\batchmode

\documentstyle{article}
\makeatletter


\newcommand{\lbl}[1]{\label{#1}}


\def\be{\begin{equation}}
\def\bea{\begin{eqnarray}}
\def\ee{\end{equation}}
\def\eea{\end{eqnarray}}
\def\half{{1 \over 2}}
\def\bF{{\bar{F}}}
\def\bD{{\bar{D}}}
\def\by{{\bar{y}}}
\def\bx{{\bar{x}}}
\def\bu{{\bar{u}}}
\def\ds{{\displaystyle}}
\def\div{{\rm div}}
\def\grad{{\bf grad}}
\def\1{{\pi}}
\def\q{{\gamma}}
\def\Q{{\Gamma}}
\def\w{{\delta}}
\def\W{{\Delta}}
\def\e{{\epsilon}}
\def\r{{\theta}}
\def\R{{\Theta}}
\def\t{{\tau}}
\def\y{{\ypsilon}}
\def\Y{{\Ypsilon}}
\def\u{{\xi}}
\def\U{{\Xi}}
\def\i{{\iota}}
\def\p{{\rho}}
\def\a{{\alpha}}
\def\A{{\nabla}}
\def\s{{\sigma}}
\def\S{{\Sigma}}
\def\d{{\phi}}
\def\D{{\Phi}}
\def\g{{\lambda}}
\def\h{{\eta}}
\def\l{{\omega}}
\def\L{{\Omega}}
\def\z{{\zeta}}
\def\Z{{\approx}}
\def\X{{\equiv}}
\def\c{{\psi}}
\def\C{{\Psi}}
\def\b{{\beta}}
\def\B{{\inf}}
\def\l{{\lambda}}
\def\n{{\nu}}
\def\N{{\nonumber \\ }}
\def\m{{\mu}}
\def\M{{\partial}}
\def\ss{{\cal }}
\def\hh{{\hat{ }}}
\def\-{{\bar}}
\def\.{{\dot}}
\def\tt{{\tilde}}
\def\inf{{\infty}}
\def\bbr{{{\Bbb{R}}}}
\def\ra{{\rightarrow}}
\def\da{{\downarrow}}
\def\ua{{\uparrow}}
\def\'{{\prime}}
\def\be{\begin{equation}}\def\bea{\begin{eqnarray}}\def\ee{\end{equation}}\def\eea{\end{eqnarray}}\def\half{{1 \over 2}}\def\bF{{\bar{F}}}\def\bD{{\bar{D}}}\def\by{{\bar{y}}}\def\bx{{\bar{x}}}\def\bu{{\bar{u}}}\def\ds{{\displaystyle}}\def\div{{\rm div}}\def\grad{{\bf grad}}\def\1{{\pi}}\def\q{{\gamma}}\def\Q{{\Gamma}}\def\w{{\delta}}\def\W{{\Delta}}\def\e{{\epsilon}}\def\r{{\theta}}\def\R{{\Theta}}\def\t{{\tau}}\def\y{{\ypsilon}}\def\Y{{\Ypsilon}}\def\u{{\xi}}\def\U{{\Xi}}\def\i{{\iota}}\def\p{{\rho}}\def\a{{\alpha}}\def\A{{\nabla}}\def\s{{\sigma}}\def\S{{\Sigma}}\def\d{{\phi}}\def\D{{\Phi}}\def\g{{\lambda}}\def\h{{\eta}}\def\l{{\omega}}\def\L{{\Omega}}\def\z{{\zeta}}\def\Z{{\approx}}\def\X{{\equiv}}\def\c{{\psi}}\def\C{{\Psi}}\def\b{{\beta}}\def\B{{\inf}}\def\l{{\lambda}}\def\n{{\nu}}\def\N{{\nonumber \\ }}\def\m{{\mu}}\def\M{{\partial}}\def\ss{{\cal }}\def\hh{{\hat{ }}}\def\-{{\bar}}\def\.{{\dot}}\def\tt{{\tilde}}\def\inf{{\infty}}\def\bbr{{{\Bbb{R}}}}\def\ra{{\rightarrow}}\def\da{{\downarrow}}\def\ua{{\uparrow}}\def\'{{\prime}}
\makeatother
\newenvironment{tex2html_wrap}{}{}
\setlength{\textheight}{10in}\begin{document}
\pagestyle{empty}
\stepcounter{section}
\newpage

{\samepage \clearpage $10^6 m^{-1}$
}


\stepcounter{section}
\newpage

{\samepage \clearpage $\d$
}


\newpage

{\samepage \clearpage $u$
}


\newpage

{\samepage \clearpage $[0,1]$
}


\newpage

{\samepage \clearpage $-1 \le  \le 1$
}


\newpage

{\samepage \clearpage $\pm
1$
}


\newpage

{\samepage \clearpage $F[,u]$
}


\newpage

{\samepage \clearpage \begin{equation}F [ \phi, u ]  = F_1 [ \phi ] + F_2[ u ] + \epsilon F_{12} [ \phi, u
].
\label{F}
\end{equation}
}


\newpage

{\samepage \clearpage \begin{equation}F_1 [ \phi]  = A\int_{\Omega}
[ w( \phi) + \frac{1}{2} \w^2 | \nabla \phi |^2 ] dv.
\label{F1}
\end{equation}
}


\newpage

{\samepage \clearpage $\L$
}


\newpage

{\samepage \clearpage $-{{\infty}} < x < {{\infty}}\}\times -h < y < h\}$
}


\newpage

{\samepage \clearpage $\L_0$
}


\newpage

{\samepage \clearpage $\{y = \pm h\}$
}


\newpage

{\samepage \clearpage $\w^2$
}


\newpage

{\samepage \clearpage $\w$
}


\newpage

{\samepage \clearpage $|\d| = 1$
}


\newpage

{\samepage \clearpage \begin{eqnarray}w(\phi) & = & {1 \over 2} (1 - \phi^2) \hbox{~~if~~} | \phi| < 1
\nonumber  
\hbox{and} & = & + \infty \hbox{~~if~~} |\phi| > 1
\label{w}
\end{eqnarray}
}


\newpage

{\samepage \clearpage $|\phi| = 1, ~w(\phi)$
}


\newpage

{\samepage \clearpage \begin{equation}F_2[u] = \int_{\Omega}W(u) dv,
\label{F2}
\end{equation}
}


\newpage

{\samepage \clearpage $W$
}


\newpage

{\samepage \clearpage \begin{equation}W(u) = \mbox{const}\cdot [u \ln u + (1-u) \ln (1-u)].
\label{W2}
\end{equation}
}


\newpage

{\samepage \clearpage ${\nabla} u$
}


\newpage

{\samepage \clearpage $F_{12}$
}


\newpage

{\samepage \clearpage \begin{equation}F_{12} = \epsilon A \int_{\Omega} p(\phi, u) dv.
\label{F12}
\end{equation}
}


\newpage

{\samepage \clearpage $\epsilon$
}


\newpage

{\samepage \clearpage $p$
}


\newpage

{\samepage \clearpage $\bar{{\epsilon}}
= \frac{{\epsilon} A}{W''(1/2)}$
}


\newpage

{\samepage \clearpage \begin{equation}|\epsilon| \ll 1;~~|\bar{{\epsilon}}| \ll 1.
\label{eps}
\end{equation}
}


\newpage

{\samepage \clearpage \begin{equation}\frac{{\partial}\d}{{\partial} t} = - \frac{{\delta} F}{{\delta} },~~\frac{{\partial} u}{{\partial} t} =
{\nabla}\cdot
D(){\nabla} u(1 +
O(\bar{{\epsilon}}))
\label{ev}\end{equation}
}


\newpage

{\samepage \clearpage $$\frac{{\delta} F}{{\delta} } = -\w^2{\Delta}+ w'() + {\epsilon} A \frac{{\partial} p}{{\partial}\d},
$$
}


\newpage

{\samepage \clearpage $D()$
}


\newpage

{\samepage \clearpage $\t$
}


\newpage

{\samepage \clearpage $D$
}


\newpage

{\samepage \clearpage $D > 0$
}


\newpage

{\samepage \clearpage $D((x,t))$
}


\newpage

{\samepage \clearpage $D(\phi) = 0$
}


\newpage

{\samepage \clearpage $|\phi| = 1$
}


\newpage

{\samepage \clearpage \begin{equation}D(\phi) = D(0)(1-\phi ^2),
\label{Dphi}
\end{equation}
}


\newpage

{\samepage \clearpage $D(0)$
}


\newpage

{\samepage \clearpage \begin{equation}\tau \frac{\partial \phi}{\partial t}  =
 \phi +  \w^2 \nabla^2 \phi - \epsilon \partial p/ \partial \phi
~~\hbox{  if  }~~| \phi| \ne 1,
\label{dphi/dt}
\end{equation}
}


\newpage

{\samepage \clearpage \begin{equation}\frac{\partial u}{\partial t} =   {\nabla}\cdot  D() {\nabla} u (1 + O(\bar{{\epsilon}})).
\label{du/dt2}
\end{equation}
}


\newpage

{\samepage \clearpage $\phi$
}


\newpage

{\samepage \clearpage ${\partial}\L$
}


\newpage

{\samepage \clearpage \begin{equation}\frac{\partial \phi}{\partial n} = 0 {\rm ~on~} \partial \Omega,
\label{phibc}
\end{equation}
}


\newpage

{\samepage \clearpage $\partial / \partial n$
}


\newpage

{\samepage \clearpage $\partial \Omega$
}


\newpage

{\samepage \clearpage $\bar{{\epsilon}}$
}


\newpage

{\samepage \clearpage \begin{equation}u = u^* \mbox{ on }{\partial}\L,
\label{vapor}
\end{equation}
}


\newpage

{\samepage \clearpage $u^*$
}


\newpage

{\samepage \clearpage $x =  \pm \B$
}


\newpage

{\samepage \clearpage \begin{equation}= \pm 1\mbox{  and  } \frac{{\partial} u}{{\partial} t} = 0
\label{inf}
\end{equation}
}


\newpage

{\samepage \clearpage $D = 0$
}


\stepcounter{section}
\newpage

{\samepage \clearpage $x$
}


\newpage

{\samepage \clearpage $x+ct$
}


\newpage

{\samepage \clearpage $c$
}


\newpage

{\samepage \clearpage $z=x+ct$
}


\newpage

{\samepage \clearpage $\phi=\phi(z,y),~ u=u(z,y)$
}


\newpage

{\samepage \clearpage $u_0$
}


\newpage

{\samepage \clearpage \begin{eqnarray}c\tau \phi_z & = &  \phi + \w^2(\phi_{zz}+\phi_{yy})
 - \epsilon (\partial p / \partial \phi ),
\label{twphi} 
cu_z & = & {\nabla}\cdot D(){\nabla} u(1 + O(\bar{{\epsilon}}));
\label{twu}
\end{eqnarray}
}


\newpage

{\samepage \clearpage \begin{eqnarray}u(z,y) = u_0,~~(z,y) = -1,~~~~z\mbox{ large negative },
\nonumber  
(z,y) = 1,~~~~z\mbox{ large positive }
\label{infbc2}
\end{eqnarray}
}


\newpage

{\samepage \clearpage $p(,u)$
}


\newpage

{\samepage \clearpage \begin{equation}g(z,y) = -\frac{\partial p}{\partial \phi}[ (z,y), u(z,y)],
\label{ar}
\end{equation}
}


\newpage

{\samepage \clearpage $\d_z$
}


\newpage

{\samepage \clearpage $z$
}


\newpage

{\samepage \clearpage $y$
}


\newpage

{\samepage \clearpage \begin{equation}c\int\int \d_z^2 dzdy =
\epsilon \int\int\d_zgdzdy.\label{ao}
\end{equation}
}


\newpage

{\samepage \clearpage ${\epsilon} g$
}


\stepcounter{section}
\newpage

{\samepage \clearpage $\label{as}$
}


\newpage

{\samepage \clearpage $\e$
}


\newpage

{\samepage \clearpage $\bar{{\epsilon}}$
}


\newpage

{\samepage \clearpage $\epsilon = 0,
~c=0$
}


\newpage

{\samepage \clearpage \begin{equation}\phi^{(0)} = S(x/\delta),
\label{phi0}
\end{equation}
}


\newpage

{\samepage \clearpage $S(x)$
}


\newpage

{\samepage \clearpage $\sin x$
}


\newpage

{\samepage \clearpage $|x| <
\pi/2$
}


\newpage

{\samepage \clearpage \begin{eqnarray}D(\phi(z)) \approx D(\phi^{(0)}(z)) & = & D(S(z/\delta))
\nonumber  
& = & \left
\begin{array}{l}
D(0) \cos^2(z/\delta) \hbox{ if } -\pi{\delta}/2 < z < \pi{\delta}/2  
0 \hbox{ otherwise }.  
\end{array}
\right.
\label{D1}
\end{eqnarray}
}


\newpage

{\samepage \clearpage \begin{equation}cu_z = (D(z)u_z)_z + D(z)u_{yy},
\label{mueqn}
\end{equation}
}


\newpage

{\samepage \clearpage $D(z)$
}


\newpage

{\samepage \clearpage $D(\phi^{(0)}(z))$
}


\newpage

{\samepage \clearpage \begin{picture}(200,100)(-50,-20)
\thicklines
\put(0,0){\line(1,0){200}}
\put(0,80){\line(1,0){200}}
\put(90,0){\line(0,1){80}}
\put(110,0){\line(0,1){80}}

\put(95,40){$\Omega_0$}
\put(0,40){$\Omega_-$}
\put(170,40){$\Omega_+$}
\put(85,85){$y = h$}
\put(80,-10){$y = -h$}
\put(115,40){$z > \pi{\delta}/2$}
\put(30,40){$z < -\pi{\delta}/2$}


\end{picture}
}


\newpage

{\samepage \clearpage $\Omega_-, \Omega_0, \Omega_+$
}


\newpage

{\samepage \clearpage $\Omega_-,~D = 0$
}


\newpage

{\samepage \clearpage $u_z=0$
}


\newpage

{\samepage \clearpage \begin{equation}u(z,y) = u_0 \hbox{ in } \Omega_-
\label{uinOm-}
\end{equation}
}


\newpage

{\samepage \clearpage $\Omega_+ $
}


\newpage

{\samepage \clearpage \begin{equation}u(z,y)=u(\pi{\delta}/2,y) \hbox{ in } \Omega_+
\label{uinOm+}
\end{equation}
}


\newpage

{\samepage \clearpage $u(\pi{\delta}/2,y)$
}


\newpage

{\samepage \clearpage $\Omega_0 = -\pi{\delta}/2<z<\pi{\delta}/2,-h<y<h\}$
}


\newpage

{\samepage \clearpage $\{z = \pm \pi{\delta}/2 \}$
}


\newpage

{\samepage \clearpage \begin{eqnarray}u = u* &  &~~~ (y=\pm h)
\nonumber  
u = u_0 & \hbox{~and~}u_z=0 & ~~~ (z=-\pi{\delta}/2)
\nonumber  
u \hbox{ continuous; } & u_z=0 & ~~~ (z=\pi{\delta}/2)
\label{bc0}
\end{eqnarray}
}


\newpage

{\samepage \clearpage $\h$
}


\newpage

{\samepage \clearpage \begin{equation}{\eta} ~{\equiv}~ \sqrt{\frac{{\delta} c}{D(0)}} \ll 1.
\label{eta}
\end{equation}
}


\newpage

{\samepage \clearpage \begin{equation}u(z,y) \approx u^{(0)}(z,y) =
u_0 + (u^* - u_0)f(z) \frac{\cosh y{\eta}/{\delta}}{\cosh h{\eta}/{\delta}},
\label{v}
\end{equation}
}


\newpage

{\samepage \clearpage $f(z)$
}


\newpage

{\samepage \clearpage \begin{eqnarray}f(z)       & \approx &  e^{-\h^2 \tan{|z|\over\delta}}
~~(-\pi{\delta}/2 \le z \le 0)
\nonumber  
\hbox{and} & \approx & 1~~~(0 \le z \le \pi{\delta}/2).
\label{fapprox}
\end{eqnarray}
}


\newpage

{\samepage \clearpage $h{\eta}/{\delta} \gg 1$
}


\newpage

{\samepage \clearpage $z \in (-\pi{\delta}/2,\pi{\delta}/2)$
}


\newpage

{\samepage \clearpage $u^{(0)}(z,y)$
}


\newpage

{\samepage \clearpage $y = \pm h$
}


\newpage

{\samepage \clearpage ${\delta}/\h$
}


\newpage

{\samepage \clearpage $h{\eta}/{\delta} \ll 1$
}


\newpage

{\samepage \clearpage \begin{equation}u(z,y)~{\approx}~u^* + \frac{\h^2}{\w^2}(u_0 - u^*)(h^2 - y^2).
\label{parab}
\end{equation}
}


\newpage

{\samepage \clearpage $u = u^{(0)}$
}


\newpage

{\samepage \clearpage $ = \d^{(0)}$
}


\newpage

{\samepage \clearpage $u = u^{(0)}$
}


\newpage

{\samepage \clearpage $\int_{-\infty}^{\infty} (\phi^{(0)}_z)^2 \,dz =
 \pi/2\w$
}


\newpage

{\samepage \clearpage \begin{eqnarray}c \approx \frac{2 \epsilon }{\pi \tau}
\int_{-\pi{\delta}/2}^{\pi{\delta}/2} \bar{g}(z) \cos{z\over \delta}\,dz,
\label{c}
\end{eqnarray}
}


\newpage

{\samepage \clearpage $\bar{g}(z)$
}


\newpage

{\samepage \clearpage $g(z,y)$
}


\newpage

{\samepage \clearpage $c
> 0$
}


\newpage

{\samepage \clearpage $g$
}


\stepcounter{section}
\newpage

{\samepage \clearpage $F_1$
}


\newpage

{\samepage \clearpage $w(\phi)$
}


\newpage

{\samepage \clearpage \begin{equation}p =  uw().
\label{Mey}
\end{equation}
}


\newpage

{\samepage \clearpage $g(u,\phi) = -\partial p/ \partial \phi = u\phi$
}


\newpage

{\samepage \clearpage $|\d| < 1$
}


\newpage

{\samepage \clearpage $ {\epsilon} > 0$
}


\newpage

{\samepage \clearpage $\epsilon A p$
}


\newpage

{\samepage \clearpage $\epsilon < 0$
}


\newpage

{\samepage \clearpage $\phi^{(0)}$
}


\newpage

{\samepage \clearpage $u^{(0)}$
}


\newpage

{\samepage \clearpage \begin{equation}c \approx {\epsilon {\delta} \h^2 \over 2 \tau}(u^*-u_0)
{\tanh { \sqrt{2/\pi} h {\eta}/{\delta}} \over  { \sqrt{2/\pi} h {\eta}/{\delta}}}.
\label{c3M}
\end{equation}
}


\newpage

{\samepage \clearpage $c \ge 0$
}


\newpage

{\samepage \clearpage $ u^*-u_0 $
}


\newpage

{\samepage \clearpage $(u^* -
u_0){\epsilon}\w^2/ D(0)$
}


\stepcounter{section}
\newpage

{\samepage \clearpage $\{z<-\pi{\delta}/2\}$
}


\newpage

{\samepage \clearpage $z {\rightarrow} -\B$
}


\newpage

{\samepage \clearpage $u_+(y)$
}


\newpage

{\samepage \clearpage $z {\rightarrow} +\B$
}


\newpage

{\samepage \clearpage $F$
}


\newpage

{\samepage \clearpage $\epsilon A P_-(u - u_0)^2$
}


\newpage

{\samepage \clearpage ${\epsilon} A P_-$
}


\newpage

{\samepage \clearpage $\epsilon A P_+(u
- u_+)^2$
}


\newpage

{\samepage \clearpage $F_{12}$
}


\newpage

{\samepage \clearpage $|\d| < \d_1$
}


\newpage

{\samepage \clearpage $\d_1 \in (0,1)$
}


\newpage

{\samepage \clearpage $\{z : (z) <
-\d_1~(> \d_1)\}$
}


\newpage

{\samepage \clearpage $\{z : | (z)| < \d_1\}$
}


\newpage

{\samepage \clearpage \begin{equation}p(z,y) =
P_-(u -u_0)^2 H(- -\d_1) + P_+(u - u_+)^2H( - \d_1),
\label{fa}
\end{equation}
}


\newpage

{\samepage \clearpage $H$
}


\newpage

{\samepage \clearpage \begin{equation}\bar{g}(z) = -\overline{\frac{{\delta} p}{{\delta} }} =
\mbox{ \boldmath $ \delta$}( + \d_1)P_-\overline{(u -u_0)^2} -
\mbox{ \boldmath $ \delta$} ( - \d_1)P_+\overline{(u -u_+)^2}
\label{fb}
\end{equation}
}


\newpage

{\samepage \clearpage $-h<y<h$
}


\newpage

{\samepage \clearpage $\d_1$
}


\newpage

{\samepage \clearpage \begin{equation}c \approx  \frac{2{\epsilon}{\delta}}{\pi \tau }P_-(u_0 - u^*)^2.
\label{c2p}
\end{equation}
}


\newpage

{\samepage \clearpage $(u^* - u_0)$
}


\stepcounter{section}
\stepcounter{section}

\end{document}