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\markboth{\hfil The limiting equation for Neumann Laplacians \hfil EJDE--2000/31}
{EJDE--2000/31\hfil Yoshimi Saito \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~31, pp.~1--25. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  The limiting equation for Neumann Laplacians on shrinking domains  
\thanks{ {\em Mathematics Subject Classifications:} 35J05, 35Q99.
\hfil\break\indent
{\em Key words and phrases:} Neumann Laplacian, tree, shrinking domains.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted  March 9, 2000. Published April 26, 2000.\hfil\break\indent
Partially supported by Deutche Forschungs Gemeinschaft grant SFB 359.} }
\date{}
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\author{ Yoshimi Saito }
\maketitle

\begin{abstract} 
  Let $\{\Omega_{\epsilon} \}_{0 < \epsilon \le1}$ be an indexed 
  family of connected open sets in ${\mathbb R}^2$, that shrinks 
  to a tree $\Gamma$ as $\epsilon$ approaches zero.
  Let $H_{\Omega_{\epsilon}}$ be the Neumann Laplacian 
  and $f_{\epsilon}$ be the restriction of an $L^2(\Omega_1)$ function 
  to $\Omega_{\epsilon} $. 
  For $z \in {\mathbb C}\backslash [0, \infty)$, set 
  $u_{\epsilon} = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon} $.
  Under the assumption that all the edges of $\Gamma$ are line segments,
  and some additional conditions on $\Omega_{\epsilon}$, 
  we show that the limit function $u_0 = \lim_{\epsilon\to 0} u_{\epsilon}$ 
  satisfies a second-order ordinary differential equation on $\Gamma$ with
  Kirchhoff boundary conditions on each vertex of $\Gamma $.
\end{abstract}  


\renewcommand{\theequation}{\thesection.\arabic{equation}}

  \section{Introduction}
  
 Let $\Omega$ be a  connected open set in ${\mathbb R}^2$. 
Consider a family of the Neumann Laplacians $H_{\Omega_{\epsilon}} $,
$0 < \epsilon \le 1 $, on the sub-domain $\Omega_{\epsilon}$ such
that $\{ \Omega_{\epsilon} \}$ shrinks to $\Gamma$ in the sense
that
\begin{eqnarray} \label{E:E1}
  &\Omega = \Omega_1 \supset \Omega_{\epsilon_2} \supset \Omega_{\epsilon_1}
  \quad (1 > \epsilon_2 > \epsilon_1 > 0), & \\
 & \lim_{\epsilon\to 0} \overline{\Omega_{\epsilon}} = \Gamma, &\nonumber
\end{eqnarray}
where the bar over a set means the closure of the set. 
We continue here the study started in \cite{SA} regarding the following
question:  In what sense does the operator
$H_{\Omega_{\epsilon}}$ converges to an operator on $\Gamma$ as
$\epsilon \to 0 $? That is, we try to find
conditions under which, given a thin domain and an
operator on the domain, an operator on an imbedded tree
or network gives a good approximation of the operator
on the domain. This investigation is part of the general question on 
replacing the study of a thin domain by the study of an imbedded tree or
network, which has been proposed in many branches of science
such as physics and chemistry. For references on this problem, 
see for example  Ruedenberg-Scherr \cite{RS}, Exner-Seba \cite{EXS}, 
Kuchment \cite{KC}, Schatzman \cite{SC},
Rubinstein-Schatzman \cite{RSC}, and Kuchment-Zeng \cite{KCZ}. 
In \cite{SC}, a family of ``fattened'' domains $\Omega_{\epsilon}$ of a $C^2$
manifold $M$ are considered. It is shown that the
$k$-th eigenvalue $\lambda_k(\epsilon)$ of the Neumann Laplacian
$H_{\Omega_{\epsilon}}$ on $\Omega_{\epsilon}$ converges asymptotically
to the $k$-th eigenvalue $\lambda_k$ of the Laplace-Beltrami
operator of $M$. In \cite{RSC} the above results are extended
to the case where the manifold $M$ is replaced by a graph
$G$; see for example \cite{KCZ}, and for a simplified proof \cite{RSC}.
  
   In \cite{SA}, we discussed the convergence of the resolvent
$(H_{\Omega_{\epsilon}} - z)^{-1}$ as $\epsilon \to 0 $. After introducing
the Hilbert spaces $L_2(\Gamma)$ and $H^1(\Gamma)$ on the tree
$\Gamma$ and defining the selfadjoint  ``Neumann
Laplacian'' operator $H_{\Gamma}$ in $L^2(\Gamma)$ as in \cite{ES}, we
presented a set of general conditions under which the
the resolvent $(H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon}$ converges as
as $\epsilon \to 0 $, where $f_{\epsilon}$ is the restriction of a function
$f \in L^2(\Omega)$ to $\Omega_{\epsilon}$ (\cite{SA}, Theorem 4.5).
Also in \cite{SA}, we studied the case
where $\Omega$ is a bounded convex set and the tree
$\Gamma$ is a straight line segment (ridge). 

Let $\Omega$ be a bounded convex set in ${\mathbb R}^2$ and suppose that 
$\Omega$ and $\Omega_{\epsilon}$ are given by
\begin{eqnarray}
& \Omega = \{ x = (x_1, x_2) : -\ell_-(x_1) < x_2 < \ell_+(x_1), \
  a < x_1 < b \}, &\nonumber \\
& \Omega_{\epsilon} = \{ x = (x_1, x_2) :
   -\epsilon\ell_-(x_1) < x_2 < \epsilon\ell_+(x_1), \
   a < x_1 < b \}, &\\
& \Gamma = \{ x = (x_1, 0) : a \le x_1 \le b \},& \nonumber
\end{eqnarray}
where $-\infty < a < b < \infty $, $0 < \epsilon \le 1$ and
$\ell_{\pm}(t)$ are positive $C^1$ functions on
$[a, b] $. Then we have, for  $f \in H^1(\Omega)$ and
$z \in {\mathbb C} \backslash [0, \infty) $,
\begin{equation}
 \lim_{\epsilon\to 0} \gamma\big[(H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon}\big]
  = (H_{\Gamma} - z)^{-1}(\gamma f),  %(1.3)
\end{equation}
in a weighted Hilbert space $L_{a_0}^2(\Gamma) $,
where $H_{\Gamma}$ is the ``Neumann Laplacian'' on
$\Gamma$ defined in \cite{ES} (see \S2), $\gamma$ is the trace
operator on $\Gamma$, and
\begin{eqnarray}
  &L_{a_0}^2(\Gamma) = L^2(\Gamma; a_0(\sigma)d\sigma),& \\
  &a_0(\sigma) = \ell_{-}(\sigma) + \ell_{+}(\sigma) &\nonumber %(1.4)
\end{eqnarray}
(\cite{SA}, Theorem 5.5).
  
   In this work, we consider the case when $\Gamma$ is a tree
such that all the edges are line segments (Assumption 4.2, (i)).
Suppose that the family $\{ \Omega_{\epsilon} \}$ is given by
\begin{equation}
  \Omega_{\epsilon} = \{ (\sigma, s) :
   -\epsilon\ell_-(\sigma) < s < \epsilon\ell_+(\sigma), \
   \sigma \in \Gamma \},   %(1.5)
\end{equation}
where $\sigma$ is the arc length along the edges of $\Gamma$
and $s$ is the arc length along the curve
$C_{\sigma} = \tau^{-1}(\sigma) $, $\tau$ being a map from
$\Omega$ into $\Omega \cap \Gamma$ which is Lipschitz continuous
almost everywhere in $\Omega$ (see \S2). Then we assume
that, for $\sigma$ belonging to the edge $e_j$ of $\Gamma $,
the curve $C_{\sigma}$ is perpendicular to the edge near
$e_j$ except its vertices (Assumption 4.2, (ii)). Set
\begin{equation}
  u_{\epsilon}(x) = u_{\epsilon}(\sigma, s) = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon},
  %(1.6)
\end{equation}
where $f \in H^1(\Omega)\cap C^1(\Omega) $. Then there exists a
subsequence $\{ u_{\epsilon_k} \}_{k=1}^{\infty}$ such that
$\{ u_{\epsilon_k}(\sigma, 0) \} $, the restriction of
$u_{\epsilon_k}$ on the tree $\Gamma $, converges to $u_0$
weakly in $L_{a_0}^2(\Gamma)$ as $k \to \infty $, and
$u_0$ satisfies the equation
\begin{equation}
  - a_0^{-1}(\sigma)\frac{d}{d\sigma}(a_0(\sigma)u') - zu = f(\sigma, 0)
  %(1.7)
\end{equation}
on each edge with the Kirchhoff boundary condition at each
vertex (Theorems 4.5 and 4.8),  where $a_0$ is given by
(1.4) and $u'$ means the derivative of $u_0$ with
respect to the arc length $\sigma$ along the edge.
  
   In \S2, after introducing the tree $\Gamma$ imbedded in the
open connected set $\Omega $, we discuss the change of variables
$x = (x_1, x_2) \to (\sigma, s) $. In \S3 some estimates of
$u_{\epsilon}(\sigma, 0)$ are given. These estimates will be used to
guarantee the weak convergence of
$\{ u_{\epsilon_k} \}_{k=1}^{\infty} $. \S4 is devoted to
showing the above convergence of $\{ u_{\epsilon_k} \}_{k=1}^{\infty}$
to a solution $u_0$ of the equation (1.6) (Theorems 4.5 and 4.8).
The main tools are Lemmas 4.1 and 4.4 whose proof will be given
in \S6. We shall discuss the continuity of the limiting
function $u_0$ at each vertex in \S5.
  
\section{ Preliminaries }  %Section 2
     In this section we are going to introduce a domain $\Omega$ in
${\mathbb R}^2 $, a tree $\Gamma$ contained in $\Omega$ and a family
$\{ \Omega_{\epsilon} \}_{0 < \epsilon \le 1}$ of sub-domains of $\Omega $.
  
   Let $\Omega$ be a domain (i.e., a connected open set) in
${\mathbb R}^2 $. Let $\Gamma \subset \overline{\Omega}$ be a tree, that is,
a connected graph without loops or cycles, where $\overline{\Gamma}$ is
the closure of $\Gamma $. Its edges $e_j $, $j \in J $, are
non-degenerate open curve such that the closure $\overline{e_j}$
is a smooth curve, where $J$ is an index set. The endpoints
$\overline{e_j} \backslash e_j$ are the vertices. Here we should note that we
allow these edges to be smooth curves, not just line segments.
We shall assume that $\Gamma$ has, at most, a countably infinite number
of edges, and hence the index set $J$ is a subset of the natural
numbers $\bf N $. We also assume that each vertex of $\Gamma$ is
of {\it finite degree}, that is, only a finite number of edges emanate
from each vertex, and that only one edge emanates from a vertex
$c$ if $c$ belongs to the boundary $\partial\Omega$ of $\Omega $.
For every $x, y \in \Gamma$ there is a unique path in $\Gamma$ joining
$x$ and $y $. Thus, by introducing the distance between $x$
and $y$ by the length of a unique path connecting $x$ and $y $,
$\Gamma$ becomes a metric space. Also, if $\Gamma$ is endowed with the
natural one-dimensional Lebesgue measure, it is a $\sigma$-finite
measure space. The tree $\Gamma$ is rooted at an arbitrary fixed point
$a \in \Gamma $. We define $t \succeq_a x$ (or equivalently
$x \preceq_a t $) to mean that $x$ lies on the path from $a$
to $t $.
  
Throughout this work we assume the following:
  \noindent {\bf(I)} Assumptions on $\Omega$ and $\Gamma $: 
\begin{description}
  \item{(1-i)} $\Omega$ be a domain (i.e., a  connected open set) in
${\mathbb R}^2$ and $\Gamma \subset \overline{\Omega}$ be a connected
tree which has at most countable number of edges $e_j $, $j \in J $,
where $\overline{\Omega}$ is the closure of $\Omega $. Each
edge $e_j$ is an open curve with finite length such that the
closure $\overline{e_j}$ is a $C^2$ curve. The endpoints
$\overline{e_j} \backslash e_j$ are called the vertices. When any
two edges are connected, they are connected only at their vertices.
Also they are not tangential at the vertex from which the two edges
emanate.
  
\item{(1-ii)} We have $E(\Gamma) \subset \Omega $, where $E(\Gamma)$ is the set
of all edges of $\Gamma $.
  
\item{(1-iii)} For $v \in V(\Gamma) \cap \partial\Omega $, only one edge emanates
from $v $, where $V(\Gamma)$ is the set of all vertices of $\Gamma $,
and $\partial\Omega$ is the boundary of $\Omega $.
\end{description}
  \noindent{\bf (II)} Assumptions on $\tau $. There exists a map $\tau$ from
$\Omega$ into $\Omega \cap \Gamma$ which satisfies the following:
\begin{description}
  \item{(2-i)} The subset $\tau^{-1}(V(\Gamma))$ is a (2-dimensional)
null set. For each $e_j \in E(\Gamma) $, set
$\Omega_j = \tau^{-1}(e_j) $. Then $\Omega_j$ is an open set and
$\tau$ is locally Lipschitz continuous on $\Omega_j $, that is,
for each $x \in \Omega_j$ there exists a neighborhood
$V(x) \subset \Omega_j$ of $x$ and a positive constant
$\gamma(x)$ such that for all $y \in V(x)$
\setcounter{equation}{0}
\begin{equation}
  d_{\Gamma}(\tau(x), \tau(y)) \le \gamma(x)|x - y|,   %(2.1)
\end{equation}
where $d_{\Gamma}$ denotes the metric on $\Gamma$ and $| \cdot |$
the Euclidean metric (for definiteness) on ${\mathbb R}^2 $.
  
\item{(2-ii)} Let $C(t) = \tau^{-1}(t)$ for $t \in E(\Gamma) $.
Then $C(t)$ is a rectifiable curve. Further,
$C(t) \cap \Gamma = \{ t \}$ and $C(t) \backslash \{ t \}$ has two
components, $C_{\pm}(t)$ say. Also we assume that $C(t)$
is not tangential to $\Gamma$ at $t $. Let $C_{+}(t)$ and
$C_-(t)$ be parameterized by arc length $s$ which
is measured from $t$ with $0 \le s \le \ell_+(t)$
on $C_+(t)$ and $-\ell_-(t) \le s \le 0$ on $C_-(t) $.
Let $\tau(x) = (\tau_1(x), \tau_2(x)) \in \Gamma$ for
$x \in \Omega $. Then, for $t \in E(\Gamma) $, there exists a null set
$e(t) \subset C(t)$ with respect to $ds $, the measure
induced by the arc length parameter $s$ on $C(t) $, such
that $\tau_1$ and $\tau_2$ are differentiable at
$x \in C(t) \backslash e(t) $.
  
\item{(2-iii)} Let
$|\nabla\tau(x)| = [|\nabla\tau_1(x)|^2 + |\nabla\tau_2(x)|^2]^{1/2} $.
For $t \in E(\Gamma)$ fixed, define $|\nabla\tau(s)|$ on
$C(t)$ by $|\nabla\tau(s)| = |\nabla\tau(x)|$ with $x \in C(t)$
and $d_{C(t)}(t, x) = s $, where $d_{C(t)}(t, x)$ is the distance
between $t$ and $x$ along $C(t) $. Then
$|\nabla\tau(s)|, |\nabla\tau(s)|^{-1} \in L^1(C(t), ds) $.
  
\item{(2-iv)} For any vertex $v \in \Omega$ the functions
$\ell_{\pm}$ are bounded below from $0$ around
$v $, i.e., for a vertex $v \in \Omega $, there exists a
neighborhood $U(v) \subset \Gamma$ of $v$ such that
\begin{eqnarray}
   &\inf_{t\in U(v)\backslash \{ v \}} \ell_-(t) > 0,& \\
   &\inf_{t\in U(v)\backslash \{ v \}} \ell_+(t) > 0\,.& \nonumber %2.2
\end{eqnarray}
\end{description}
     Some examples of the triples $(\Omega, \Gamma, \tau)$
are given in \cite{EHA, EHB, ES, SA} including horn-shaped
domains, room and passages domains and fractal domains.
  
   For $j \in J$ let the edge $e_j$ have
the vertices $a_j$ and $b_j$ such that
$b_j \succeq_a a_j $, where the tree $\Gamma$ is rooted at
$a $. Then we parameterize $e_j$ by
$\sigma_j(t) = \mathop{dist}(a_j, t) $, where
$\mathop{dist}(a_j, t)$ is the arc length from $a_j$ to
$t \in e_j$ along $e_j $. From now on we may drop the
subscript $j$ in $\sigma_j$ if there is no danger of
misunderstanding. If
$x = (x_1, x_2) \in \Omega_j = \tau^{-1}(e_j) \subset \Omega$ is
such that $\tau(x) = t(\sigma)$ and $\mathop{dist}(t(\sigma), x) = s $,
where $\mathop{dist}(t(\sigma), x)$ is the distance between $x$
and $t(\sigma)$ along the curve $C_{t(\sigma)} $, then a co-ordinate
system on $\Omega_j$ is defined by
\begin{equation}
  x = x(\sigma, s), \quad \tau(x) = t(\sigma), \quad
  s \in (-\ell_-(\sigma), \ \ell_+(\sigma)),   %(2.3)
\end{equation}
where $\ell_{\pm}(\sigma) = \ell_{\pm}(t(\sigma)) $.
  A family $\{ \Omega_{\epsilon} \}_{0 < \epsilon \le 1}$ of sub-domains of
$\Omega$ is defined as follows:
  \paragraph{Definition} For $j \in J$ and
$0 < \epsilon \le 1$ let
\begin{equation}
  \Omega_j^{(\epsilon)} =
  \{ x = x(\sigma, s) \ / \ \sigma \in e_j, \
  - \epsilon\ell_{-}(\sigma) < s < \epsilon\ell_{+}(\sigma) \}   %(2.4)
\end{equation}
and
\begin{equation}
 \Omega_{\epsilon} = \big[\cup_{j\in J}
  \overline{\Omega_j^{(\epsilon)}}\big]^{\circ}, %(2.5)
\end{equation}
where $A^{\circ}$ is the interior of $A $. By definition
we have $\Omega = \Omega_1 $.
  
\paragraph{Definition} For each $0 < \epsilon \le 1$ let
$H_{\Omega_{\epsilon}}$ be the Neumann Laplacian on $\Omega_{\epsilon} $.
  
   It is known ((2.4) in \cite{ES}) that
\begin{equation}
   \frac{\partial(x_1, x_2)}{\partial(\sigma, s)} = \frac1{|\nabla\tau(\sigma, s)|}.
  %(2.6)
\end{equation}
Let $I \in e_j $. Then we have, for $f \in L^1(\tau^{-1}(I)) $,
\begin{equation}
  \int_{\tau^{-1}(I)} f(x) dx
   = \int_{I}\,d\sigma
   \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  f(\sigma, s)|\nabla\tau(\sigma,  s)|^{-1} \, ds. \label{E:inta}
\end{equation}   %(2.7)
Note that we have again simplified the notation by writing
$(\sigma, s)$ for $x(\sigma, s) $. Of particular importance is
the case when $f = F\circ\tau$ in (2.7) with
$F \in L^1(I) $:
\begin{eqnarray}
  {\int_{\tau^{-1}(I)} F\circ\tau(x) \, dx } 
  &=& {\int_{I} F(\sigma)\, d\sigma
   \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  |\nabla\tau(\sigma,  s)|^{-1} \, ds } \\
&=:& {\int_{I} F(\sigma)\alpha_{\epsilon}(\sigma)\, d\sigma, }\nonumber
\end{eqnarray}   %(2.8)
where
\begin{equation}
  \alpha_{\epsilon}(\sigma) := 
 \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \frac1{|\nabla\tau(\sigma, s)|}\, ds.
\end{equation}
If $I = e_j$ in (2.7) and (2.8), then
$\tau^{-1}(I)$ should be replaced by $\Omega_j^{(\epsilon)} $.
  
\section{Evaluation of $u_{\epsilon} = (H_{\Omega_{\epsilon}} - z)^{-1}f$ 
on $\Gamma $}   %Section 3
     We shall start with an additional assumption on the tree
$\Gamma$ and the family $\{ \Omega_{\epsilon} \}_{0 < \epsilon \le 1} $.
Then we shall show some evaluation for the restriction of
\setcounter{equation}{0}
\begin{equation}
   u_{\epsilon} = u_{\epsilon}(f, z)) = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon}  %(3.1)
\end{equation}
on $\Gamma $, where $z \in {\bf C} \backslash [0, \infty)$ and $f_{\epsilon}$
is the restriction of $f \in L^2(\Omega)$ on $\Omega_{\epsilon} $.
  
\paragraph{\bf Assumption 3.1.} (i) For each $j \in J$
$\ell_{\pm}(\sigma)$ are positive $C^1$ function on $e_j$ and
are continuously extended on $\overline{e_j} $. Also
$\ell_{\pm}(\sigma)$ satisfy
\begin{eqnarray}
& \sup_{e_j} (\ell_{-}(\sigma) + \ell_{+}(\sigma)) \equiv L_j < \infty,& \\
& \sup_{e_j} (|\ell_{-}'(\sigma)| + |\ell_{+}'(\sigma)|) \equiv R_j
   < \infty, &\nonumber  %(3.2)
\end{eqnarray}
where ${\ell_{\pm}'(\sigma) = \frac{d}{d\sigma}\ell_{\pm}(\sigma) } $.
  
   (ii) For $j \in J$ there exists $\epsilon_j \in (0, 1]$
such that $|\nabla\tau(\sigma,  s)|$ is continuous on
$\Omega_j^{(\epsilon_j)} $,
\begin{eqnarray}
& 0 < m_j \equiv \inf_{x(\sigma, s)\in\Omega_j^{(\epsilon_j)}}
  |\nabla\tau(\sigma, s)|
   \le \sup_{x(\sigma, s)\in\Omega_j^{(\epsilon_j)}} |\nabla\tau(\sigma, s)|
  \equiv M_j < \infty,&  \\
&\sup_{x(\sigma, s)\in\Omega_j^{(\epsilon_j)}}
  \big|{\frac{\partial x(\sigma, s)}{\partial s} } \big|
   \equiv K_j < \infty.&\nonumber %(3.3)
\end{eqnarray}
  
   Now we introduce a positive function on each
$e_j$ which will play an important role.
  
\paragraph{\bf Definition 3.2.} For each $j \in J $, set
\begin{equation}
  a_0^{(j)}(\sigma) = \frac{\ell_{-}(\sigma) + \ell_{+}(\sigma)}
   {|\nabla\tau(\sigma, 0)|}
  \quad (\sigma \in e_j).  %(3.4)
\end{equation}
  
   Note that $a_0^{(j)}$ is a bounded positive function
on $e_j $. From now on we may drop the superscript
$j$ in $ a_0^{(j)}$ if there is no risk of
misunderstanding, i.e. $a_0(\sigma) = a_0^{(j)}(\sigma) $.
Also note that
\begin{equation}
   a_0(\sigma) = \lim_{\epsilon\to 0} \epsilon^{-1}\alpha_{\epsilon}(\sigma)
  \quad (\sigma \in e_j),   %(3.5)
\end{equation}
where $\alpha_{\epsilon}$ is given by (2.9).
  
   For a subset $\Omega'$ of $\Omega $,
$\| f \|_{\Omega'}$
denotes the $L^2$ norm of $f$ on $\Omega' $. Also
we set
\begin{eqnarray}
& \|\psi \|_{e_j, a_o}^2
  = {\int_{e_j} |\psi(\sigma)|^2a_0(\sigma) \, d\sigma, } &\\
& \|\psi \|_{e_j}^2 = {\int_{e_j} |\psi(\sigma)|^2 \, d\sigma, 
}&\nonumber  %3.6
\end{eqnarray}
  
\paragraph{Lemma 3.3.} {\it We have
\begin{equation}
  \| u(\cdot, 0) \|_{e_j, a_0}^2
  \le 2\big(\frac{M_j}{m_j}\big)
  \big\{\epsilon L_j^2K_j^2\|\nabla u \|_{\Omega_j^{(\epsilon)}}^2
  + \epsilon^{-1}\| u \|_{\Omega_j^{(\epsilon)}}^2\big\}, %3.7
\end{equation}
and
\begin{equation}
\| u \|_{\Omega_j^{(\epsilon)}}^2 \le 2\epsilon \big(\frac{M_j}{m_j}\big)
   \big\{\epsilon L_j^2K_j^2\|\nabla u \|_{\Omega_j^{(\epsilon)}}^2
  + \| u(\cdot, 0) \|_{e_j, a_0}^2\big\}  %3.8
\end{equation}
for $u \in H^1(\Omega_j^{(\epsilon)})\cap C^1(\Omega_j^{(\epsilon)}) $,
where $\|\ \|_A $, $A \subset {\mathbb R}^2 $, is the norm
of $L^2(A) $.}
  
  {\bf Proof.} (I) Let $I$ be a closed subset of $e_j $.
Since $I \subset e_j \subset \Omega_j^{(\epsilon)} $, $u_{\epsilon}(\sigma, 0)$
and $u_{\epsilon}'(\sigma, 0)$ are bounded on $I $. Then we have
\begin{eqnarray}
\| u(\cdot, 0) \|_{I, a_0}^2  
   &=& \int_{I} \epsilon^{-1}|\nabla\tau(\sigma, 0)|^{-1}
   \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   |u(\sigma, 0)|^2 \, ds\,d\sigma  \nonumber\\
 &\leq& {2\epsilon^{-1}m_j^{-1}\int_{I}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   |u(\sigma, 0) - u(\sigma, s)|^2 \, ds\,d\sigma } \nonumber  \\
&& {+ 2\epsilon^{-1}m_j^{-1}\int_{I}
   \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   |u(\sigma, s)|^2 \, ds\,d\sigma } \\
&\equiv& {2\epsilon^{-1}m_j^{-1}(I_1 + I_2). }\nonumber
\end{eqnarray}   %3.9
Using the second inequality of (3.3), wee see that
\begin{equation}
  \big|\frac{\partial u}{\partial s}\big| \le K_j|\nabla u|,  %3.10
\end{equation}
which is combined with (2.7) to give
\begin{eqnarray}
I_1 &\le& { \int_{I}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)} 
\big|\int_0^s \big|\frac{\partial u}{\partial s}(\sigma, \eta)\big|\, 
d\eta\big|^2 \,ds\,d\sigma} \nonumber \\
&\leq& { \int_{I}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \big(\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \big|\frac{\partial u}{\partial s}(\sigma, \eta)\big|\,
   d\eta\big)^2 \, ds\,d\sigma} \nonumber\\
&\leq& {L_j\epsilon \int_{I} \big(\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \big|\frac{\partial u}{\partial s}(\sigma, s)\big|\,
   ds\big)^2 \, d\sigma} \\
&\leq& {(L_j\epsilon)^2 \int_{I} \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \big|\frac{\partial u}{\partial s}(\sigma, s)\big|^2 \, ds\,d\sigma} \nonumber\\
&\leq& {(L_j\epsilon)^2K_j^2M_j \int_{I} \int_{-\epsilon\ell_{-}
(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  |\nabla u(\sigma, s)|^2\frac{ds\,d\sigma}{|\nabla\tau(\sigma,  s)|} } \nonumber\\
&\leq& {(L_j\epsilon)^2K_j^2M_j \|\nabla u \|_{\Omega_j^{(\epsilon)}}^2, }
\nonumber
\end{eqnarray}  %3.11
where we have used the fact that $\tau^{-1}(I) \subset \Omega_j^{(\epsilon)} $.
As for $I_2$ we have
\begin{equation}
  I_2 \le M_j \int_{I}
 \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   |u(\sigma, s)|^2\frac{ds\,d\sigma}{|\nabla\tau(\sigma,  s)|}
   \le M_j\| u \|_{\Omega_j^{(\epsilon)}}^2.  %3.12
\end{equation}
Thus we have from (3.11) and (3.12)
\begin{equation}
 \| u(\cdot, 0) \|_{I, a_0}^2
  \le 2\big(\frac{M_j}{m_j}\big)
   \big\{L_j^2K_j^2\epsilon\|\nabla u \|_{\Omega_j^{(\epsilon)}}^2
   + \epsilon^{-1}\| u \|_{\Omega_j^{(\epsilon)}}^2\big\}.
  %3.13
\end{equation}
Since $I \subset e_j$ is arbitrary, (3.7) follows from (3.13).
  
   (II) As in (I), we have
\begin{eqnarray}
  \| u \|_{\Omega_j^{(\epsilon)}}^2 
&\leq& { 2\int_{e_j}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)} |u(\sigma, s) - u(\sigma, 0)|^2
   |\nabla\tau(\sigma,  s)|^{-1}\, ds\,d\sigma } \nonumber\\
&& { + 2\int_{e_j}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  |u(\sigma, 0)|^2|\nabla\tau(\sigma,  s)|^{-1}\, ds\,d\sigma }  \\
& \equiv& 2(J_1 + J_2). \nonumber
\end{eqnarray}   %3.14
Then we can proceed as in evaluation $I_j$ to obtain
\begin{eqnarray}
  J_1 &\le& \big(\frac{M_j}{m_j}\big)L_j^2K_j^2\epsilon^2
  \|\nabla u \|_{\Omega_j^{(\epsilon)}}^2\,,  \\ %3.15
 J_2 &\le& \big(\frac{M_j}{m_j}\big)\epsilon
   \| u(\cdot, 0) \|_{e_j, a_0}^2, %3.16
\end{eqnarray}
which completes the proof. \hfil$\diamondsuit$
  
\paragraph{Proposition 3.4.} {\it Suppose that  Assumptions
{\rm 2.1} and {\rm 3.1} hold. Let $f \in H^1(\Omega) $. Set
\begin{equation}
  u_{\epsilon} = u_{\epsilon}(f_{\epsilon}, z) = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon},
  %3.17
\end{equation}
where $z \in {\mathbb C} \backslash [0, \infty)$ and $f_{\epsilon}$ is the
restriction of $f$ on $\Omega_j^{(\epsilon)}$.  Suppose that
\begin{equation}
 \limsup_{\epsilon\to 0} \sum_{k\in J}
   \big(\frac{M_k}{m_k}\big)\big\{\epsilon L_k^2K_k^2
   \| \nabla f \|_{\Omega_k^{\epsilon}}^2
   + \| f(\cdot, 0) \|_{e_k, a_0}^2 \big\} < \infty,
   %3.18
\end{equation}
where $f(\sigma, 0)$ on each edge $e_j$ is given by the trace
of $f$ on $e_j $, Then, for sufficiently small
$\epsilon \in (0, 1]$ and $j \in J $,}
\begin{eqnarray}
\| u_{\epsilon}(\cdot, 0) \|_{e_j, a_0}^2 
 &\le& 4\big(\frac{M_j}{m_j}\big)|z|^{-1}
   \Big[L_j^2K_j^2\| f \|_{\Omega_{\epsilon}}^2  \\
&&+ |z|^{-1}\sum_{k\in J}
   \big(\frac{M_k}{m_k}\big)\big\{\epsilon L_k^2K_k^2
   \| \nabla f \|_{\Omega_k^{\epsilon}}^2
  + \| f(\cdot, 0) \|_{e_k, a_0}^2 \big\}\Big]. \nonumber
\end{eqnarray}  %3.19
  
\paragraph{Remark 3.5} It has been
known (see,e.g., Gilbarg-Trudinger
\cite{GT}, Theorem 8.10) that the condition $f \in H^1(\Omega)$ implies
$u_{\epsilon} \in H^3(\Omega)_{\rm{loc}} $, since $u_{\epsilon} \in H^1(\Omega) $.
Then, by the Sobolev imbedding theorem (see, e.g. Adams \cite{AD},
Theorem 5.4), we have $u_{\epsilon} \in C^1(\Omega) $.
  
\paragraph{Proof of Proposition 3.4.} (I) It is easy to see that
\begin{eqnarray} 
& {\| u_{\epsilon} \|_{\Omega_j^{(\epsilon)}}^2
   \le \| u_{\epsilon} \|_{\Omega_{\epsilon}}^2
   \le |z|^{-2}\| f \|_{\Omega_{\epsilon}}^2, } & \\
& {\| \nabla u_{\epsilon} \|_{\Omega_j^{(\epsilon)}}^2
   \le \| \nabla u_{\epsilon} \|_{\Omega_{\epsilon}}^2
   \le |z|\| u \|_{\Omega_{\epsilon}}^2
   + \| f \|_{\Omega_{\epsilon}}
  \| u_{\epsilon} \|_{\Omega_{\epsilon}}
  \le 2|z|^{-1}\| f \|_{\Omega_{\epsilon}}^2. }&\nonumber
\end{eqnarray} %3.20
     (II) It follows from (3.20) and (3.7) with $u$ replaced by
$u_{\epsilon}$ that
\begin{equation}
   \| u_{\epsilon}(\cdot, 0) \|_{e_j, a_0}^2
  \le 2\big(\frac{M_j}{m_j}\big)
  \big\{2\epsilon L_j^2K_j^2|z|^{-1}\| f \|_{\Omega_j^{(\epsilon)}}^2
   + \epsilon^{-1}|z|^{-2}
   \| f \|_{\Omega_j^{(\epsilon)}}^2\big\}. %3.21
\end{equation}
The inequality (3.19) is obtained from (3.21) and (3.8) with
$u$ replaced by $f $. \hfil$\diamondsuit$
  \section{The limiting equation} %Section 4
  Let $u_{\epsilon} = (H_{\Omega_{\epsilon}} - z)^{-1}f$ be as in (3.17). 
Since
$\| u_{\epsilon}(\cdot, 0)\|_{e_j, a_0}$ is uniformly bounded for
$\epsilon \in (0, 1]$ by Proposition 3.4 if $f \in H^1(\Omega)$
satisfies (3.18), $u_{\epsilon}$ converges
weakly along some subsequence $\{ \epsilon_m \}_{m=1}^{\infty}$
with the limiting function $u_0 $. In this section we shall
prove that $u_0$ is a solution of a second-order ordinary
differential equation on $\Gamma$ with the Kirchhoff boundary
condition on each vertex (Theorems 4.5 and 4.8). The equation
is independent from choice of the subsequence
$\{ \epsilon_m \}_{m=1}^{\infty}$.
   First we shall state two lemmas (Lemmas 4.1 and 4.4) which will
play crucial roles in this section. These lemmas will be shown in
\S6. In order to prove Lemma 4.4, we need another important
assumption (Assumption 4.2).
     Let $\Gamma_0$ be a measurable subset of the tree $\Gamma$
and let $a$ be a positive measurable function defined on
$\Gamma_0 \cap \cup_{j\in J} e_j $. Then the Hilbert space
$L^2(\Gamma_0, a)$ is a weighted $L^2$ space with inner product
\setcounter{equation}{0} \begin{equation}
 { (F, \ G)_{\Gamma_0,a} =
   \sum_{j\in J} \int_{\Gamma_0 \cap e_j}
   F(\sigma)\overline{G(\sigma)} \, a(\sigma)d\sigma }   %(4.1)
\end{equation}
and norm $\| F \|_{\Gamma_0} = [(F, \ F)_{\Gamma_0,a}]^{1/2} $.
We denote $L^2(\Gamma_0, 1)$ by $L^2(\Gamma_0) $.
  \paragraph{Lemma 4.1.} {\it Suppose that {\rm Assumptions 2.1} and
{\rm 3.1} are satisfied. Let $j \in J$ and let 
$\{ u_{\epsilon} \}_{0<\epsilon\le1}$ be a family of functions such that
\begin{equation}
  u_{\epsilon} \in H^1(\Omega_j^{(\epsilon)})\bigcap C^1(\Omega_j^{(\epsilon)})
  \quad (0 < \epsilon \le 1).  %4.2
\end{equation}
Let $F \in L^2(e_j, a_0) $, i.e., $\| F \|_{e_j, a_0} < \infty $.
Set $v(x) = F(\tau(x)) $. Then there exists
$C_j = C_j(M_j, m_j, L_j, K_j) $, a positive constant depending
only on $M_j, m_j, L_j, K_j $, such that
\begin{eqnarray}
\lefteqn{\big|{\frac1{\epsilon} }  {\int_{\Omega_j^{(\epsilon)}}
  u_{\epsilon}(x)\overline{v(x)} \, dx
  - \int_{e_j} u_{\epsilon}(\sigma, 0)\overline{F(\sigma)}a_0(\sigma) \,
   d\sigma \big| } }\\
&\leq & {C_j\big\{\sqrt{\epsilon}
   \| \nabla u_{\epsilon} \|_{\Omega_j^{(\epsilon)}} \| F \|_{e_j, a_0} }   
  {+ \| u_{\epsilon}(\cdot, s) \|_{e_j, a_0}
 \big[ \int_{e_j} |F(\sigma)|^2 \psi_{\epsilon}(\sigma)^2a_0(\sigma)
\, d\sigma   \big]^{1/2} \big\},} \nonumber
\end{eqnarray}   %4.3
where}
\begin{equation}
   \psi_{\epsilon}(\sigma) = \frac1{\epsilon a_0(\sigma)}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
 (|\nabla\tau(\sigma,  s)|^{-1} - |\nabla\tau(\sigma, 0)|^{-1}) \, ds. %4.4
\end{equation}
  
Here we need another assumption.
  \paragraph{Assumption 4.2.}  (i) All the edges $e_j$ of the
tree $\Gamma$ are finite (non-degenerate) {\it line segments.}
   (ii) Let $j \in J $. Let $E$ be a closed subset of the
(open) edge $e_j $. Then there exists a positive number
$\epsilon_j(E) \in (0, 1] $, depending only on $j$ and $E $,
such that, for any $t \in E$, the portion of the curve
$C_t\cap\Omega_j^{(\epsilon_j(E))}$ is a line segment which
is {\it perpendicular} to $e_j $.
  \paragraph{Remark 4.3.} (i) Roughly speaking, (ii) of the above
assumption claims that the curve $C_t$ is
perpendicular to $e_j$ near $e_j$ except the vertices.
    (ii) Assumption 4.2, (ii) also implies
\begin{eqnarray} 
 & {|\nabla\tau(\sigma,  s)| = 1 \quad
 ((\sigma, s) \in \tau^{-1}(E)\cap\Omega_j^{(\epsilon_j(E))}), }&\nonumber \\
 &{|\nabla\tau(\sigma, 0)| = 1 \quad (\sigma \in e_j),  } &  \\
 &{a_0(\sigma) = \ell_{-}(\sigma) + \ell_{+}(\sigma) \quad 
(\sigma \in e_j). } &\nonumber %4.5
\end{eqnarray} 
  \paragraph{Lemma 4.4.} {\it Suppose that {\rm Assumptions 2,1, 3.1}
and {\rm 4.2} hold. Let $j \in J$ and let
$\{ u_{\epsilon} \}_{0<\epsilon\le1}$ be a family of functions such that
\begin{equation}
  u_{\epsilon} \in H^1(\Omega_j^{(\epsilon)})\bigcap C^1(\Omega_j^{(\epsilon)})
   \quad (0 < \epsilon \le 1).   %4.6
\end{equation}
Let $F \in C^2(e_j)$ with
$F' \in C_0^1(e_j) $, where $F'$ is the derivative of $F$
with respect to $\sigma $. Let $\epsilon_0 = \epsilon_j(\mathop{supp} F') $.
Then, by setting $v(x) = F(\tau(x)) $, the inequality
\begin{eqnarray}
\lefteqn{\big| \frac1{\epsilon} \int_{\Omega_j^{(\epsilon)}} 
   \nabla u_{\epsilon} \cdot \overline{\nabla v} \, dx
  - \int_{e_j} \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, 0)
   \overline{F'(\sigma)}a_0(\sigma) \, d\sigma \big| } \nonumber   \\
 &\le& \sqrt{\epsilon} C(L_j, R_j) (\| F' \|_{e_j, a_0}
  + \| F'' \|_{e_j, a_0})
  \| \nabla u_{\epsilon} \|_{\Omega_j^{(\epsilon)}}
\end{eqnarray} %4.7
holds for $\epsilon \in (0, \epsilon_0) $, where $C(L_j, R_j)$
is a positive constant depending only on $L_j$ and
$R_j $.}
  
\paragraph{Theorem 4.5.} {\it Suppose that Assumptions
{\rm 2,1,3.1} and {\rm 4.2} hold. Let $f \in H^1(\Omega)$ which
satisfies} (3.17). {\it Let $f_{\epsilon}$ be the restriction of
$f$ on $\Omega_j^{(\epsilon)}$ for $\epsilon \in (0, 1] $. Let
$u_{\epsilon} = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon}$ be as in} (3.16).
{\it Let $j \in J $. Let $\{ \epsilon_m \}_{m=1}^{\infty} \subset (0, 1]$
be a decreasing sequence such that $\{ \epsilon_m \}_{m=1}^{\infty}$
converges to $0$ and the sequence
$\{ u_{\epsilon_m}(\cdot, 0) \}_{m=1}^{\infty}$ converges weakly in
$L^2(e_j, a_0) $. Then the limit function $u_0$ satisfies
\begin{equation}
 \int_{e_j} u_0(\sigma) \big\{-(a_0(\sigma)\overline{F'(\sigma)})'
  - z\overline{F(\sigma)}a_0(\sigma)
   - f(\sigma, 0)\overline{F(\sigma)}a_0(\sigma)\big\} \,d\sigma = 0  %4.8
\end{equation}
for any $F \in C_0^2(e_j) $, i.e., $u_0$ is a weak solution
of the equation
\begin{equation}
   - \frac{1}{a_0}(a_0u')' - zu = f(\cdot, 0)  %4.9
\end{equation}
on $e_j $.}
  \paragraph{Remark 4.6.} (i) The sequence
$\{ \epsilon_m \}_{m=1}^{\infty}$ which satisfies the conditions
in the above theorem does exist since
$\| u_{\epsilon}(\cdot, 0) \|_{e_j, a_0}$ is
uniformly bounded for $\epsilon \in (0, 1]$ by Proposition 3.4.
     (ii) Thus, the limit function $u_0$ is, not only a weak
solution of (4.9), but also a strong solution with
$u_0 \in C^2(e_j) $.
  \paragraph{Proof of Theorem 4.5.} (I) Let $v(x) = F(\tau(x)) $.
We extend $F$ on $\Gamma$ by setting $F = 0$ outside
$e_j $. Then we have $v \in H^1(\Omega_{\epsilon})$ and $v = 0$
outside $\Omega_j^{(\epsilon)} $. Let $\epsilon_0$ be as in Lemma 4.1.
Note that $\psi_{\epsilon}(\sigma) = 0$ on $e_j$ for
$\epsilon \in (0, \epsilon_0] $, where $\psi_{\epsilon}(\sigma)$ is given
by (4.4). Therefore, replacing $u_{\epsilon}$ by $zu_{\epsilon}$ in
Lemma 4.1 and using the second inequality in (3.20), we obtain
\begin{eqnarray} 
\lefteqn{\big| {\frac1{\epsilon} \int_{\Omega_j^{(\epsilon)}}
   zu_{\epsilon}(x)\overline{v(x)} \, dx
   - \int_{e_j} zu_{\epsilon}(\sigma, 0)\overline{F(\sigma)}
  a_0(\sigma) \, d\sigma \big| }} \nonumber\\
 & \le& C_j|z|\sqrt{\epsilon}\| \nabla u_{\epsilon} \|_{\Omega_j^{(\epsilon)}}
  \| F \|_{e_j, a_0}   \\
 & \le& 2C_j\sqrt{\epsilon}\| f \|_{\Omega_{\epsilon}}
  \| F \|_{e_j, a_0}\hspace{4cm} \nonumber
\end{eqnarray}  %4.10
for $\epsilon \in (0, \epsilon_0] $, which implies that
\begin{equation}
 \frac1{\epsilon} \int_{\Omega_j^{(\epsilon)}} zu_{\epsilon}(x)\overline{v(x)} \, dx
   = \int_{e_j} zu_{\epsilon}(\sigma, 0)\overline{F(\sigma)}a_0(\sigma) \, d\sigma
  + O(\sqrt{\epsilon}) \quad (\epsilon \to 0),   %4.11
\end{equation}
when $F \in C_0^2(e_j)$ is fixed. Next we set $u_{\epsilon} = f_{\epsilon}$
in Lemma 4.1 to obtain
\begin{equation}
 \frac1{\epsilon} \int_{\Omega_j^{(\epsilon)}} f_{\epsilon}(x)\overline{v(x)} \, dx
   = \int_{e_j} f(\sigma, 0)\overline{F(\sigma)}a_0(\sigma) \, d\sigma
  + O(\sqrt{\epsilon}) \quad (\epsilon \to 0),   %4.12
\end{equation} 
     (II) Similarly we have from Lemma 4.4
\begin{equation}
   \frac1{\epsilon} \int_{\Omega_j^{(\epsilon)}} \nabla u_{\epsilon}(x) \cdot
  \overline{\nabla v(x)} \, dx
= \int_{e_j} \frac{\partial u_{\epsilon}}{\partial \sigma}(\sigma, 0)
   \overline{F'(\sigma)}a_0(\sigma) \, d\sigma
  + O(\sqrt{\epsilon}) \quad (\epsilon \to 0),   %4.13
\end{equation} 
     (III) It follows from (4.11), (4.12) and (4.13) that
\begin{eqnarray}
\lefteqn{  {\frac1{\epsilon} \int_{\Omega_j^{(\epsilon)}}}
  {\big\{ \nabla u_{\epsilon} \cdot \overline{\nabla v}
   - zu_{\epsilon} \overline{v} - f_{\epsilon}
   \overline{v} \big\} \, dx } }\nonumber\\
& =& {\int_{e_j}
  \big\{ \frac{\partial u_{\epsilon}}{\partial \sigma}(\sigma, 0)
  \overline{F'(\sigma)}a_0(\sigma) - zu_{\epsilon}(\sigma, 0)
  \overline{F(\sigma)}a_0(\sigma) }  \\
&& - f(\sigma, 0)\overline{F(\sigma)}a_0(\sigma) \big\} \, d\sigma
   + O(\sqrt{\epsilon}). \nonumber
\end{eqnarray}  
  Noting that the domain of integration in the left-hand side
  of (4.13) can be extended to $\Omega_{\epsilon}$ and that
  $v \in H^1(\Omega_{\epsilon}) $, by the definition of the
  Neumann Laplacian $H_{\Omega_{\epsilon}}$
\begin{eqnarray}
 0 &=& \frac1{\epsilon} \int_{\Omega_{\epsilon}} 
  \big\{ \nabla u_{\epsilon} \cdot \overline{\nabla v}
  - zu_{\epsilon} \overline{v} - f_{\epsilon}
   \overline{v} \big\} \, dx  \nonumber\\
& =& {\int_{e_j}
  \big\{ \frac{\partial u_{\epsilon}}{\partial \sigma}(\sigma, 0)
  \overline{F'(\sigma)}a_0(\sigma)
  - zu_{\epsilon}(\sigma, 0)\overline{F(\sigma)}a_0(\sigma)  }  \\
&& - f(\sigma, 0)\overline{F(\sigma)}a_0(\sigma) \big\} \,
   d\sigma + O(\sqrt{\epsilon}).\nonumber
\end{eqnarray}   %4.15
Thus, by using partial integration, we have
\begin{equation}
   {0 = \int_{e_j} u_{\epsilon}(\sigma) \big\{-(a_0(\sigma)\overline{F'(\sigma)})'
  - z\overline{F(\sigma)}a_0(\sigma)
  - f(\sigma, 0)\overline{F(\sigma)}a_0(\sigma)\big\} \, d\sigma
   + O(\sqrt{\epsilon}).  }   %4.16
\end{equation}
Set $\epsilon = \epsilon_m$ in (4.15) and let $m \to \infty $.
Then we have (4.7), where we should note that, since $a_0$
is positive and bounded below from zero on any closed subset
of, $u_{\epsilon_m}(\cdot, 0)$ converges to $u_0$ weakly in
$L^2(I)$ as well as in $L^2(I, a_0)$ with any closed
subset $I$ of $e_j $. This completes the proof. \hfil$\diamondsuit$
  
\paragraph{Corollary 4.7.} {\it Let $u_0$ be as in} Theorem
4.5. {\it Then following limits exist}
$$\lim_{\sigma\to a_j+0} a_0(\sigma)u_0'(\sigma),\quad
\lim_{\sigma\to b_j-0} a_0(\sigma)u_0'(\sigma) \,.$$
  
\paragraph{Proof.} The proof is obvious. With $\sigma, \sigma_0 \in e_j $
we have
 \begin{equation}
  a_0(\sigma)u_0'(\sigma)
  = - \int_{\sigma_0}^{\sigma} \big(zu_{\epsilon}(\eta, 0) - f(\eta, 0)\big) \,
   d\eta  + a_0(\sigma_0)u_0'(\sigma_0)  %4.17
\end{equation}
\quad\hfil$\diamondsuit$
     Noting that $\Gamma$ consists of at most countably infinite
edges, we may assume that there exists a sequence
$\{ \epsilon_m \}_{m=1}^{\infty}$ such that there exists
$u_0 \in L^2(\Gamma, a_0)_{\rm{loc}}$ such that
\begin{eqnarray} 
& \epsilon_m \to 0  \quad (m \to \infty), & \\
& u_{\epsilon_m}(\cdot, 0) \to u_0  \quad \mbox{in  } L^2(e_j)
  \quad (j \in J), &\nonumber  %4.18
\end{eqnarray} 

  \paragraph{Theorem 4.8.} {\it Suppose that {\rm Assumptions 2,1,
3.1} and {\rm 4.2} hold. Let
$u_{\epsilon_m} = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon_m}$
be as in} (4.18), {\it where $f$ is as in} Theorem 4.5. {\it
Let $c$ be a vertex of $\Gamma$ and set
\begin{equation}
   J(c) = \{ j \in J : a_j = c \ \mbox{\rm or  } b_j = c \},   %4.19
\end{equation}
where $a_j$ and $b_j$ are the endpoints of $e_j$ with
$b_j \succeq_a a_j $. Then it follows that
\begin{equation}
   \sum_{j\in J(c)} \eta(j)a_0(c)u_0'(c) = 0,   %4.20
\end{equation}
i.e., The Kirchhoff boundary condition is satisfied at each vertex
of $\Gamma $, where
\begin{equation}
\eta(j) = \left\{\begin{array}{ll}
   1  & \mbox{if } c = b_j, \\
   -1 & \mbox{if } c = a_j, \end{array}\right.
\end{equation}   %4.21
and}
\begin{equation}
a_0(c)u_0'(c) =\left\{ \begin{array}{ll}
  \lim_{\sigma\to b_j, \sigma\in e_j} a_0(\sigma)u_0'(\sigma) 
  & \mbox{if } \eta(j) = 1,  \\
  \lim_{\sigma\to a_j, \sigma\in e_j} a_0(\sigma)u_0'(\sigma) 
  & \mbox{if } \eta(j) = -1. \end{array} \right.
\end{equation} %4.22

\paragraph{Proof.} (I) Let $F$ be a function defined on
$\Gamma$ such that
\begin{eqnarray}
 &\mathop{\rm supp} F \subset (\cup_{j\in J(c)} e_j)\cup \{ c \}, &\nonumber\\
 & F = 1 \ \mbox{ in a neighborhood of  } c, &\\
 & F \mbox{ is }  C^2 \mbox{ on each $e_j$ with $j \in J(c)$}. &\nonumber %4.23 
\end{eqnarray}
Let $F_j$ be the restriction of $F$ on $\overline{\epsilon_j} $.
Then $F_j \in C^2(\Gamma_j)$ and $F_j' \in C_0^1(e_j) $. Set
$v_j(x) = F_j(\tau(x)) $. Then,
using Lemmas 4.1 and 4.4, and proceeding as in the proof of
Theorem 4.5, we have
\begin{eqnarray}
 \lefteqn{\frac1{\epsilon} \int_{\Omega_j^{(\epsilon)}} 
  \big\{ \nabla u_{\epsilon} \cdot
 \overline{\nabla v_j} - zu_{\epsilon} \overline{v_j} - f_{\epsilon}
   \overline{v_j} \big\} \, dx }\nonumber \\
&=& { \int_{e_j}
  \big\{ \frac{\partial u_{\epsilon}}{\partial \sigma}(\sigma, 0)
  \overline{F_j(\sigma)}a_0(\sigma)
   - zu_{\epsilon}(\sigma, 0)\overline{F_j(\sigma)}a_0(\sigma) }   \\
&&  - f(\sigma, 0)\overline{F_j(\sigma)}a_0(\sigma) \big\} \, d\sigma
   + O(\sqrt{\epsilon}). \nonumber
\end{eqnarray}  %4.24
By summing up both sides of (4.24) with respect to
$j \in J(c) $, which is a finite set since $\Gamma$ is
of finite degree, we obtain, as in (4.14),
\begin{eqnarray}
   0 &= & {\sum_{j\in J(c)} \int_{e_j}
  \big\{ \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, 0)
   \overline{F'(\sigma)}a_0(\sigma)
  - zu_{\epsilon}(\sigma, 0)\overline{F(\sigma)}a_0(\sigma)  }  \\
&=&  - f(\sigma, 0)\overline{F(\sigma)}a_0(\sigma) \big\} \, d\sigma
   + O(\sqrt{\epsilon}), \nonumber
\end{eqnarray}  %4.25
where, and in the sequel, we shall use $F(\sigma)$ in place of
$F_j $. Here, by partial integration,
\begin{equation}
  \int_{e_j} \frac{\partial u_{\epsilon}}{\partial \sigma}(\sigma, 0)
  \overline{F'(\sigma)}a_0(\sigma) \, d\sigma
 = - \int_{e_j} u_{\epsilon}(\sigma, 0)(a_0(\sigma)\overline{F'(\sigma)})' \, d\sigma,
   %4.26
\end{equation}
where we should note that $F'$ has a compact support in
$e_j $. Combine (4.25) and (4.26) and let $\epsilon \to 0$
along $\epsilon_m$ to give
\begin{eqnarray}
   0 &= & {\sum_{j\in J(c)} \int_{e_j}
  \big\{ - u_0(\sigma, 0)(\overline{F'(\sigma)}a_0(\sigma))'
  - z u_0(\sigma, 0)\overline{F(\sigma)}a_0(\sigma) }  \\
  &&  - f(\sigma, 0)\overline{F(\sigma)}a_0(\sigma) \big\} \,
   d\sigma + O(\sqrt{\epsilon}). \nonumber
\end{eqnarray}  %4.27

     (II) Suppose that $c = a_j $. Then, repeating partial
integration, and noting that $F = 1$ near $c $, we
obtain
\begin{eqnarray}
\lefteqn{- \int_{e_j}  u_0(\sigma, 0)(\overline{F'(\sigma)}a_0(\sigma))' \,
   d\sigma }\nonumber \\ 
 &=& {\int_{e_j} u_0'(\sigma, 0)\overline{F'(\sigma)}a_0(\sigma) \,
   d\sigma } \nonumber\\
 &=& {\lim_{\sigma\to a_j} \int_{\sigma}^{b_j} u_0'(\sigma)
   \overline{F'(\sigma)}a_0(\sigma) \, d\sigma  } \\
 &=& {- \int_{e_j} (a_0(\sigma)u_0'(\sigma, 0))'\overline{F'(\sigma)} \,
   d\sigma - a_0(c)u_0'(c) }  \nonumber\\
 &=& {- \int_{e_j} (a_0(\sigma)u_0'(\sigma, 0))'\overline{F'(\sigma)} \,
   d\sigma + \eta(j) a_0(c)u_0'(c) }  \nonumber
\end{eqnarray}   %4.28
Similarly, we have, for $c = b_j $,
\begin{eqnarray}
 \lefteqn{- \int_{e_j}  u_0(\sigma, 0)(\overline{F'(\sigma)}a_0(\sigma))' \,
  d\sigma } \\
  \phantom{{- \int_{e_j} }}
   & =& - \int_{e_j} (a_0(\sigma)u_0'(\sigma, 0))'\overline{F'(\sigma)} \,
   d\sigma + \eta(j) a_0(c)u_0'(c), \nonumber
\end{eqnarray}  %4.29
where we should note that $u_0 \in C^2(e_j)$ ((ii) of
Remark 4.6). \smallskip
  
     (III) It follows from (4.26), (4.27) and (2.28) that
\begin{eqnarray}
 0 &=& \sum_{j\in J(c)} \int_{e_j}  \big\{ -(a_0(\sigma)u_0'(\sigma, 0))'
   - z u_0(\sigma, 0)a_0(\sigma, 0) \\
&& - f(\sigma, 0)a_0(\sigma, 0) \big\}\overline{F(\sigma)} \, d\sigma
   + \sum_{j\in J(c)} \eta(j)a_0(c)u_0'(c) \nonumber
\end{eqnarray}  %4.30
Since $u_0$ is now a strong solution of the equation (4.9),
the first term of the left-hand side of (4.30) is zero, and
hence the Kirchhoff boundary condition (4.20) follows from
(4.30). \hfil$\diamondsuit$
  
  
  \section{Continuity of the limit function}  %Section 5
  
   Let $u_0$ be a limit function on $\Gamma$ given by (4.17).
Since $u_0$ is a solution of the differential equation on each
$e_j $, $j \in J $, $u_0$ is smooth on each $e_j$
(Remark 4.6, (ii)). In this section, we shall show, under some
additional conditions, that $\{ u_{\epsilon} \}$ converges to $u_0$
in stronger senses, and that $u_0$  is continuous at the
vertices of $\Gamma $.
  
   The proof of the following proposition will be given in \S6.
  \setcounter{equation}{0}
  
\paragraph{Proposition 5.1.} {\it Suppose that {\rm Assumptions
2,1, 3.1} and {\rm 4.2} hold. Let
$u_{\epsilon} = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon} $,
where $z \in {\bf C} \backslash [0, \infty) $, $f \in H^1(\Omega) $,
and $f_{\epsilon}$ is the restriction of $f$ on $\Omega_{\epsilon} $.
Let $f$ satisfy} (3.18) {\it Then there exists a positive
constants $C_j = C_j(K_j, K_j', L_j, M_j, m_j, z) $,
depending only on $K_j, L_j, M_j, m_j$ and $z $, such that
\begin{eqnarray}
 \lefteqn{ \| u_{\epsilon}'(\cdot, 0) \|_{e_j,a_0}^2 }\\
 &\le &  C_j\big[\epsilon\| u_{\epsilon} \|_{2,\Omega_{\epsilon}}^2   
  + \sum_{k\in J} \big(\frac{M_k}{m_k}\big)\big( \epsilon L_k^2K_k^2
   \| \nabla f \|_{\Omega_{\epsilon}^{(k)}}^2
  + \| f(\cdot, 0) \|_{e_k, a_0}^2 \big) \big], \nonumber
\end{eqnarray}  %5.1
where $\|\cdot \|_{2,\Omega_{\epsilon}}$ is the norm of the
second-order Sobolev space $H^2(\Omega_{\epsilon})$ on $\Omega_{\epsilon} $.}
  
\paragraph{Theorem 5.2.} {\it Suppose that {\rm Assumptions 2,1,
3.1} and {\rm 4.2} hold. Suppose that
\begin{equation}
  \limsup_{\epsilon\to0} \sqrt{\epsilon} \| u_{\epsilon} \|_{2,\Omega_{\epsilon}}
   < \infty.  %5.2
\end{equation}
Let $f \in H^1(\Omega)$ which satisfies} (3.18), {\it and let
$u_{\epsilon} = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon} $, where
$z \in \bf C \backslash [0, \infty] $, and $f_{\epsilon}$ is the restriction
of $f$ on $\Omega_j^{(\epsilon)} $. Let
$\{ \epsilon_{m} \}_{m=1}^{\infty}$ such that $\epsilon_m \to 0$ as
$m \to \infty$ and $u_{\epsilon_m}(\cdot, 0)$ converges weakly
in each $L^2(e_j, a_0) $. Then the limit function $u_0$ is
continuous at any vertex $c$ of $\Gamma$ such that
$c \in \Omega $.}
  
\paragraph{Example 5.3.} Let 
\begin{eqnarray}
 &\Omega = \{ x = (x_1, x_2) : -\ell_-(x_1) < x_2 < \ell_+(x_1), \
   a < x_1 < b \}, &\nonumber\\
& \Omega_{\epsilon} = \{ x = (x_1, x_2) :
   -\epsilon\ell_-(x_1) < x_2 < \epsilon\ell_+(x_1), \ a < x_1 < b \}, &\\
&\Gamma = \{ x = (x_1, 0) : a \le x_1 \le b \},&\nonumber
\end{eqnarray}   %5.3
where $-\infty < a < b < \infty $, $0 < \epsilon \le 1$ and
$\ell_{\pm}(t)$ are positive $C^1$ functions on
$[a, b] $. and the map $\tau$ is given by
\begin{equation}
  \tau(x_1, x_2) = (x_1, 0)
   \quad ((x_1, x_2) \in \Omega).  %5.4
\end{equation}
Suppose that $\Omega$ is a bounded convex set. Note that
each $\Omega_{\epsilon}$ is a convex set, too. Then it has been known
(\cite{SA}) that
\begin{equation}
  \| u_{\epsilon} \|_{2,\Omega_{\epsilon}}
  \le C(z)\| f_{\epsilon} \|_{\Omega_{\epsilon}}
  \le C(z)\| f \|_{\Omega},  %5.5
\end{equation}
where $C(z)$ is a positive constant depending only on
$z $. Thus the condition (5.2) is satisfied in this case.
  
\paragraph{Proof of Theorem 5.2.} Let
\begin{equation}
   J(c) = \{ j \in J : a_j = c \ \mbox{\rm or } b_j = c \},
  %5.6
\end{equation}
For each $j \in J(c)$
let $c_j \in e_j$ and let $\overline{e_j}'$ be all points on
$\overline{e_j}$ between $c$ and $c_j \in e_j$ (including
$c$ and $c_j $). Set
$\Gamma(c)' = \cup_{j \in J(c)} \overline{e_j}' $.
Since $a_0$ is bounded below from $0 $, it follows from
(3.19) and (5.2) that there exists $\epsilon_0 \in (0, 1]$ such
that the sequences \linebreak
$\{ \| u_{\epsilon}(\cdot, 0) \|_{e_j} \}_{0<\epsilon<\epsilon_0}$ and
$\{ \| u_{\epsilon}'(\cdot, 0) \|_{e_j} \}_{0<\epsilon<\epsilon_0}$ are
uniformly bounded, where the norm $\|\ \ \|_{e_j}$ is given
in (3.6). Therefore there exist a sequence
$\{ \epsilon_m \} $, $\epsilon_m \to 0$ ($ m \to \infty $), and
$c_j' \in \overline{e_j}' $, ($ j \in J(c) $), such that
$\lim_{m\to\infty} u_{\epsilon_m}(c_j', 0)$ exists for each
$j \in J(c) $. Then we see from
\begin{eqnarray}
 &u_{\epsilon_m}(\sigma, 0)
   = \int_{c_j'}^{\sigma} u_{\epsilon}'(\eta, 0) \, d\eta
   + u_{\epsilon_m}(c_j', 0) \quad (\sigma \in \Gamma_j'), &\\
&u_{\epsilon_m}(\sigma, 0) - u_{\epsilon_m}(\sigma', 0)
  = \int_{\sigma'}^{\sigma} u_{\epsilon}'(\eta, 0) \, d\eta
   \quad (\sigma, \sigma' \in \Gamma_j')  \,,&\nonumber
\end{eqnarray}   %5.7
(3.19) and (5.1) that $\{ u_{\epsilon_m} \}$ is uniformly
bounded and equicontinuous on $\Gamma(c)' $. Therefore there
exists a subsequence of $\{ \epsilon_m \} $, which will be denoted
again by $\{ \epsilon_m \} $, such that $\{ u_{\epsilon_m} \}$
converges to $u_0$ uniformly on $\Gamma(c)' $, and hence
$u_0$ is continuous on $\Gamma(c)' $. This completes the proof.
\hfil$\diamondsuit$
  
\paragraph{Example 5.4} (Rooms and passages domain). 
     Let $\{h_k\} $, $\{\delta_{2k}\} $, $k = 1, 2, \cdots $, be
infinite sequences of positive numbers such that
\begin{equation}
 \sum_{k=1}^{\infty} h_k = b \le \infty, \ \
   0 < \mbox{\rm const. } \le \frac{h_{k+1}}{h_k} \le 1, \ \
  0 < \delta_{2k} \le h_{2k+1},  %5.8
\end{equation}
and let $H_k := \sum_{j=1}^k h_j, k = 1, 2, \cdots $. Then
$\Omega \subset {\mathbb R}^2$ is defined as the union of the rooms $R_k$
and passages $P_{k+1}$ given by
\begin{eqnarray} 
& R_k = (H_k - h_k, H_k) \times (-\frac{h_k}2, \frac{h_k}2),&  \\
& P_{k+1} = [H_k, H_k + h_{k+1}] \times
   (-\frac{\delta_{k+1}}2, \frac{\delta_{k+1}}2),  &\nonumber
\end{eqnarray}   %5.9
for $k = 1, 3, 5,\dots $. In \S6.1 of \cite{EHA}, this was analyzed
as an example of a generalized ridged domain with generalized ridge
$\Gamma = [0, b]$  $(b < \infty)$ or $\Gamma = [0, \infty)$ $(b = \infty) $.
In order to make $\Gamma$ a tree, each point on $\Gamma$ which connects
a room and the adjacent passage can be called a vertex
($V_0, V_1, V_2, \dots$ in Fig. 1)
  
\begin{figure}[ht] 
\unitlength=0.3mm
\begin{center}
\begin{picture}(300,100)(-30,15)



\put(-30,0){\line(0,1){100}}

\put(-30,100){\line(1,0){100}}

\put(-30,0){\line(1,0){100}}



\put(70,100){\line(0,-1){30}}

\put(70,0){\line(0,1){30}}



\put(70,30){\line(1,0){60}}

\put(70,70){\line(1,0){60}}



\put(130,70){\line(0,1){20}}

\put(130,30){\line(0,-1){20}}



\put(130,90){\line(1,0){80}}

\put(130,10){\line(1,0){80}}



\put(210,10){\line(0,1){25}}

\put(210,90){\line(0,-1){25}}



\put(210,35){\line(1,0){40}}

\put(210,65){\line(1,0){40}}



\put(250,35){\line(0,-1){15}}

\put(250,65){\line(0,1){15}}



\put(250,20){\line(1,0){20}}

\put(250,80){\line(1,0){20}}

\end{picture}

\end{center}
\caption{A domain of Rooms and passages}
\label{room-passage}
\end{figure}

     A mapping $\tau$ is defined as follows:
     (i) in a passage $P $: $\tau(x_1, x_2) = x_1 $;
     (ii) in the first half of the room $R$ succeeding the passage
$P $:
\begin{equation}
  \tau(x_1, x_2) = \max(x_1, |x_2|-\frac{\delta}2), \quad
   {0 \le x_1 \le \frac{h}2},   %5.10
\end{equation}
where $P$ is of width $\delta$ and $0 \le x_1 \le h$
in $R$ after translation. Hence $\tau$ is Lipschitz and
$|\nabla\tau| = 1$ almost everywhere in $\Omega $. It is easy to
see that $a_0(x_1)$ in this case is a bounded continuous function
on $\Gamma $, and hence
the Kirchhoff boundary condition will be imposed only at
$x_1 = 0, b \ (b < \infty)$ or $x_1 = 0 \ (b = \infty) $.
Since $a_0$ is positive on $\Gamma $, the limit function $u_0$
is continuous on $\Gamma $, and the differential equation for
$u_0$ can be explicitly written using $h_k$ and
$\delta_{2k} $.
  
\begin{figure}[ht] 
\unitlength=0.3mm
\begin{center}
\begin{picture}(230,130)(-10,-225)
\put(-10,-170){\line(1,0){230}}



\put(-10,-200){\thicklines{\line(1,0){50}}}

\put(-10,-140){\thicklines{\line(1,0){50}}}



\put(40,-240){\thicklines{\line(1,0){140}}}

\put(40,-100){\thicklines{\line(1,0){140}}}



\put(40,-240){\thicklines{\line(0,1){40}}}

\put(40,-100){\thicklines{\line(0,-1){40}}}



\put(180,-240){\thicklines{\line(0,1){50}}}

\put(180,-100){\thicklines{\line(0,-1){50}}}



\put(180,-190){\thicklines{\line(1,0){40}}}

\put(180,-150){\thicklines{\line(1,0){40}}}



\put(40,-140){\line(1,1){40.7}}

\put(40,-200){\line(1,-1){40.7}}



\put(130,-100){\line(1,-1){48.5}}

\put(130,-240){\line(1,1){48.5}}



\put(120,-100){\vector(0,-1){70}}

\put(110,-240){\vector(0,1){70}}



\put(40, -120){\line(1,0){20}}

\put(60,-120){\vector(0,-1){50}}



\put(180,-220){\line(-1,0){30}}

\put(150,-220){\vector(0,1){50}}
\end{picture}
\end{center}
\caption{$C_t$ for a domain of rooms and passages}
\label{curve}
\end{figure}
  
   Finally we are going to show, under some additional conditions.
that $u_{\epsilon}(\cdot, 0)$ \linebreak converges as $\epsilon \to 0$
without taking a subsequence. We are now in a position to introduce
another weighted $L^2$ spaces on the tree $\Gamma $.
  
\paragraph{Definition 5.5.} Suppose that a tree $\Gamma$ satisfies
(I) of Assumption 2.1. Let $a(\sigma)$ and $b(\sigma)$ be positive
functions defined on $\cup_{j\in J} e_j$ such that
$a(\sigma)$ and $b(\sigma)$ are bounded from $0$ near each
vertex $v \in \Omega $. Then let $H^1(\Gamma, a, b)$ be a subspace
of $L^2(\Gamma, a)$ such that $F \in H^1(\Gamma, a, b)$ satisfies
the following conditions.
\begin{description}  
\item{(a)} $F$ is continuous on $\Gamma \cap \Omega $.
  
\item{(b)} $F$ is absolutely continuous on each
$e_j \cap \Omega $.
  
\item{(c)} $F$ satisfies
\begin{equation}
   \| F \|_{\Gamma, a, b, 1}^2
  = \sum_{j\to J} \int_{e_j} |F'(\sigma)|^2 \, b(\sigma)d\sigma
  + \| F \|_{\Gamma, a}^2 < \infty,  %5.11
\end{equation}
where $F'$ denotes the derivative of $F$ with respect
to $\sigma $.
\end{description}
  
Note that it is assumed that $\Gamma \cap \partial\Omega$ consists
only of the vertices of the tree $\Gamma $. The next lemma
guarantees that $H^1(\Gamma, a, b)$ is a Hilbert space with
inner product
\begin{equation}
  (F, \ G)_{\Gamma, a, b, 1}
  = \sum_{j\in J} \int_{e_j} F'(\sigma)\overline{G'(\sigma)} \, b(\sigma)d\sigma
   + (F, G)_{\Gamma, a}.  %5.12
\end{equation}
and norm
\begin{equation}
\| F \|_{\Gamma, a, b, 1} = [(F, \ F)_{\Gamma, a, b, 1}]^{1/2}
  %(5.13)
\end{equation} 

  \paragraph{Lemma 5.6.} {\it Suppose that $H^1(\Gamma, a, b)$ be
as in} Definition 5.5. {\it Then $H^1(\Gamma, a, b)$ is a
Hilbert space with its norm and inner product given by}
(5.12) {\it and} (5.13). \smallskip
  
The proof of this lemma will be given in \S6.
  Set $H^1(\Gamma, a_0) = H^1(\Gamma, a_0, a_0) $. Then,
under Assumption 3.1, it follows from Lemma 5.6 that
$H^1(\Gamma, a_0)$ is a Hilbert space.
  
   Let $H_{\Gamma,0}$ be the selfadjoint operator in
$L^2(\Gamma, a_0)$ associated with the sesquilinear form
\begin{equation}
\ell_0[F, G] = \int_{\Gamma} F'(\sigma)\overline{G'(\sigma)} \, a_0(\sigma)d\sigma
  \quad (F, G \in H^1(\Gamma, a_0))  %5.14
\end{equation} 
  \paragraph{Theorem 5.7.} {\it Suppose that {\rm Assumptions 2,1,
3.1, 4.2} hold. Suppose that
\begin{equation}
   \limsup_{\epsilon\to0} \sqrt{\epsilon} \| u_{\epsilon} \|_{2,\Omega_{\epsilon}}
  < \infty. %5.15
\end{equation}
Suppose that the tree $\Gamma$ has a finite number of edges.
Let $f \in H^1(\Omega) $, and let
$u_{\epsilon} = (H_{\Omega_{\epsilon}} - z)^{-1}f_{\epsilon} $,
where $z \in {\bf C} \backslash [0, \infty] $, and $f_{\epsilon}$ is the
restriction of $f$ on $\Omega_j^{(\epsilon)} $. Then we have
$u_{\epsilon}(\cdot, 0) \in H^1(\Gamma, a_0)$ and
\begin{equation}
  u_{\epsilon}(\cdot, 0) \to (H_{\Gamma, 0} - z)^{-1}f(\cdot,0)
  \quad (\epsilon \to 0)  %5.16
\end{equation}
weakly in $H^1(\Gamma, a_0) $.}
  
\paragraph{Proof.} (I) Let $N$ be the number of the edges of
$\Gamma $. Then it follows from Propositions 3.4 and 5.1 that
\begin{eqnarray}
\lefteqn{ {\sum_{j=1}^N \big(\| u_{\epsilon}(\cdot, 0) \|_{e_j, a_0}^2
  +  \| u_{\epsilon}'(\cdot, 0) \|_{e_j,a_0}^2 \big) }}\nonumber  \\
&\leq&  4|z|^{-1} \Big[\big(\sum_{j=1}^N
  \frac{M_jK_j^2L_j^2}{m_j}\big) \| f \|_{\Omega_{\epsilon}}^2  \\
&&+ |z|^{-1}\big\{\sum_{j=1}^N
   \big(\frac{M_j}{m_j}\big)
  \big(\sum_{k=1}^N
   \big(\frac{M_k}{m_k}\big)\big(\epsilon L_k^2K_k^2
   \|\nabla f \|_{\Omega_{\epsilon}^{(k)}}^2   
   + \| f(\cdot, 0) \|_{e_k, a_0}^2 \big)\big\}\Big] \nonumber\\
&& +  \big(\sum_{j=1}^N C_j\big)
   \Big[ \epsilon\| u_{\epsilon} \|_{2, \Omega_{\epsilon}}^2  
 + \sum_{k=1}^N  \big(\frac{M_k}{m_k}\big)\big( \epsilon L_k^2K_k^2
   \| \nabla f \|_{\Omega_{\epsilon}^{(k)}}^2
   + \| f(\cdot, 0) \|_{e_k, a_0}^2 \big) \Big], \nonumber
\end{eqnarray}  %5.17
which, together with the fact that $u_{\epsilon} \in C^1(\Omega) $, implies
that $u_{\epsilon}(\cdot, 0) \in H^1(\Gamma, a_0)$ for each
$0 < \epsilon \le 1 $. Also, by replacing
$\| \nabla f \|_{\Omega_{\epsilon}^{(k)}}$ by
$\| \nabla f \|_{\Omega}$ in (5.15), we see that
$\| u_{\epsilon}(\cdot, 0) \|_{\Gamma, a_0, 1}$
is uniformly bounded for $\epsilon \in (0, 1] $.
  
     (II) Let $\{ \epsilon_n \}_{n=1}^{\infty} \subset (0, 1]$ be a
sequence such that $\epsilon_n \to 0$ as $n \to \infty$ and
$u_{\epsilon_n}$ converges weakly in $H^1(\Gamma, a_0) $.  Let
$u_0 \in H^1(\Gamma, a_0)$ be the limit function. Since
$u_{\epsilon_n}$ is a bounded sequence in $L^2(\Gamma, a_0)$
and  $H^1(\Gamma, a_0)$ is dense in $L^2(\Gamma, a_0) $, $u_0$
is also the weak limit of  $u_{\epsilon_n}$ in each
$L^2(e_j, a_0)$ $(j \in J) $. Therefore from Remark 4.6, (ii)
and Theorem 4.7 we see that $u_0$ is a solution of the equation
$- a_0(\sigma)^{-1}(a_0(\sigma)u')' - zu = f(\sigma, 0)$ with the Kirchhoff
boundary condition at each vertex of $\Gamma $. Let
$F \in H^1(\Gamma, a_0) $. Then, by using partial integration and
the fact that $u_0$ satisfies the above equation with Kirchhoff
boundary condition, we have
\begin{eqnarray}
\ell_0[u_0, F] & =& {\sum_{n=1}^N \int_{e_j} u_0'(\sigma)
   \overline{F'(\sigma)} \, a_0(\sigma)d\sigma }  \nonumber \\
&=& - \int_{\Gamma} (a_0(\sigma)u_0(\sigma)')'\overline{F(\sigma)} \, d\sigma \\
&=& (zu_0 - f(\cdot, 0), F)_{\Gamma, a_0}, \nonumber
\end{eqnarray}  %5.18
where $\ell_0[u_0, F]$ is the sesquilinear form used to define
the operator $H_{\Gamma, 0}$ ((5.15)). Then, by the definition of
$H_{\Gamma, 0} $, $u_0$ belongs to the domain of $H_{\Gamma, 0}$
and
\begin{equation}
   u_0 = (H_{\Gamma, 0} - z)^{-1}f(\cdot,0).   %5.19
\end{equation}
Since the limiting function $u_0$ is now independent of the
subsequence $u_{\epsilon_n} $, we can conclude that the sequence
$\{ u_{\epsilon}(\cdot, 0) \}$ itself converges to
$(H_{\Gamma, 0} - z)^{-1}f(\cdot,0)$ weakly in $H^1(\Gamma, a_0) $.
This completes the proof. \hfil$\diamondsuit$
  
\paragraph{Remark 5.8.} Let $(\Omega, \Gamma, \tau)$ be as in
Example 5.3. Then not only Theorem 5.7 can be applied to this
case, but also it has been shown in \cite{SA}, \S5 that the sequence
$\{ u_{\epsilon}(\cdot, 0) \}$ converges to
$(H_{\Gamma, 0} - z)^{-1}f(\cdot,0)$ strongly in $L^2(\Gamma, a_0) $.
  

\section{Proofs}  \setcounter{equation}{0}
  
\paragraph{Proof of Lemma 4.1.} (I) By (2.6) and (2.7) we have
 \begin{eqnarray} 
 \lefteqn{\int_{\Omega_j^{(\epsilon)} u_{\epsilon}(x)}\overline{v(x)} \, dx }
 \nonumber \\ 
 &=&  {\int_{e_j} \overline{F(\sigma)}
   \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   u_{\epsilon}(\sigma, s)|\nabla\tau(\sigma,  s)|^{-1} \, ds\,d\sigma }\nonumber \\
&=&  {\int_{e_j} \overline{F(\sigma)}
  \big\{ \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  u_{\epsilon}(\sigma, 0)|\nabla\tau(\sigma, 0)|^{-1} \,ds } \\
&& + \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \big(u_{\epsilon}(\sigma, s)|\nabla\tau(\sigma,  s)|^{-1}  
 - u_{\epsilon}(\sigma, 0)|\nabla\tau(\sigma, 0)|^{-1}
   \big) \, ds \big\}\, d\sigma \nonumber\\
&\equiv &\epsilon
   \int_{e_j} u_{\epsilon}(\sigma, 0)\overline{F(\sigma)}a_0(\sigma) \, d\sigma
  + G_1 + G_2, \nonumber
\end{eqnarray}  %A.1
where
\begin{eqnarray}
 &G_1 = \int_{e_j} \overline{F(\sigma)}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   (u_{\epsilon}(\sigma, s) - u_{\epsilon}(\sigma, 0))|
   \nabla\tau(\sigma,  s)|^{-1}) \,ds\,d\sigma,&  \\
 &G_2 =  \int_{e_j} u_{\epsilon}(\sigma, 0)\overline{F(\sigma)}
   \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   (|\nabla\tau(\sigma,  s)|^{-1} - |
   \nabla\tau(\sigma, 0)|^{-1}) \,ds\,d\sigma. &\nonumber \\
\end{eqnarray}   %A.2
 

     (II) Proceeding as in (3.11), we obtain
\begin{eqnarray}
 \frac1{\epsilon}|G_1| 
 &\le& \frac1{\epsilon} \int_{e_j} |F(\sigma)|
   \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \big(\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \big|\frac{\partial u_{\epsilon}(\sigma, \eta)}{\partial s}\big|\,
  d\eta \big) |\nabla\tau(\sigma,  s)|^{-1} \, ds\,d\sigma  \nonumber  \\
 &\le& \frac{M_j}{m_j} \int_{e_j} |F(\sigma)|a_0(\sigma)
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \big|\frac{\partial u_{\epsilon}(\sigma, s)}{\partial s}\big|\,
   ds\,d\sigma     \\
& \le& \frac{M_j}{m_j} \int_{e_j} |F(\sigma)|a_0(\sigma) \big(
  \big[\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \big|\frac{\partial u_{\epsilon}(\sigma, s)}{\partial s}\big|^2
   |\nabla\tau(\sigma,  s)|^{-1}\, ds \big]^{1/2} \times  \nonumber  \\
&& \big[\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  |\nabla\tau(\sigma,  s)| \, ds
   \big]^{1/2} \big) \, d\sigma \nonumber 
\end{eqnarray}   %A.3
Using (3.3) and the first inequality of (3.2) we see that
\begin{eqnarray*} 
\lefteqn{\big[\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \big|\frac{\partial u_{\epsilon}(\sigma, s)}{\partial s}\big|^2
   |\nabla\tau(\sigma,  s)|^{-1}\, ds \big]^{1/2} }\\   
 &\le& K_j \big[\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  |\nabla u_{\epsilon}(\sigma, s)|^2
  |\nabla\tau(\sigma,  s)|^{-1}\, ds \big]^{1/2},  
  \end{eqnarray*}
  and
\begin{eqnarray} 
&\big[\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   |\nabla\tau(\sigma,  s)| \, ds \big]^{1/2} \le
  \sqrt{\epsilon L_jM_j}, & \\
&\sqrt{a_0(\sigma)} \le \sqrt{\frac{L_j}{m_j}},  &\nonumber
\end{eqnarray}   %A.4
and hence we have
\begin{eqnarray}
 \frac1{\epsilon}|G_1| & \le&
   \big(\frac{M_j}{m_j}\big)^{3/2}L_jK_j\sqrt{\epsilon}
  \int_{e_j} |F(\sigma)| \sqrt{a_0(\sigma)} \times  \nonumber\\
&&\big[\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   |\nabla u_{\epsilon}(\sigma, s)|^2
   |\nabla\tau(\sigma,  s)|^{-1}\, ds \big]^{1/2} \, d\sigma  \\
&\le& \big(\frac{M_j}{m_j}\big)^{3/2}L_jK_j\sqrt{\epsilon}
  \| F \|_{e_j, a_0}
  \|\nabla u_{\epsilon} \|_{\Omega_j^{(\epsilon)}}. \nonumber
\end{eqnarray}   %A.5

     (III) As for $G_2 $, we have
\begin{equation}
  {\frac1{\epsilon}|G_2| \le
\int_{e_j} |u_{\epsilon}(\sigma, 0)||F(\sigma)|a_0(\sigma)\psi_{\epsilon}(\sigma) \,d\sigma,}
   %(A.6)
\end{equation}
where $\psi_{\epsilon}(\sigma)$ is given by (4.3). Thus, we obtain
\begin{equation}
\frac1{\epsilon}|G_2| \le \| u_{\epsilon}(\cdot, 0) \|_{e_j, a_0}
   \big[ \int_{e_j} |F(\sigma)|^2 \psi_{\epsilon}(\sigma)^2a_0(\sigma)\, d\sigma
  \big]^{1/2},  %A.7
\end{equation}
which, together with (6.5), completes the proof. \hfil$\diamondsuit$
  
\paragraph{Proof of Lemma 4.4.} (I) Let
$a_j$ and $b_j$ be the vertices of $e_j$ such that
$b_j \succeq_a a_j $. Since $\nabla u_{\epsilon} \cdot \overline{\nabla v}$ is
invariant under the shift and rotation of the coordinate system,
we may assume that our coordinate system has the origin at
$a_j$ and the $x_1$-axis in the direction of $e_j $.
According to the change of coordinates system, the constant
$K_j$ in (3.3) in Assumption 3.1 may have to be replaced
another (finite) positive constant which will be denoted again
by $K_j $, while all other constants in Assumption 3.1
do not need to be changed. Then, since
$v(x) = F(\tau(x)) = F(x_1)$
 in $\Omega_j^{(\epsilon)} $, $\epsilon \in (0, \epsilon_0) $, we have 
\begin{eqnarray}
\int_{\Omega_j^{(\epsilon)}} \nabla u_{\epsilon} \cdot \overline{\nabla v} \, dx 
&=& \int_{\Omega_j^{(\epsilon)}}
  \frac{\partial u_{\epsilon}}{\partial x_1}
  \cdot \overline{F'(x_1)} \,dx \nonumber \\
&=& \int_{e_j} \overline{F'(\sigma)}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, s) \, ds\,d\sigma  \nonumber\\
&=& \int_{e_j} \overline{F'(\sigma)}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, 0) \, ds\,d\sigma   \\
&&+ \int_{e_j} \overline{F'(\sigma)}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \big(\frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, s)
  - \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, 0) \big) \,
   ds\,d\sigma \nonumber \\
&=& \epsilon\int_{e_j} \frac{\partial u_{\epsilon}}
  {\partial\sigma}(\sigma, 0)
   \overline{F'(\sigma)}a_0(\sigma) \, d\sigma + H. \nonumber
\end{eqnarray}  %A.8
Here we should note that we may assume, by (4.5), that
$|\nabla\tau(\sigma,  s)| = 1$ in the change of variable formula (2.8).
  
   (II) By noting that
\begin{eqnarray}
\lefteqn{\frac{d}{d\sigma} \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  (u_{\epsilon}(\sigma, s) - u_{\epsilon}(\sigma, 0)) \,ds}\nonumber \\
&=& \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \big(\frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, s)
  -  \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, 0)\big) \, ds  \\
&& + \epsilon\ell_{+}'(\sigma)(u_{\epsilon}(\sigma, \epsilon\ell_{+}(\sigma)) - u_{\epsilon}
   (\sigma, 0))  
+ \epsilon\ell_{-}'(\sigma)(u_{\epsilon}(\sigma, -\epsilon\ell_{-}(\sigma))
  - u_{\epsilon}(\sigma, 0)), \nonumber
\end{eqnarray}   %A.9
we have
\begin{eqnarray}
 H &=& \int_{e_j} \overline{F'(\sigma)} \frac{d}{d\sigma}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   (u_{\epsilon}(\sigma, s) - u_{\epsilon}(\sigma, 0)) \, ds\,d\sigma \nonumber \\
&& - \epsilon\int_{e_j} \overline{F'(\sigma)}
   \ell_{+}'(\sigma)(u_{\epsilon}(\sigma, \epsilon\ell_{+}(\sigma))
  - u_{\epsilon}(\sigma, 0)) \, d\sigma  \\
&& - \epsilon\int_{e_j} \overline{F'(\sigma)}
   \ell_{-}'(\sigma)(u_{\epsilon}(\sigma, -\epsilon\ell_{-}(\sigma))
   - u_{\epsilon}(\sigma, 0)) \, d\sigma \nonumber \\
&\equiv& H_{1} - H_{2} - H_{3}. \nonumber
\end{eqnarray}   %A.10
     Using partial integration and noting that
$F'(\sigma) \in C_0^1(e_j) $, we obtain
\begin{equation}
  H_{1} = - \int_{e_j} \overline{F''(\sigma)}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   (u_{\epsilon}(\sigma, s) - u_{\epsilon}(\sigma, 0)) \, ds\,d\sigma,
  %A.11
\end{equation}
and hence, by proceeding as in (3.11),
\begin{eqnarray}
    |H_{1}| &\le& \int_{e_j} |F''(\sigma)|
   \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)} \big|\int_0^{s}
   \big|\frac{\partial u_{\epsilon}}{\partial s}(\sigma, \eta)\big|
   \, d\eta\big| \, ds\,d\sigma \nonumber \\
& \le& \epsilon \int_{e_j} |F''(\sigma)|a_0(\sigma)
   \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \big|\frac{\partial u_{\epsilon}}{\partial s}(\sigma, s)\big|
   \, ds\,d\sigma  \\
& \le& \epsilon^{3/2} L_j \| F'' \|_{e_j, a_0}
  \| \nabla u_{\epsilon} \|_{\Omega_j^{(\epsilon)}}. \nonumber
\end{eqnarray}  %A.12
     As for the term $H_2 $, we have
\begin{eqnarray}
|H_2| &\le &\epsilon\int_{e_j} |F'(\sigma)||\ell_{+}'(\sigma)|
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \big|\frac{\partial u_{\epsilon}}{\partial s}(\sigma, s)\big|
  \, ds\,d\sigma  \nonumber   \\
& \le& \epsilon^{3/2}R_j\int_{e_j} |F'(\sigma)|
   \sqrt{a_0(\sigma)}
   \big[\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \big|\frac{\partial u_{\epsilon}}{\partial s}(\sigma, s)\big|^2
   \, ds \big]^{1/2} \, d\sigma   \\
& \le& \epsilon^{3/2}R_j \| F' \|_{e_j, a_0}
  \| \nabla u_{\epsilon} \|_{\Omega_j^{(\epsilon)}}. \nonumber
\end{eqnarray}   %A.13
     Similarly,
\begin{equation}
 |H_3| \le \epsilon^{3/2}R_j \| F' \|_{e_j, a_0}
   \|\nabla u_{\epsilon} \|_{\Omega_j^{(\epsilon)}}.
  %A.14
\end{equation}
The inequality (4.6) is obtained from (6.12), (6.13) and (6.14).
\hfil$\diamondsuit$
  
\paragraph{Proof of Proposition 5.1.} (I) As
in the proof of Lemma 4.4, we may assume that our coordinate
system has the origin at $a_j$ and the $x_1$-axis in the
direction of $e_j $. Let $I \subset e_j$ be a closed
interval and set
\begin{equation}
  T \equiv \int_I \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, 0)
  \overline{F'(\sigma)} a_0(\sigma)\, d\sigma   %A.15
\end{equation}
for $F \in C^(I) $, where we should note that
$u_{\epsilon}(\sigma, 0) \in C^(I) $, too, and hence $T$ is well-defined.
Then, we have
\begin{eqnarray}
 T &=& \frac1{\epsilon}\int_I \overline{F'(\sigma)}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, 0) \, ds\,d\sigma \nonumber \\
&=& \frac1{\epsilon}\int_I \overline{F'(\sigma)}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \big\{ \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, 0)
   - \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, s) \big\}
   \, ds\,d\sigma  \\
& & + \frac1{\epsilon}\int_I \overline{F'(\sigma)}
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, s) \, ds\,d\sigma 
  \nonumber \\
& \equiv& T_1 + T_2 \nonumber \\
\end{eqnarray}  %A.16
 
     (II) We have
\begin{eqnarray}
|T_1| &\le& \frac1{\epsilon}\int_I |F'(\sigma)|
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)} \big|\int_0^s
  \big|\frac{\partial^2u_{\epsilon}}{\partial\sigma\partial s}(\sigma, \eta)\big| \,
   d\eta \big| \, ds\,d\sigma  \nonumber \\  
&\le& \int_I |F'(\sigma)| a_0(\sigma)
  \int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
  \big|\frac{\partial^2u_{\epsilon}}{\partial\sigma\partial s}(\sigma, s)\big| \,
   ds\,d\sigma     \\
&\le& \sqrt{\epsilon}L_j \int_I |F'(\sigma)|
   \sqrt{a_0(\sigma)} \big[\int_{-\epsilon\ell_{-}(\sigma)}^{\epsilon\ell_{+}(\sigma)}
   \big|\frac{\partial^2u_{\epsilon}}{\partial\sigma\partial s}(\sigma, s)\big|^2 \,
   ds \big]^{1/2} \, d\sigma. \nonumber 
\end{eqnarray}  %A.17
Since $\sigma = x_1 $, we have from (3.3)
\begin{equation}
   {\big|\frac{\partial^2u_{\epsilon}}{\partial\sigma\partial s}\big|^2
  = \big|\frac{\partial}{\partial s}\big( \frac{\partial u_{\epsilon}}{\partial x_1}
  \big)\big|^2
  \le K_j^2 \big(\big|\frac{\partial^2u_{\epsilon}}{\partial x_1^2} \big|^2
   + \big|\frac{\partial^2u_{\epsilon}}{\partial x_1\partial x_2} \big|^2
   \big),}   %A.18
\end{equation}
which is combined with (6.17) to give
\begin{equation}
  |T_1| \le \sqrt{\epsilon}L_jK_j\sqrt{M_j}\| F' \|_{I, a_0}
   \| u_{\epsilon} \|_{2, \Omega_{\epsilon}},
  %A.19
\end{equation}
where we have used the change of variable formula (2.6) and
Assumption 3.1, (ii).
  
   (III) As for $T_2 $, we can proceed as in (II) to obtain
\begin{equation}
  |T_2| \le \frac1{\sqrt{\epsilon}}K_j\sqrt{M_j}
  \| F' \|_{I, a_0}
  \| \nabla u_{\epsilon} \|_{\Omega_{\epsilon}},  %A.20
\end{equation}
which, together with the second inequality of (3.19), yield
\begin{equation}
|T_2| \le \frac1{\sqrt{\epsilon}}|z|^{-1/2}\sqrt{2}K_j\sqrt{M_j}
  \| F' \|_{I, a_0} \| f \|_{\Omega_{\epsilon}}.
   %A.21
\end{equation}
  
   (IV) It follows from (6.16), (6.19) and (6.21) that
\begin{equation}
\big| \int_I \frac{\partial u_{\epsilon}}{\partial\sigma}(\sigma, 0)
  \overline{F'(\sigma)} a_0(\sigma)\, d\sigma \big|
   \le C_j\big(\sqrt{\epsilon}\| u_{\epsilon} \|_{2, \Omega_{\epsilon}}
   + \frac1{\sqrt{\epsilon}} \| f \|_{\Omega_{\epsilon}}\big)
   \| F' \|_{I, a_0}   %A.22
\end{equation}
Setting $F(\sigma) = u_{\epsilon}(\sigma, 0)$ in (6.22) and noting that
$I \subset e_j$ is arbitrary, we obtain
\begin{equation}
 \| u_{\epsilon}'(\cdot,0) \|_{e_j, a_0}
   \le \mbox{\rm const. }\big(
  \sqrt{\epsilon}\| u_{\epsilon} \|_{2, \Omega_{\epsilon}}
   + \frac1{\sqrt{\epsilon}} \| f \|_{\Omega_{\epsilon}}\big).
  %A.23
\end{equation}
As in the proof of Proposition 3.4, we can estimate
$\| f \|_{\Omega_{\epsilon}}$ by using (3.8) with $u$
replaced by $f $. Thus we have (5.1). \hfil$\diamondsuit$
  
\paragraph{Proof of Lemma 5.6.} Since it is easy
to see that $H^1(\Gamma, a, b)$ is a pre-Hilbert space, we have only
to prove the completeness of $H^1(\Gamma, a, b) $. Let
$\{ F_n \}_{n=1}^{\infty}$ be a Cauchy sequence of
$H^1(\Gamma, a, b) $. Let $\Gamma_0$ be a connected compact set of
$\Gamma$ such that $\Gamma_0 \subset \Gamma \cap \Omega $. Since
$\Gamma_0$ is closed and meets only a finite number of edges
(\cite{EHB}, Lemma 2.1), it follows that
\begin{equation} 
\inf_{\sigma \in \Gamma_0 \backslash V(\Gamma)} a(\sigma) > 0, \quad 
   \inf_{\sigma \in \Gamma_0 \backslash V(\Gamma)} b(\sigma) > 0
\end{equation}
and hence $\{ F_n \}_{n=1}^{\infty}$ is a Cauchy sequence with
respect to the norm
\begin{equation}
  ||| F |||_{\Gamma_0,1} = \int_{\Gamma_0} \big\{
   \big|\frac{dF}{d\sigma} \big|^2
  + |F(\sigma)|^2\big\} \, d\sigma.   %A.25
\end{equation}
Since a connected compact set $\Gamma_0 \subset \Gamma \cap \Omega$ can
be chosen arbitrarily, we see that there is a function $F$ on
$\Gamma$ such that
\begin{equation}
   ||| F - F_n |||_{\Gamma_0, 1} \to 0 \quad (n \to \infty) %A.26
\end{equation}
for any connected compact $\Gamma_0 \subset \Gamma \cap \Omega $. We may
assume by taking a subsequence of $\{ F_n \}_{n=1}^{\infty}$
if necessary that $F_n(\sigma)$ converges to $F(\sigma)$ for almost
all $\sigma$ in $\Gamma \cap \Omega $. Then, by using the inequality
\begin{eqnarray*}
|F_n(\sigma) - F_m(\sigma)|
  & \le& \big| \int_{\sigma_0}^{\sigma}
  \big|\frac{dF_n(\eta)}{d\eta}
  - \frac{dF_m(\eta)}{d\eta}\big| \, d\eta \big|
   + |F_n(\sigma_0) - F_m(\sigma_0)|   \\
& \le& \int_{\Gamma_0}
  \big|\frac{dF_n(\eta)}{d\eta}
  - \frac{dF_m(\eta)}{d\eta}\big| \, d\eta
   + |F_n(\sigma_0) - F_m(\sigma_0)|  \to 0 
\end{eqnarray*} 
as $n, m \to \infty$ for $\sigma \in \Gamma_0 $, where $\{ F_n \}$
 converges at
$\sigma = \sigma_0 $, we see that $\{ F_n \}$ converges uniformly
on $\Gamma_0 $, and hence $\{ F_n \}$ converges to a
continuous function $F$ on $\Gamma \cap \Omega $. This proves that
$F$ is continuous on $\Gamma \cap \Omega $. Let
$\sigma, \sigma' \in e_j $. Then, by letting $n \to \infty$ in
$$
  F_n(\sigma') - F_n(\sigma)
  = \int_{P(\sigma, \sigma')} \frac{dF_n}{d\eta} \, d\eta,
$$
we have
$$ 
  F(t') - F(t)
  = \int_{P(\sigma, \sigma')} \frac{dF_n}{d\eta} \, d\eta,
$$    %A.29
$F$ is locally absolute continuous on each $e_j $.
By noting that $\Gamma \backslash \Gamma \cap \Omega = \Gamma \cap \partial\Omega$
is a countable set, it is easy to see that $\{ F_n \}$
converges to $F$ in the norm $\|\ \ \|_{\Gamma, a_0, 1} $,
which completes the proof. \hfil$\diamondsuit$
  
  \paragraph{Acknowledgments}
   This work was started during the author's visit to the
University of Heidelberg in the summer of 1999 under the support of
SFB (Sonderforschungsbereich) 359. I would like to express my
gratitude to Professor Willi J\"ager for his hospitality and useful
suggestions.

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\noindent {\sc Yoshimi Saito }\\
Department of mathematics \\
University of Alabama at Birmingham\\
Birmingham, AL 35294, USA. \\
e-mail: saito@math.uab.edu
  
\end{document}