{lemma}

For a given function $\vec f=(f^1,\ldots,f^N)^T\in L^1_{loc}(\Omega)^N$,
we shall say that
$\vec u=(u^1,\ldots,u^N)^T$ is a weak solution in $Y^{1,2}_0(\Omega)^N$
of $L\vec u=-\vec f$ if $\vec u \in Y^{1,2}_0(\Omega)^N$ and
\begin{equation}
\label{eq:P08c}
\int_\Omega A^{\alpha\beta}_{ij} D_\beta u^j D_\alpha \phi^i=
\int_\Omega f^i\phi^i
\quad \forall \vec \phi\in C^\infty_c(\Omega)^N.
\end{equation}
It is routine to check that if $\Omega$ is a Green domain and
$\vec u$ is a weak solution in $Y^{1,2}_0(\Omega)^N$ of $L\vec u=0$, then
$\vec u\equiv 0$. Therefore, a weak solution in $Y^{1,2}_0(\Omega)^N$ of
$L\vec u = -\vec f$ is unique.

\subsection{Main results}
Let us state our main results.
First, we consider domains with either finite volume or finite width.
We shall denote by $|\Omega|$ the Lebesgue measure of $\Omega$
and by $\delta(\Omega)$ the width of $\Omega\subset\mathbb R^2$;
more precisely, we define
\begin{equation}
\delta(\Omega)\coloneqq\inf\set{\operatorname{dist}(\ell_1,\ell_2):\Omega
\text{ lies between two parallel lines }\ell_1,\ell_2};\quad
\inf \emptyset=\infty.
\end{equation}
\begin{theorem} \label{thm1}
Let the operator $L$ satisfy the conditions \eqref{eqP-02} and \eqref{eqP-03}.
Assume that $\Omega\subset \mathbb R^2$ is an open connected set with either
finite volume or finite width so that
\begin{equation}
\label{eq001}
\gamma=\gamma(\Omega)\coloneqq\max\big(|\Omega|^{-1},\delta(\Omega)^{-2}\big)>0.
\end{equation}
Then, there exists a Green's matrix
$\vec{G}(x,y)=(G_{ij}(x,y))_{i,j=1}^N$ defined on
$\{(x,y)\in\Omega\times\Omega:x\neq y\}$ satisfying the properties that
\begin{equation}
\label{eqG-54}
\int_{\Omega}A^{\alpha\beta}_{ij} D_\beta G_{jk}(\cdot,y)
D_\alpha \phi^i = \phi^k(y)
\quad \forall \vec{\phi}\in C^\infty_c(\Omega)^N
\end{equation}
and that for all $\eta\in C^\infty_c(\Omega)$ satisfying
$\eta\equiv 1$ on $B_r(y)$ for some $r<d_y$,
\begin{equation}
\label{eqG-55}
(1-\eta)\vec{G}(\cdot,y)\in Y^{1,2}_0(\Omega)^{N\times N}.
\end{equation}
The Green's matrix $\vec G(x,y)$ in $\Omega$ is unique in the following sense:
\begin{enumerate}
\item
$\vec G(x,y)$ is continuous in $\set{(x,y)\in\Omega\times\Omega:x\neq y}$.
\item
$\vec G(x,\cdot)$ is locally integrable for all $x\in\Omega$.
\item For any
$\vec{f}=(f^1,\ldots,f^N)^T \in C^\infty_c(\Omega)^N$,
the function $\vec u=(u^1,\ldots,u^N)^T$ given by
\begin{equation}
\label{eqZ-70}
\vec{u}(x)\coloneqq\int_{\Omega} \vec{G}(x,y)\vec{f}(y)\,dy \quad
\left(i.e.,\, u^i(x)\coloneqq\int_{\Omega} G_{ij}(x,y) f^j(y)\,dy\right)
\end{equation}
is a unique weak solution in $Y^{1,2}_0(\Omega)^N$ of $L\vec{u}=-\vec{f}$.
\end{enumerate}
Moreover, $\vec G(x,y)$ satisfies the following pointwise estimate:
\begin{equation}
\label{eqG-56}
|\vec G(x,y)| \leq C\left(\frac{1}{\gamma R^2}+\ln \frac{R}{|x-y|}\right)
\quad\text{if } |x-y|<R\coloneqq\tfrac{1}{2}\max(d_x,d_y),
\end{equation}
where $d_x\coloneqq\operatorname{dist}(x,\partial\Omega)$ and
$C=C(\lambda,\Lambda,N)<\infty$.
Consequently, $\vec G(\cdot,y)$ and $\vec G(x,\cdot)$
belong to $L^p(B_r(y)\cap\Omega)$ and $L^p(B_r(x)\cap\Omega)$, respectively,
for all $r>0$ and $p\in [1,\infty)$.
Furthermore, $D \vec G(\cdot,y)$ and $D \vec G(x,\cdot)$
belong to $L^p(B_r(y)\cap\Omega)$ and $L^p(B_r(x)\cap\Omega)$, respectively,
for all $r>0$ and $p\in [1,2)$.
Finally, we have the following symmetry relation:
\begin{equation}
\label{eq:E23}
\vec G(y,x)= {}^t\vec G(x,y)^T\qquad
(\,\text{i.e, }\,G_{ij}(y,x)={}^tG_{ji}(x,y)\,),
\end{equation}
where ${}^t\vec G(x,y)$ is the Green's matrix of the transpose operator
${}^tL$ in $\Omega$.
\end{theorem}

\begin{remark}
When $|\Omega|<\infty$, we have global $L^p$ estimates
for the Green's matrix and its derivatives.
In that case, it will be evident from the proof of Theorem~\ref{thm1} that
$\vec G(\cdot,y)$ and $\vec G(x,\cdot)$ belong to $L^p(\Omega)$
for all $p\in [1,\infty)$ and that $D \vec G(\cdot,y)$ and $D \vec G(x,\cdot)$
belong to $L^p(\Omega)$ for all $p\in [1,2)$.
\end{remark}
Next, we consider a domain above a Lipschitz graph.
Let $\Omega$ be given by
\begin{equation}
\label{eq:LD01}
\Omega=\{(x_1,x_2)\in \mathbb R^2: x_2>\varphi(x_1)\},
\end{equation}
where $\varphi:\mathbb R\to \mathbb R$ is a Lipschitz function with a Lipschitz
constant $M\coloneqq \norm{\varphi'}_\infty<\infty$.

\begin{theorem} \label{thm2}
Let the operator $L$ satisfy the conditions \eqref{eqP-02} and \eqref{eqP-03}.
Assume that $\Omega$ is given by \eqref{eq:LD01}.
Then, there exists a unique Green's matrix
$\vec{G}(x,y)$ satisfying all the properties of Theorem~\ref{thm1} except
\eqref{eqG-56}.
Instead of \eqref{eqG-56} of Theorem~\ref{thm1}, we have
\begin{equation}
\label{eq:H01j}
|\vec G(x,y)|
\leq C\min\left\{1+\ln_+(d_{x,y}/|x-y|),\,d_{x,y}^\mu |x-y|^{-\mu}\right\}
\quad\forall x,y\in\Omega,\quad x\neq y,
\end{equation}
where $d_{x,y}\coloneqq\min(d_x,d_y)$, $d_x\coloneqq \operatorname{dist}(x,\partial\Omega)$,
$\ln_+t\coloneqq\max(\ln t,0)$, $C=C(\lambda,\Lambda,N,M)<\infty$, and
$\mu=\mu(\lambda,\Lambda,M)\in (0,1)$.
In particular, \eqref{eq:H01j} implies $\vec G(x,y)\to 0$ as $|x-y|\to \infty$.
\end{theorem}
\mysection{Proof of Theorem~\ref{thm1}} \label{sec3}
Throughout this section, we employ the letter $C$ to denote a constant depending
on $\lambda, \Lambda, N$ while we use $C(\alpha,\beta,\ldots)$ to denote
a constant depending on quantities $\alpha,\beta,\ldots,$ as well as
$\lambda, \Lambda, N$.
It should be understood that $C$ may vary from line to line.

Let us recall the following version of Poincar\'e inequality
(see e.g., \cite{GT} for the proof).

\begin{lemma}
\label{lem:poincare}
If $\Omega\subset \mathbb R^2$ is an open connected set with either finite volume
or finite width.
Let $\gamma$ be given as in \eqref{eq001} of Theorem~\ref{thm1}.
Then
\begin{equation}
\label{eq002v}
\norm{u}_{L^2(\Omega)}\leq
(2\gamma)^{-1/2}\norm{D u}_{L^2(\Omega)}
\quad \forall u\in W^{1,2}_0(\Omega).
\end{equation}
\end{lemma}

By using the above lemma, one can show that if $\Omega$ has either
finite volume or finite width, then $\Omega$ is a Green domain and
$Y^{1,2}_0(\Omega)=W^{1,2}_0(\Omega)$
(see e.g. \cite[\S 1.3.4]{MZ}).
In the rest of this section we shall identify $Y^{1,2}_0(\Omega)$ with
$W^{1,2}_0(\Omega)$.

\subsection{Construction of the Green's matrix}
Let $\vec \Gamma(t,x,s,y)$
($x,y\in \Omega$ and $t,s\in \mathbb R$) be
the parabolic Green's matrix given as in \cite[Corollary~2.9]{CDK}.
Note that we have $\vec\Gamma(t,x,s,y)=\vec\Gamma(t-s,x,0,y)$.
Throughout the paper, we shall denote
\begin{align}
\label{eq:k01}
\vec K(t,x,y)&\coloneqq\vec \Gamma(t,x,0,y),\\
\label{eq:k02}
\bar{\vec K}(t,x,y)&\coloneqq\int_0^t\vec K(s,x,y)\,ds.
\end{align}
We record here some properties of $\vec K(t,x,y)$ derived in \cite[Corollary~2.9]{CDK}
for the reference.
Recall that $d_x\coloneqq\operatorname{dist}(x,\partial\Omega)$ for $x\in \Omega$.
\begin{align}
\label{eq5.15c}
&\sup_{t\in (r^2,\infty)} \int_\Omega |\vec K(t,x,y)|^2 \,dx
\leq Cr^{-2} \quad \forall r<d_y,\\
\label{eq5.18}
&\iint_{(0,\infty)\times\Omega\setminus (0,r^2)\times B_r(y)}
|\vec K(t,x,y)|^4 \,dx\,dt \leq Cr^{-4} \quad \forall r<d_y,\\
\label{eq5.15b}
&\iint_{(0,\infty)\times\Omega\setminus (0,r^2)\times B_r(y)}
|D_x\vec K(t,x,y)|^2 \,dx\,dt \leq Cr^{-2} \quad \forall r<d_y,\\
\label{eq5.15}
&\iint_{(0,r^2)\times B_r(y)} |\vec K(t,x,y)|^p \,dx\,dt
\leq C(p) r^{-2p+4} \quad \forall r<d_y \quad\forall p\in [1,2),\\
\label{eq5.19}
&\iint_{(0,r^2)\times B_r(y)} |D_x\vec K(t,x,y)|^p \,dx\,dt \leq C(p) r^{-3p+4}
\quad \forall r<d_y \quad\forall p\in [1,4/3),\\
\label{eq5.18a}
&|\vec K(t,x,y)| \leq C\left\{\max\big(\sqrt{|t|},|x-y|\big)\right\}^{-2}
\quad \text{if } \max\big(\sqrt{|t|},|x-y|\big)<\tfrac{1}{2}\max(d_x,d_y).
\end{align}

We define the Green's matrix $\vec G(x,y)$ as follows:
\begin{equation}
\label{eq:g01}
\vec G(x,y)\coloneqq\lim_{t\to \infty} \bar{\vec K}(t,x,y)
=\int_0^\infty \vec K(s,x,y)\,ds
\quad \forall x,y \in \Omega,\quad x \neq y.
\end{equation}
The next lemma will show that $\vec G(x,y)$ is well defined.

\begin{lemma}
\label{lem:E01}
For any $x,y\in \Omega$ with $x\neq y$, we have
$\int_0^\infty \abs{\vec K(s,x,y)}\,ds<\infty$.
\end{lemma}
\begin{proof}
By \cite[Theorem~2.7]{CDK}, we know that
$t\mapsto \vec K(t,x,y)$ is continuous in $t\in \mathbb R$ for $x\neq y$.
Therefore, we only need to show that
$\int_a^\infty \abs{\vec K(t,x,y)}\,dt<\infty$ for some $a>0$.
Let $\vec u$ be the $k$-th column of $\vec K(\cdot,\cdot,y)$.
Then, by the local boundedness estimate (see \cite{Kim})
\begin{equation}
\label{eq:E13}
\abs{\vec u(t,x)}\le C\left(\fint_{t-\rho^2}^t\fint_{B_\rho(x)}
\abs{\vec u(s,y)}^2\,dy\,ds\right)^{1/2}\quad\forall t>\rho^2\
\quad\forall\rho<d_x.
\end{equation}
By \eqref{eq002v} of Lemma~\ref{lem:poincare},
$I(t)\coloneqq\int_\Omega \abs{\vec u(t,\cdot)}^2$ satisfies
\begin{equation*}
I'(t)= -2\int_\Omega A^{\alpha\beta}_{ij}D_\beta u^j D_\alpha u^i(t,\cdot)
\le -2\lambda \int_\Omega \abs{D \vec u(t,\cdot)}^2
\le -4\lambda\gamma I(t)\quad \forall t>0.
\end{equation*}
Therefore, by using \eqref{eq5.15c} we obtain
\begin{equation}
\label{eq:E15}
\int_\Omega \abs{\vec K(t,\cdot,y)}^2
\leq C e^{-4\lambda\gamma(t-r^2)} r^{-2}\quad\forall t>r^2 \quad \forall r<d_y.
\end{equation}
By combining \eqref{eq:E13} and \eqref{eq:E15}
we have
\begin{equation}
\label{eq:E15h}
\abs{\vec K(t,x,y)}\le Ae^{-2\lambda\gamma t}
\quad \forall t> a,
\end{equation}
where $A=A(d_x,d_y)<\infty$ and $a=a(d_x,d_y)<\infty$. The lemma is proved.
\end{proof}

Next, we show that $\vec G(\cdot,y)$ is continuous in
$\Omega\setminus\set{y}$ for any $y\in\Omega$.
We need the following lemma the proof of which can be found in
\cite[Theorem~3.3]{Kim} (c.f. (2.20) and (2.21) in \cite{CDK}).

\begin{lemma}
\label{lem:P01}
Let $L$ satisfy \eqref{eqP-02} and \eqref{eqP-03}.
If $\vec u(t,x)$ is a weak solution of $\vec u_t-L\vec u=0$ in
$Q_{2r}^{-}\coloneqq Q_{2r}^{-}\big((t_0,x_0)\big)$
($\coloneqq (t_0-4r^2,t_0)\times B_{2r}(x_0)$),
then for all $x,x'\in B_r(x_0)$ and $t\in (t_0-r^2,t_0)$,
\begin{align*}
|\vec u(x,t)-\vec u(x',t)|&\le C |x-x'|^\mu r^{-(1+\mu)}
\norm{D_x\vec u}_{L^2(Q_{2r}^{-})},\\
|\vec u(x,t)-\vec u(x',t)|&\le C |x-x'|^\mu r^{-(2+\mu)}
\norm{\vec u}_{L^2(Q_{2r}^{-})},
\end{align*}
where $\mu=\mu(\lambda,\Lambda)\in (0,1)$.
\end{lemma}

Let $\vec u(t,x)$ be the $k$-th column of $\vec K(t,x,y)$.
Fix $x_0\in \Omega$ with $x_0\neq y$ and choose $r>0$ such that
$r<d_y$ and $B_{2r}(x_0)\subset \Omega\setminus B_r(y)$.
By \cite[Therem~2.7]{CDK} and \eqref{eq:k01}, we find that
$\vec u(t,x)$ is a weak solution of $\vec u_t-L \vec u=0$
in $Q_{2r}^-\big((t_0,x_0)\big)$ for any $t_0\in\mathbb R$.
Therefore, by using Lemma~\ref{lem:P01}, \eqref{eq5.15c}, \eqref{eq5.15b},
and \eqref{eq:E15} we have (recall that $\vec K(t,x,y)\equiv 0$ for $t<0$)
\begin{align}
\label{eq:E21a}
|\vec K(t,x,y)-\vec K(t,x_0,y)|&\le C |x-x_0|^\mu r^{-(2+\mu)}
\quad\forall x\in B_r(x_0) \quad \forall t\in \mathbb R,\\
\label{eq:E21b}
|\vec K(t,x,y)-\vec K(t,x_0,y)|&\le C |x-x_0|^\mu r^{-(2+\mu)}
e^{-2\lambda\gamma(t-r^2)}\quad\forall x\in B_r(x_0)\quad \forall t> 5r^2.
\end{align}
Then, for any $x\in B_r(x_0)$, we have
\begin{align}
\nonumber
|\vec G(x,y)-\vec G(x_0,y)|&\le\int_0^{5r^2} |\vec K(t,x,y)-\vec K(t,x_0,y)|\,dt
+\int_{5r^2}^\infty |\vec K(t,x,y)-\vec K(t,x_0,y)|\,dt\\
\label{eq:E22}
&\le C|x-x_0|^\mu r^{-\mu}(1+r^{-2}\gamma^{-1}).
\end{align}
Therefore, we find that $\vec G(\cdot,y)$ is locally H\"older
continuous in $\Omega\setminus\set{y}$.
Let ${}^t\vec G(x,y)$ be the Green's matrix of the transpose
operator ${}^tL$ in $\Omega$, i.e.,
\begin{equation}
\label{eq:g02}
{}^t\vec G(x,y)\coloneqq \lim_{t\to \infty} {}^t\bar{\vec K}(t,x,y)
=\int_0^\infty {}^t\vec K(s,x,y)\,ds
\quad \forall x,y \in \Omega,\quad x \neq y,
\end{equation}
where ${}^t \vec K(t,x,y)$ and ${}^t\bar{\vec K}(t,x,y)$ are defined similarly
as in \eqref{eq:k01} and \eqref{eq:k02}.
Let ${}^t\vec \Gamma(s,y,t,x)$ be the parabolic
Green's matrix of $\cLt\coloneqq -\partial_t-{}^tL$ constructed
as in \cite{CDK}. Then by \cite[Lemma~3.5]{CDK}
\begin{equation}
\label{eq:E43t}
{}^t\vec K(t,x,y)
={}^t\vec \Gamma(-t,x,0,y)=\vec \Gamma(0,y,-t,x)^T=\vec K(t,y,x)^T,
\end{equation}
and thus we conclude that
\begin{equation}
\label{eq:E43x}
{}^t\bar{\vec K}(t,x,y) = \bar{\vec K}(t,y,x)^T\quad\text{and}\quad
{}^t\vec G(x,y) = \vec G(y,x)^T.
\end{equation}
In particular, we proved \eqref{eq:E23}.
Since ${}^tL$ satisfies \eqref{eqP-02} and \eqref{eqP-03} with
the same $\lambda, \Lambda$, we find as in \eqref{eq:E22} that
${}^t\vec G(\cdot,x)$ is locally H\"older continuous in
$\Omega\setminus\{x\}$ for all $x\in\Omega$.
Therefore, by \eqref{eq:E23} we conclude that $\vec G(x,y)$ is continuous in
$\set{(x,y)\in\Omega\times\Omega:x\neq y}$.

Next, we prove that $\vec G(x,\cdot)$ is locally integrable for all $x\in\Omega$
and $\vec u$ defined by \eqref{eqZ-70} is a weak solution in
$W^{1,2}_0(\Omega)^N$ of $L\vec u = -\vec f$.

\begin{lemma}
\label{lem:E02}
The following estimates hold uniformly for all $t>0$:
\begin{align}
\label{eq:E25}
&\norm{\bar{\vec K}(t,\cdot,y)}_{L^p(B_\rho(y))}\le C(p) \gamma^{-1}\rho^{2/p-2}
\quad\forall p\in [1,2), \quad \text{where }\rho=d_y.\\
\label{eq:E26}
&\norm{\bar{\vec K}(t,\cdot,y)}_{L^4(\Omega\setminus B_r(y))}
\le C \gamma^{-1}r^{-3/2}
\quad \forall r \leq d_y.\\
\label{eq:E27}
&\norm{D\bar{\vec K}(t,\cdot,y)}_{L^p(B_\rho(y))}\le C(p) \gamma^{-1}\rho^{2/p-3}
\quad \forall p\in [1,4/3),\quad \text{where }\rho=d_y.\\
\label{eq:E28}
&\norm{D\bar{\vec K}(t,\cdot,y)}_{L^2(\Omega\setminus B_r(y))}\le
C \gamma^{-1}r^{-2} \quad \forall r \leq d_y.
\end{align}
\end{lemma}

\begin{proof}
We begin by proving \eqref{eq:E25}.
Fix $p\in [1,2)$. By Minkowski's inequality, we have
\begin{align}
\nonumber
\left(\int_{B_\rho(y)}|\bar{\vec K}(t,x,y)|^p\,dx\right)^{1/p}
&\leq \int_0^t\left(\int_{B_\rho(y)} |\vec K(s,x,y)|^p\,dx\right)^{1/p}ds\\
\label{eq:E29c}
&\leq \int_0^{\rho^2} + \int_{\rho^2}^\infty
\left(\int_{B_\rho(y)} |\vec K(s,x,y)|^p\,dx\right)^{1/p}ds \coloneqq I_1+I_2.
\end{align}
We estimate $I_1$ by using H\"older's inequality and \eqref{eq5.15}
as follows:
\begin{equation}
\label{eq:E31a}
I_1 \leq
\left(\int_0^{\rho^2}\!\!\!\int_{B_\rho(y)}
|\vec K(s,x,y)|^p\,dx\,ds\right)^{1/p} \rho^{2(1-1/p)} \leq C(p) \rho^{2/p}.
\end{equation}
To estimate $I_2$, observe that H\"older's inequality and \eqref{eq:E15} yield
(recall $1\leq p<2$)
\begin{align*}
\left(\int_{B_\rho(y)} |\vec K(s,x,y)|^p\,dx\right)^{1/p}
&\leq \left(\int_{B_\rho(y)} |\vec K(s,x,y)|^2\,dx\right)^{1/2}
\abs{B_\rho(y)}^{1/p-1/2}\\
&\leq C(p) \rho^{2/p-2} e^{-2\lambda\gamma (s-\rho^2)}\quad \forall s>\rho^2.
\end{align*}
Therefore, we obtain
\begin{equation}
\label{eq:E31b}
I_2 \leq C(p) \rho^{2/p-2}
\int_{\rho^2}^\infty e^{-2\lambda\gamma (s-\rho^2)}\,ds
\leq C(p) \rho^{2/p-2}\gamma^{-1}.
\end{equation}
Since $1\leq \rho^{-2}\gamma^{-1}$ in any case,
we obtain \eqref{eq:E25} by combining \eqref{eq:E31a} and \eqref{eq:E31b}.

Next, we prove \eqref{eq:E26}.
By using Minkowski's inequality as in \eqref{eq:E29c}, we have
\begin{equation}
\nonumber
\left(\int_{\Omega\setminus B_r(y)}|\bar{\vec K}(t,x,y)|^4\,dx\right)^{1/4}
\leq \int_0^{r^2}
+ \int_{r^2}^\infty\left(\int_{\Omega\setminus B_r(y)}
|\vec K(s,x,y)|^4\,dx\right)^{1/4}ds
\coloneqq I_3+I_4.
\end{equation}
By proceeding as in \eqref{eq:E31a} but using \eqref{eq5.18} instead, we obtain
\begin{equation}
\label{eq:E31c}
I_3 \leq
\left(\int_0^{r^2}\!\!\!\int_{\Omega\setminus B_r(y)}
|\vec K(s,x,y)|^4\,dx\,ds\right)^{1/4}
r^{2(1-1/4)} \leq C r^{1/2}.
\end{equation}
By a well known embedding theorem (see e.g., \cite[\S II.3]{LSU} or
\cite[Theorem~6.9]{Lieberman}), the energy inequality, and \eqref{eq:E15},
we have for $t>r^2$
\begin{align}
\nonumber
\int_t^\infty\!\!\!\int_{\Omega}|\vec K(s,x,y)|^4\,dx\,ds
&\leq C\left(\sup_{t\leq s}\int_{\Omega}|\vec K(s,x,y)|^2\,dx\right)
\int_t^\infty\!\!\!\int_{\Omega}|D_x\vec K(s,x,y)|^2\,dx\,ds\\
\label{eq:E54a}
&\leq C\left(\int_{\Omega}|\vec K(t,x,y)|^2\,dx\right)^2
\leq C r^{-4} e^{-8\lambda\gamma(t-r^2)}.
\end{align}
Then, by using H\"older's inequality and \eqref{eq:E54a} we estimate
\begin{align}
\nonumber
I_4 &\leq \sum_{j=1}^\infty \left(\int_{jr^2}^{(j+1)r^2}\!\!\!\int_\Omega
|\vec K(s,x,y)|^4\,dx\,ds\right)^{1/4} r^{3/2}
\leq C r^{1/2}\sum_{j=1}^\infty e^{-2\lambda\gamma(j-1)r^2}\\
\label{eq:E54b}
&\leq C r^{1/2}(1+r^{-2}\gamma^{-1}).
\end{align}
By combining \eqref{eq:E31c} and \eqref{eq:E54b}, we get \eqref{eq:E26}.

We now turn to the proof of \eqref{eq:E27}.
Fix $p\in [1,4/3)$.
As in \eqref{eq:E29c}, we have
\begin{equation*}
\left(\int_{B_\rho(y)}|D_x\bar{\vec K}(t,x,y)|^p\,dx\right)^{1/p}
\leq \int_0^{\rho^2}
+ \int_{\rho^2}^\infty\left(\int_{B_\rho(y)}
|D_x\vec K(s,x,y)|^p\,dx\right)^{1/p}ds \coloneqq I_5+I_6.
\end{equation*}
By H\"older's inequality and \eqref{eq5.19}, we find
\begin{equation}
\label{eq:E31g}
I_5 \leq
\left(\int_0^{\rho^2}\!\!\!\int_{B_\rho(y)}
|D_x\vec K(s,x,y)|^p\,dx\,ds\right)^{1/p}
\rho^{2(1-1/p)} \leq C(p) \rho^{-1+2/p}.
\end{equation}
To estimate $I_6$, note that H\"older's inequality implies
(recall $1 \leq p < 4/3$)
\begin{equation}
\label{eq:E71a}
\left(\int_{B_\rho(y)}|D_x\vec K(s,x,y)|^p\right)^{1/p} \leq
\left(\int_{B_\rho(y)}|D_x\vec K(s,x,y)|^2\right)^{1/2}|B_\rho(y)|^{1/p-1/2}.
\end{equation}
As in \eqref{eq:E54a}, the energy inequality and \eqref{eq:E15} yield
\begin{equation}
\label{eq:E71b}
\int_t^\infty\!\!\!\int_{\Omega}|D_x\vec K(s,x,y)|^2\,dx\,ds
\leq C \rho^{-2} e^{-4\lambda\gamma(t-\rho^2)}\quad \forall t>\rho^2.
\end{equation}
Then, as in \eqref{eq:E54b}, we estimate
$I_6$ by combining \eqref{eq:E71a} and \eqref{eq:E71b}
\begin{equation}
\label{eq:E31h}
I_6 \leq C(p) \rho^{2/p}\sum_{j=1}^\infty
\left(\int_{j\rho^2}^{(j+1)\rho^2}\!\!\!\int_\Omega
|D_x\vec K(s,x,y)|^2\,dx\,ds\right)^{1/2}
\leq C(p) \rho^{2/p-1}(1+\rho^{-2}\gamma^{-1}).
\end{equation}
We obtain \eqref{eq:E27} by adding \eqref{eq:E31g} and \eqref{eq:E31h}.

Finally, we prove \eqref{eq:E28}.
By using Minkowski's inequality again, we have
\begin{equation}
\nonumber
\left(\int_{\Omega\setminus B_r(y)}|D_x\bar{\vec K}(t,x,y)|^2\,dx\right)^{1/2}
\leq \int_0^{r^2}
+ \int_{r^2}^\infty\left(\int_{\Omega\setminus B_r(y)}
|D_x \vec K(s,x,y)|^2\,dx\right)^{1/2}ds
\coloneqq I_7+I_8.
\end{equation}
We estimate $I_7$ by using H\"older's inequality and \eqref{eq5.15b}:
\begin{equation}
\label{eq:E31v}
I_7 \leq \left(\int_0^{r^2}\!\!\!\int_{\Omega\setminus B_r(y)}
|D_x\vec K(s,x,y)|^2\,dx\,ds\right)^{1/2} r \leq C.
\end{equation}
Also, by using \eqref{eq:E71b} and proceeding as in \eqref{eq:E31h}, we obtain
\begin{equation}
\label{eq:E29q}
I_8 \leq
r\sum_{j=1}^\infty \left(\int_{jr^2}^{(j+1)r^2}\!\!\!\int_\Omega
|D_x\vec K(s,x,y)|^2\,dx\,ds\right)^{1/2} \leq C(1+r^{-2}\gamma^{-1}).
\end{equation}
Therefore, \eqref{eq:E28} follows from \eqref{eq:E31v} and \eqref{eq:E29q}.
The lemma is proved.
\end{proof}

Fix $p_0\in(1,2)$ and $r\leq d_y$. By \eqref{eq:g01} and  \eqref{eq:E25},
there exists a sequence
$\{t_m\}_{m=1}^\infty$ tending to infinity such that
$\bar{\vec K}(t_m,\cdot,y)\wto \vec G(\cdot,y)$ weakly in $L^{p_0}(B_r(y))$,
and thus we have
\begin{equation*}
\norm{\vec G(\cdot,y)}_{L^{p_0}(B_r(y))}\leq C(p_0,\gamma,d_y,r)<\infty
\quad \forall r\leq d_y.
\end{equation*}
By a similar reasoning, \eqref{eq:E26} yields that
\begin{equation*}
\norm{\vec G(\cdot,y)}_{L^4(\Omega\setminus B_r(y))}\leq
C \gamma^{-1}r^{-3/2} \quad \forall r\in (0,d_y].
\end{equation*}
The above inequalities together with \eqref{eq:E23} imply that
$\vec G(x,\cdot)$ is locally integrable for any $x\in\Omega$.
Therefore, the integral in \eqref{eqZ-70} is absolutely convergent
for any $\vec f\in C^\infty_c(\Omega)$, and thus $\vec u$ is well defined
in \eqref{eqZ-70}.
Moreover, \eqref{eq:E25} and \eqref{eq:E26} together with \eqref{eq:E43x} imply
\begin{equation}
\label{eq:E29x}
\vec v(t,x)\coloneqq \int_\Omega \bar{\vec K}(t,x,y)\vec f(y)\,dy
\end{equation}
is well defined.
By the dominated convergence theorem, we also find that
\begin{equation}
\label{eq:E29a}
\lim_{t\to\infty}\vec v(t,x)=\int_\Omega \vec G(x,y)\vec f(y)\,dy=\vec u(x).
\end{equation}
Also, by the definition of $\bar{\vec K}(t,x,y)$ in \eqref{eq:k02},
it is easy to verify
\begin{equation}
\label{eq:E29i}
\vec v_t(t,x)=\int_\Omega \vec K(t,x,y)\vec f(y)\,dy\quad \forall t>0.
\end{equation}
Then, as in the proof of Lemma~\ref{lem:E01}, we have
\begin{equation}
\label{eq:E29b}
\norm{\vec v_t(t,\cdot)}_{L^2(\Omega)}^2\leq
C e^{-4\lambda\gamma t}\norm{\vec f}_{L^2(\Omega)}^2\quad \forall t>0.
\end{equation}
We need the following lemma to show that $\vec u$ is a weak solution in
$W^{1,2}_0(\Omega)^N$ of $L\vec u=-\vec f$.
The readers are asked to consult \cite{CDK} or \cite{LSU}
for the definition of $\rV^{1,0}_2((0,T)\times\Omega)$, etc.
\begin{lemma}
\label{lem:E02b}
For all $T>0$, the function $\vec v$ defined in \eqref{eq:E29x}
is the unique weak solution in $\rV^{1,0}_2((0,T)\times\Omega)^N$
of the problem
\begin{equation}
\label{eq:E32}
\vec v_t- L\vec v=\vec f,\quad
\vec v=0\,\text{ on }\,(0,T)\times\partial\Omega,\quad \vec v(0,\cdot)= 0.
\end{equation}
\end{lemma}
\begin{proof}
Let $\vec w$ be the weak solution in $\rV^{1,0}_2((0,T)\times\Omega)^N$
of the problem \eqref{eq:E32},
the existence and uniqueness of which can be found in \cite{LSU}.
We only need to show that $\vec v\equiv \vec w$ in $(0,T)\times\Omega$.
Fix $t\in (0,T)$ and $x\in\Omega$.
Let ${}^t\vec{\Gamma}(s,y,t,x)$ be the parabolic Green's matrix of
$\cLt\coloneqq -\partial_t-{}^tL$ constructed as in \cite{CDK}.
Then, by proceeding similarly as in the proof of \cite[Theorem~2.7]{CDK},
we obtain (c.f. \cite[Lemma~3.1]{CDK})
\begin{align*}
w^k(t,x)&=
\int_0^T\!\!\!\int_\Omega {}^t\Gamma_{ik}(s,y,t,x)f^i(y)\,dy\,ds=
\int_0^T\!\!\!\int_\Omega \Gamma_{ki}(t,x,s,y)f^i(y)\,dy\,ds\\
&=\int_0^t\!\!\!\int_\Omega K_{ki}(t-s,x,y)f^i(y)\,dy\,ds
=\int_0^t\!\!\!\int_\Omega K_{ki}(s,x,y)f^i(y)\,dy\,ds
=v^k(t,x),
\end{align*}
where we have used $\vec K(t,x,y)\equiv 0$ for $t<0$.
The lemma is proved.
\end{proof}

Note that \eqref{eq:E29b} particularly implies
$\vec v_t(t,\cdot)\in L^2(\Omega)^N$, and thus it is not hard to verify
\begin{equation}
\label{eq:E35}
\int_\Omega v^i_t(t,\cdot)\phi^i+
\int_\Omega A^{\alpha\beta}_{ij}Dv^j(t,\cdot)D_\alpha\phi^i=
\int_\Omega f^i\phi^i\quad \forall\vec\phi\in W^{1,2}_0(\Omega)^N
\quad\forall t>0.
\end{equation}
Then, by setting $\vec\phi=\vec v(t,\cdot)$ above, we find that
for almost all $t>0$,
\begin{align*}
\lambda\norm{D \vec v(t,\cdot)}_{L^2(\Omega)}^2 &\leq
\left(\norm{\vec f}_{L^2(\Omega)}+\norm{\vec v_t(t,\cdot)}_{L^2(\Omega)}\right)
\norm{\vec v(t,\cdot)}_{L^2(\Omega)} \\
&\leq C\gamma^{-1/2}\norm{\vec f}_{L^2(\Omega)}
\norm{D\vec v(t,\cdot)}_{L^2(\Omega)},
\end{align*}
where we have used \eqref{eq:E29b} and Lemma~\ref{lem:poincare}.
Therefore, for almost all $t>0$,
\begin{equation}
\norm{D \vec v(t,\cdot)}_{L^2(\Omega)} \leq
C\gamma^{-1/2}\norm{\vec f}_{L^2(\Omega)}.
\end{equation}
Then, by the weak compactness and \eqref{eq:E29a}, we find that there exists
an increasing sequence $\set{t_m}_{m=1}^\infty$ tending to infinity such that
\begin{equation}
\label{eq:E36}
\lim_{m\to\infty}\int_\Omega A^{\alpha\beta}_{ij}Dv^j(t_m,\cdot)D_\alpha\phi^i=
\int_\Omega A^{\alpha\beta}_{ij}Du^jD_\alpha\phi^i
\quad \forall\vec\phi\in W^{1,2}_0(\Omega)^N.
\end{equation}
Therefore, it follows from
\eqref{eq:E29b}, \eqref{eq:E35}, and \eqref{eq:E36} that $\vec u$ defined in
\eqref{eqZ-70} is a weak solution in $W^{1,2}_0(\Omega)^N$ of $L\vec u=-\vec f$.

Now, we prove the uniqueness. Suppose that there exists another
matrix valued function $\tilde{\vec G}(x,y)$ such that
$\tilde{\vec G}(x,y)$ is continuous
in $\set{(x,y)\in\Omega\times\Omega:x\neq y}$,
$\tilde{\vec G}(x,\cdot)$ is locally integrable in $\Omega$
for all $x\in\Omega$, and for all
$\vec f \in C^\infty_c(\Omega)^N$, the function
$\tilde{\vec u}(x)\coloneqq \int_\Omega \tilde{\vec G}(x,y)\vec f(y)\,dy$
is a weak solution of $L\tilde{\vec u}= -\vec f$ in $W^{1,2}_0(\Omega)^N$.
Then, the difference $\vec w\coloneqq \vec u-\tilde{\vec u}$ is a weak solution
of $L\vec w=0$ in $W^{1,2}_0(\Omega)^N$, and thus $\vec w\equiv 0$.
Therefore, we have
\begin{equation}
\label{eq:E50w}
\int_\Omega (\vec G-\tilde{\vec G})(x,y)\vec f(y)\,dy=0
\quad \forall \vec f\in C^\infty_c(\Omega)^N.
\end{equation}
We conclude from \eqref{eq:E50w} that
$\vec G(x,\cdot)\equiv \tilde{\vec G}(x,\cdot)$ in $\Omega\setminus\{x\}$
 for all $x\in\Omega$, and thus $\vec G(x,y)=\tilde{\vec G}(x,y)$
for all $x,y\in\Omega$ with $x\neq y$. We have proved the uniqueness.

\subsection{Proof of identities \eqref{eqG-54} and \eqref{eqG-55}}
\label{sec3.2}
Let us first prove \eqref{eqG-54}.
From \eqref{eq:k01}, \eqref{eq:k02} and
the construction of parabolic Green's matrix
$\vec{\Gamma}(t,x,s,y)$ in \cite{CDK}, it follows (c.f. (3.3) in \cite{CDK})
\begin{equation}
\label{eq:E01}
\int_\Omega K_{ki}(t,\cdot,y)\phi^i+
\int_\Omega A^{\alpha\beta}_{ij}D_\beta\bar K_{jk}(t,\cdot,y)
D_\alpha\phi^i=\phi^k(y)\quad\forall \vec\phi\in C^\infty_c(\Omega)^N.
\end{equation}
We note that \eqref{eq:E15} yields
\begin{equation}
\label{eq:E02}
\lim_{t\to\infty}\int_\Omega K_{ki}(t,\cdot,y)\phi^i=0\quad\forall k=1,\ldots,N.
\end{equation}
If we write $\vec\phi=\zeta\vec\phi +(1-\zeta)\vec\phi$, where
$\zeta\in C^\infty_c(B_r(y))$ such that $\zeta\equiv 1$ on $B_{r/2}(y)$, then
\begin{align}
\nonumber
\int_\Omega A^{\alpha\beta}_{ij}D_\beta\bar K_{jk}(t,\cdot,y) D_\alpha\phi^i
&=\int_{B_r(y)} A^{\alpha\beta}_{ij}D_\beta\bar K_{jk}(t,\cdot,y)
D_\alpha(\zeta\phi^i)\\
\label{eq:E03}
&\quad +\int_{\Omega\setminus B_{r/2}(y)}
A^{\alpha\beta}_{ij}D_\beta\bar K_{jk}(t,\cdot,y) D_\alpha((1-\zeta)\phi^i)
\coloneqq I_1(t)+I_2(t).
\end{align}
By Lemma~\ref{lem:E02} and \eqref{eq:g01}, we find that there exists
an increasing sequence $\set{t_m}_{m=1}^\infty$ tending to infinity such that
\begin{align}
\label{eq:E04}
\lim_{m\to\infty} I_1(t_m)
&=\int_{B_r(y)} A^{\alpha\beta}_{ij}D_\beta G_{jk}(\cdot,y)
D_\alpha(\zeta\phi^i),\\
\label{eq:E05}
\lim_{m\to\infty} I_2(t_m)
&=\int_{\Omega\setminus B_{r/2}(y)}
A^{\alpha\beta}_{ij}D_\beta G_{jk}(t,\cdot,y) D_\alpha((1-\zeta)\phi^i).
\end{align}
Therefore, by combining \eqref{eq:E01}--\eqref{eq:E05},
we obtain \eqref{eqG-54}.

Next, we prove \eqref{eqG-55}.
We claim
\begin{equation}
\label{eq:E18}
\norm{(1-\eta)\bar{\vec K}(t,\cdot,y)}_{W^{1,2}(\Omega)}\le C(\eta,\gamma)
<\infty \quad \forall t>0,
\end{equation}
where $\eta\in C^\infty_c(\Omega)$ is such that
$\eta\equiv 1$ on $B_r(y)$ for some $r<d_y$.
Assume for the moment that the claim is true.
Then, by the weak compactness and \eqref{eq:g01}, there exists
an increasing sequence $\{t_m\}_{m=1}^\infty$ tending to infinity such that
\begin{equation*}
(1-\eta)\bar{\vec K}(t_m,\cdot,y)\wto (1-\eta)\vec G(\cdot,y)
\quad\text{weakly in }W^{1,2}(\Omega).
\end{equation*}
On the other hand, by \cite[Theorem~2.7]{CDK}, we find that
$(1-\eta)\bar{\vec K}(t,\cdot,y) \in W^{1,2}_0(\Omega)$ for all $t>0$.
Since $W^{1,2}_0(\Omega)$ is weakly closed in $W^{1,2}(\Omega)$,
we have $(1-\eta)\vec G(\cdot,y)\in W^{1,2}_0(\Omega)$ as desired.
To complete the proof of \eqref{eqG-55}, it remains to prove the claim
\eqref{eq:E18}.
In fact, by Lemma~\ref{lem:poincare}, it is enough to show
\begin{equation}
\label{eq:E20}
\norm{D((1-\eta)\bar{\vec K}(t,\cdot,y))}_{L^2(\Omega)}\le C(\eta)
<\infty \quad \forall t>0.
\end{equation}
Let us prove \eqref{eq:E20}.
Assume that $\eta$ is supported in a ball $B\subset \mathbb R^2$.
Then
\begin{align}
\nonumber
\norm{D\big((1-\eta)\bar{\vec K}(t,\cdot,y)\big)}_{L^2(\Omega)}
&\le \norm{1-\eta}_{L^\infty}
\norm{D\bar{\vec K}(t,\cdot,y)}_{L^2(\Omega\setminus B_r(y))}
+\norm{D\eta}_{L^\infty}
\norm{\bar{\vec K}(t,\cdot,y)}_{L^2(B \setminus B_r(y))}\\
\nonumber
&\le C(\eta) \norm{D\bar{\vec K}(t,\cdot,y)}_{L^2(\Omega\setminus B_r(y))}
+C(\eta)|B|^{1/4}
\norm{\bar{\vec K}(t,\cdot,y)}_{L^4(\Omega\setminus B_r(y))}\\
\label{eq:E21v}
&\leq C(\eta) C(\gamma,r)=C(\eta,\gamma)<\infty\qquad\forall t>0,
\end{align}
where we have used Lemma~\ref{lem:E02} in the last step.
This completes the proof of \eqref{eq:E20}, and thus \eqref{eqG-55} is proved.
\subsection{Proof of logarithmic bound \eqref{eqG-56} and $L^p$ estimates}
\label{sec3.3}
Without loss of generality, we may assume $d_y \geq d_x$;
otherwise, in light of \eqref{eq:E23}, we may exchange the role of $x$ and $y$.
Note that if $|x-y|<R=d_y/2$, then \eqref{eq5.18a} yields
\begin{equation}
\label{eq5.18y}
|\vec K(t,x,y)| \leq C\left\{\max\big(\sqrt{t},|x-y|\big)\right\}^{-2}
\quad \forall t \in (0,R^2).
\end{equation}
On the other hand, if we set $\rho=R/2$
in \eqref{eq:E13} and $r=\sqrt{3}R/2$
in \eqref{eq:E15} (note that $\rho<d_x$, $r<d_y$, and $\rho^2+r^2=R^2$),
then \eqref{eq:E15h} becomes
\begin{equation}
\label{eq:E15j}
|\vec K(t,x,y)|\leq C R^{-2} e^{-2\lambda\gamma(t-R^2)}\quad \forall t>R^2.
\end{equation}
Then, by using \eqref{eq5.18y} and \eqref{eq:E15j},
we obtain (recall $|x-y|<R$)
\begin{align}
\nonumber
|\vec G(x,y)|
&\leq C\left(\int_0^{|x-y|^2}|x-y|^{-2}\,dt+ \int_{|x-y|^2}^{R^2} t^{-1}\,dt
+ \int_{R^2}^\infty R^{-2}e^{-2\lambda\gamma(t-R^2)}\,dt \right)\\
\nonumber
&=C\big(1+2\ln (R/|x-y|)+ R^{-2}(2\lambda\gamma)^{-1}\big)\\
\label{eq:R12p}
&\leq C\big(R^{-2}\gamma^{-1}+ \ln(R/|x-y|)\big).
\end{align}
We have thus proved \eqref{eqG-56}.
We now turn to the proof of local $p$-summability of $\vec G(\cdot,y)$
and $D\vec G(\cdot,y)$.
Note that \eqref{eq:R12p} particularly implies that
\begin{equation}
\label{eq:R12q}
\norm{\vec G(\cdot,y)}_{L^p(B_\rho(y))} \leq C(p,d_y,\gamma)<\infty
\quad \forall \rho\in (0,d_y/2] \quad
\forall p \in[1,\infty).
\end{equation}
We claim that
$|D\vec G(\cdot,y)|\in L^p(B_\rho(y))$ for all $0<\rho<d_y$ and $1\leq p<2$.
Let $\vec u$ be the $k$-th column of $\vec G(\cdot,y)$.
Then, by \eqref{eqG-55}, we have
\begin{equation*}
\vec u \in W^{1,2}(\Omega\setminus B_\rho(y))^N\quad \forall \rho \in (0,d_y)
\end{equation*}
and thus, by \eqref{eqG-54}, we find that $\vec u$ is a weak solution of
$L \vec u = 0$ in $\Omega\setminus B_\rho(y)$ for any $\rho<d_y$.
It follows from \eqref{eqG-56} that there is $r_0=r_0(\gamma,d_y)<1$ and
$C_0=C_0(\gamma,d_y)<\infty$ such that
\begin{equation}
\label{eqG-56b}
|\vec G(x,y)| \leq C_0 \ln(1/|x-y|)
\quad \forall x\in B_{r_0}(y).
\end{equation}
Fix $r<r_0$ and let $\zeta\in C^\infty_c(B_r(y))$ be a cut-off function
satisfying $\zeta\equiv 1$ on $\overline{B}_{r/2}(y)$ and $|D\zeta| \leq C/r$.
Then, by \eqref{eqG-55} we find
\begin{equation}
\label{eqn:V01a}
(1-\zeta)^2\vec u \in W^{1,2}_0(\Omega')^N\quad
\text{where } \Omega'\coloneqq \Omega\setminus\overline{B}_{r/2}(y).
\end{equation}
Since $\vec u$ is a weak solution in $W^{1,2}(\Omega')^N$ of $L\vec u=0$,
by using \eqref{eqn:V01a} we have
\begin{equation*}
0=\int_{\Omega'}(1-\zeta)^2 A^{\alpha\beta}_{ij} D_\beta u^j D_\alpha u^i
-\int_{\Omega'} 2(1-\zeta) A^{\alpha\beta}_{ij}D_\beta u^j D_\alpha\zeta\,u^i.
\end{equation*}
Therefore, by using the bound \eqref{eqG-56b} we estimate
\begin{equation*}
\int_{\Omega\setminus B_r(y)}\abs{D \vec u}^2
\le C r^{-2}\int_{B_r(y)\setminus B_{r/2}(y)}\abs{\vec u}^2
\le C C_0^2(\ln(r/2))^2.
\end{equation*}
Therefore, we have
\begin{equation}
\label{eqE-34}
\int_{\Omega\setminus B_r(y)}\abs{D \vec G(\cdot,y)}^2
\le C C_0^2(\ln(r/2))^2
\quad\forall r<r_0.
\end{equation}
Next, let $A_t=\set{x\in\Omega:\abs{D_x\vec G(x,y)}>t}$ and choose $r=2/t$.
Then by \eqref{eqE-34}
\begin{equation*}
\abs{A_t\setminus B_r(y)}\le t^{-2}
\int_{A_t\setminus B_r(y)}\abs{D\vec G(\cdot,y)}^2 \le C C_0^2 t^{-2}(\ln t)^2
\quad \forall t> 2/r_0.
\end{equation*}
and $\abs{A_t\cap B_r(y)}\leq |B_r(y)|\leq Ct^{-2}$.
Therefore, we conclude that for any $y\in\Omega$, there exist
$C_1=C_1(\gamma,d_y)<\infty$ and $t_0=t_0(\gamma,d_y)>0$ such that
\begin{equation}
\label{eqE-36}
\abs{\set{x\in\Omega:\abs{D_x \vec G(x,y)}>t}}\le
C_1 t^{-2}(\ln t)^2\quad \forall t>t_0.
\end{equation}
From the estimates \eqref{eqE-36}, it follows that
$|D\vec G(\cdot,y)|\in L^p(B_r(y))$ for all $r< d_y$ and for all $p\in [1,2)$
as we shall demonstrate below. Let $r<d_y$ be given and choose $\tau>t_0$.
Note that
\begin{align*}
\int_{B_r(y)} |D\vec G(\cdot,y)|^p&=
\int_{B_r(y)\cap \{\,|D\vec G(\cdot,y)|\le \tau\}} |D\vec G(\cdot,y)|^p+
\int_{B_r(y)\cap \{\,|D\vec G(\cdot,y)|>\tau\}} |D \vec G(\cdot,y)|^p\\
&\le  \tau^p\abs{B_r(y)}+
\int_{\{\,|D\vec G(\cdot,y)|> \tau\}} |D\vec G(\cdot,y)|^p.
\end{align*}
By using \eqref{eqE-36}, we estimate (recall $\tau>t_0$)
\begin{align*}
\int_{\{\,|D\vec G(\cdot,y)|> \tau\}} |D\vec G(\cdot,y)|^p
&=\int_0^\infty p t^{p-1} |\{\abs{D\vec G(\cdot,y)}> \max(t,\tau)\}|\,dt\\
&\le C_1 \tau^{-2}(\ln \tau)^2\int_0^\tau p t^{p-1}\,dt
+ C_1 \int_\tau^\infty p t^{p-3}(\ln t)^2\,dt.
\end{align*}
Note that the above integrals converge if $0<p<2$,
and thus we have shown that
\begin{equation}
\label{eq:R12w}
\int_{B_r(y)}|D\vec G(\cdot,y)|^p \leq C(p,\gamma,d_y,r)<\infty\quad
\forall r\in (0,d_y) \quad \forall p\in [1,2).
\end{equation}
On the other hand, \eqref{eq:E28} yields
\begin{equation}
\label{eq:T28a}
\norm{D\vec G(\cdot,y)}_{L^2(\Omega\setminus B_r(y))}\le
C \gamma^{-1}r^{-2} \quad \forall r \in (0, d_y).
\end{equation}
By combining \eqref{eq:R12w} and \eqref{eq:T28a}, we find
\begin{equation}
\label{eq:T28b}
\norm{D \vec G(\cdot,y)}_{L^p(B_r(y)\cap\Omega)}<C(p,\gamma,d_y,r)<\infty
\quad \forall r>0\quad \forall p\in [1,2).
\end{equation}
Next, for $r\geq d_y/2$, fix a cut-off function
$\zeta\in C^\infty_c(B_{2r}(y)\setminus \overline B_{d_y/4}(y))$
such that $\zeta\equiv 1$ on
$\overline B_r(y)\setminus B_{d_y/2}(y)$ and $|D\zeta|\leq C/d_y$.
By a similar computation as in \eqref{eq:E21v}, we have
\begin{equation*}
\norm{D(\zeta \vec G(\cdot,y))}_{L^2(B_{2r}(y)\cap\Omega)}
\leq C(\gamma,d_y,r)<\infty.
\end{equation*}
Since $\zeta \vec G(\cdot,y)\in W^{1,2}_0(B_{2r}(y)\cap \Omega)$,
the Sobolev inequality yields
\begin{equation}
\label{eq:T29c}
\norm{\vec G(\cdot,y)}_{L^p(B_r(y)\cap\Omega\setminus\overline B_{d_y/2}(y))}
<C(p,\gamma,d_y,r)<\infty \quad \forall p\in [1,\infty).
\end{equation}
Then by combining \eqref{eq:R12q} and \eqref{eq:T29c}, we obtain
\begin{equation}
\label{eq:T30d}
\norm{\vec G(\cdot,y)}_{L^p(B_r(y)\cap\Omega)}<C(p,\gamma,d_y,r)<\infty
\quad \forall r>0\quad \forall p\in [1,\infty).
\end{equation}
Finally, from \eqref{eq:E23}, \eqref{eq:T28b}, and \eqref{eq:T30d},
it follows that $|D\vec G(x,\cdot)|$ belongs to $L^p(B_r(x)\cap\Omega)$
for all $r>0$ and $1\leq p<2$ and that $|\vec G(x,\cdot)|$ belongs to
$L^p(B_r(x)\cap\Omega)$ for all $r>0$ and $1\leq p<\infty$.
This completes the proof of the theorem.

\mysection{Proof of Theorem~\ref{thm2}} \label{sec4}
Throughout this section, we employ the letter $C$ to denote a constant depending
on $\lambda, \Lambda, N$, and $M\coloneqq\norm{\varphi'}_\infty$.
We use $C(\alpha,\beta,\ldots)$ to denote a constant depending on quantities
$\alpha,\beta,\ldots,$ as well as $\lambda, \Lambda, N, M$.

For $x=(x_1,x_2)\in \Omega$, where $\Omega$ is as in \eqref{eq:LD01},
we shall denote $\bar x\coloneqq (x_1,\varphi(x_1))\in \partial\Omega$.
Note that $d_x$ is comparable to $|x-\bar x|$; more precisely, we have
\begin{equation}
\label{eq:H01t}
d_x\leq |x-\bar x| \leq  \sqrt{1+M^2}\,d_x\quad \forall x\in\Omega.
\end{equation}
We shall use the following notations.
\begin{align}
P_r^-(t_0,x_0)&\coloneqq \{(t,x)\in \mathbb R\times\Omega: t_0-r^2<t<t_0,\, |x-x_0|<r\},\\
S_r^-(t_0,x_0)&\coloneqq \mathbb R\times\partial\Omega\,\cap\,\partial P_r^-(t_0,x_0).
\end{align}

We ask the readers to consult \cite{CDK} or \cite{LSU}
for the definition of the space $V_2$.
\begin{lemma}
\label{lem:H01}
Assume that the operator $L$ satisfy the conditions \eqref{eqP-02}
and \eqref{eqP-03}.
Let $\Omega$ be given as in \eqref{eq:LD01} and let
$\bar x_0\in \partial\Omega$.
Assume that $\vec u(t,x)$ is a weak solution in $V_2(P_{2R}^-(t_0,\bar x_0))$
of $\vec u_t- L \vec u=0$ and vanishes on $S_{2R}^-(t_0,\bar x_0)$.
Then, for any $y_0\in B_R(\bar x_0)\cap \Omega$, we have
\begin{equation}
\label{eq6.18v}
\int_{P_\rho^-(t_0,y_0)}|D_x \vec u|^2 \leq
C\left(\frac{\rho}{r}\right)^{2+2\mu}
\int_{P_r^-(t_0,y_0)}|D_x \vec u|^2
\quad \forall \rho<r\leq R,
\end{equation}
where $\mu=\mu(\lambda,\Lambda,M)\in (0,1)$.
As a consequence, for all $t\in (t_0-R^2,t_0)$, we have
\begin{equation}
\label{eq6.18w}
|\vec u(t,x)-\vec u(t,x')| \leq C|x-x'|^\mu R^{-(1+\mu)}
\left(\int_{P_{2R}^-(t_0,\bar x_0)}|D_x\vec u|^2\right)^{1/2}
\quad \forall x,x'\in B_R(\bar x_0).
\end{equation}
\end{lemma}
\begin{proof}
Let $\vec v(x)$ be a weak solution in $W^{1,2}(\Omega\cap B_{2R}(\bar x_0))$
of $L\vec v =0$ which vanishes on $\partial\Omega\cap B_{2R}(\bar x_0)$.
Let $y_0\in B_R(\bar x_0)\cap \Omega$.
By a well-known boundary regularity theory for weak solutions
of elliptic systems in two dimensional Lipschitz domains
(see e.g. \cite{Morrey}), we have
\begin{equation}
\label{eq6.18a}
\int_{B_\rho(y_0)\cap \Omega}|D \vec v|^2 \leq
C\left(\frac{\rho}{r}\right)^{2\mu}
\int_{B_r(y_0)\cap \Omega}|D \vec v|^2
\quad \forall \rho<r\leq R,
\end{equation}
where $\mu=\mu(\lambda,\Lambda,M) \in(0,1)$.
By a routine adjustment of an argument in \cite{Kim},
one can deduce \eqref{eq6.18v} and \eqref{eq6.18w} from \eqref{eq6.18a}.
\end{proof}

Let $\Omega$ be given as in \eqref{eq:LD01}.
It is rather tedious but routine to check that the estimate \eqref{eq6.18v}
allows us to treat $\Omega$ as if $\Omega=\mathbb R^2$ in the proof of
\cite[Theorem~2.7]{CDK}.
Consequently, we have the following estimates:
\begin{align}
\label{eq618a}
&|\vec K(t,x,y)| \leq C \{\max(|t|^{1/2},|x-y|)\}^{-2},\\
\label{eq615c}
&\int_\Omega |\vec K(t,x,y)|^2 \,dx \leq Ct^{-1}
\quad \forall t>0,\\
\label{eq615b}
&\iint_{(0,\infty)\times\Omega\setminus (0,r^2)\times B_r(y)}\!\!\!
|D_x\vec K(t,x,y)|^2 \,dx\,dt \leq Cr^{-2} \quad \forall r>0,\\
\label{eq619}
&\iint_{(0,r^2)\times (B_r(y)\cap \Omega)}\!\!\!
|D_x\vec K(t,x,y)|^p \,dx\,dt \leq C(p) r^{-3p+4}
\quad \forall r>0 \quad\forall p\in [1,4/3).
\end{align}

To show the convergence of the integral in \eqref{eq:g01},
we need the following lemma.
\begin{lemma}
\label{lem:H03}
Let $\Omega$ be given as in \eqref{eq:LD01}.
There exists $\mu=\mu(\lambda,\Lambda,M)\in (0,1)$ such that
\begin{align}
\label{eq:H01r}
\abs{\vec K(t,x,y)}&\leq C d_x^{\mu} \{\max(|t|^{1/2},|x-y|)\}^{-2-\mu}
\quad\forall x,y\in\Omega,\quad x\neq y.
\end{align}
\end{lemma}
\begin{proof}
Denote $r\coloneqq \max(|t|^{1/2},|x-y|)$.
We may assume that $d_x< r/(10\sqrt{1+M^2})$; otherwise, \eqref{eq:H01r} is
an easy consequence of \eqref{eq618a}.
Let $\vec u(t,x)$ be the $k$-th column of $\vec K(t,x,y)$ and set $R=r/4$.
Then by \eqref{eq:H01t}, \eqref{eq6.18w}, and \eqref{eq615b}, we have
\begin{align}
\nonumber
|\vec u(t,x)|&=|\vec u(t,x)-\vec u(t,\bar x)| \leq C |x-\bar x|^\mu R^{-1-\mu}
\left(\iint_{(0,\infty)\times\Omega\setminus
(0,R^2)\times B_R(y)}\!\!\!|D_x \vec u|^2\right)^{1/2}\\
\label{eq:H02u}
&\leq  C |x-\bar x|^\mu R^{-2-\mu} \leq C d_x^\mu r^{-2-\mu}.
\end{align}
We obtain \eqref{eq:H01r} from \eqref{eq:H02u}. The lemma is proved.
\end{proof}
Then,  it follows from \eqref{eq:H01r} that for all $x,y\in\Omega$
with $x\neq y$, we have
\begin{align}
\label{eq:H15a}
\int_0^\infty|\vec K(t,x,y)|\,dt
=\int_0^{|x-y|^2} + \int_{|x-y|^2}^\infty|\vec K(t,x,y)|\,dt
\leq C d_x^\mu |x-y|^{-\mu}<\infty.
\end{align}
Therefore, $\vec G(x,y)$ given the same as in \eqref{eq:g01}
is well defined and satisfies
\begin{equation}
\label{eq:H01k}
\abs{\vec G(x,y)}\leq C d_x^{\mu}|x-y|^{-\mu} \quad\forall x,y\in\Omega,
\quad x\neq y.
\end{equation}
In fact, by using \eqref{eq618a} and \eqref{eq:H01r} together,
we may obtain a better bound
\begin{align}
\nonumber
|\vec G(x,y)|
&\leq C\left(\int_0^{|x-y|^2}|x-y|^{-2}\,dt+ \int_{|x-y|^2}^{d_x^2} t^{-1}\,dt
+ \int_{d_x^2}^\infty d_x^\mu t^{-1-\mu/2}\,dt \right)\\
\label{eq:H01l}
&\leq C\big(1+\ln(d_x/|x-y|)\big)\qquad\text{if }\, |x-y|<d_x.
\end{align}
Then by combining \eqref{eq:H01k} and \eqref{eq:H01l}, we derive
(recall $\ln_+t\coloneqq \max(\ln t,0)$)
\begin{equation}
\label{eq:H01m}
|\vec G(x,y)| \leq C\min\left\{1+\ln_+(d_x/|x-y|),d_x^\mu |x-y|^{-\mu}\right\}
\quad \forall x,y\in\Omega,\quad x\neq y.
\end{equation}
Recall that \eqref{eq:E23} is a consequence of \eqref{eq:E43t}, which
remain valid here.
Therefore, \eqref{eq:H01j} follows from \eqref{eq:E23} and \eqref{eq:H01m}.
Note that \eqref{eq:H01j} implies that for any $r>0$ and $p\in[1,\infty)$
\begin{align}
\label{eq:H01w}
\norm{\vec G(x,\cdot)}_{L^p(B_r(x)\cap \Omega)}\leq C(p,d_x,r)<\infty;
\quad
\norm{\vec G(\cdot,y)}_{L^p(B_r(y)\cap \Omega)}\leq C(p,d_y,r)<\infty.
\end{align}
In particular, we have shown that $\vec G(x,\cdot)$ is locally integrable
for all $x\in\Omega$.
Next, we show that $\vec G(\cdot,y)$ is locally H\"older continuous
in $\Omega\setminus\{y\}$.
Fix $x_0\in \Omega$ with $x_0\neq y$ and choose $r>0$ such that $r<d_y$ and
$B_{2r}(x_0)\subset \Omega\setminus B_r(y)$.
Similarly as in \eqref{eq:H02u}, Lemma~\ref{lem:H01} yields
\begin{equation}
\label{eq:H21b}
|\vec K(t,x,y)-\vec K(t,x_0,y)| \le C |x-x_0|^\mu t^{-(1+\mu/2)}
\quad \forall x\in B_r(x_0)
\quad \forall t> t_0,
\end{equation}
where $t_0\coloneqq 8(r+\sqrt{1+M^2}d_{x_0})^2$.
Notice that \eqref{eq:E21a} still remains true here.
Therefore, by using \eqref{eq:H21b} instead of \eqref{eq:E21b} and
proceeding as in \eqref{eq:E22} we obtain
\begin{equation}
\label{eq:H22d}
|\vec G(x,y)-\vec G(x_0,y)|\leq C|x-x_0|^\mu
\quad\forall x\in B_r(x_0).
\end{equation}
Then, it follows from \eqref{eq:E23} and \eqref{eq:H22d}
that $\vec G(x,y)$ is continuous in $\set{(x,y)\in\Omega\times\Omega:x\neq y}$.

Now let us prove that $\vec u$ defined as in \eqref{eqZ-70} is a unique
weak solution in $Y^{1,2}_0(\Omega)^N$ of $L\vec u=-\vec f$.
First observe that $\Omega$ is a Green domain.
Let $\vec v(t,x)$ be defined the same as in \eqref{eq:E29x}.
Then as in \eqref{eq:E29a}, we have $\lim_{t\to\infty} \vec v(t,x)=\vec u(x)$.
Also, $\vec v_t(t,x)$ has the same representation as in \eqref{eq:E29i}.
Then, by \eqref{eq615c} and Minkowski's inequality, we have
\begin{equation}
\label{eq:H29y}
\norm{\vec v_t(t,\cdot)}_{L^2(\Omega)}
\leq C t^{-1/2}\norm{\vec f}_{L^1(\Omega)}\quad\forall t>0
\end{equation}
and thus, by Lemma~\ref{lem:E02b}, we estimate
\begin{equation}
\label{eq:H29w}
\norm{\vec v(t,\cdot)}_{L^2(\Omega)}
\leq C \norm{\vec f}_{L^1(\Omega)} \int_0^t s^{-1/2}\,ds\leq C t^{1/2}
\norm{\vec f}_{L^1(\Omega)}\quad\forall t>0.
\end{equation}
Assume that $\vec f$ is supported in a ball $B\subset \mathbb R^2$.
Then by setting $\vec \phi=\vec v(t,\cdot)$ in \eqref{eq:E35}, we get
\begin{align}
\nonumber
\lambda\norm{D\vec v(t,\cdot)}_{L^2(\Omega)}^2 &\leq
\norm{\vec v_t(t,\cdot)}_{L^2(\Omega)} \norm{\vec v(t,\cdot)}_{L^2(\Omega)}+
\norm{\vec f}_{L^2(\Omega\cap B)} \norm{\vec v(t,\cdot)}_{L^2(\Omega\cap B)}\\
\label{eq:H36e}
&\leq C\norm{\vec f}_{L^1(\Omega\cap B)}^2+ C(\Omega,B)\,
\norm{\vec f}_{L^2(\Omega\cap B)} \norm{D\vec v(t,\cdot)}_{L^2(\Omega)},
\end{align}
where we have used \eqref{eq:H29y}, \eqref{eq:H29w}, and \eqref{eq:P178}.
Then by applying Cauchy-Schwarz inequality to \eqref{eq:H36e}, we find
\begin{equation}
\label{eq:H36g}
\norm{D\vec v(t,\cdot)}_{L^2(\Omega)}^2
\leq C(\Omega,B)\left(\norm{\vec f}_{L^1(\Omega\cap B)}^2+
\norm{\vec f}_{L^2(\Omega\cap B)}^2\right)
\leq C(\Omega,B)\norm{\vec f}_{L^2(\Omega)}^2 \quad\forall t>0.
\end{equation}
Therefore, by the weak compactness and \eqref{eq:E29a},
we conclude that there exists an increasing sequence
$\set{t_m}_{m=1}^\infty$ tending to infinity such that
$D\vec v(t_m,\cdot) \wto D\vec u$ weakly in $L^2(\Omega)^N$
so that \eqref{eq:E36} holds.
Also, by \eqref{eq:H36g} we find $\norm{D\vec u}_{L^2(\Omega)}<\infty$.
By using \eqref{eq:P178} and \eqref{eq:H36g}
it is not hard to verify that for any $\zeta\in C^\infty_c(\mathbb R^2)$,
\begin{align*}
\norm{\zeta \vec v(t,\cdot)}_{W^{1,2}(\Omega)}\leq
C(\zeta,\Omega,B)\norm{\vec f}_{L^2(\Omega)}<\infty \quad\forall t>0.
\end{align*}
Therefore, we conclude that $\zeta \vec u \in W^{1,2}_0(\Omega)^N$
for all $\zeta\in C^\infty_c(\Omega)$ and thus,
$\vec u \in Y^{1,2}_0(\Omega)^N$.
Consequently, it follows from \eqref{eq:E35}, \eqref{eq:E36}, \eqref{eq:H29y},
and Lemma~\ref{lem:P04} that $\vec u$ is a unique weak solution in
$Y^{1,2}_0(\Omega)^N$ of $L\vec u= -\vec f$.

By arguing the same as in the proof of Theorem~\ref{thm1}, we
get the uniqueness of the Green's matrix in $\Omega$.

We need the following lemma to prove \eqref{eqG-54} and \eqref{eqG-55}.
\begin{lemma}
\label{lem:H04}
Let $\Omega$ be given as in \eqref{eq:LD01}.
Then, for all $y\in\Omega$ and for all $t>0$, we have
\begin{align}
\label{eq:H17n}
&\norm{D \bar{\vec K}(t,\cdot,y)}_{L^p(B_\rho(y)\cap\Omega)}
\leq C(p)\rho^{2/p-1} \quad \forall \rho>0 \quad \forall p\in [1,4/3),\\
\label{eq:H18p}
&\norm{D \bar{\vec K}(t,\cdot,y)}_{L^2(\Omega\setminus B_r(y))}
\leq C(1+d_y^\mu r^{-\mu})\quad \forall r>0.
\end{align}
\end{lemma}
\begin{proof}
We proceed similarly as in the proof of Lemma~\ref{lem:E02}.
Let us begin by proving \eqref{eq:H17n} first.
By Minkowski's inequality, we have
\begin{equation*}
\left(\int_{B_\rho(y)\cap \Omega}\!\!\!
|D_x\bar{\vec K}(t,x,y)|^p dx\right)^{1/p}
\leq \int_0^{\rho^2}+\int_{\rho^2}^\infty
\left(\int_{B_\rho(y)\cap \Omega}\!\!\!|D_x \vec K(s,x,y)|^p dx\right)^{1/p}ds
\coloneqq I_1+I_2.
\end{equation*}
Then, by H\"older's inequality and \eqref{eq619}, we have
\begin{equation}
\label{eq:H15d}
I_1\leq \left(\int_0^{\rho^2}\!\!\!\int_{B_\rho(y)\cap \Omega}
|D_x \vec K(s,x,y)|^p\,dx\,ds\right)^{1/p} \rho^{2(1-1/p)}
\leq C(p) \rho^{-1+2/p}.
\end{equation}
On the other hand, by using H\"older's inequality
\begin{equation}
\label{eq:H16m}
I_2 \leq C(p) \rho^{2/p}\sum_{j=1}^\infty
\left(\int_{j\rho^2}^{(j+1)\rho^2}\!\!\!\int_{B_\rho(y)\cap \Omega}
|D_x \vec K(s,x,y)|^2\,dx\,ds\right)^{1/2}
\coloneqq C(p)\rho^{2/p} \sum_{j=1}^\infty I_{2,j}.
\end{equation}
By setting $r=\sqrt{(j+1)/2}\,\rho$ in \eqref{eq6.18v}
and using \eqref{eq615b}, we estimate
\begin{align}
\nonumber
(I_{2,j})^2 &=\int_{P_\rho^-(2r^2,y)} |D_x \vec K(t,x,y)|^2 dx\,dt
\leq C\left(\frac{\rho}{r}\right)^{2+2\mu}
\int_{P_r^-(2r^2,y)}|D_x \vec K(t,x,y)|^2\,dx\,dt\\
\nonumber
&\leq C\left(\frac{\rho}{r}\right)^{2+2\mu}
\int_{(0,\infty)\times\Omega\setminus(0,r^2)\times B_r(y)}
|D_x \vec K(t,x,y)|^2\,dx\,dt\\
\label{eq:H17b}
&\leq C\rho^{2+2\mu} r^{-4-2\mu}= C \rho^{-2}(j+1)^{-2-\mu}.
\end{align}
Therefore, by combining \eqref{eq:H16m} and \eqref{eq:H17b}, we find
\begin{equation}
\label{eq:H18c}
I_2\leq C(p) \rho^{2/p-1}\sum_{j=1}^\infty (j+1)^{-1-\mu/2}=C(p)\rho^{2/p-1},
\end{equation}
and thus, \eqref{eq:H17n} follows from \eqref{eq:H15d} and \eqref{eq:H18c}.

Next, we turn to the proof of \eqref{eq:H18p}.
As before, Minkowski's inequality yields
\begin{align*}
\left(\int_{\Omega\setminus B_r(y)}\!\!\!
|D_x\bar{\vec K}(t,x,y)|^2 dx\right)^{1/2} &\leq \int_0^{r^2}+\int_{r^2}^\infty
\left(\int_{\Omega\setminus B_r(y)}\!\!\!
|D_x \vec K(s,x,y)|^2 dx\right)^{1/2}ds\\
&\coloneqq I_3+I_4.
\end{align*}
Then, by H\"older's inequality and \eqref{eq615b}, we have
\begin{equation}
\label{eq:H23f}
I_3\leq \left(\int_0^{r^2}\!\!\!\int_{\Omega\setminus B_r(y)}
|D_x \vec K(s,x,y)|^2\,dx\,ds\right)^{1/2} r
\leq C.
\end{equation}
We need the following inequality to estimate $I_4$:
\begin{equation}
\label{eq:H23e}
I_5(t)\coloneqq
\int_t^\infty\!\!\!\int_\Omega |D_x \vec K(s,x,y)|^2 dx\,ds
\leq C d_y^{2\mu} t^{-1-\mu}\quad \forall r>0\quad \forall t>0.
\end{equation}
Let us momentarily assume that \eqref{eq:H23e} holds and proceed
similarly as in \eqref{eq:H17b} to get
\begin{align}
\nonumber
I_4 \leq
\left(\int_{r^2}^\infty\!\!\!\int_{\Omega\setminus B_r(y)}
\!\!\!|D_x \vec K(s,x,y)|^2 dx\,ds\right)^{1/2}
&\leq \sum_{j=0}^\infty 2^{j/2}r
\left(\int_{2^j r^2}^{2^{j+1} r^2}\!\!\!\int_\Omega
|D_x \vec K(s,x,y)|^2 dx\,ds\right)^{1/2}\\
\label{eq:H24e}
&\leq Cd_y^\mu r^{-\mu} \sum_{j=0}^\infty 2^{-j\mu/2} \leq C d_y^\mu r^{-\mu}.
\end{align}
By combining \eqref{eq:H23f} and \eqref{eq:H24e}, we obtain \eqref{eq:H18p}.
It only remains to prove \eqref{eq:H23e}.
Note that by \eqref{eq:H01r} and \eqref{eq:E43t} we have
\begin{equation}
\label{eq:H31o}
|\vec K(s,x,y)|\leq C d_y^\mu\{\max(s^{1/2},|x-y|)\}^{-2-\mu}\quad
\forall x\neq y\quad\forall s>0.
\end{equation}
Let $\zeta\in C^\infty(\mathbb R)$ be such that $0\leq \zeta\leq 1$,
$\zeta\equiv 1$ on $[t,\infty)$, $\zeta\equiv 0$ on $(-\infty,t/2]$,
and $|\zeta'| \leq 4/t$.
Then, by the energy inequality (see e.g., \cite[\S III.2]{LSU})
and \eqref{eq:H31o}, we have
\begin{align*}
I_5(t)&\leq
\int_0^\infty\!\!\!\int_\Omega \zeta(s)|D_x \vec K(s,x,y)|^2\,dy\,ds
\leq C \int_0^\infty\!\!\!\int_\Omega |\zeta'(s)| |\vec K(s,x,y)|^2\,dy\,ds\\
&\leq Ct^{-1}d_y^{2\mu} \int_{t/2}^t \left(\int_{|x-y|<\sqrt{s}} s^{-2-\mu}\,dy
+\int_{|x-y|\geq \sqrt{s}} |x-y|^{-4-2\mu}\,dy\right)\,ds\\
&\leq Ct^{-1}d_y^{2\mu} \int_{t/2}^t s^{-1-\mu}\,ds \leq C d_y^{2\mu}
t^{-1-\mu}.
\end{align*}
This completes the proof of the lemma.
\end{proof}

We now prove \eqref{eqG-54} and \eqref{eqG-55}.
To prove \eqref{eqG-54}, first recall that \eqref{eq:E01} holds.
By \eqref{eq615c}, we find that \eqref{eq:E02} remains valid.
Assume that $\vec \phi\in C^\infty_c(\Omega)^N$ is supported in
$B_R(y)\cap\Omega$.
By \eqref{eq:g01} and \eqref{eq:H17n}, we find that there is a sequence
$\{t_m\}_{m=1}^\infty$ tending to infinity such that
\begin{equation*}
D\vec{\bar K}(t_m,\cdot,y)\wto D \vec G(\cdot,y)\quad
\text{weakly in } L^p(B_R(y)\cap\Omega)^{N\times N}\text{ for some }p>1.
\end{equation*}
Therefore, we find
\begin{equation}
\label{eq:H23w}
\lim_{m\to\infty} \int_\Omega A^{\alpha\beta}_{ij}D_\beta
\bar K_{jk}(\cdot,y) D_\alpha \phi^i
=\int_\Omega A^{\alpha\beta}_{ij}D_\beta G_{jk}(\cdot,y) D_\alpha \phi^i.
\end{equation}
By combining \eqref{eq:E01}, \eqref{eq:E02}, and \eqref{eq:H23w}, we obtain
\eqref{eqG-54}.
To prove \eqref{eqG-55}, first observe that \eqref{eq:H18p} yields
\begin{equation}
\label{eq:H35p}
\norm{D\vec G(\cdot,y)}_{L^2(\Omega\setminus B_r(y))} \leq C(d_y,r)<\infty
\quad \forall r>0.
\end{equation}
By using \eqref{eq:H01w} and \eqref{eq:H35p} and proceeding
similarly as in \eqref{eq:E21v}, we obtain
\begin{equation*}
\norm{D((1-\eta)\vec G(\cdot,y))}_{L^2(\Omega)}\leq
C(\eta,d_y)<\infty.
\end{equation*}
It follows from \cite[Theorem~2.7]{CDK} that for any
$\zeta\in C^\infty_c(\mathbb R^2)$, we have
\begin{equation*}
\vec F(t,\cdot)\coloneqq\zeta(1-\eta)\bar{\vec K}(t,\cdot,y)
\in W^{1,2}_0(\Omega)^{N\times N}\quad\forall t>0.
\end{equation*}
Clearly, $\lim_{t\to\infty}\vec F(t,\cdot)=\zeta(1-\eta)\vec G(\cdot,y)$.
Moreover, by utilizing \eqref{eq:E43x}, \eqref{eq:H01r}, and \eqref{eq:H18p},
it is not hard to verify
$\norm{\vec F(t,\cdot)}_{W^{1,2}(\Omega)}\leq C(\zeta,\eta,d_y)<\infty$
for all $t>0$.
Then, by a similar argument as in Section~\ref{sec3.2}, we get
$\zeta(1-\eta)\vec G(\cdot,y)\in W^{1,2}_0(\Omega)^{N\times N}$.
We have proved \eqref{eqG-55}.

Finally, notice that with \eqref{eqG-55} at hand,
we may proceed similarly as in Section~\ref{sec3.3} to conclude that
that $D\vec G(\cdot,y)$ and $D\vec G(x,\cdot)$ belong to
$L^p(B_r(y)\cap\Omega)$ and $L^p(B_r(x)\cap\Omega)$, respectively,
for all $r>0$ and $p\in [1,2)$.
We have already seen in \eqref{eq:H01w} that
that $\vec G(\cdot,y)$ and $\vec G(x,\cdot)$ belong to
$L^p(B_r(y)\cap\Omega)$ and $L^p(B_r(x)\cap\Omega)$, respectively,
for all $r>0$ and $p\in [1,\infty)$.
This completes the proof of the theorem.

\mysection{Remark on fundamental matrices} \label{sec5}
In this section, we introduce a result of Auscher et al. \cite{AMcT}
regarding construction of a fundamental matrix in $\mathbb R^2$.
Let $\mathcal H^1(\mathbb R^2)$ be the usual Hardy space in $\mathbb R^2$ and
$C_0(\mathbb R^2)$ be the space of continuous functions on $\mathbb R^2$
vanishing at infinity.
For $x,y\in \mathbb R^2$, $x\neq y$, define
\begin{equation}
\vec \Gamma(x,y)\coloneqq
\int_0^1 \vec K(t,x,y)\,dt+ \int_1^\infty (\vec K(t,x,y)-\vec K(t,x,x))\,dt.
\end{equation}
The following theorem appears in \cite{AMcT} as Theorem~3.16, where $L$ is
assumed to be an elliptic operator with complex coefficients.
With appropriate changes, the same proof carries over.
\begin {theorem}[Auscher-McIntosh-Tchamitchian]
Let the operator $L$ satisfy \eqref{eqP-02} and \eqref{eqP-03}.
Then for all $x\in \mathbb R^2$, $\vec\Gamma(x,\cdot) \in BMO$ and
for $\vec f=(f^1,\ldots,f^N)^T\in \mathcal H^1(\mathbb R^2)^N$,
the function defined by
\begin{equation*}
T\vec f(x)\coloneqq \int_{\mathbb R^2} \vec\Gamma(x,y)\vec f(y)\,dy
\end{equation*}
belongs to $C_0(\mathbb R^2)^N$.
The linear map thus defined is continuous from $\mathcal H^1(\mathbb R^2)^N$
into $C_0(\mathbb R^2)^N$.
Moreover, for all $\vec f\in \mathcal H^1(\mathbb R^2)^N$,
$\vec u(x)\coloneqq T \vec f(x)$ satisfies
$\norm{D \vec u}_{L^2(\mathbb R^2)}<\infty$ and is a weak solution of
$L\vec u= -\vec f$ in the sense of \eqref{eq:P08c}.
\end{theorem}

redpajama_set_name: RedPajamaArXiv


Mato Paha or "Bear Mountain" is the Lakota name given to this site. To the Cheyenne, it is "Noahvose." This geological formation is one of several intrusions of igneous rock in the Black Hills that formed millions of years ago. The mountain is sacred to many American Indian tribes who come here to hold religious ceremonies. Please be respectful of worshippers and their religious practices.
Many American Indians see Bear Butte as a place where the creator has chosen to communicate with them through visions and prayer.
During your visit, you will see colorful pieces of cloth and small bundles or pouches hanging from the trees. These prayer cloths and tobacco ties represent the prayers offered by individuals during their worship. Please respect these offerings and leave them undisturbed.
The Bear Butte Education Center highlights the mountain's geology, history and the cultural beliefs of the Northern Plains Indians. An on-site interpreter is available during the summer months. Please call for group arrangements.
Pets are not allowed on the Summit Trail. It is too narrow to safely accommodate a hiker and pets. Pets may be taken across Highway 79 to the horse camp area and allowed to exercise while on a leash.
Please stay on the trail and respect those participating in religious activities. Do not disturb or photograph prayer cloths and tobacco ties.
A buffalo herd roams the base of the butte. Buffalo are dangerous, please do not approach.
Horseback riding is allowed west of Highway 79 ONLY. Riders can use the Centennial Trail west of the horse camp.
Possession or consumption of alcohol at Bear Butte State Park east of Highway 79 is prohibited. Bear Butte is a sacred site for many people and will be respected as such.
Leaving human remains (ashes) at Bear Butte State Park is prohibited. Bear Butte is not a burial site and is not to be treated as one.
Uncased firearms or bows are prohibited at Bear Butte year-round east of Highway 79. This reduces danger and disturbance to those who gather at the site to worship.


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Note R = retiring councillor. Footnote explains if this was for a different ward.
Con Maj 948 32.8%. Retiring 1980, 1979 & 1978.
Hodder & Rowling were retiring councillors for Cherry Hinton, Fitch for Coleridge. No change, Con Maj 906 40.5%.
No change, Con Maj 463 20.3% Retiring 1980. No change, Con Maj 1118 24.8%.
No change, Con Maj 345 11.7%. No change, Con Maj 169 5.1%. No change, Con Maj 386 11.1%.
No change, Con Maj 239 7.2%. No change, Con Maj 158 4.8%. No change, Con Maj 227 6.4%.
No change, Con Maj 560 17.7%. No change, Con Maj 428 13.0%. No change, Con Maj 514 17.4%.
No change, Con Maj 244 8.1%. LDm gain, Maj 790 23.7%. LDm gain, Maj 617 21.0%.
No change, Con Maj 18 0.5%. No change, LDm Maj 592 23.3%. No change, LDm Maj 433 18.0%.
LDm gain, Maj 571 24.5%. No change, LDm Maj 951 38.1%. Second place retiring 2003. No change, LDm Maj 1102 47.0%.
Queen Edith's gained the area between Hills Road and the railway from Trumpington and lost Cherry Hinton Road to Coleridge and Cherry Hinton.
No change, LDm Maj 913 32.7%. Retiring in 2008, 2007 & 2006. No change, LDm Maj 754 28.7%.
No change, LDm Maj 604 23.6%. No change, LDm Maj 412 16.5%. No change, LDm Maj 908 20.9%.
No change, LDm Maj 488 15.2%. Lab gain, Maj 121 4.4%. No change, LDm Maj 411 12.9%.
No change, LDm Maj 400 9.0%. No change, LDm Maj 388 14.0%. No change, LDm Maj 432 15.2%.
County results in Queen Edith's E.D.
Copyright © Colin Rosenstiel & Keith Edkins 2018.


text:
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The coins are Sealed inside a transparent plastic film and presented in a keepsake envelope. Each set contains penny, nickel, dime, twenty-five cent, fifty-cent and dollar coins. Specifications:...
1999 NUNAVUT UNCIRCULATED PROOF LIKE SET
The coins are sealed inside a transparent plastic film and presented in a keepsake envelope. Accompanied by an enhancement card. Specifications: Mintage: 74,821
2000 SPECIAL EDITION 3-BEAR TOONIE UNCIRCULATED PROOF LIKE SET
The coins are sealed inside a transparent plastic film and presented in a keepsake envelope. Accompanied by an enhancement card. Specifications: Mintage: 186,985 (includes both types of 2000 PL sets)
1998 PROOF DOUBLE DOLLAR SET
1998 Proof SetThe Royal Canadian Mint is proud to present the 1998 proof set of Canadian coinage. This is teh only set available that features both the commemorative silver dollar and the aureate...
2012 REGULAR PROOF SET WITH PURE SILVER DOLLAR - 200TH ANNIVERSARY OF THE WAR OF 1812
Building unity-The seeds of a nationFor Canadians, the War of 1812 conjures a rich historical moment steeped in both fact and myth, full of heroes, villains, disastrous defeats, and glorious...
2004 COMMEMORATIVE PROOF DOUBLE DOLLAR SET - 400TH ANNIVERSARY OF THE FIRST FRENCH SETTLEMENT IN NORTH AMERICA
In June 1604, Samuel de Champlain and Pierre du Gua, sieur de Mints were exploring the Atlantic coast in order to establish the first settlement that would solidify France's rights to this...
2001 PROOF DOUBLE DOLLAR SET - NATIONAL BALLET OF CANADA
2001 Proof SetThis silver dollar commemorates the 50th anniversary of the National Ballet of Canada.Specifications:Finish: ProofMintage: 74,194
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A FIRST! Lucky Loonie, 100th Grey Cup $1 coin, and a 2012 Loonie and Toonie with new security features! This special edition uncirculated set represents a wonderful opportunity for you and your...
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The Royal Canadian Mint is proud to present the 2011 Special Edition Uncirculated Set. Included in this 8 coin set are the coloured and non-coloured 25-cent Wood Bison, Orca Whale and Peregrine...
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redpajama_set_name: RedPajamaCommonCrawl


Do you find it hard to figure out how to get all your responsibilities -- home and work -- accomplished in a day? Are you someone who wants a career, a family, friends, AND time for yourself? You are not alone. Many people seek the perfect balance that will give them enough time for everything and still a few hours to sleep. Many times when the topic of work-life balance comes up, working mothers are the focus of the discussion. How can women work outside the house while maintaining a warm home, deep friendships, a loving family and a sense of self? What about those women who are not mothers? What about men who may or may not be fathers? Many people struggle to find a balance that makes them happy and it is important to acknowledge that we all can benefit from some help achieving and maintaining a healthy, balanced lifestyle that includes anything we find important, whether it is family, or friendships, or a bicycle club. Kelly Wallace explores the struggle of work-life balance and seeks to expand the discussion of who that struggle affects in her piece "Work-Life Balance Not Just a Women's Issue" on cnn.com.


text:
Often asked: Why Is It Important For A Fire Department To Have A Budget Before An Emergency?
by Rodney Franklin 0 comments on Often asked: Why Is It Important For A Fire Department To Have A Budget Before An Emergency?
1 How do I cut my fire department budget?
2 What role does the fire chief play in the decision concerning the fire department's budget?
3 How are fire departments funded?
4 Why is the fire department the first to respond?
5 Do fire departments make money?
6 What is AFG grant?
7 Are firefighters state or federal employees?
8 How are emergency services funded?
9 How do volunteer fire departments make money?
10 Do police come when an ambulance is called?
11 Do firefighters sleep at the station?
12 Why is a fire truck sent with an ambulance?
How do I cut my fire department budget?
Almost 85 percent to 90 percent of a fire department budget is personnel so the only way to make significant reductions is to reduce personnel. It seems the trend for the quick fix is to reduce fire prevention and other municipal staff by consolidating inspection personnel with the building department.
What role does the fire chief play in the decision concerning the fire department's budget?
Now the chief or their designee has information on the current year's budget's strengths and weaknesses, ideas from members within the department and similar documentation. They will take all of this information and begin compiling the draft budget for the following fiscal year.
How are fire departments funded?
As such, funding for firefighters is provided mostly by state and local governments. During the 1990s, shortfalls in state and local budgets, coupled with increased responsibilities of local fire departments, led many in the fire community to call for additional financial support from the federal government.
You might be interested: Who To Contact To Make Donation To South Bend Fire Department?
Why is the fire department the first to respond?
Why are firefighters the first on-scene of a medical call? Firefighters respond to medical calls because they are trained and staffed to do so. In many areas, there are more fire engines in a given area than ambulances, so firefighters will usually be the first to arrive and give medical care in an emergency.
Do fire departments make money?
According to the Firefighters Support Alliance, grants are distributed largely due to population size, and the number of stations in each state. California and New York, two of the most populous states, were each awarded over $15 million in grants in 2017. These states have 3,332 and 2,297 departments, respectively.
What is AFG grant?
The Assistance to Firefighters Grants ( AFG ) program is designed to enhance the health and safety of public and firefighting personnel and to provide support for emergency responders regarding fire, medical, and all hazard events. Funds may be used for: Operations and safety activities such as: Training. Equipment.
Are firefighters state or federal employees?
A federal agency hires a federal firefighter, while a municipal firefighter works for a city, county, or state. Government -controlled positions have regulated salaries throughout the country while municipal wages vary depending on the demand, cost of living, and other locally related aspects.
How are emergency services funded?
EMS response is reported to be at the intersection of healthcare, public health, and public safety, yet reimbursement by health insurance providers is often the only source of funding. Local government funding of EMS and ambulance service varies widely across the United States and is subject to change annually.
You might be interested: Often asked: How High Do Fire Department Ladders Reach?
How do volunteer fire departments make money?
Municipalities are the support of volunteer fire departments. Additional funding may include, for example, contributions from support organizations, donations made in fundraising or income from various events.
Do police come when an ambulance is called?
Often, the police department will answer 911 initially, but then transfer the call to a medical dispatch center once they figure out you are having a medical emergency.
Do firefighters sleep at the station?
Firefighters do (sometimes) sleep at the station or firehouse. Stations usually are equipped with sleeping quarters, but depending on how busy the firefighters are, they may not have time to sleep while on duty.
Why is a fire truck sent with an ambulance?
The reason we send a fire engine or a fire truck along with the ambulance is if the patient is critically ill or not able to walk and must be carried to our gurney, the additional people staffing the fire engine or truck ensure that patient care needs are met in a timely and safe manner.
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Step into a world where igloos serve as residence for the statuesque and textile clad. Gyunel's autumn/winter 2014 Galanthus collection warped us into a winter wonderland of artful luxe. Palettes spanned from deep teals, rich orchids, up-town greys to hi-definition metallics.
Utilising garments as canvasses, the Central Saint Martin's graduate made great use and clear evidence of her artistic background. Seasons past have introduced many menswear trends into our wardrobes, but Gyunel's perspective allows us to maintain the same sharp tailoring through a female guise.
Every strong shoulder was matched with a flush of chiffon, transitioning us into the happy medium that sits between lady and gentleman. Gorgeously transmitting oil paintings to digital prints, we were wowed by cunning melanges of silks, lamb nappa leathers and mohair. Strong shape play and innovative takes on classic silhouettes crystallised this collection as the epitome of feminine power dressing, and quite frankly, the office wardrobe of our dreams. Structured teal short suits and mauve long line jackets with textured shoulder panelling gave us a nice take on our favourite season staples, while wispy, feminine, silk and chiffon dresses gave great context and silver lining to an otherwise aggressively chic collection.
The show-stopper: an apropos cocoon shaped parka embellished with iridescent lights. Day one of fashion week, Gyunel has set a great foundation for what will surely be known as the framework for wearable runway that is as bold as it is business.


IDEAS IN ACTION - DANCE EDITION is an end-of-quarter sharing of student choreographic ideas and works in progress that have arisen out of classes in WACD, either this quarter, last quarter, or even last year. The sharing takes place in Room 200 on Thursday, March 14 at 6:15pm. This will be a "work-light" showing (no theatrical lights), and will include feedback and discussion from the gathered audience.


This entry was posted in Holiday Survival with Bayard & Holmes and tagged Bayard & Holmes, Dysfunction, family, holiday, relationships, Self-Help, Thanksgiving.
Great inclusion of everybody. Happy Thanksgiving!
Happy Thanksgiving for you and yours.
And never forget that dysfunctional wouldn't be the same without some fun in it.
I hope you folks at Bayard & Holmes had an enjoyable Thanksgiving Weekend.
We did, thank you. Hope you did, as well.


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Patriots reportedly aggressively pursued TE Jared Cook before he signed with Saints
This past week, tight end Jared Cook signed with the Saints, which meant the Patriots once again had a player choose another team over them.
According to ESPN's Mike Reiss, the Patriots aggressively courted Cook before he signed with New Orleans. The report noted Rob Gronkowski's uncertainty for 2019 factored into his decision, as Cook has a better chance of rising to the top of the depth chart with the Saints.
The 31-year-old is coming off his first Pro Bowl season with the Raiders where he recorded 68 catches for 896 yards and six touchdowns.
Cook is entering his 11th season in the league and has played for four teams -- the Titans, Rams, Packers and Raiders.
Related: Sunday 7: Tom Brady's offseason plans should be factored into what Patriots do at wide receiver


Just in time for the love month, Belle De Jour also released it's very first EXCLUSIVE BDJ box! Instead of getting items from different brands, they are giving us five of the most beloved beauty products from Shiseido.
The concept of this exclusive BDJ box is to help girls lighten, brighten and let us be in our most radiant self - just like we've been bitten by the love bug!
I was really excited when I received my exclusive Shiseido Belle De Jour box! I wanted to open it right away because I wanted to see what was inside the box.
And what greeted me were the Shiseido items inside. Not only do they come in big sizes, but the items inside are also some of Shiseido's best products.
1.) Shiseido Benefiance Wrinkle Resist 24 - Visibly reduces wrinkles and promotes silky smooth skin with breakthrough mukurossi extract which inhibits wrinkle-trigerring enzymes.
2.) Shiseido Benefiance Wrinkle Resist 24 Night Emulsion - Offers visible reduction in the appearance of wrinkles, and helps promote silky smooth skin condition while encouraging recovery by morning.
3.) Shiseido White Lucent Intensive Brightening Mask - Reduces the apearance of dark spots for a flawless, even complexion.
4.) Shiseido White Lucent Intensive Spot Targeting Serum - Dark spots, age spots, acne scars and uneven skin tone begone!
5.) Shiseido Perfect Rouge Lipstick - Contains Hyaluronic acid which holds moisture while Super Hydro-Wrap Vitalizing DE locks it in. Improves lips' condition with regular use.
So all the items I mentioned above are included in the Belle De Jour Shiseido Beauty Box. How amazing it is to get all those products, not to mention the original price of each item, for a very reasonable price care of BDJ!
How do you get your own Belle De Jour Beauty Box? There are 4 options to choose from. 1 time purchase for only P480. 3 months for P1,400. 6 months for P2,800 and 1 year subscription for only P5,280.
There are also different payment methods for your convenience. Over the counter and online banking, mobile cash or non-traditional outlets like LBC or SM payment counters.
For more information and to know more about the Belle De Jour Beauty Box, please visit bdjbox.com.
Thank you for reading! Stay beautiful and in love!


Brand loyalty: It's the biggest factor of your brand's success. If you don't have a loyal customer base that purchases from you time and time again, you have to compete with other brands on factors like price and convenience.
Consumer attitudes can make or break your brand's success. Download our comprehensive brand research guide to learn how to measure your brand power.
• How convenient is our company to use?
• Compared to our competitors, is our product quality better, worse, or about the same?
• How well do our customer service representatives answer your questions?
• How likely are you to recommend us to others?
• Do you trust our brand?
• How did we earn your trust?
• How do we keep your trust?
• How reliable would you consider our brand?
• How would you rate the quality of our product?
• How valuable is [brand or product] to you?
• How likely would it be for you to switch brands if an alternative brand was sold in a more convenient location?
• How likely would it be for you to switch brands if an alternative brand was cheaper?
Finding out how loyal customers feel toward your brand is important, but digging deeper for insights on what specifically contributes to that loyalty is important too. Read more about creating and using a brand attributes survey to see how your company meets customer expectations across the board.
Download our comprehensive brand research guide to learn how to measure your brand power.


You won't find your mother's hot milk and cocoa in this year's batch of Vancouver Hot Chocolate Festival offerings. Ingredients like caramelized bananas, vanilla bean, chai spices and the highest quality chocolate take these hot chocolates up a few notches.
Until February 14, 23 of the city's finest cafes, chocolate shops and bakeries are offering up to 60 unique hot chocolate drinks to warm you up during the cold, grey days of winter.
Partial funds from festival sales will be donated to restaurateurMark Brand's new charity project A Better Life Foundation, which aims to create a self-sustaining meal program for people in assisted living and low-income housing on Vancouver's Downtown Eastside.
And be sure to tweet a picture of your hot chocolate for a chance to win gift certificates from all participating vendors, valued at $500.
Not sure where to start imbibing? Try these 7 wonderful, wild hot chocolates, each paired perfectly with a tiny sweet treat to nibble on.
An ode to Elvis, this hot chocolate at Soirette Macarons and Tea is a rockabilly combo of peanut butter and caramelized banana, with French milk and dark chocolate. Served with a peanut biscotti.
Named after the neighbourhood where the shop is located (the Armory District), this hot chocolate from Chocolate Arts is made with a single origin chocolate plus hints of raspberry. It's rich but not too sweet.
This delectable drink from Secret Location contains a surprisingly delicious combination of roasted white chocolate, candy cap, and cognac. (Candy cap is a unique small mushroom that, when dried, has the aroma and flavour of earthy maple syrup.) Served with a brown butter Caramelia financier.
There's nothing vanilla about this vanilla bean white hot chocolate with rose, made with Valrhona chocolate at Beaucoup Bakery.
Beer in hot chocolate? It's a combination that works when you infuse rich Cocoa Barry cocoa with local R&B Brewing Company's Dark Star Oatmeal Stout, top it with a stout marshmallow, and serve with an oatmeal raisin cookie. Try it at Terra Breads.
As the name suggests, Gem Chocolates' offering makes this a pseudo-healthy choice with its blend of high anti-oxidant-content cocoa and blueberries.
Reminiscent of India in name and in flavour, this rich, dark hot chocolate from Bel Café is flavoured with chai spices.


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Mack Law: Sherman
North Oaks Primary Care
Mack Law: Glenn
North Oaks Rehab
Catalano returns to coach STA Falcons Baseball team
Posted: Tuesday, June 18, 2019 | Views: 1669
St. Thomas Aquinas Regional Catholic High School has selected Cole Catalano as the next Head Baseball Coach.
This will be Catalano's third stint with the STA baseball program.
Catalano was a non-faculty coach from 2010-2012. He then joined the faculty in 2015 and served as assistant coach until 2018, winning a state championship in 2016. Last year, Catalano served as the Head Baseball Coach at Hammond High Magnet School.
Catalano prepped at Fountainebleau High School in Mandeville and played baseball at Southeastern Louisiana University while completing his undergraduate degree in 2010. In 2015 he joined the faculty at St. Thomas as a Theology teacher and during that time he took over the role of Campus Minister.
In addition to his role as Head Baseball Coach, Catalano will serve as Advancement Director. He will oversee the Annual Appeal, the STA newsletter, The Flame, and manage the donor and alumni databases.
Catalano said, "I am thrilled to be back where I started my high school coaching career. The Catholic identity of the school has had a major impact on my life. I look forward to rekindling past relationships, while building new ones, both as a Head Baseball Coach and Advancement Director."
STA Principal Pat Sanguinetti added, "We are excited to have Cole back on campus. When I came to STA, it was clear that he made a positive impact on the STA community and we needed him back on our staff. These two positions were the perfect opportunity to make that happen."


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Morin Lake is a small lake north-west of Prince Albert, Saskatchewan, Canada.  Saskatchewan Highway 55 provides access to the regional park which is via a turn off 15 kilometres south of the highway. It is a popular summer destination, with its campgrounds and cabins. Nearby communities include Victoire and Debden.

See also
List of lakes of Saskatchewan
Tourism in Saskatchewan

References

Lakes of Saskatchewan

redpajama_set_name: RedPajamaWikipedia