\documentclass[mathpazo]{jpde}

%%%%% author macros %%%%%%%%%
% place your own macros HERE
%%%%% end %%%%%%%%%

\def\x{{\bf x}}
\def\y{{\bf y}}
\def\R{{\bf R}}

\begin{document}
%%%%% title : short title may not be used but TITLE is required.
% \title{TITLE}
% \title[short title]{TITLE}
\title{A Demonstration of the \LaTeXe \ Class File for the
Journal of Partial Differential Equations}

%%%%% author(s) :
% single author:
% \author[name in running head]{AUTHOR\corrauth}
% [name in running head] is NOT OPTIONAL, it is a MUST.
% Use \corrauth to indicate the corresponding author.
% Use \email to provide email address of author.
% \footnote and \thanks are not used in the heading section.
% Another acknowlegments/support of grants, state in Acknowledgments section
% \section*{Acknowledgments}
\author[M.~Wong, C.~Lai and J.~Smith]{Mike Wong\affil{1}, Chris Lai\affil{1}\comma\corrauth and John Smith\affil{2}}
\address{\affilnum{1}\ Address 1\\
\affilnum{2}\ Address 2} \emails{{\tt jpde@global-sci.org}
(M.~Wong), {\tt xxx@xxx} (C.~Lai), {\tt xxx@xxx} (J.~Smith)}

% multiple authors:
% Note the use of \affil and \affilnum to link names and addresses.
% The author for correspondence is marked by \corrauth.
% use \emails to provide email addresses of authors
% e.g. below example has 3 authors, first author is also the corresponding
%      author, author 1 and 3 having the same address.
% \author[Zhang Z R et.~al.]{Zhengru Zhang\affil{1}\comma\corrauth,
%       Author Chan\affil{2}, and Author Zhao\affil{1}}
% \address{\affilnum{1}\ School of Mathematical Sciences,
%          Beijing Normal University,
%          Beijing 100875, P.R. China. \\
%           \affilnum{2}\ Department of Mathematics,
%           Hong Kong Baptist University, Hong Kong SAR}
% \emails{{\tt zhang@email} (Z.~Zhang), {\tt chan@email} (A.~Chan),
%          {\tt zhao@email} (A.~Zhao)}
% \footnote and \thanks are not used in the heading section.
% Another acknowlegments/support of grants, state in Acknowledgments section
% \section*{Acknowledgments}


%%%%% Begin Abstract %%%%%%%%%%%
\begin{abstract}
This paper describes the use of the \LaTeXe \ {\sf jpde.cls} class
file for setting papers for the {\it Journal of Partial Differential
Equations}.
\end{abstract}
%%%%% end %%%%%%%%%%%

%%%%% AMS/Chinese Library Classification/Keywords %%%%%%%%%%%
\ams{52B10, 65D18, 68U05, 68U07} \clc{O175.27} \keywords{\LaTeXe}

%%%% maketitle %%%%%
\maketitle


%%%% Start %%%%%%
\section{Introduction}
This paper is described how to use the {\sf jpde.cls} class file for
publication in the {\it Journal of Partial Differential Equations}.
The {\sf jpde.cls} class file preserves much of the standard \LaTeXe
\ interface so that authors can easily convert their standard
\LaTeXe \ {\sf article} style files to the {\sf jpde} style.

\section{Preparation of Manuscript}
The Title Page should contain the article title, authors' names and complete affiliations,
and email addresses of all authors. The Abstract should provide a brief summary of the main findings of the paper.

\medskip

References should be cited in the text by a number in square brackets. Literature cited should appear on a separate page at the
end of the article and should be styled and punctuated using standard abbreviations for journals
(see Thomson ISI list of journal abbreviations).
 For unpublished lectures of symposia,
include title of paper, name of sponsoring society in full, and date. Give titles of unpublished reports with ''(unpublished)''
following the reference. Only articles that have been published or are in press should be included in the references.
Unpublished results or personal communications should be cited as such in the text.
Please note the sample at the end of this paper.



Equations should be typewritten whenever possible and the number placed in parentheses at the right margin.
Reference to equations should use the form ''Eq. (2.1)'' or simply ''(2.1).''
Superscripts and subscripts should be typed or handwritten clearly above and below the line, respectively.



Figures should be in a finished form suitable for publication.
Number figures consecutively with Arabic numerals.
Lettering on drawings should be of professional quality or generated by high-resolution
computer graphics and must be large enough to withstand appropriate reduction for publication.
For example, if you use {\sf MATLAB} to do figure plots,
axis labels should be at least point 18. Title should be 24 points or above. Tick marks labels
better have 14 points or above. Line width should be 2 (or above).


Illustrations in color in most cases can be accepted only if the authors defray the cost.
At the Editor's discretion a limited number of color figures each year of special interest
will be published at no cost to the author.

\section{Mathematical Formulas (Examples)}

In \cite{cjgc-ofdft} it was claimed that there  always exists a
minimizer; however, the statement of Theorem 2.1 is incomplete. In
this note we present the full statement, with a detailed proof.

The theorem stated in \cite{cjgc-ofdft} holds as long as the number
of electrons is below a certain critical value. The correct
statement for the theorem in \cite{cjgc-ofdft} is:
\begin{theorem}[Existence of minimizers]
\label{existence} Given $v \in C^{\infty}(\overline{\Omega})$, and
$K_{WT}\in L^2_{loc} (\mathbb{R}^3)$, consider the problem
\begin{equation}
\inf _{u \in {\cal B}} F[u],
\end{equation}
where $F$ and ${\cal B}$ are
\begin{eqnarray}
&& F[u] = \frac{1}{2} \int_\Omega |\nabla u|^2 - \frac{7C_{TF}
N^{2/3}}{25} \int_\Omega  u^{10/3} + \frac{4C_{TF}N^{2/3}}{5}
\int_\Omega |u| ^{5/3} \left (K_{WT} \ast |u|^{5/3}\right ) \nonumber\\
&&\qquad\quad + \frac{N}{2} \int_\Omega u^2\,\left (\frac{1}{|\x|}
\ast u^2\right ) -\frac{3}{4} \left ( \frac{3N}{\pi} \right )^{1/3}
\int_\Omega u^{8/3} \nonumber\\
&&\qquad\quad + \int_\Omega u^2 \varepsilon( Nu^2 ) + \int_\Omega
v(\x) u^2(\x)\,d\x,\label{energy}
\end{eqnarray}
and
\begin{equation}
\label{constraint} {\cal B} = \left \{ u \in H_0^1(\Omega) \bigg | u
\ge 0,\ \int_\Omega u^2 = 1 \right \}.
\end{equation}
In $(\ref{energy})$, the set $\Omega$ is open and bounded, and
star-shaped with respect to $0$; $\varepsilon$ is defined as
\begin{equation}
\label{def:epsilon} \varepsilon(Nu^2) = \left \{ \begin{array}{l}
\displaystyle \frac{\gamma}{1 + \beta_1 \sqrt{r_s}+\beta_2 r_s},\ \ \ r_s \ge 1, \\
A\ln (r_s) + B + C r_s \ln(r_s) + Dr_s,\ \ \ r_s \le 1, \end{array}
\right .
\end{equation}
where $r_s = \left ( {4\pi N u^2}/3\right )^{-\frac{1}{3}}$; the
parameters used are $\gamma = -0.1423$, $\beta_1 = 1.0529$, $\beta_2
= 0.3334$, $A = 0.0311$, $B=-0.048$, and $C=2.019151940622\times
10^{-3}$ and $D=-1.163206637891\times 10^{-2}$ are chosen so that
$\varepsilon(r)$ and $\varepsilon'(r)$ are continuous at $r=1$.

Then, there exists $N_0>0$ such that:
\begin{enumerate}\addtolength{\itemsep}{-0.15cm}\vspace{-0.15cm}
\item If $N < N_0$ then
$\exists u^* \in {\cal B}$ such that
\begin{equation}
F[u^*] = \min _{u \in {\cal B}} F[u].
\end{equation}
\item If $N > N_0$ then
\begin{equation}
\inf _{u \in {\cal B}} F[u] = - \infty.
\end{equation}
\end{enumerate}
\end{theorem}

{\it Proof.} The second part of the theorem was proved in
\cite{BlancCances:2005,BlancCances:Preprint}. We outline the proof
here for completeness. Since $0 \in \Omega$, $\exists \delta_0>0$
such that $B(0,\delta_0) \subset \Omega$. Consider a compactly
supported function $u_0 \in C_0^\infty\left ( B(0,1)\right )$, such
that
\begin{equation}
\int_ {\mathbb{R}^3} u_0^2 = 1,
\end{equation}
and consider  the  rescaling
\begin{equation}
u_\delta(\x) = \frac{1}{\delta^{3/2}} u_0 \left ( \frac{\x}{\delta}
\right ),\ \ \ 0<\delta<\delta_0.
\end{equation}
Then $u_{\delta} \in {\cal B}$, and
\begin{equation}
\label{leading-term} F[u_\delta] = \frac{1}{\delta^2} \left (
\frac{1}{2} \int_\Omega |\nabla u_0|^2 - \frac{7C_{TF} N^{2/3}}{25}
\int_\Omega u_0^{10/3} \right ) + \mathcal O\left ( \frac{1}{\delta}
\right ).
\end{equation}
Define
\begin{equation}
A_0 = \inf _{u \in H_0^1(\Omega),\ \|u\|_2=1} \frac{\int_{\Omega}
|\nabla u|^2}{\int_{\Omega} u^{10/3}} > 0.
\end{equation}
Then if $A_0/2 <  {7C_{TF} N^{2/3}}/{25}$, we can choose $u_0$ so
that the leading term in (\ref{leading-term}) is negative, and when
$\delta \to 0$, the desired result follows.

For the existence of minimizers, assume that $N$ is such that $A_0/2
> {7C_{TF} N^{2/3}}/{25}$. By Lemma 3.1,
there exist $C>0$, $\delta>0$ such that
\begin{eqnarray}
&&\quad F[u] \ge \frac{1}{2} \int_{\Omega} |\nabla u|^2 - \left
(\frac{7C_{TF} N^{2/3}}{25}+\delta \right ) \int _{\Omega} u^{10/3}
- C \nonumber\\
&& \ge \left ( \frac{1}{2} - \frac{1}{A_0}\left ( \frac{7C_{TF}
N^{2/3}}{25} +\delta \right ) \right )\int_{\Omega} |\nabla u|^2 \ge
\tau \int_{\Omega} |\nabla u|^2 - C,
\end{eqnarray}
where $\tau > 0$. Therefore the functional is coercive, and the
result follows from now from standard arguments in the Calculus of
Variations, involving the Sobolev Embedding, and the
Rellich-Kondrachov compactness theorem. $\hfill \Box$ \@


\section{Header Information}
The heading for any file using {\sf jpde.cls} is like this;
\begin{verbatim}

\documentclass[mathpazo]{jpde}

\begin{document}

\title{Make the Title in Title Case}

\author[An Author et.~al]{First Author\affil{1},
Second Author\affil{2}\comma\corrauth
and Third Author\affil{1}}

\address{\affilnum{1}\ Address for first and third authors \\
\affilnum{2}\ Address for second author}

\emails{{\tt jpde@global-sci.org} (A.~Author), {\tt
second@author.email} (S.~Author), {\tt third@author.email}
(T.~Author)}

\begin{abstract}
Text here, no equation, no citation, no reference.
\end{abstract}

\ams{list here}
\clc{list here}
\keywords{list here}

\maketitle

\section{First Section}

\end{document}

\end{verbatim}


\section{Some Remarks}
\subsection{Mathematics}
{\sf jpde.cls} makes the full functionality of \AmS\TeX \ available.
We encourage the use of the {\sf align}, {\sf gather} and {\sf
multline} environments for displayed mathematics.



\subsection{Cross-referencing}
The use of the \LaTeX cross-reference system for figures, tables, equations
and citations is encouraged.


%%%% Acknowledgments %%%%%%%%
\section*{Acknowledgments}
The author would like to thank  ....

%%%% Bibliography  %%%%%%%%%%
\begin{thebibliography}{99}
\bibitem{cjgc-ofdft}
C. J. Garc{\'\i}a-Cervera, An efficient real space method for
orbital-free density-functional theory, Commun. Comput. Phys., 2(2)
(2007), 334-357.
\bibitem{BlancCances:2005}
X.~Blanc and E.~Canc\`es, Nonlinear instability of
density-independent orbital-free
  kinetic-energy functionals,
J. Chem. Phys., 122 (2005), 214106.
\bibitem{BlancCances:Preprint}
X.~Blanc and E.~Canc\`es, Technical report,
  http:/\!/www.ann.jussieu.fr/publications/2005/ R05014.pdf, 2005.
\bibitem{Goossens} Michel Goossens, Frank Mittelbach and Alexander Samarin,
The LaTeX Companion,
Addison-Wesley, 1994.
\bibitem{Kopka}Helmut Kopka and Patrick W.~Daly, A Guide to LaTeX,
Addison-Wesley, 1999.
\bibitem{Knuth}Donald E. Knuth, The TeXbook,
Addison-Wesley, 1992.
\bibitem{Other}A.~N.~Other, A demonstration of the LaTeX2e class file for
the International Journal for Numerical Methods in Engineering,
Int.~J.~Numer.~Meth.~Engng,  00 (2000), 1-6.
\bibitem{Yin}Z.~Yin, H.~J.~H.~Clercx and D.~C.~Montgomery,
An easily implemented task-based parallel scheme for the Fourier
pseudospectral solver applied to 2D Navier-Stokes turbulence,
Computers \& Fluids, 33 (2004), 509-520.
\end{thebibliography}

\end{document}
