数理解析研究所講究録 1208 巻 2001 年 7-23

7

EIGENVALUE PROBLEMS ON DOMAINS WITH CRACKS
Kazushi Yoshitomi
Graduate School of Mathematics, Kyushu University,
Hakozaki, Fukuoka 812-8581, Japan(yositomi@math.kyushu-u.ac.jp)
0. Introdu ctio
Let$\Omega$be abounded smooth domain in:$\gamma$$\mathrm{R}^{2}$$n$

and let$(t_{0}>0)$$[0, t_{0}]\ni t\mapsto\gamma(t)\in \mathrm{R}^{2}$

be asmooth curve without self-intersection. We impose the following hypotheses on.
, and, $7(0)=0$ ,(H.2) There exists $r_{0}\in(0,t_{0})$ such thaton$\gamma(t)=(t, 0)$$[0, r_{0}]$,$\mathrm{n}\{x\in \mathrm{R}^{2}; |x|<r_{0}\}=\{x=(x_{1}, x_{2})\in \mathrm{R}^{2} ; x_{1}>0, |x|<r_{0}\}$$\Omega$an.
transversely atintersectsconsists of two connected components. Let.
which contains, we set(H.3) The curve
The set$\gamma$.

$\gamma(t\mathrm{o})$$\Omega\pm \mathrm{b}\mathrm{e}$$\Omega\backslash \gamma((0, t_{0}))$the connected component of$(r\mathrm{o}/2, \pm r\mathrm{o}/2)$$\Omega\backslash \gamma((0,t_{0}))$For

and 7.

$\gamma((0,t_{0}))\subset\Omega$$7(\mathrm{t}0)\in\partial\Omega$$0\in\partial\Omega$$(\mathrm{H}\mathrm{i})$$\Omega$$\epsilon\in(0,t_{0})$$\Omega_{\epsilon}=\Omega\backslash \gamma([\epsilon, t_{0}))${$Q_{\epsilon}=$$u\in H^{1}(\Omega_{\epsilon})$;,on CM)},$u=0$$q_{\epsilon}(u, v)=(\nabla u, \nabla v)_{L^{2}(\Omega)}$,$u,v\in Q_{\epsilon}$.

is the negative Laplacianbe the self-adjoint operator associated with the form . The operatoron (( , to)) We
boundary
condition
Neumann’sandonconditionboundary
with Dirichlet’sonis monotonefunctionThe
multiplicity.
withcountedofeigenvalue-ththedenote by
set
0.
We
to
tends
as
and
of
behavior
asymptotic
non-decreasing. We study theLet$L_{\epsilon}$an$L_{\epsilon}$$\Omega_{\epsilon}$$\epsilon$$\gamma$$\lambda_{j}(\cdot)$$L_{\epsilon}$$j$$\lambda_{j}(\epsilon)$$q_{\epsilon}$$\lambda_{2}(\epsilon)$$\lambda_{1}(\epsilon)$$Q_{0}^{\pm}=${$u\in H^{1}(\Omega_{\pm})$;$q_{0}^{\pm}(u, v)--(\nabla u, \nabla v)_{L^{2}(\Omega)}\pm$

Let
of

on

$u=0$

an

$\epsilon$$\cap\partial\Omega\pm$$u,v\in Q_{0}^{\pm}$

’

. Let
be the self-adjoint operator associated with the form
thatassumefurtherWe
repeated according to multiplicity.
$q_{0}^{\pm}$},.

$\lambda_{1}^{\pm}<\lambda_{2}^{\pm}\leq\cdots$$L_{0}^{\pm}$$L_{0}^{\pm}$(H.4)$\lambda_{1}^{+}=\lambda_{1}^{-}$

be the eigenvalues.

which is normalized by the conditions
associated with the eigenvalueisin aneighborhood of 0and
analytic
is real. We claim thaton
given by aconvergent power series expansion:Let

be the eigenvector of$\varphi_{1}^{\pm}$$\lambda_{1}^{\pm}$$L_{0}^{\pm}$$\varphi_{1}^{\pm}$$\varphi_{1}^{\pm}>0$$\Omega\pm \mathrm{a}\mathrm{n}\mathrm{d}$$\varphi_{1}^{\pm}$$||\varphi_{1}^{\pm}||_{L^{2}(\Omega)}\pm=1$$\varphi_{1}^{\pm}(x)=\sum_{j=1}^{\infty}\sum_{k=1}^{j}C_{j,k}^{\pm}r^{2j-1}\cos(2k-1)\theta$

where$(r, \theta)$

is the polar coordinate of$x$$c_{1,1}^{\pm}>0$,

centered at 0. Let$K=\{j\geq 2$ ;We define $\nu=\min K$ if and only if

in aneighborhood of 0 with

$K\neq\emptyset$. We set$\frac{C_{j,j}^{+}}{c_{1,1}^{+}}\neq\frac{C_{j,j}^{-}}{c_{1,1}^{-}}\}$$\lambda_{0}=\lambda_{1}^{+}$$(=\lambda_{1}^{-})$.
. Our main result is the following(1)

8
THEOREM 0.1. The function$\lambda_{2}(\epsilon)$

If K$\neq\emptyset$$\lambda_{2}(\epsilon)$

admits the asymptotic expansion:as$\sim\lambda_{0}+\sum_{m=1}^{\infty}\sum_{n=0}^{m-1}\lambda_{m,n}\epsilon^{2m}(\log\epsilon)^{n}$then the function$\lambda_{1}(\epsilon)$$=\emptyset$then$\lambda_{1}(\epsilon)=\lambda\circ$

$/or$

,

$\lambda_{1,0}>0$.

(2)admits the asymptotic expansion:$\lambda_{1}(\epsilon)\sim\lambda_{0}+\sum_{\dot{|}=2\nu-1}^{\infty}\sum_{j=0}^{[(:-2\nu+1)/3]}\mu_{\dot{l},j}\epsilon^{2:}(\log\epsilon)^{j}$

If K$\epsilonarrow 0$

sufficiently small$\epsilon>0$as$\epsilonarrow 0$,$\mu 2\nu-1,0>0$

.

(3)

.

This work is inspired and motivated by that of Dauge and Helffer [2]. In amore general setting,, wherethey proved thatXj(e)for all
are the rearrangement ofrepeated according to multiplicity. This result interests us in the asymptotic behaviorofas tends to 0. They also suggested the following problem called geometric tunneling. The
problem they suggested is to estimate A2
in the case that is symmetric with respect to the-axis and 7is aline segment on the -axis. Our Theorem 0.1 also solves this problem.
There are many works on asymptotics of eigenvalues of elliptic differential equations on singularly
perturbed domains. R. Gadyl’shin and A. M. Il
considered adomain with anarrow slit. S. Ozawa[9] and V. Maz’ya, S. Nazarov, and B. Plamenevskii [8] considered adomain with asmall hole. S. Jimbo[7], R. Gadyl’shin [5], and J. Arrieta, J. Hale, and Q. Han [1] considered adumbbel domain with ashrinking handle. R. Gadyl’shin $[3, 4]$ considered aproblem to change the boundary condition on asmall part of the boundary. In [3, 4, 5, 6], R. Gadyl’shin and A. M.
used the method of matched
expansion: they decomposed the region into overlapping subregions and constructed asymptotic solutions
on the respective subregions such that the solutions are asymptotically same on the intersection of thesubregions. In [8], V. Maz’ya, S. Nazarov, and B. Plamenevskii used the method of compound expansion
which is somewhat similar to the method of matched expansion.
We use the method of matched expansion to construct an approximate solution in the proof of Theorem
0.1. Our procedure to construct the approximate solution is somewhat similar to the procedure used in[6]. But the form of the approximate solution in this paper differs from that in [3, 4, 5, 6, 8]. In thepresent PaPer, we prove only (2) for simplicity.
$\lim_{\epsilonarrow 0}$$j\in \mathrm{N}$$=\kappa_{j}$$\kappa_{1}\leq\kappa_{2}\leq\cdots$$\{\lambda_{j}^{+}\}_{j=1}^{\infty}\cup\{\lambda_{j}^{-}\}_{j=1}^{\infty}$$\lambda_{j}(\epsilon)$$\epsilon$$\Omega$$(\epsilon)-\lambda_{1}(\epsilon)$$x_{1}$$x_{1}$$’.\mathrm{n}[6]$$\mathrm{I}1’ \mathrm{n}$1. Eigenvector of the limit problem
We begin with introducing some notations and conventions which we use throughout this paper. Forxand r $>0$ , we denote by $D(x,$r) the open disk of radius r centered at x. For r $>0$ , we define$\in \mathrm{R}^{2}$$D(r)$ $=D(0,r)$

,

D (r)

$=\{(x_{1},x_{2})\in \mathrm{D}(\mathrm{r}) \pm x_{1}>0\}$,$D_{+\pm}(r)=\{(x_{1},x_{2})\in D_{+}(r); \pm x_{2}>0\}$.

exists, then we. If the principal valueLet
To avoid cumbersome classification, we use the following conventions about sumdenote it by. We regard an undefined termandmations and sequences: if $p>q$ , we define
defined,then the formula $b_{1}+b_{2}=b_{3}$aredefinedandisnotexample,Forif
and
as 0in formulae.
$f,g\in \mathrm{n}_{\delta\in(0,t\mathrm{p})}L^{2}(\Omega\pm\backslash D_{+\pm}(\delta))$$\lim_{\deltaarrow+0(f,g)_{L^{2}(\Omega\backslash D(\delta))}}\pm+\pm$$(f, g)_{\Omega}\pm\cdot$$\sum_{\dot{|}=p}^{q}a_{t}=0$$b_{3}$$b_{1}$

means$b_{1}=b_{3}$$\{a:\}_{\dot{l}=p}^{q}=\emptyset$$b_{2}$.

andIn order to construct the asymptotic expansion ofctor
near 0. Let us show the following.
behavior of the$\lambda_{1}(\epsilon)$$\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{v}\dot{\mathrm{e}}$$\lambda_{2}(\epsilon)$, we first analyze the asymptotic$\varphi_{1}^{\pm}$

PROPOSITION 1.1. The function
convergent power-series expansion:$\varphi_{1}^{\pm}$is real analytic in a neighborhood of 0and$\varphi_{1}^{\pm}(x)=\sum_{j=1}^{\infty}\sum_{k=1}^{j}C_{j,k}^{\pm}r^{2j-1}\cos(2k-1)\theta$

in a neighborhood of 0urith$\varphi_{1}^{\pm}$is given by$c_{1,1}^{\pm}>0$

.

$a$

9
Proof. We prove the assertion only formethod of reflection. We set

tPi because that forforfor$\varphi_{1}^{+}(x_{1},x_{2})$$\psi(x_{1},x_{2})=\{$$\varphi_{1}^{+}(x_{1}, -x_{2})$The function$\psi(x)$is even with respect to$\psi$$x_{2}$$\in H^{1}(D_{+}(r_{0}))$. Because,$\psi$$\ovalbox{\tt\small REJECT}?\ovalbox{\tt\small REJECT}$$x\in D_{++}(r_{0})$,$x\in D_{+-}(r_{0})$.

$\varphi_{1}^{+}\in Q_{0}^{+}$, we haveon {0}$=0$

is similar. In the proof, we use the$\mathrm{x}$$(-r_{0},r_{0})$.

(4)Moreover we get$-\Delta\psi=\lambda_{0}\psi$

in the sense of distribution on$D_{+}(r_{0})$.

(5)Using (4), (5), and the regularity estimate for elliptic differential equations, we obtain
We set,for$-\psi(-x_{1}, x_{2})$$x\inD_{-}(r_{0}/2)$
for.
$\psi(x_{1}, x_{2})$$\psi$$\in H^{2}(D_{+}(r_{0}/2))$.

$x\in \mathrm{D}(\mathrm{r}\mathrm{o}/2)$$\tilde{\psi}(x_{1}, x_{2})=\{$

The function$H^{1}(D(r_{0}/2))$

is even with respect to. Besides we get$\tilde{\psi}(x)$$-\Delta\tilde{\psi}=\lambda_{0}\overline{\psi}$$x_{2}$and odd with respect toin the sense of distribution on$x_{1}$. Prom (4), we have$D(r_{0}/2)$$\tilde{\psi}\in$.

(6)

Thus the regularity estimate for elliptic differential equations implies that
. Moreover,
the analytic hypoellipticity for elliptic differential equations with analytic coefficients implies that is real
analytic in aneighborhood of 0. Combining this with the fact thaton
, we obtainin aneighborhood of 0. Rewriting this in thepolar coordinate, we infer that there exists
such that$\tilde{\psi}\in C^{\infty}(D(r_{0}/2))$$\tilde{\psi}$$\tilde{\psi}(x_{1}, x_{2})=-\tilde{\psi}(-x_{1}, x_{2})=\tilde{\psi}(x_{1}, -x_{2})$$D(r\mathrm{o}/2)$$\tilde{\psi}(x)=\sum_{j=1}^{\infty}\sum_{k=0}^{\infty}d_{j,k}x_{1}^{2j-1}x_{2}^{2k}$$\tilde{r}_{0}\in(0, r_{0}/2)$$\tilde{\psi}(x)=\sum_{j=1}^{\infty}\sum_{k=1}^{j}C_{j,k}^{+}r^{2j-1}$cooe(2k–1)\mbox{\boldmath $\theta$} on$D(\tilde{r}_{0})$.

(7)

Let us show that $c_{1,1}^{+}>0$ by acontradiction. Note thatin an-neighborhood of 0. Becauseinand $\cos\theta>0$ on $(0, \pi/2)$ , we have
. We assume that $C_{1,1}^{+}=0$ . Using (6), (7), andthe analiticity of near 0, we obtain$\tilde{\psi}=\varphi_{1}^{+}$$\varphi_{1}^{+}>0$$\Omega_{+}$$\Omega_{+}$$C_{1,1}^{+}\geq 0$$\tilde{\psi}$(8)$c_{j+1,k}^{+}=- \frac{\lambda_{0}}{4(j+k)(j+1-k)}c_{j,k}^{+}$

for $j\geq 1$ and $1\leq k\leq j$ . Let us show that for all$c_{j,k}^{+}=0$$j\in \mathrm{N}$

for

,
(9)

$1\leq k\leq j$

by using induction on . (9) is valid for $j=1$ by the assumption. Let. Suppose that (9) is valid
for $j\leq m$ . Then (8) implies that $C_{m+1,k}^{+}=0$ for $1\leq k\leq m$ . So, (7) implies that$j$$m\in \mathrm{N}$$\overline{\psi}(x)=C_{m+1,m+1}^{+}r^{2m+1}$

as. Becausecooe(2m $+1$ )$\theta+\mathcal{O}(r^{2m+3})$

inand $\cos(2m+1)\theta$ takes both positive and negative sign on $(0, \pi/2)$ , we
get $c_{m+1,m+1}^{+}=0$ . Hence (9) is valid for $j\leq m+1$ . Thus (9) holds for all. This together with theanaliticity of at 0 implies thatin aneighborhood of 0. But this violates the fact thatin
. Therefore we get $c_{1,1}^{+}>0$ . This completes the proof of Proposition 1.1.
$rarrow \mathrm{O}$$\varphi_{1}^{+}>0$$\Omega_{+}$$j\in \mathrm{N}$$\overline{\psi}$$\Omega_{+}$$\tilde{\psi}=0$$\varphi_{1}^{+}>0$$\square$10
2. Preliminaries for the construction of the asymptotic expansion of the second eigenvalue
In order to analyze the asmptotic behaviour of the eigenvalues, we use the method of matched expansion(see [6]). We define$\xi=\epsilon^{-1}x$.

the polar coordinate of centered at 0.
The variable is called the inner variable. We denote by
and the associated eigenvector in the
We seek the asymptotic expansions of the second eigenvalue offollowing form.
$\xi$$(\mathrm{p}, \theta)$$\xi$$L_{\epsilon}$$\lambda_{\epsilon}:=\lambda_{0}+\sum_{m=1}^{\infty}\sum_{n=0}^{m-1}\epsilon^{2m}(\log\epsilon)^{n}\lambda_{m,n}$$\varphi_{\epsilon}^{\mathrm{o}\mathrm{u}\mathrm{t},\pm}(x):=\varphi_{0,0}^{\pm}(x)+\sum_{\dot{|}=1}^{\infty}\sum_{j=0}^{i-1}\epsilon^{2:}(\log\epsilon)^{j}\varphi_{i,j}^{\pm}(x)$$\varphi_{\epsilon}^{\mathrm{i}\mathrm{n}}(x):=\sum_{k=1}^{\infty}\sum_{l=0}^{k-1}\epsilon^{2k-1}(\log\epsilon)^{l}v_{k,l}(\xi)$

on

,

(10)

on

$\Omega_{\pm}\backslash D_{+\pm}(\epsilon^{1/2})$$D_{+}(2\epsilon^{1/2})\cap\Omega_{\epsilon}$

asymptotically coincides withsuch thatandWe shall construct(11) into the equation(10)
Inserting.
andregion,equation:
get
we
the
outer
identifying the powers of and$\varphi_{\epsilon}^{\mathrm{o}\mathrm{u}\mathrm{t},\pm}$$\varphi_{\epsilon}^{\mathrm{i}\mathrm{n}}$$\varphi_{\epsilon}^{\mathrm{i}\mathrm{n}}$$D_{+}(2\epsilon^{1/2})\backslash D_{+\pm}(\epsilon^{1/2})$

,

(11)

.

(12)

$\varphi_{\epsilon}^{\mathrm{o}\mathrm{u}\mathrm{t},\pm}$in the intermediate$-\Delta\varphi_{\epsilon}^{\mathrm{o}\mathrm{u}\mathrm{t},\pm}=\lambda_{\epsilon}\varphi_{\epsilon}^{\mathrm{o}\mathrm{u}\mathrm{t},\pm}$and$\log\epsilon$$\epsilon$on$( \Delta+\lambda_{0})\varphi_{p,q}^{\pm}=-\sum_{m=1}^{p}\sum_{j=\max\{0,q-m+1\}}^{\min\{p-m,q\}}\lambda_{m,q-j}\varphi_{p-m,j}^{\pm}$$\Omega\pm$,$(\mathrm{O}\mathrm{U}\mathrm{T})_{p,q}^{\pm}$on$\varphi_{p,q}^{\pm}=0$$\partial\Omega\cap\partial\Omega_{\pm}$,$\frac{\partial}{\partial n}\varphi_{p,q}^{\pm}=0$on$\Gamma:=\gamma((0,t_{0}))$We put$\Pi=\{(\xi_{1},\xi_{2})\in \mathrm{R}^{2}; \xi_{1}>0\}\backslash ([1, \infty)\mathrm{x}\{0\})$Inserting (10) and (12) into theget the inner equation:$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}-\Delta\varphi_{\epsilon}^{\mathrm{i}\mathrm{n}}=\lambda_{\epsilon}\varphi_{\epsilon}^{\mathrm{i}\mathrm{n}}$.

.

and identifying the powers ofon$\Delta v_{p+1,q}=-\lambda_{0}v_{p,q}-\sum_{m=1}^{p-1}\sum_{I=\max\{0,q-m+1\}}^{\min\langle q,p-m-1\}}\lambda_{m,q-l}v_{p-m,l}$

$\Pi$

and

$\epsilon$$\log\epsilon$, we,$(\mathrm{I}\mathrm{N})_{p,q}$$v_{p+1,q}=0$

on {0}$\mathrm{x}\mathrm{R}$,$\frac{\partial}{\partial\xi_{2}}v_{p+1,q(}\cdot,$on$\pm 0)=0$$(1, \infty)$.

We introduce some notations and function spaces. We put$P=(0, \infty)\mathrm{x}(0, \infty)$,$P_{1}^{\delta}=P\backslash (D(0, \delta)\cup D((1,0),$ $\delta))$,$\delta$$\in(0, \frac{1}{2})$.

be the polar coordinate centered at $(1,0)$ . Let $A$ denote the class of function $f\in C^{\infty}(P)$satisfying the following (i) and (ii).

Let$(\tilde{r}, s)$(i).

$f \in C^{1}(D_{++}(\frac{\overline 1}{2}))\cap\bigcap_{\delta\in(0,1/2)}C^{\infty}(\overline{P_{1}^{\delta}})$(ii) There exists aconstant $C>0$ such that$\tilde{r}^{-1/2}|f(\tilde{r}, s)|+\tilde{r}^{-1/2}|\frac{\partial}{\partial s}f(\tilde{r}, s)|+\tilde{r}^{1/2}|\frac{\partial}{\partial\tilde{r}}f(\tilde{r}, s)|\leq C$

on$D((1,0),$.

$\frac{1}{2})\cap P$

11
We identify$\mathbb{R}^{2}$with$\mathbb{C}$by the map. For$\mathrm{R}^{2}\ni(y_{1},y_{2})\mapsto y_{1}+\sqrt{-1}y_{2}\in \mathrm{C}$$\xi(\eta)=\sqrt{\eta^{2}+1}$$\eta\in \mathbb{C}$, we put,where the branch cut of the square root is the positive real axis. Let$g\in C^{\infty}(P)$ such that there exists afunction(ii), and (iii).

$B$denote the class of function
satisfying the following (i),$h\in C^{\infty}(\mathrm{R}^{2}\backslash \{\sqrt{-1}, -\sqrt{-1}\})$

(i)

on .
$P$

$h( \eta)=\frac{|\eta|^{2}}{|\eta^{2}+1|}g(\xi(\eta))$(ii)(iii) The function is bounded near
For $i\geq 0$ and $r>0$ , we define$h(\eta_{1}, \eta_{2})=\mathrm{h}(\mathrm{r}\mathrm{j})-\varphi)=-h(-\eta_{1}, \eta_{2})$$J_{i}(r)=$We choose{ $f\in\dot{C}(D(r))|$.

$\chi\in C^{\infty}([0, \infty))$on$\{\sqrt{-1}\}$$h$$\mathrm{R}^{2}\backslash \{\sqrt{-1}, -\sqrt{-1}\}$and $\{-\sqrt{-1}\}$ .

$\mathrm{f}(\mathrm{x}\mathrm{u}\mathrm{x}2)=\mathrm{f}(\mathrm{x}\mathrm{u}$.

$-\mathrm{x}_{2})=-f(-x_{1}$,on$x_{2})$$\mathrm{X}(\mathrm{r})$}.

such thatfor$\mathrm{X}(\mathrm{r})=1$$r\leq 1$

and

for

$\mathrm{X}(\mathrm{r})=0$$r\geq 2$.

Instead of giving acomplete proof of (2), we prove the following lemma which plays acentral role in
the proof of (2).

LEMMA 2.1. There exist, (IN), OUT
satisfying$\{\varphi_{i,0}^{\pm}\}_{i\geq 0}\cup\{\varphi_{N+:+1,N}^{\pm}\}_{N\geq 1,i\geq 0}$$\lambda_{1,0}>0$$($$)^{\pm}$, and the following, $\{v_{N+:+1,N}\}_{N\geq 0,i\geq 0}$ , and$(P1)\triangleleft P9)$$\{\lambda_{N+:,N}\}_{N\geq\circ,j\geq 1}$.

$(P1)_{i,N}$$v_{N+:+1,N}(\xi)=v_{N+i+1,N}^{+}(\xi)+v_{N+i+1,N}^{-}(\xi)$,$\xi\in\Pi$.

$(P2)_{i,N}$.

$v_{N+i+1,N}^{+}( \xi)=\sum_{j=1}^{i+1}\sum_{k=1}^{j}C_{i,j,k,N}\rho^{2j-1}\cos(2k-1)\theta$

The function$(P4)_{i,N}$ The
functionby $term$ two times.
$(P3)_{i,N}$

is odd, and $v_{\overline{N}+i+1,N}(\xi)\in A\cap B$ .
has the following asymptotic expansion which can be differentiated term$v_{\overline{N}+i+1,N}(\xi_{1}, \cdot)$$v_{\overline{N}+i+1,N}$$v_{\overline{N}+i+1,N}( \xi)\sim(\sum_{l=0}^{[(i+1)/2]}\sum_{j=l-i}^{0}+\sum_{l=[(i+1)/2]+1}^{i}\sum_{j=1}^{l}+\sum_{l=\cdot+1}^{\infty}.\sum_{j=l-:}^{l})C_{\dot{1},l,j,N}^{-}\rho^{2:-2l+1}$

cooe(2j-1)\mbox{\boldmath $\theta$}$+ \sum\sum d_{l,j,l,N}\rho^{2j+2l-1}\log\rho\cos(2j-1)\theta[(i+1)/2]:-2j+1$
$j=1$

$l=0$

, $\xi\in P$ .
as
The function
has the folloing asymptotic expansion which can be differentiated$term$
many
infinitelybyterm
times.
$\rhoarrow\infty$$(P5)_{p,q}$$\varphi_{q+p,q}^{\pm}\in C^{\infty}(\Omega\pm)$$p-1$$\varphi_{q+p,q}^{\pm}(x)$$2p-2$

$p$

$p$

oo

0

$\sim(\sum \sum +\sum\sum+ \sum\infty \sum)k_{s,p,j,q}^{\pm}r^{2s-2p+1}$$s=0j=p-s$$s=pj=1$

cooe(2j–1)\mbox{\boldmath $\theta$}$s=2p-1j=p-s$$+ \sum\sum^{\min\{p,s-p+1,[(s+1)/2]\}}l_{s,j,s-p-j+1,q}^{\pm}r^{2s-2p+1}\log r\infty$

$j=1$

$s=p$

,
as
The function$rarrow \mathrm{O}$$(P6)_{p,q}$$x\in\Omega\pm\cdot$

Besides,$(\varphi_{q+p,q}^{\pm}(x), \varphi_{1}^{\pm})_{\Omega}\pm is$well defined.

,$\varphi_{q+p,q}(x):=\{\begin{array}{l}\varphi_{q+p,q}^{+}(x)\varphi_{q+p,q}^{-}(x)\end{array}$$forfor$$x\in\Omega_{-}x\in\Omega_{+}$

satisfies$(1-\chi(\epsilon^{-1/2}r))\varphi_{q+p,q}(x)\in D(L_{\epsilon})$$/or$

sufficiently small$\epsilon>0$

.

cooe

$(2\mathrm{j} -1)\theta$

12
We

$r$

ewrite the asymptotic expansions of $v_{N+:+1,N}$ andas follows.

$\varphi_{N+:,N}^{\pm}$$v_{N+:+1,N(\xi)\sim\sum_{l=0}^{\infty}h_{\dot{1},l,N}^{\pm}(\rho,\theta)+\sum_{l=1}^{\dot{|}}m_{\dot{1},l,N}^{\pm}(\rho,\theta)\log\rho}$$(\rhoarrow\infty, \xi\in\Pi, \pm\xi_{2}>0)$$\varphi_{N+:,N}^{\pm}(x)\sim\sum_{s=0}^{\infty}t_{s,:,N}^{\pm}(r, \theta)+\sum_{s=}^{\infty}.\cdot u_{s,:,N}^{\pm}(r, \theta)\log r$

where

,

$(rarrow 0, x\in\Omega_{\pm})$

,

(13)

,

(14)

,
and
are homogeneous polynomials of degree $2s-2l+1$ .
$(P7)_{p,q}$ For $p\geq 1$ and $q\geq 0$ , there exists $r_{p,q}>0$ suchthat for all $M\geq 2p-1$ , the function$h_{s,l,q}^{\pm}(\cdot, \theta)$$m_{s,l,q}^{\pm}(\cdot, \theta)$$t_{s,l,q}^{\pm}(\cdot, \theta)$$u_{s,l,q}^{\pm}(\cdot, \theta)$$\varphi_{q+p,q}^{\pm}(x)-(\sum_{s=0}^{M}t_{s,p,q}^{\pm}(r, \theta)+\sum_{s=\dot{l}}^{M}u_{s,p,q}^{\pm}(r, \theta)\log r)$(PS)

can be extended to a function which belongs to $J_{2M-2p+3}(r_{p,q})$ .
The following ’matching conditions’ are satisfied. For $0\leq q$ ,$1\leq l$$u_{s,l,q}^{\pm}(\cdot,\theta)=m_{s,l,q}^{\pm}(\cdot, \theta)$, and $l\leq s$ , we have.

(M.

$1)_{s,l,q}^{\pm}$

For $0\leq q$, $0\leq s$ , and $0\leq l$ , we have.

$t_{s,l,q}^{\pm}(\cdot,\theta)=h_{s,l,q}^{\pm}(\cdot, \theta)-m_{s+1,l+1,q-1}^{\pm}(\cdot, \theta)$

(P9)

For N

$\geq 0$and M$\geq 1$$(M.2)_{s,l,q}^{\pm}$, The following ‘compatibility condition’ holds.

$(C_{1,1}^{+})^{-1} \{-\sum_{s=0}^{M-1}\frac{\pi}{2}(2M-2s-1)C_{s,M,M-s}^{-},{}_{NM-s,M-s}C^{+}$-$\sum_{s=1}^{N+M-1}\sum_{j=N-s+1}^{N+M-s}\lambda_{s,N-j}(\varphi_{N+M-s,j}^{+}, \varphi_{1}^{+})_{\Omega_{+}}\}$$=-(C_{1,1}^{-})^{-1} \{\sum_{s=0}^{M-1}\frac{\pi}{2}(2M-2s-1)C_{s,M,M-s}^{-},{}_{NM-s,M-s}C^{-}$-REMARK. The number$\lambda_{M+N,N}$$\sum_{s=1}^{N+M-1}\sum_{j=N-s+1}^{N+M-s}\lambda_{s,N-j}(\varphi_{N+M-s,j}^{-}, \varphi_{1}^{-})_{\Omega_{-}}\}$.

$(C)_{M,N}$is equal to the both sides of (C) $)_{M,N}$ . Besides, it can be proved thatfor $N\geq 0$ .

$\lambda_{N+1,N}=2^{-2N-3}\pi\lambda 0^{N}\{(C_{1,1}^{+})^{2}+(C_{1,1}^{-})^{2}\}$

REMARK. The matching conditions in (PS) are derived as follows. Inserting (14) into (11) we
obtain aformal power series of andwhose coefficients are polynomials of . Inserting (13) into(12) and using the coordinate change, we also obtain aformal power series of andwhose
coefficients are polynomials of . Identifying the powers of andof these two formal power series, we
getand. Similarly, we obtainandby using (11) (12), (13)
(14)
and
$,$

$\epsilon$$\log\epsilon$$r$$\xi=\epsilon^{-1}x$$r$$(M.1)_{s,l,q}^{+}$$(M.1)_{s,l,q}^{+}$$\log\epsilon$$\epsilon$$\epsilon$$(M.1)_{s,l,q}^{-}$$\log\epsilon$$(M.1)_{s,l,q}^{-}$

$,,$

$,$

$.$

Lemma 2.1 can be proved by induction. For simplicity, we give aconstruction only for the first two
terms of (11)\pm and (12). We first give the precise form of, which are the coefficients of theandleading terms of (ll)\pm and (12) respectively. To this end, we introduce some harmonic functions. For, we defin$U_{j}(\xi)=\Re((\sqrt{\xi^{2}-1})^{2j-1})$ ,,$\varphi_{0,0}^{\pm}$$v_{1,0}$$j\in \mathrm{N}$$\xi\in\Pi$13
is harmonic on n,
where the branch cut of the square root is the positive real axis. We claim that
expansion which
asymptotic
followingtheMoreover,has
odd
function.
an
is
,
and
c
many
times.
infinitely
term
by
term
can be differentiated$U_{\ovalbox{\tt\small REJECT}}$$Ar^{\ovalbox{\tt\small REJECT}_{*}}I\mathit{3}$$U_{\ovalbox{\tt\small REJECT}}$$U_{\ovalbox{\tt\small REJECT}}(\mathrm{C}_{\mathrm{Z}:}\ovalbox{\tt\small REJECT})\ovalbox{\tt\small REJECT}$$U_{\ovalbox{\tt\small REJECT}}$$U_{j}( \xi)\sim\pm\sum_{l=0}^{\infty}\tau_{j,l}\rho^{2j-2l-1}\cos(2j-2l-1)\theta$

where$\tau_{1,1}=-\frac{1}{2}$,$\mathrm{T}2,\mathrm{i}=-\frac{3}{2}$, and $\tau_{j,0}=1$ for$j\in \mathrm{N}$as

Then we get$\varphi_{0,0}^{\pm}(x)=\pm(C_{1,1}^{\pm})^{2}r\cos\theta+O(r^{3})$$\varphi_{0,0}^{\pm}$satisfies$(\mathrm{O}\mathrm{U}\mathrm{T})_{0,0}^{\pm}$,$\pm\xi_{2}>0$

,

(15)

.

as

$rarrow \mathrm{O}$

,

(16)

$x\in\Omega\pm\cdot$:$(\Delta+\lambda_{0})\varphi_{0,0}^{\pm}=0$on$\varphi_{0,0}^{\pm}=0$$\xi\in\Pi$. We recall Proposition 1.1. We put$\varphi_{0,0}^{\pm}=\pm C_{1,1}^{\pm}\varphi_{1}^{\pm}$

We claim that,$\rhoarrow\infty$$\partial\Omega\cap\partial\Omega_{\pm}$,on$\Omega\pm$,on$\frac{\partial}{\partial n}\varphi_{0,0}^{\pm}=0$$\Gamma$.

We put$v_{1,0}^{+}( \xi)=\frac{1}{2}((C_{1,1}^{+})^{2}-(C_{1,1}^{-})^{2})\rho\cos\theta$

The functions

and$v_{1,0}^{+}(\xi_{1}, \cdot)$,,$v_{1,0}^{-}(\xi_{1}, \cdot)$

are even and odd respectively. We claim that$\Delta v_{1,0}--0$$v_{1,0}=0$

Hence, the functions$\varphi_{0,0}^{\pm}$andon {0}$v_{1,0}$$v_{1,0}(\xi)=v_{1,0}^{+}(\xi)+v_{1,0}^{-}(\xi)$$v_{1,0}^{-}( \xi)--\frac{1}{2}((C_{1,1}^{+})^{2}+(C_{1,1}^{-})^{2})U_{1}(\xi)$$\mathrm{x}\mathrm{R}$

,

on

$\Pi$

$v_{1,0}$satisfies.

$(\mathrm{I}\mathrm{N})_{1,0}$

:

,

on

$\frac{\partial}{\partial\xi_{2}}v_{1,0}(\cdot, \pm 0)=0$$(1, \infty)$.

satisfy $(P1)_{0,0}-(P7)_{0,0}$ . From (15), we obtain$v_{1,0}(\xi)=\pm(C_{1,1}^{\pm})^{2}\rho\cos\theta+\mathcal{O}(\rho^{-1})$as$\rhoarrow\infty$(1) and (15) allow us to write the asymptotic expansions of$\varphi_{0,0}^{\pm}(x)$”$\sum_{s=0}^{\infty}t_{s,0,0}^{\pm}(r, \theta)$$v_{1,0}( \xi)\sim\sum_{l=0}^{\infty}h_{0,l,0}^{\pm}(\rho, \theta)$asas$\rhoarrow\infty$,$\xi\in\Pi$and$\varphi_{0,0}^{\pm}$$rarrow 0$,,$v_{1,0}$

,

,

.

(17)

as follows.

$x\in\Omega_{\pm}$$\xi\in\Pi$$\pm\xi_{2}>0$,$\pm\xi_{2}>0$,

are homogeneous polynomials of degree $(2s+1)$ and $(-2l+1)$ respectively.
and
where
(17)
that
(16)
and
It follows from$t_{s,0,0}^{\pm}(\cdot, \theta)$$h_{0,l,0}^{\pm}(\cdot, \theta)$$t_{0,0,0}^{\pm}(\cdot, \theta)=h_{0,0,0}^{\pm}(\cdot, \theta)$.

holds. Moreover, (C)1,0 holds.
So. We shall prove the following.
, and,Let us construct$(M.2)_{0,0,0}^{\pm}$$\varphi_{1,0}^{\pm}$$\lambda_{1,0}$$v_{2,0}$,, and $\lambda_{1,0}>0$ satisfying $(IN)_{1,0}$ ,LEMMA 2.2. There exist$0\leqs\leq
1,0\leq
l\leq 1;(C)_{2,0}$ .
;
for$\varphi_{1,0}^{\pm}$$(M.1)_{1,1,0}^{\pm}$$\mathrm{v}2\mathrm{f}\mathrm{o}$$(M.2)_{s,l,0}^{\pm}$

We need the following proposition in the proof of Lemma 2.2.

$(OUT)_{1,0}^{\pm}$, $(P1)_{1,0}-(P7)_{1,0}$ ,14PROPOSITION 2.3. Suppose that gE B has the following asymptotic expansion which can be differentiated term by term two times.
$g( \xi)\sim\sum_{l=0}^{\infty}a0,\iota,\iota\rho^{-2l+1}$ascooe(21-1)\mbox{\boldmath $\theta$}$\rhoarrow\infty$,$\xi\in P$

.

(18)

Then the problem$-\Delta v=g$

on P,$\frac{\partial}{\partial\xi_{2}}v=0$on$\Gamma_{1}:=(1, \infty)$x{0},

v $=0$

on(19)$\Gamma_{2}:=\partial P\backslash \Gamma_{1}$has a solution v $\in A\cap B$ having the following asymptotic expansion which can be differentiated term byterm two times.
$v( \xi)\sim(\sum 1\sum 0+\sum\infty\sum l)a_{1,l,j}\rho^{-2l+3}$$l=0j=l-1$

coe $(2j-1)\theta+b_{1,0,0}\rho\log\rho\cos\theta$as$\rhoarrow\infty$,.

$\xi\in P$$l=2j=l-1$

To prove this proposition, we need the following.

PROPOSITION 2.4. Suppose that the function

satisfies the following

$f$

(i)
.
(ii) The function $f$ is bounded nearand $\{-\sqrt{-1}\}$ .
(iii) $f(y_{1}, \eta_{2})=f(y_{1}, -y_{2})=-f(-y_{1}, y_{2})$ on.
(iv) There exists an integer $N\geq 10$ such that for any multi-index
such thatfor $|y|\geq 2$ .

$(\mathrm{i})-(\mathrm{i}\mathrm{v})$.

$f\in C^{\infty}(\mathrm{R}^{2}\backslash \{\sqrt{-1}, -\sqrt{-1}\})$$\{\sqrt{-1}\}$$\mathrm{R}^{2}\backslash \{\sqrt{-1}, -\sqrt{-1}\}$$\alpha$$C_{\alpha}>0$

satisfying$|\alpha|\leq 2$, there exists$|\partial^{\alpha}f(y)|\leq C_{\alpha}|y|^{-N-|\alpha|}$We put$u( \eta)=\frac{1}{2\pi}\int_{\mathrm{R}^{2}}\log|\eta-y|f(y)dy$.

Then we have the following.
$u\in C^{\infty}(\mathrm{R}^{2}\backslash \{\sqrt{-1}, -\sqrt{-1}\})\cap C^{1}(\mathrm{R}^{2})$-Au $=f$ on$u(\eta_{1}, \mathrm{b})$Moreover,$u$$\mathrm{R}^{2}\backslash \{\sqrt{-1}, -\sqrt{-1}\}$$=u(\eta_{1}, -m)=-u(-\eta_{1}, m)$

.

.

on

.

$\mathrm{R}^{2}$

has the following asymptotic expansion which can be differentiated term by term tetto times.
$+\mathcal{O}(\tilde{\rho}^{-[N/2]+3})$$u(\eta)=1\leq$$j \equiv 1\mathrm{m}\mathrm{o}\mathrm{d} 2\sum_{j\leq[N/2]-4},C_{j}\tilde{\rho}^{-j}\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{e}j\omega$as$\tilde{\rho}arrow\infty$, where$(\tilde{\rho}, \omega)$is the polar coordinate of$\eta$.

Since this proposition is derived by astraightforward computation, we omit the proof. By using this
proposition, we shall prove the following.

PROPOSITION 2.5. Let $g(\xi)\in B$ . Suppose that there eists an integer $N\geq 10$ such that for any
multi-index satisfying, there eists
such that$\alpha$$|\alpha|\leq 2$$C_{\alpha}>0$$|\partial_{\xi}^{\alpha}g(\xi)|\leq C_{\alpha}|\xi|^{-N-|\alpha|}$for,$\xi\in P$$|\xi|\geq 2$

.

(20)

Then the boundary value problem$-\Delta v(\xi)=g(\xi)$

on

,

$P$

(21)
$\frac{\partial}{\partial\xi_{2}}v=0$on$\Gamma_{1}$

,

$v=0$

on

$\Gamma_{2}$15
has a unique solution vE A
term by term two times.

having the following asymptotic expansion which can be differentiated$f^{\ovalbox{\tt\small REJECT}*}B$as$+\mathcal{O}(\rho^{-[N/2]+3})$$v(\xi)=1\leq$$\rho=|\xi|arrow\infty$

.

(22)

$j \equiv 1\mathrm{m}\mathrm{o}\mathrm{d} 2\sum_{j\leq[N/2]-4},C_{j}\rho^{-j}\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{e}\mathrm{j}\theta$

Proof. It is easy to see that

and$(0, \infty)$$\mathrm{x}\{0\}$onto$\Gamma_{2}$andis aconformal map ffom $P$ onto $P$ and that maps {0}respectively. Thus the problem (21) is equivalent to the problem$\xi=\xi(\eta)$$\Gamma_{1}$onWe setexists $C_{\alpha}>0$ such thatsatisfies the assumptions$\mathrm{f}(\mathrm{n})=\overline{|}^{\frac{|\eta|^{2}}{\eta+1|}g(\xi(\eta))}$$(0, \infty)$$\mathrm{x}\{0\}$

,

,

in

$- \Delta u(\eta)=\frac{|\eta|^{2}}{|\eta^{2}+1|}g(\xi(\eta))$$\frac{\partial}{\partial\eta_{2}}u=0$$P$

on {0}$u=0$.
$|\alpha|\leq 2$$\alpha$$f$$|\eta|\geq 2$$|\partial_{\eta}^{\alpha}f(\eta)|\leq C_{\alpha}|\eta|^{-N-|\alpha|}$$(\mathrm{i})-(\mathrm{i}\mathrm{i}\mathrm{i})$such thaton$-\Delta u=f$$\mathrm{R}^{2}\backslash \{\sqrt{-1}, -\sqrt{-1}\}$$u(\eta_{1}, \eta_{2})=u(\eta_{1}, -\eta_{2})=-u(-\eta_{1}, \eta_{2})$$u$$\mathrm{x}(0, \infty)$, there. From (20), we claim that for any multi-index satisfying$g\inB$$\eta\inP$, the function,. Becauseforin Proposition 2.4. From Proposition 2.4, we claim that there exists$u\in C^{\infty}(\mathrm{R}^{2}\backslash \{\sqrt{-1}, -\sqrt{-1}\})\cap C^{1}(\mathrm{R}^{2})$Moreover, the function$\mathrm{x}[0, \infty)$

$\xi$

,

on

$\mathrm{R}^{2}$.

has the following asymptotic expansion.

as$=\mathcal{O}(\tilde{\rho}^{-[N/2]+3-m})$(23)$\tilde{\rho}arrow\infty$$\partial_{\overline{\rho}}^{m}\partial_{\omega}^{n}(u(\eta)-1\leq j\equiv 1\mathrm{m}\mathrm{o}\mathrm{d} 2\sum_{j\leq[N/2]-4},C_{j}\tilde{\rho}^{-j}\cos j\omega)$

being the polar coodinate of . We put $\eta(\xi)=\sqrt{\xi^{2}-1}$ , where the branch cut of thefor $m+n\leq 2$ ,. We set $v(\xi)=u(\eta(\xi))$ .
square root is the positive real axis. This function is the inverse function of
$v\in
A\cap
B$ . Letbe theThen we infer that is asolution of (21). Besides, it can be verified thatpolar coordinate of . Then we have$(\mathrm{p}, \omega)$$\eta$$\xi(\eta)$$(\rho, \theta)$

$v$

$\xi$

$\tilde{\rho}^{-j}\cos j\omega$(24)$= \Re(\eta(\xi)^{-j})=\sum_{k=0}^{\infty}\tau_{j,k}\rho^{-j-2k}\cos(j-2k)\theta$, where the above summation can be differentiated term by term infinitely many times. By afor
direct computation, we infer that there exists $C>0$ such that$\rho\geq 2$$|\eta(\xi)|\leq C|\xi|$

,

,

and

$| \frac{d}{d\xi}\eta(\xi)|\leq c$$| \frac{d^{2}}{d\xi^{2}}\eta(\xi)|\leq C|\xi|^{-3}$for$|\xi|\geq 2$

,

(25)

stands for the differentiation by the complex variable . Using (23), (24), and (25), we obtain
where
(22).
Let us show the uniqueness of the solution of (21) which has the asymptotic expansion of the form
(22). Let , $v_{2}\in A$ be the solutions of (21) which have the asymptotic expansion of the form (22). We
set $h=v_{1}-v_{2}$ . Then we get$\xi$$\frac{d}{d\xi}$$\mathrm{v}\mathrm{i}$$-\Delta h=0$

on

,

$P$

$\frac{\partial}{\partial\xi_{2}}h=0$

From (22), we infer that for any multi-index$\alpha$onsatisfying$|\partial_{\xi}^{\alpha}h(\xi)|\leq C_{\alpha}|\xi|^{-1-|\alpha|}$$\Gamma_{2}$

,

$h=0$

$|\alpha|\leq 2$

for

on

$\Gamma_{1}$

.

(26)

, there exists $C_{\alpha}>0$ such that$|\xi|\geq 2$

.

(22)

16
For

$\delta$$\in(0.’1)\pi$

and $R>2$ , we put$Q_{\delta,R}=D_{++}(R)\backslash \overline{D(1,\delta)}$

We have
Green’s formula, we obtain$h\in C^{\infty}(Q_{\delta,R})\cap C^{1}(\overline{Q_{\delta,R}})$

,

. Let

$\Lambda_{\delta}=\partial D(1, \delta)\cap\partial Q_{\delta,R}$$(\tilde{r}, s)$,$\mathrm{A}^{R}=\partial D(R)\cap\partial Q_{\delta,R}$

be the polar coodinate centered at $(1, 0)$ . Using (26) and$0= \int_{Q_{\delta.R}}(-\Delta h)hd\xi=\int_{Q_{\delta,R}}|\nabla h|^{2}d\xi+\int_{\Lambda_{\delta}}\frac{\partial}{\partial\tilde{r}}h\cdot hdS+\int_{\Lambda^{R}}\frac{\partial}{\partial\rho}h\cdot hdS$

From (27), we get$| \int_{\mathrm{A}^{R}}\frac{\partial}{\partial\rho}h\cdot hdS|\leq CR^{-2}arrow 0$$\mathit{6}arrow 0$$Rarrow\infty$

.

(28)

. Since $h\in A$, we get$=0$ . Because $h=0$ onin (28), we getas$Rarrow\infty$as a. Taking the limit
andhave $h=0$ on $P$ . This completes the proof of Proposition 2.5.
Proof of Proposition 2.3. By using formulae$arrow 0$.

$| \int_{\Lambda_{\delta}\pi}\partial h\cdot hdS|\leq C\deltaarrow 0$$\int_{P}|\nabla h|^{2}\not\in$$\Gamma_{1}$, we$\square$$\Delta(\rho^{2j+2m-1}\cos(2j-1)\theta)=4m(2j+m-1)\rho^{2j+2m-3}\cos(2j-1)\theta$,(29)

A $(\rho^{2j+2m-1}\log\rho\cos(2j-1)\theta)=4m(2j+m-1)\rho^{2j+2m-3}\log\rho\cos(2j-1)\theta$$+2(2j+2m-1)\rho^{2j+2m-3}\cos(2j-1)\theta$,$j$,$m\in \mathbb{Z}$

,

(30)

we claim that there exists aformal power series$( \sum_{l=0j}^{1}\sum_{=l-1}^{0}+\sum_{l=2j}^{\infty}\sum_{=l-1}^{l})\tilde{a}_{1,l,j}\rho^{-2l+3}$

cooe(2j-1)$\theta+\tilde{b}_{1,0,0}\rho\log\rho$coe$\theta$such that the function$v_{N}( \xi):=(\sum_{l=0j}^{1}\sum_{=l-1}^{0}+\sum_{l=2j}^{N}\sum_{=l-1}^{l})\tilde{a}_{1,l,j}\rho^{-2l+3}\omega \mathrm{s}(2j-1)\theta+\tilde{b}_{1},0,0\rho\log\rho\cos\theta$satisfies$- \Delta v_{N}(\xi)=\sum_{l=0}^{N}a0,\iota,\iota\rho^{-2l+1}$$\mathrm{c}\mathrm{o}\mathrm{e}(2l-1)\theta=$:$g_{N}(\xi)$for all $N\geq 2$ .
Let $N$ be an arbitrary integer satisfying $-2N-1\leq-10$ . Prom (18), we have$O(\rho^{2s-2N-1-|\alpha|})$ asfor. We choosesuch thatfor $s\leq 2$ ,for $s\geq 3$ . We put. We seek asolution of (19) which takes the form$\partial_{\xi}^{\alpha}(g(\xi)-g_{N}(\xi))=$$\rhoarrow \mathrm{o}\mathrm{o}$$\tilde{\chi}_{1}(s)=1$$|\alpha|\leq 2$$\tilde{\chi}_{1}\in C^{\infty}([0, \infty))$$\tilde{\chi}_{1}(s)=0$$\chi_{1}(\xi)=\tilde{\chi}_{1}(|\eta(\xi)|)$$v=\tilde{v}_{N}+\chi_{1}v_{N}$Inserting (31) into (19), we derive the equation for$\tilde{v}_{N}$

.

(31)

:

on

$-\Delta\tilde{v}_{N}=(1-\chi_{1})g+x1(g-g_{N})+2\nabla_{X1}\cdot\nabla v_{N}+v_{N}\Delta_{X1}=:h_{N}$

,

$P$

(32)
$\frac{\partial}{\partial\xi_{2}}\tilde{v}_{N}=0$on$\Gamma_{1}$We have $h_{N}\in B$ andthat the problem (32) has asolution
be differentiated term by term two times.

$\partial_{\xi}^{\alpha}h_{N}(\xi)=O(\rho^{-2N-1-|\alpha|})$$\tilde{v}_{N}\in A\cap B$$\tilde{v}_{N}=$$j \equiv 1\mathrm{m}\mathrm{o}’ \mathrm{d}2\sum_{j=1}^{N-4}c_{N\dot{\rho}}\rho^{-j}$coe,$\tilde{v}_{N}=0$on$\mathrm{r}_{2}$.

for. Thus, Proposition 2.5 implies
having the folowing asymptotic expansion which canas$\rhoarrow\infty$$|\alpha|\leq 2$$j\theta+\mathcal{O}(\rho^{-N+s+3})$as$\rhoarrow\infty$

.

(33)

We show that the right side of (31) is independent of the choice of $N$ . Letandbe arbitrary
integers satisfying $-2N_{j}-1\leq-10(j=1,2)$ . We set.
Prom the definition ofand (33), we getasfor. As in the proof of
Proposition 2.5, we obtain $w=0$ . This completes the proof of Proposition 2.3. 0$N_{1}$$N_{2}$$w=(\tilde{v}_{N_{1}}+\chi_{1}v_{N_{1}})-(\tilde{v}_{N_{2}}+\chi_{1}v_{N_{2}})(\in A\cap B)$$v_{N}$$\partial_{\xi}^{\alpha}w=\mathcal{O}(\rho^{-1-|\alpha|})$$\rhoarrow\infty$$|\alpha|\leq 2$

17
3. Construction of the asymptotic expansion of the second eigenvalue
In this section, we complete the proof of Lemma 2.2.
Proof 0F LEMMA 2.2. Let us constructformal solutionof:$\varphi_{1,0}^{\pm}$. By using formulae (29) and (30), one can construct a$(\Delta+\lambda_{0})\varphi=0$$\check{\varphi}_{1,0}^{\pm}$$\check{\varphi}_{1,0}^{\pm}(x)=\sum_{s=0}^{\infty}\tilde{h}_{s,1,0}^{\pm}(r, \theta)+\sum_{s=1}^{\infty}\tilde{m}_{s,1,0}^{\pm}(t, \theta)\log r$$= \tilde{k}_{0,1,1,0}^{\pm}r^{-1}\cos\theta+\sum_{s=1j}^{\infty}\sum_{=1-s}^{0}\tilde{k}_{s,1,j,0}^{\pm}r^{2s-1}\cos(2j-1)\theta+\sum_{s=1}^{\infty}\tilde{l}_{s,1,s-1,0}^{\pm}r^{2s-1}\log r\mathrm{c}\mathrm{o}s\theta$(34)

satisfying$\tilde{h}_{0,1,0}^{\pm}(r, \theta)=h_{0,1,0}^{\pm}(r, \theta)$

where$\tilde{h}_{s,1,0}^{\pm}(\cdot, \theta)$

and

,

(35)

are homogeneous polynomials of degree $2s-1$ . For$\tilde{m}_{s,1,0}^{\pm}(\cdot, \theta)$$\check{\varphi}_{1,0}^{N\pm}(x)=\sum_{s=0}^{N}\tilde{h}_{s,1,0}^{\pm}(r, \theta)+\sum_{s=1}^{N}\tilde{m}_{s,1,0}^{\pm}(r, \theta)\log r$

We choosesolutionofsuch that $\chi_{0}=1$ on
which takes the form$\chi_{0}\in C^{\infty}([0, \infty))$$\check{\varphi}_{1,0}^{N\pm}$$(\mathrm{O}\mathrm{U}\mathrm{T})_{1,0}^{\pm}$$\check{\varphi}_{1,0}^{N\pm}(x)=\chi_{0}(r)\varphi_{1,0}^{N\pm}(x)+\tilde{\varphi}_{1,0}^{N\pm}(x)$,$[0, \tilde{r}_{0}/4]$,, we put.

and $\chi_{0}=0$ on$\tilde{\varphi}_{1,0}^{N\pm}\in D(L_{0}^{\pm})$$N\in \mathrm{N}$(36)$[\tilde{r}_{0}/2, \infty)$. We first seek a.

(37)$(\check{\varphi}_{1,0}^{N\pm}, \varphi_{1}^{\pm})_{\Omega}\pm=0$

We have.

(38)$(\Delta+\lambda_{0})\check{\varphi}_{1,0}^{N\pm}=(\Delta+\lambda_{0})\tilde{\varphi}_{1,0}^{N\pm}+\chi \mathrm{o}(r)(\Delta+\lambda_{0})\varphi_{1,0}^{N\pm}+\varphi_{1,0}^{N\pm}\Delta\chi_{0}(r)+2\nabla\chi_{0}(r)\cdot\nabla\varphi_{1,0}^{N\pm}$

Substitutingofinand using (37), (38), and the fact that
, we get the equations for$\check{\varphi}_{1,0}^{N\pm}$$(\Delta+\lambda_{0})\varphi=0$

for$\varphi_{1,0}^{\pm}$$(\mathrm{O}\mathrm{U}\mathrm{T})_{1,0}^{\pm}$$\check{\varphi}_{1,0}^{\pm}$is aformal solution$\tilde{\varphi}_{1,0}^{N\pm}:$on$(\Delta+\lambda_{0})\tilde{\varphi}_{1,0}^{N\pm}=f_{N,1}^{\pm}-\lambda_{1,0}\varphi_{0,0}^{\pm}$$\Omega_{\pm}$,(39)$\tilde{\varphi}_{1,0}^{N\pm}=0$on$\partial\Omega\cap\partial\Omega_{\pm}$,on$\frac{\partial}{\partial n}\tilde{\varphi}_{1,0}^{N\pm}=0$$\Gamma$,$(\tilde{\varphi}_{1,0}^{N\pm}, \varphi_{1}^{\pm})_{L^{2}(\Omega)}\pm---(\chi \mathrm{o}(r)\varphi_{1,0}^{N\pm}, \varphi_{1}^{\pm})_{\Omega}\pm$

’

where

.

$f_{N,1}^{\pm}:=-\varphi_{1,0}^{N\pm}\Delta\chi_{0}-2\nabla\chi_{0}\cdot\nabla\varphi_{1,0}^{N\pm}-\chi_{0}(r)\lambda_{0}(\overline{h}_{N,1,0}^{\pm}(r, \theta)+\tilde{m}_{N,1,0}^{\pm}(r, \theta)\log r)$

Let $N\geq 2$ . Let us show that
. Sincebelonging to$f_{N,1}^{\pm}\in L^{2}(\Omega_{\pm})$

and that the function

can be extended to afunction$f_{N,1}^{\pm}$$J_{2N-3}(\tilde{r}_{0}/4)$$\tilde{h}_{N,1,0}^{\pm}(r, \theta)=\sum_{j=1-N}^{0}\tilde{k}_{N,1,j,0}^{\pm}r^{2s-1}\cos(2j-1)\theta$and,$\tilde{m}_{N,1,0}^{\pm}(r, \theta)\log r=\tilde{l}_{N,1,N-1,0}^{\pm}r^{2N-1}\log r\cos\theta$

we have$\tilde{h}_{N,1,0}^{\pm}(r, \theta)\in\bigcap_{i=0}^{\infty}\bigcap_{r>0}J_{\dot{l}}(r)$Because Axo $=0$ and$\nabla\chi_{0}=0$and$\tilde{m}_{N,1,0}^{\pm}(r, \theta)\log r\in\bigcap_{r>0}J_{2N-3}(r)$

.

(40)

in aneighborhood of 0, we have$- \varphi_{1,0}^{N\pm}\Delta\chi_{0}-2\nabla\chi 0^{\cdot}\nabla\varphi_{1,0}^{N\pm}\in\bigcap_{i=0}^{\infty}J_{i}(\tilde{r}0/4)$.

Thus we obtain$f_{N,1}^{\pm}\in J_{2N-3}(\tilde{r}0/4)$

.

(41)

18
Moreover we obtain.

$-\varphi_{1,0}^{N\pm}\Delta\chi_{0}-2\nabla\chi_{0}\cdot\nabla\varphi_{1,0}^{N\pm}-\chi_{0}(r)\lambda_{0}(\tilde{h}_{N,1,0}^{\pm}(r, \theta)+\tilde{m}_{N,1,0}^{\pm}(r, \theta)\log r)\in C^{\infty}(\mathrm{R}^{2}\backslash D(\tilde{r}_{0}/16))$

Thus, we get

.

(42)

$f_{N,1}^{\pm}\in L^{2}(\Omega_{\pm}\backslash D(\tilde{r}0/8))$

Using (41) and (42), we get.
Let us consider the following equations derived by substituting$f_{N,1}^{\pm}\in L^{2}(\Omega_{\pm})$$\lambda_{1,0}^{N\pm}$on$(\Delta+\lambda_{0})\tilde{\varphi}_{1,0}^{N\pm}=f_{N,1}^{\pm}-\lambda_{1,0}^{N\pm}\varphi_{0,0}^{\pm}$for Aito in (39).

$\Omega_{\pm}$

,
(43)

on

$\tilde{\varphi}_{1,0}^{N\pm}=0$

Because$\lambda\circ$$\partial\Omega\cap\partial\Omega_{\pm}$,$\frac{\partial}{\partial n}\tilde{\varphi}_{1,0}^{N\pm}=0$

is asimple eigenvalue of$L_{0}^{\pm}$on$\Gamma$,$(\tilde{\varphi}_{1,0}^{N\pm}, \varphi_{1}^{\pm})_{L^{2}(\Omega)}\pm=-(\chi_{0}(r)\varphi_{1,0}^{N\pm}, \varphi_{1}^{\pm})_{\Omega}\pm\cdot$, the equation (43) has asolution$\tilde{\varphi}_{1,0}^{N\pm}\in D(L_{0}^{\pm})$

if and only if

.

(44)

$\lambda_{1,0}^{N\pm}=\frac{(f_{N,1}^{\pm},\varphi_{1}^{\pm})_{L^{2}(\Omega)}\pm}{(\varphi_{0,0}^{\pm},\varphi_{1}^{\pm})_{L^{2}(\Omega)}\pm}$by (44). Then (43) has aunique solution which belongs to
We define. We denote it by.
,
Now we compute the asymptotic expansion ofas
For this purpose, we firstcompute the asymptotic expansion ofby the reflection argument used in the proof of Proposition
1.1. Prom (41), we claim that there exists
such that$D(L_{0}^{\pm})$$\lambda_{1,0}^{\pm}$$xarrow \mathrm{O}$$\check{\varphi}_{1,0}^{N\pm}(x)$$\tilde{\varphi}_{1,0}^{N\pm}$$x\in\Omega\pm\cdot$$\tilde{\varphi}_{1,0}^{N\pm}$$\tilde{f}_{N,1}^{+}\in C^{2N-3}(D(\tilde{r}0/4))$$\tilde{f}_{N,1}^{+}=f_{N,1}^{+}$

Let$\Psi_{1,0}^{N+}(x)$on$D_{++}(\tilde{r}0/4)$,$\tilde{f}_{N,1}^{+}(x_{1}, x_{2})=\tilde{f}_{N,1}^{+}(x_{1}, -x_{2})=-\tilde{f}_{N,1}^{+}(-x_{1},x_{2})$on$D(\tilde{r}0/4)$.

be such that$\Psi_{1,0}^{N+}(x)=\tilde{\varphi}_{1,0}^{N+}(x)$on$D_{++}(\tilde{r}0/8)$,$\Psi_{1,0}^{N+}(x_{1},x_{2})=\Psi_{1,0}^{N+}(x_{1}, -x_{2})=-\Psi_{1,0}^{N+}(-x_{1}, -x_{2})$on$D(\tilde{r}0/8)$.

As in the proof of Proposition 1.1, we obtain$\Psi_{1,0}^{N+}\in H^{1}(D(\tilde{r}0/8))$$(\Delta+\lambda_{0})\Psi_{1,0}^{N+}=\tilde{f}_{N,1}^{+}-\lambda_{1,0}^{N+}\varphi_{0,0}^{+}$

,

on

.

$D(\tilde{r}0/8)$

From (41), we get

(45)

.

(46)

$\tilde{f}_{N,1}^{+}-\lambda_{1,0}^{N+}\varphi_{0,0}^{+}\in C^{2N-3}(D(\tilde{r}0/8))\subset H^{2N-3}(D(\tilde{r}0/16))$

By using (45), (46), the regularity estimate for elliptic differential equations, and Sobolev’s imbedding
theorem, we obtain$\Psi_{1,0}^{N+}\in H^{2N-1}(D(\tilde{r}0/32))\subset C^{2N-3}(D(\tilde{r}0/32))$.

Hence we have$\partial_{x}^{\beta}(\Psi_{1,0}^{N+}(x)-\sum_{|\alpha|\leq 2N-4}\frac{1}{\alpha!}\partial_{x}^{\alpha}\Psi_{1,0}^{N+}(0)x^{\alpha})=\mathcal{O}(r^{2N-3-|\beta|})$for $|\beta|\leq 2N-4$ . Because$\Psi_{1,0}^{N+}(x)$is even with respect to$x_{2}$and odd with respect to.

$\sum_{|\alpha|\leq 2N-4}\frac{1}{\alpha!}\partial_{x}^{\alpha}\Psi_{1,0}^{N+}(0)x^{\alpha}=\sum_{j=1}^{N-2}\sum_{k=1}^{j}\gamma_{j,k,N}^{+}r^{2j-1}\cos(2k-1)\theta$$x_{1}$, we get19
Thus, we claim thatterm (2N-4) times.

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{o}$has the following asymptotic expansion which can be differentiated term by$\check{\varphi}_{1,0}^{N+}(x)=\varphi_{1,0}^{N+}(x)+\sum_{j=1}^{N-2}\sum_{k=1}^{j}\gamma_{j,k,N}^{+}r^{2j-1}\cos(2k-1)\theta+\mathcal{O}(r^{2N-3})$

In asimilar fashion,
times.

$\check{\varphi}_{1,0}^{N-}$as$rarrow \mathrm{O}$,$x\in\Omega_{+}$

Next we show that
$N_{1}=2$ . Let $N_{2}>N_{1}$ . Becausecooe(2k $-1$ ) $\theta+O(r^{2N-3})$$\check{\varphi}_{1,0}^{N\pm}=\chi_{0}(r)\varphi_{1,0}^{N\pm}+\tilde{\varphi}_{1,0}^{N\pm}$$\check{\varphi}_{1,0}^{N_{1}\pm}$

and

as

$rarrow \mathrm{O}$,$x\in\Omega_{-}$$(\mathrm{O}\mathrm{U}\mathrm{T})_{1,0}^{\pm}$on

for $j=1,2$ ,$\Omega_{\pm}$.

Thus we obtain$(\Delta+\lambda_{0})(\check{\varphi}_{1,0}^{N_{1}\pm}-\check{\varphi}_{1,0}^{N_{2}\pm})=(\lambda_{1,0}^{N_{2}\pm}-\lambda_{1,0}^{N_{1}\pm})\varphi_{0,0}^{\pm}$

(49)

,

(50)

.

$(\check{\varphi}_{1,0^{\pm}}^{N_{1}}-\check{\varphi}_{1,0^{\pm}}^{N_{2}}, \varphi_{1}^{\pm})_{\Omega}\pm=0$

Note that.

(51)$\check{\varphi}_{1,0}^{N_{1}\pm}-\check{\varphi}_{1,0}^{N_{2}\pm}=\chi_{0}(r)(\varphi_{1,0}^{N_{1}\pm}-\varphi_{1,0}^{N_{2}\pm})+(\tilde{\varphi}_{1,0}^{N_{1}\pm}-\tilde{\varphi}_{1,0}^{N_{2}\pm})$, we have$\tilde{\varphi}_{1,0}^{N_{1}\pm}-\tilde{\varphi}_{1,0}^{N_{2}\pm}\in D(L_{0}^{\pm})$$\chi_{0}(r)(\varphi_{1,0}^{N_{1}\pm}-\varphi_{1,0}^{N_{2}\pm})\in J_{3}(\tilde{r}0/2)$(48)$\lambda_{1,0}^{N\pm}$$(\check{\varphi}_{1,0}^{N_{1}\pm}, \varphi_{1}^{\pm})_{\Omega}\pm=(\check{\varphi}_{1,0}^{N_{2}\pm}, \varphi_{1}^{\pm})_{\Omega}\pm=0$$\tilde{\varphi}_{1,0}^{N_{1}\pm},\tilde{\varphi}_{1,0}^{N_{2}\pm}\in D(L_{0}^{\pm})$.

are independent of the choice of $N\geq 2$ . We set, we getare the solutions ofand$\check{\varphi}_{1,0}^{N_{2}\pm}$$(\Delta+\lambda_{0})\check{\varphi}_{1,0}^{N_{j}\pm}=-\lambda_{1,0}^{N_{f}\pm}\varphi_{0,0}^{\pm}$and. From (40), we have$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\chi_{0}(r)(\varphi_{1,0}^{N_{1}\pm}-\varphi_{1,0}^{N_{2}\pm}))\subset\overline{D_{+\pm}(\tilde{r}0/2)}$, we have $u\in D(L_{0}^{\pm})$ if and only if there exists $w\in L^{2}(\Omega\pm)$ such that$=(w, v)_{L^{2}(\Omega)}\pm$ for all. Combining (52) with Green’s identity:

Note that for(47)

has the following asymptotic expansion which can be differentiated $(2N-4)$$\check{\varphi}_{1,0}^{N-}(x)=\varphi_{1,0}^{N-}(x)+\sum_{j=1}^{N-2}\sum_{k=1}^{j}\gamma_{j,k,N}^{-}r^{2j-1}$

Because.

$u\in Q_{0}^{\pm}$

.

(52)

$(\nabla u, \nabla v)_{L^{2}(\Omega)}\pm$$v\in Q_{0}^{\pm}$$u\in H^{2}(\Omega_{\pm})$$( \nabla u, \nabla v)_{L^{2}(\Omega)}\pm=\int_{\partial\Omega}\pm\frac{\partial}{\partial n}u\cdot vdS-(\Delta u, v)_{L^{2}(\Omega)}\pm$’

we have$\chi 0(r)(\varphi_{1,0}^{N_{1}\pm}-\varphi_{1,0}^{N_{2}\pm})\in D(L_{0}^{\pm})$,$v\in H^{1}(\Omega_{\pm})$,. So (51) implies that$\check{\varphi}_{1,0}^{N_{1}\pm}-\check{\varphi}_{1,0}^{N_{2}\pm}\in D(L_{0}^{\pm})$

.

(53)

associated with the simple eigenvalue Ao. Thus (49) and (53)is the eigenvector of
Note that. This together. So we havein
imply that the right side of (49) is orthogonal toareand the number. Thus the function
with (49), (50), and (53) imply that
respectively.
andindependent of the choice of $N\geq 2$ , which we denote by
are given by the left side and the right side of $(C)_{1,0}$ respectively. Noteand
Let us show that
that
(54)
.
on
$L_{0}^{\pm}$$\varphi_{1}^{\pm}$$\varphi_{1}^{\pm}$$L^{2}(\Omega_{\pm})$$\lambda_{1,0}^{N_{1}\pm}=\lambda_{1,0}^{N_{2}\pm}$$\check{\varphi}_{1,0}^{N\pm}$$\check{\varphi}_{1,0}^{N_{1}\pm}=\check{\varphi}_{1,0}^{N_{2}\pm}$$\check{\varphi}_{1,0}^{\pm}$$\lambda_{1,0}^{+}$$\lambda_{1,0}^{-}$$(\Delta+\lambda_{0})\check{\varphi}_{1,0}^{\pm}=-\lambda_{1,0}^{\pm}\varphi_{0,0}^{\pm}$Multiplying (54) by$\varphi_{1}^{\pm}$$\lambda_{1,0}^{N\pm}$$\lambda_{1,0}^{\pm}$, and integrating over$\Omega_{\pm}$$\Omega_{\pm}$, we obtain$((\Delta+\lambda_{0})\check{\varphi}_{1,0}^{\pm}, \varphi_{1}^{\pm})_{\Omega}\pm=-\lambda_{1,0}^{\pm}(\varphi_{0,0}^{\pm}, \varphi_{1}^{\pm})_{L^{2}(\Omega)}\pm\cdot$(55)

Prom (37), we get$((\Delta+\lambda_{0})\check{\varphi}_{1,0}^{+}, \varphi_{1}^{+})_{\Omega}+=((\Delta+\lambda_{0})(\chi \mathrm{o}(r)\varphi_{1,0^{+}}^{N_{1}}), \varphi_{1}^{+})_{\Omega}++((\Delta+\lambda_{0})\tilde{\varphi}_{1,0}^{N_{1}+}, \varphi_{1}^{+})_{\Omega}+\cdot$

(56)

20
Because$\tilde{\varphi}_{1,0}^{N_{1}+}$,$\varphi_{1}^{+}\in D(L_{0}^{+})$

and$(\Delta+\lambda\circ)\varphi_{1}^{+}=0$, we have.

(57)$((\Delta+\lambda_{0})\tilde{\varphi}_{1,0}^{N_{1}+}, \varphi_{1}^{+})_{\Omega_{+}}=(\tilde{\varphi}_{1,0}^{N_{1}+}, (\Delta+\lambda_{0})\varphi_{1}^{+})_{\Omega_{+}}=0$

Let$\delta$$\in(0,\tilde{r}_{0}/4)$. We put$g(r, \theta)=\chi \mathrm{o}(r)\varphi_{1,0}^{N_{1}+}$,$S_{\delta}=D_{++}(\tilde{r}0/2)\backslash D_{++}(\delta)$$w_{2,\delta}=\partial D_{++}(\delta)\cap\partial D(\delta)$

We have,$\mathrm{d}\mathrm{S}\mathrm{s}=w_{1}\cup \mathrm{w}2\mathrm{y}\mathrm{s}\cup w_{3,\delta}\cup w_{4,\delta}$g$\in C^{\infty}(\overline{S_{\delta}})$,Notice that$r_{n}\varphi_{1}^{+}=0\partial$on$w_{3,\delta}=[\delta,\tilde{r}_{0}/2]\mathrm{x}\{0\}$onand$w_{3,\delta}$$w_{1}$$w_{1}=\partial D_{++}(\tilde{r}0/2)\cap\partial D(\tilde{r}0/2)$,

and$w_{4,\delta}=\{0\}\mathrm{x}[\delta,\tilde{r}_{0}/2]$,.

. Using (34) and (36), we get$g(r, \theta)=0$$g= \frac{\partial}{\partial n}g=0$

,

,

for r

$\geq\tilde{r}_{0}/2$

on

$g=0$

$\varphi_{1}^{+}=0$on$w_{4,\delta}$,on$g(\delta, \theta)=\varphi_{1,0}^{N_{1}+}(\delta, \theta)$$w_{4,\delta}$

,

and

on

$\frac{\partial}{\partial n}g=0$$w_{2,\delta}$$w_{3,\delta}$,.

. Thus we obtain$((\Delta+\lambda_{0})(\chi \mathrm{o}(r)\varphi_{1,0}^{N_{1}+}), \varphi_{1}^{+})_{\Omega_{+}}$$= \lim_{\deltaarrow+0}\int_{\Omega\backslash D_{+}(0,\delta)}\varphi_{1}^{+}(\Delta+\lambda_{0})g(r, \theta)dx+$$= \lim_{\deltaarrow+0}\int_{S_{\delta}}\varphi_{1}^{+}(\Delta+\lambda_{0})g(r,\theta)dx$.

$= \lim_{\deltaarrow+0}\{\int_{\partial S}$$( \varphi_{1}^{+}\frac{\partial}{\partial n}g-g\frac{\partial}{\partial n}\varphi_{1}^{+})dS-\mathit{1}_{l}g(\Delta+\lambda_{0})\varphi_{1}^{+}dx\}$$= \lim_{\deltaarrow+0}\int_{0}^{\pi/2}(-\frac{\partial}{\partial r}\varphi_{1,0}^{N_{1}+}(\delta, \theta)\varphi_{1}^{+}(\delta, \theta)+\varphi_{1,0}^{N_{1}+}(\delta, \theta)\frac{\partial}{\partial r}\varphi_{1}^{+}(\delta, \theta))\delta d\theta$

.

(58)

From (34) and (36), we obtain$\varphi_{1,0}^{N_{1}\pm}(x)=\tilde{k}_{0,1,1,0}^{\pm}r^{-1}$

coe $\theta+O(r|\log r|)$ as r

$arrow \mathrm{O}$

.

(59)

From (1), we have$\varphi_{1}^{\pm}(x)=C_{1,1}^{\pm}r$

coe$\theta+O(r^{3})$as r$arrow \mathrm{O}$

.

(60)

Using (58)-(60), we get

.

(61)

$(( \Delta+\lambda_{0})(\chi_{0}(r)\varphi_{1,0}^{N_{1}+}), \varphi_{1}^{+})_{\Omega}+=\frac{\pi}{2}\tilde{k}_{0,1,1,0}^{+}C_{1,1}^{+}$

Using (55)-(57) and (61), we get. In asimilar fashion, we obtain. From(35), we have
. So we get.
We aet. From (34), (36), (37), (47), and (48), we claim that the function,
which is asolution of, has the following asymptotic expansion which can be differentiated termby term infinitely many times.
$\lambda_{1,0}^{+}=-\frac{\pi}{2}\tilde{k}_{0,1,1,0}^{+}$$\tilde{k}_{0,1,1,0}^{\pm}=\mp\frac{1}{4}((C_{1,1}^{+})^{2}+(C_{1,1}^{-})^{2})$$\lambda_{1,0}^{-}=\frac{\pi}{2}\tilde{k}_{0,1,1,0}^{-}$$\lambda_{1,0}^{+}=\lambda_{1,0}^{-}=\frac{\pi}{8}((C_{1,1}^{+})^{2}+(C_{1,1}^{-})^{2})$$\lambda_{1,0}=\lambda_{1,0}^{+}(=\lambda_{1,0}^{-})$$\check{\varphi}_{1,0}^{\pm}$$(\mathrm{O}\mathrm{U}\mathrm{T})_{1,0}^{\pm}$$\check{\varphi}_{1,0}^{\pm}(x)\sim\check{k}_{0,1,1,0}r^{-1}$as$rarrow \mathrm{O}$,$x\in 1_{\pm}$coe$\theta+\sum_{s=1j}^{\infty}\sum_{=1-s}^{0}\check{k}_{s,1,j,0}^{\pm}r^{2s-1}\infty \mathrm{s}(2j-1)\theta+\sum_{s=1}^{\infty}\check{l}_{s,1,s-1,0}^{\pm}r^{2s-1}\log r\cos(2j-1)\theta$. We rewrite this as$\check{\varphi}_{1,0}^{\pm}(x)\sim\sum_{s=0}^{\infty}\check{t}_{s,1,0}^{\pm}(r, \theta)+\sum_{s=1}^{\infty}\check{u}_{s,1,0}^{\pm}(r, \theta)\log r$$(rarrow 0, x\in\Omega_{\pm})$

,

(62)

21
where

and

$\check{t}_{s,1,0}^{\pm}(\cdot, \theta)$$\check{u}_{s,1,0}^{\pm}(r, \theta)$(48), we haveare homogeneous polynomials of degree $2s-1$ . From (34)-(36), (47) and$\check{t}_{0,1,0}^{\pm}($.,$\theta)=\tilde{h}_{0,1,0}^{\pm}(\cdot, \theta)=h_{0,1,0}^{\pm}(\cdot, \theta)$

.

(63)

Let us show that there exists asolutionofsatisfying the following $(a)-(e)$ .
(a)
for
.
(b)
cooe(2k-1)\mbox{\boldmath $\theta$}.
(c) The function
is odd, and.
(d) The function
has the following asymptotic expansion which can be differentiated term byterm two times.
$(\mathrm{I}\mathrm{N})_{1,0}$$\check{v}_{2,0}$$\check{v}_{2,0}(\xi)=\check{v}_{2,0}^{+}(\xi)+\check{v}_{2,0}^{-}(\xi)$$\xi\in\Pi$$\check{v}_{2,0}^{+}(\xi)=\sum_{j=1}^{2}\sum_{k=1}^{j}\check{C}_{2,j,k,0}^{+}\rho^{2j-1}$$\check{v}_{2,0}^{-}(\xi)\in A\cap B$$\check{v}_{2,0}^{-}(\xi_{1}, \cdot)$$\check{v}_{2,0}^{-}(\xi)$$\check{v}_{2,0}^{-}(\xi)\sim(\sum 1\sum 0+\sum\infty \mathrm{I} )\check{c}_{1,l,j,0\rho^{2M-2l+1}\cos(2j-1)\theta+\check{d}_{1,1,0,0}\rho\log\rho\cos\theta}$

The conditions $(a)-(d)$ allow us to write the asymptotic expansion ofas$\check{v}_{2,0}(\xi)\sim\sum_{l=0}^{\infty}\check{h}_{1,l,0}^{\pm}(\rho, \theta)+\check{m}_{1,1,0}^{\pm}(\rho, \theta)\log\rho$

where$\check{h}_{1,l,0}^{\pm}(\cdot, \theta)$as$\rhoarrow\infty$,.

$\xi\in P$$l=2j=l-1$$l=0j=l-1$

and$\check{m}_{1,l,0}^{\pm}(\cdot, \theta)$

as follows.

$\# 2,0$$\rhoarrow\infty$, $\xi\in\Pi,$ $\pm\xi_{2}>0$ ,(64)

are homogeneous polynomials of degree $3-2l$ .

(e) We have$\check{h}_{1,1,0}^{\pm}(\cdot, \theta)=\check{t}_{1,1,0}^{\pm}(\cdot, \theta)$

,

(65)

$\check{h}_{1,0,0}^{\pm}(\cdot, \theta)=t_{1,0,0}^{\pm}(\cdot, \theta)$

,

(66)
,

$\check{m}_{1,1,0}^{\pm}(\cdot, \theta)=\check{u}_{1,1,0}^{\pm}(\cdot, \theta)$

We seek asolution of$(\mathrm{I}\mathrm{N})_{1,0}$(67)

which takes the form$\hat{v}_{2,0}(\xi)=\hat{v}_{2,0}^{+}(\xi)+\hat{v}_{2,0}^{-}(\xi)$,$\xi\in\square$

,

(68)

where
and
are even and odd respectively. Inserting (68) into
the terms of the both sides which are even with respect to , we get$\hat{v}_{2,0}^{+}(\xi_{1}, \cdot)$$\hat{v}_{2,0}^{-}(\xi_{1}, \cdot)$$(\mathrm{I}\mathrm{N})_{1,0}$

and identifying$\xi_{2}$$\Delta\hat{v}_{2,0}^{+}=\lambda_{0}\tilde{v}_{1,0}^{+}$

on

,

$P$

(69)
$\hat{v}_{2,0}^{+}=0$

Inserting (68) into
we get$(\mathrm{I}\mathrm{N})_{1,0}$

on {0}$\mathrm{x}(0, \infty)$,$\frac{\partial}{\partial\xi_{2}}\hat{v}_{2,0}^{+}=0$

on$(0, \infty)$$\mathrm{x}\{0\}$.

and identifying the terms of the both sides which are odd with respect to$\Delta\hat{v}_{2,0}^{-}=\lambda_{0}\tilde{v}_{1,0}^{-}$

on$\xi_{2}$

,

,

$P$

(70)
$\hat{v}_{2,0}^{-}=0$

on$\Gamma_{2}$,$\frac{\partial}{\partial\xi_{2}}\hat{v}_{2,0}^{-}=0$

on$\Gamma_{1}$.

First, we consider the equation (69). Using (29), we infer that there exists asolution of (69) whichtakes the form.
(71)$\hat{v}_{2,0}^{+}(\xi)=\hat{C}_{1,2,1}^{+}\rho^{3}\cos\theta$

Next, we consider the equation (70). From Proposition 2.3, we claim that there exists asolutionof (70) having the following asymptotic expansion which can be differentiated term by term
two times.
$\hat{v}_{2,0}^{-}\in A\cap$ $B$$\hat{v}_{2,0}^{-}(\xi)\sim(\sum_{l=0j}^{1}\sum_{=l-1}^{0}+\sum_{l=2j}^{\infty}\sum_{=l-1}^{l})\hat{C}_{1,l,j}^{-}\rho^{3-2l}\cos(2j-1)\theta+\hat{d}_{1,1,0}\rho\log\rho\cos\theta$$(72\rangle$

22
as$\ovalbox{\tt\small REJECT}+\mathrm{o}\mathrm{o},$$\ovalbox{\tt\small REJECT}$/. EP. We extend

again. We define$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\mathrm{Z}_{?}0}(4)$to an odd function with respect toby (63), (71), and (72). Then we can write$\ovalbox{\tt\small REJECT} 7\mathrm{o}(\mathrm{C})$as$\hat{v}_{2,0}(\xi)\sim\sum_{l=0}^{\infty}\hat{h}_{1,l,0}^{\pm}(\rho, \theta)+\hat{m}_{1,1,0}^{\pm}(\rho, \theta)\log\rho$

where
and
harmonic function to$\hat{h}_{M,l,0}^{\pm}(\cdot, \theta)$$\rhoarrow\infty$,$\alpha_{2}^{\pm}$

are some constant. From$(\mathrm{I}\mathrm{N})_{1,0}$,,$(\Delta+\lambda_{0})\check{\varphi}_{1,0}^{\pm}=-\lambda_{1,0}\varphi_{0,0}^{\pm}$$\Omega\pm$, we haveon$\Delta t_{1,0,0}^{\pm}=-\lambda_{0}t_{0,0,0}^{\pm}$

,

,

(76)

,

.

(77)

. Since$\Delta(\check{u}_{1,1,0}^{\pm}(r,\theta)\log r)=-\lambda_{0}\check{t}_{0,1,0}^{\pm}(r, \theta)$$=-\lambda_{0}(h_{0,1,0}^{\pm}-\check{t}_{0,1,0}^{\pm})$$\Delta(\hat{h}_{1,0,0}^{\pm}-t_{1,0,0}^{\pm})=-\lambda_{0}(h_{0,0,0}^{\pm}-t_{0,0,0}^{\pm})$$(M.2)_{0,0,0}^{\pm}$

(73)

(75)

, we get. So, we obtain$\Omega_{\pm}$$\Delta\{(\hat{m}_{1,1,0}^{\pm}(r, \theta)-\check{u}_{1,1,0}^{\pm}(r, \theta))\log r\}$

From

,

(74)

$\Delta\hat{h}_{1,0,0}^{\pm}=-\lambda_{0}h_{0,0,0}^{\pm}$$\hat{h}_{1,1,0}^{\pm}(r, \theta)=\pm\hat{C}_{1,1,0}r\cos\theta$on$\pm\xi_{2}>0$, (29), (30), and (71)-(73), we get$\Delta(\hat{m}_{1,1,0}^{\pm}(r, \theta)\log r)=-\lambda_{0}h_{0,1,0}^{\pm}$

Since

,

$\tilde{v}_{2,0}$$\hat{h}_{1,1,0}^{\pm}(r, \theta)-\check{t}_{1,1,0}^{\pm}(r, \theta)=\alpha_{2}^{\pm}r\cos\theta$and$\xi\in\Pi$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\mathrm{o}(\mathrm{C})$are homogeneous polynomials of degree $2M-2l+1$ . By adding some
, we shall constructsatisfying (e) as well as $(a)-(d)$ . Let us show that$\hat{h}_{1,0,0}^{\pm}(r, \theta)-t_{1,0,0}^{\pm}(r, \theta)=\alpha_{1}^{\pm}r^{3}\cos 3\theta$$\alpha_{1}^{\pm}$which we denote by$\hat{m}_{M,l,0}^{\pm}(\cdot, \theta)$$\hat{v}_{M+1,0}$$\hat{m}_{1,1,0}^{\pm}(r, \theta)=\check{u}_{1,1,0}^{\pm}(r, \theta)$where$42_{\mathrm{t}}$$(\Delta+\lambda_{0})\varphi_{0,0}^{\pm}=0$,.

and (63), we get$\Delta\{\hat{m}_{1,1,0}^{\pm}(r, \theta)-\check{u}_{1,1,0}^{\pm}(r, \theta))\log r\}$$\Delta(\hat{h}_{1,0,0}^{\pm}-t_{1,0,0}^{\pm})=0$

$=0$

,

(78)

.

(79)

Note that$\hat{m}_{1,1,0}^{\pm}(r, \theta)-\check{u}_{1,1,0}^{\pm}(r, \theta)=\pm\hat{d}_{1,1,0}r\cos\theta-\check{l}_{1,1,0,0}^{\pm}r\cos\theta$,$\hat{h}_{1,0,0}^{\pm}(r, \theta)-t_{1,0,0}^{\pm}(r, \theta)=\hat{C}_{1,2,1}^{+}r^{3}\cos\theta\pm\sum_{j=-1}^{0}\hat{C}_{1,0,j}^{-}r^{3}\cos(2j-1)\theta\mp C_{1,1}^{\pm}\sum_{k=1}^{2}C_{2,k}^{\pm}r^{3}$$\check{t}_{1,1,0}^{\pm}(r,\theta)=\check{k}_{1,1,0,0^{f}}^{\pm}$(80)

cooe(2k-1)\mbox{\boldmath $\theta$}, (81)

coe .

(82)

$\theta$

From (77) and (82), we get (76). (75) follows from (79) and (81). We have (74) ffom (78) and (80). We

put$\check{v}_{2,0}(\xi)=\hat{v}_{2,0}(\xi)-\frac{1}{2}(\alpha_{1}^{+}+\alpha_{1}^{-})\rho^{3}$coe$3 \theta-\frac{1}{2}(\alpha_{1}^{+}-\alpha_{1}^{-})U_{2}(\xi)$$- \frac{1}{2}(\alpha_{2}^{+}+\alpha_{2}^{-})\rho\cos\theta-\{\frac{3}{4}(\alpha_{1}^{+}-\alpha_{1}^{-})+\frac{1}{2}(\alpha_{2}^{+}-\alpha_{2}^{-})\}U_{1}(\xi)$.

Then it follow from (15) and (74)-(76) that the function
satisfies $(a)-(e)$ .
We modify
andsuch that the resulting functions satisfy the compatibility condition as wellas the matching condition. We put,(83)$\# 2,0$$\check{\varphi}_{1,0}^{\pm}$$\mathrm{V}2,0$$\varphi_{1,0}^{\pm}=\kappa^{\pm}\varphi_{1}^{\pm}+\check{\varphi}_{1,0}^{\pm}$$v_{2,0}( \xi)=\check{v}_{2,0}(\xi)+\frac{1}{2}(\kappa^{+}C_{1,1}^{+}+\kappa^{-}C_{1,1}^{-})\rho\cos\theta+\frac{1}{2}(\kappa^{+}C_{1,1}^{+}-\kappa^{-}C_{1,1}^{-})U_{1}(\xi)$

,

(84)

23
being aconstant which we shall determine later. From (62), (83), and Proposition 1.1, we claim thatsatisfying $(P5)_{1,0}$ and $(P7)_{1,0}$ , has the asymptotic, which is asolution of
the function
expansion :$\kappa^{\pm}$$(\mathrm{O}\mathrm{U}\mathrm{T})_{1,0}^{\pm}$$\varphi_{1,0}^{\pm}$$\varphi_{1,0}^{\pm}(x)\sim\sum_{s=0}^{\infty}t_{s,1,0}^{\pm}(r, \theta)+\sum_{s=1}^{\infty}u_{s,1,0}^{\pm}(r, \theta)\log r$

where

and

$t_{s,1,0}^{\pm}(\cdot, \theta)$$u_{s,1,0}^{\pm}(\cdot, \theta)$$(rarrow 0, x\in\Omega_{\pm})$,are homogeneous polynomials of degree $2s-1$ . Besides we have$t_{0,1,0}^{\pm}(r, \theta)=\check{t}_{0,1,0}^{\pm}(r, \theta)$

(85)

,

(86)

,

$t_{1,1,0}^{\pm}(r, \theta)=\check{t}_{1,1,0}^{\pm}(r, \theta)+\kappa^{\pm}C_{1,1}^{\pm}r\cos\theta$.

$u_{1,1,0}^{\pm}(r, \theta)=\check{u}_{1,1,0}^{\pm}(r, \theta)$

From (15), (64), and (84), we claim that the function(P1), has the asymptotic expansion:$v_{2,0}$(87), which is asolution of$(\mathrm{I}\mathrm{N})_{1,0}$

satisfying$\mathrm{i},\mathrm{O}-(\mathrm{P}7)\mathrm{i},0$as$v_{2,0}( \xi)\sim\sum_{l=0}^{\infty}h_{1,l,0}^{\pm}(\rho, \theta)+m_{1,1,0}^{\pm}(\rho, \theta)\log\rho$where$h_{1,l,0}^{\pm}(\cdot, \theta)$, and$m_{1,l,0}^{\pm}(\cdot, \theta)$$\rhoarrow\infty,$$\pm\xi_{2}>0$,

are homogeneous polynomials of degree $3-2l$ . Besides we get$h_{1,0,0}^{\pm}(\rho, \theta)=\check{h}_{1,0,0}^{\pm}(\rho, \theta)$

(88)

,

$h_{1,1,0}^{\pm}(\rho, \theta)=\check{h}_{1,1,0}^{\pm}(\rho, \theta)+\kappa^{\pm}C_{1,1}^{\pm}\rho\cos\theta$$m_{1,1,0}^{\pm}(\rho, \theta)=\check{m}_{1,1,0}^{\pm}(\rho, \theta)$

(89)

,

.

(90)

Rom (67), (87), and (90), we have$u_{1,1,0}^{\pm}(\cdot, \theta)=m_{1,1,0}^{\pm}(\cdot, \theta)$

.

(91)

It follows from (63) and (85) that$t_{0,1,0}^{\pm}(\cdot, \theta)=h_{0,1,0}^{\pm}(\cdot, \theta)$

.

(92)

From (65), (66), (86), (88), and (89), we get$t_{1,l,0}^{\pm}(\cdot, \theta)=h_{1,l,0}^{\pm}(\cdot, \theta)$

for$0\leq l\leq 1$

.

(93)

holds by the assumption. Combining this with (91), (92), and (93), we claim that
holds for $0\leq s\leq 1,0\leq l\leq 1$ . Besides $(P1)_{1,0}-(P7)_{1,0}$ hold. Let us show
holds andsatisfying $(C)_{2,0}$ . Notice that $(C)_{2,0}$ is equivalent toandthat there exist
Note that$(M.2)_{0,0,0}^{\pm}$$(M.2)_{s,l,0}^{\pm}$$(M.1)_{1,1,0}^{\pm}$$\kappa^{+}$$\kappa^{-}$$- \frac{3}{2}\pi(C_{1,1}^{+})^{-1}C_{0,2,2,0}^{-}C_{2,2}^{+}+\frac{3}{2}\pi(C_{1,1}^{-})^{-1}C_{0,2,2,0}^{-}C_{2,2}^{-}=(C_{1,1}^{+})^{-1}\lambda_{1,0}(\varphi_{1,0}^{+}, \varphi_{1}^{+})_{\Omega_{+}}+(C_{1,1}^{-})^{-1}\lambda_{1,0}(\varphi_{1,0}^{-}, \varphi_{1}^{-})_{\Omega_{-}}$and the left side of the above equality is independent of. Becauseis equal to$\lambda_{1,0}((C_{1,1}^{+})^{-1}\kappa_{+}+(C_{1,1}^{-})^{-1}\kappa-)$$\kappa^{+}$$\lambda_{1,0}>0$,. The right side of the above equality
andsatisfying $(C)_{2,0}$ .
, there exist
and$\kappa^{-}$$\kappa^{+}$$\kappa^{-}$$\square$

References

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