数理解析研究所講究録
第 1901 巻 2014 年 79-91

79

On profiles of critical eigenfunctions for linearized
problems of bistable reaction diﬀusion equations1
Tohru Wakasa
Kyushu Institute of Technology
wakasa@mns.kyutech.ac.jp

Introduction and main results

1

Let us consider the boundary value problem(1.1)$\{\begin{array}{ll}\epsilon^{2}u_{xx}(x)+f(u(x))=0, in (0,1) ,u_{x}(0)=u_{x}(1)=0, \end{array}$Here$\epsilon$

is a positive parameter. The corresponding parabolic PDE to (1.1) is given by$\{\begin{array}{l}u_{t}(x, t)=\epsilon^{2}u_{xx}(x, t)+f(u(x, t (x, t)\in(0,1)\cross(0, +\infty) ,u_{x}(0, t)=u_{x}(1, t)=0, t\in(0, +\infty) ,u(x, 0)=u_{0}(x) , x\in(O, 1) .\end{array}$(1.2)

We are interested in the case that the function $f\in C^{1}$ is a bistable nonlinearity:
(A1)

$f$

has exactly three zeros $u_{-}<0<u+$ with f $(0)>0,$$f_{u}(u_{\pm})<$O.

As a typical example of , we refer $f(u)=u-u^{3}$ with $u_{\pm}=\pm 1$ . Then (1.2) is the simplest
model which describes phase transition of materials; both $u\pm and0$ represent two stablestates and one unstable state. Under (A1) the reaction-diﬀusion problem (1.2) defines a
gradient system on Sobolev space $H^{1}(0,1)$ , and in particular, the solution $u(x, t)$ exists
globally in time and converges to a solution of (1.1) as $tarrow+\infty.$
Main interests to (1.1) and (1.2) are to understand the existence and stability of(stationary) solutions of (1.1), and to understand the tranisient and asymptotic dynamics
in (1.2). Moreover, it is interesting to consider these problems in a situation that issmall, in view of pattern formation.
We summarize the classical results on (1.1) (see Propositions 2.1-2.3, and also [1], [3]and [5]). Any nontrivial solution of (1.1) is characterized by the number of zeros in thesatisfies (1.1) and it
interval $(0,1)$ . We sayis the -mode solution of (1.1) ifexactly admit -zeros
in the interval $(0,1)$ . Here we also assume that$f$$\epsilon$$n$$u_{n,\epsilon}$

$n$

(A2)

$f_{u}$(A3)$f$$z_{1},$$\cdots,$$z_{l},$$\cdots,$$u_{n,\epsilon}$$z_{n}$

is decreasing for $u\in(O, u_{+})$ .
is odd; $u_{-}=-u_{+}.$

lThis article is based on ajointwork with Shoji Yotsutani (Ryukoku Univ

80
(with
It is well known that for arbitrary $n\in N$ , exactly two -mode solutions$u_{n,\epsilon}(0)>0)$ exists if and only if $\epsilon\in(0, \sqrt{f_{u}(0)}/(n\pi))$ .
that
The -mode solution$-u_{-}<u_{n,\epsilon}(x)<u_{+}$ for every $x\in[0$ , 1 , and it can be extended as a periodic function
on $R$ because of the boundary condition. Moreover, the concept of bifurcation theory isuseful to understand the solution structure of (1.1): the two bifurcation curves$n$$\pm u_{n,\epsilon}$$n$$u_{n,\epsilon}$$]$$S_{n}^{\pm}:=\{(\epsilon, u)\in R_{+}\cross H^{1}(0,1)|(\epsilon, u)=(\epsilon,\pm u_{n,\epsilon}(x)), \epsilon\in(0, \sqrt{f_{u}(0)}/n\pi)\}$appears from the line of unstable solution $u=0$ at the point $(\epsilon, u)=((\sqrt{f_{u}(0)}/n\pi)^{-1},0)$ .
See Figs. 1 and 2.

Figure 1: Profiles of$u_{3,\epsilon}(x)$and$f_{u}(u_{3,\epsilon})$for $f(u)=u-u^{3}$ when

Figure 2: Bifurcation diagram of (1.1) for $f(u)=u-u^{3}$ in$(\epsilon^{-1}, \alpha)$$\epsilon$

is small.

-plane, where $\alpha=u(0)$ .

Let us introduce$z_{\ell}^{n}= \frac{2\ell-1}{2n},$$\ell=1$, . . . , $n,$$x_{\ell}^{n}= \frac{\ell}{n},$$\ell=0$, . . . , $n.$

It follows from (A3) that the set of zeros toand the period (resp.
is given by
$1/n$
$2/n$
anti-period) of). If is suﬃciently small, then(resp.
is close tois$u_{n,\epsilon}$$\epsilon$$u_{n,\epsilon}$$\{z_{\ell}^{n}\}_{\ell=1}^{n}$$u_{n,\epsilon}(x)$81appears in theaway from zeros and the transition layers connecting between
eitheris characterized by. The transition layer atneighborhoods of$u_{\pm}$$u\pm$$x=z_{\ell}^{n}$$\{z_{\ell}^{n}\}$$u_{n,\epsilon}(x) \sim(-1)^{\ell}U(\frac{x-z_{\ell}^{n}}{\epsilon})$in a neighborhood of
rescaled problem$z_{\ell}^{n}$,, where $U=U(z)$ is the (unique) heteloclinic solution of the$\{\begin{array}{ll}U_{zz}(z)+f(U(z))=0 in R,U(-\infty)=u_{-}, U(+\infty)=u+, U(0)=0.

(1.3)\end{array}$small enough (with respect to ). Let
Now we fix $n\in N$ arbitrarily and choose
us consider the linearized eigenvalue problem associated with$\epsilon>0$$n$$u_{n,\epsilon}$$\{\begin{array}{ll}\epsilon^{2}\varphi_{xx}(x)+f_{u}(u_{n,\epsilon}(x))\varphi(x)+\lambda\varphi(x)=0 in (0,1) ,\varphi_{x}(0)=\varphi_{x}(1)=0.

(1.4)\end{array}$, the $(j+1)$ -th eigenvalue
and
For $j\in N\cup\{0\}$ , we denote by
and the corresponding eigenfunction. Fig. 3 displays some profiles of eigenfunctions in
the case $f(u)=u-u^{3}.$It follows from the Sturm-Liouville theory with (A2) and (A3) thatis unstable.
(see e.g., [1]). This fact concludes thatis known metastable in the following sense: for $0\leq j<n,$
On the other hand,. This fact is closely related with thewith some $d>0$ as
super slow dynamics arrising in (1.2) (see Carr-Pego [2], Fusco-Hale [4]).
$\varphi_{j}(x)=\varphi_{j}^{n,\epsilon}(x)$$\lambda_{j}=\lambda_{j}^{n,\epsilon}$$\lambda_{0}^{n,\epsilon}<$$\lambda_{n-1}^{n,\epsilon}<0<\lambda_{n}^{n,\epsilon}<\cdots<+\infty$$<$$u_{n,\epsilon}$$u_{n,\epsilon}$$\epsilonarrow 0$$\lambda_{j}^{n,\epsilon}=O(\exp(-d/\epsilon))$Figure 3: Profiles of$\varphi_{j}^{3,\epsilon}(j=0, \ldots, 8)$

for $f(u)=u-u^{3}.$of (1.4) in theandIn this article we are interested in the -dependence ofsituation that is fixed and is given small enough. More precisely, we will investigate the$\epsilon$$n$$\epsilon$$\lambda_{j}^{n,\epsilon}$$\varphi_{j}^{n,\epsilon}$

82. The eigenfunctions and eigenvalues arewith $0\leq j<n$ asasymptotic profiles
often called as critical eigenfunctions and critical eigenvalues, and they play an essential. Moreover, the profiles of critical eigenfunctions give us
role for the stability ofsignificant information on the dynamical behavior of (1.2) near the stationary solutions,and it has been investigated by the previous works [2] and [4].
The purpose of this article is to show the following conjecture:,“For the case $0\leq j<n,$has spikes in neighborhoods ofproportional
spike
to
is
and the height of each
which is proposed by E. Yanagida. In order to justify Yanagida’s conjecture, our problems
are formulated as follows:$\epsilonarrow 0$$\varphi_{j}^{n,\epsilon}$$u_{n,\epsilon}$$z_{\ell}^{n}(\ell=1, \cdots, n)$$\varphi_{j}^{n,\epsilon}$$\cos j\pi z_{\ell}^{n}$$\bullet$$\bullet$to characterize the locallized patterns (spike) for the critical eigenfunctions as$\epsilonarrow 0,$

to give some “symmetric” property arising in critical eigenfunctions.

, which is chosen as positive function, some fundamenFor the principal eigenfunction
tal results can be easily obtained (see Proposition 2.5-2.7). In particular, it has exactly-spikes with the same height atarbitrarily. Our main results are given by the following two
Fix the height oftheorems.
$\varphi_{0}^{n,\epsilon}$$x=z_{\ell}^{n}.$$n$$\varphi_{0}^{n,\epsilon}$

Theorem 1. Assume $(A1)-(A3)$ and the normalized condition

For any$\delta$$\lim_{\epsilonarrow 0}\frac{\varphi_{j}^{n,\epsilon}(0)}{\varphi_{0}^{n,\epsilon}(0)}=\cos j\pi z_{1}^{n}.$with $0<\delta<1/(4n)$ ,(i)$\lim_{\epsilonarrow 0}\frac{\varphi_{j}^{n,\epsilon}(x)}{\varphi_{0}^{n,\epsilon}(x)}=\cos j\pi z_{1}^{n}$, uniformly in$[0, x_{1}^{n}-\delta],$(ii)$\lim_{\epsilonarrow 0}\frac{\varphi_{j}^{n,\epsilon}(x)}{\varphi_{0}^{n,\epsilon}(x)}=\cos j\pi z_{\ell}^{n}$, uniformly in$[x_{\ell-1}^{n}+\delta, x_{\ell}^{n}-\delta],$(iii)$\lim_{\epsilonarrow 0}\frac{\varphi_{j}^{n,\epsilon}(x)}{\varphi_{0}^{n,\epsilon}(x)}=\cos j\pi z_{n}^{n}$, uniformly in$[x_{n-1}^{n}+\delta, 1].$$\ell=2$, . . . , $n-1,$

Theorem 2. Suppose the assumptions in Theorem 1 holds. Then for $\ell=1$ , 2, . . . , $n-1,$$\lim_{\epsilonarrow 0}\frac{\varphi_{j}^{n,\epsilon}(x_{\ell}^{n}+\epsilon z)}{\varphi_{0}^{n,\epsilon}(x_{p}^{n}+\epsilon z)}=\mathcal{C}_{j)}^{n}\ell+d_{j,\ell}^{n}\tanh\sqrt{-f_{u}(u_{\pm})}z,$

uniformly in any compact subset in $R,$$c_{j,l}^{n}= \frac{1}{2}(\cos j\pi z_{l+1}^{n}+\cos j\pi z_{\ell}^{n}) , d_{j,\ell}^{n}=\frac{1}{2}(\cos j\pi z_{\ell+1}^{n}-\cos j\pi z_{\ell}^{n})$.

To this end, we remark that in the previous work in $[7]-[9]$ we have obtained more
precise result on every eigenfunction in the two cases of typical $f:f(u)=\sin u$ and$f(u)=u-u^{3}.$832Preliminaries

2.1

$n$

-mode solutions

We recall well known results on (1.1) under assumptions $(A1)-(A3)$ . Let us introduce$F(u):= \int_{0}^{u}f(s)ds.$

Proposition 2.1. Suppose that $f\in C^{1}$ satisfies $(A1)-(A3)$ . Let
equation$\epsilon>0$

and $n\in N$ . The(2.1)$\int_{-\alpha}^{\alpha}\frac{1}{\sqrt{2(F(\alpha)-F(w))}}dw=\frac{1}{n\epsilon}, \alpha\in(0, u_{+})$

has a solution
if and only if $\epsilon\in(0, \sqrt{f_{u}(0)}/n\pi)$ . Moreover,decreasing with respect to$a=\alpha_{n,\epsilon}$$\alpha_{n,\epsilon}$is unique,$\epsilon,$$\lim_{\epsilonarrow 0}\alpha_{n,\epsilon}=u+$, and$\epsilonarrow\sqrt{f_{u}(0)}/n\pi\lim\alpha_{n,\epsilon}=0.$

and $n\in$ N.
Proposition 2.2. Suppose that $f\in C^{1}$ satisfies $(A1)-(A3)$ . Let$\epsilon\in(0, f_{u}(0)/(n\pi))$ , there exists two-mode solutionsof (1.1) satisfying$\epsilon>0$$\int_{u_{n,\epsilon}(x)}^{\alpha_{n,\epsilon}}\frac{1}{\sqrt{2(F(\alpha_{n\epsilon})-F(w))}}dw=\frac{x}{n\epsilon}$

where
(i)
(ii)

$\alpha_{n,\epsilon}$$u_{n,\epsilon}$is as in Proposition 2.1. Moreover,is monotone decreasing in$u_{n,\epsilon}(x+ \frac{1}{n})=-u_{n,\epsilon}(x)$

and

(iii)

$u_{n,\epsilon}(x_{\ell}^{n})=(-1)^{\ell}\alpha_{n,\epsilon}$(iv)$u_{n,\epsilon}( \frac{1}{n}-x)=-u_{n,\epsilon}(x)$(v)$u_{n,\epsilon}(z_{\ell}^{n})=0$

If$\pm u_{n,\epsilon}(x)$$n$$u_{n,\epsilon}$(2.2)

for $x\in[0, 1/n]$ ,satisfies the following properties:$[0, x_{1}^{n}],$for $x\in R,$$(u_{n,\epsilon})_{x}(x_{\ell}^{n})=0$

for$\ell=0$, 1, . . . , $n,$

for $x\in R,$, for $\ell=1$ , . . . , $n.$Conversely, any -mode solutions of (1.1) is given either$n$$u_{n,\epsilon}(x)$$or-u_{n,\epsilon}(x)$.

Proposition 2.3. Assume $(A1)-(A3)$ . Let $U=U(z)$ be the (unique) heteloclinic solutionof the rescaled problem (2.5). Then, for each $\ell=1$ , . . . ,$n,$$\lim_{\epsilonarrow 0}u_{n,\epsilon}(z_{p}^{n}+\epsilon z)=(-1)^{l}U(z)$in the topology of$C_{loc}^{2}(R)$,.

Remark 2.1. For a wider class of the bistable , Propositions 2.1-2.3 can be generalized;(A2) and (A3) are to be replaced in more general assumptions
$f$

84
(A2’)

$f_{u}(u) \leq\frac{f(u)}{u}$for $u\in(O, u_{+})$ ,(A3) $\int_{u-}^{u_{+}}f(s)ds=0,$respectively. Roughly speaking, (A2’) coinsides with the monotonicity ofand also. The assumption (A3’) proves thethe degeneracy of the linearized oparators around
existence of $U(z)$ of (2.5); it characterizes the limiting profiles of -mode solutions as$\alpha_{n,\epsilon}$$u_{n,\epsilon}$$n$$\epsilonarrow 0.$

2.2

Linearized eigenvalue problem

We rewrite (1.4) as follows:(2.3)$\{\begin{array}{ll}\epsilon^{2}(\varphi_{xx}(x)+(g_{n,\epsilon}(x)+\lambda)\varphi(x)=0, in (0,1),\varphi_{x}(0)=\varphi_{x}(1)=0, \end{array}$where$g_{n,\epsilon}(x)=f_{u}(u_{n,\epsilon}(x))$.

On the other hand, we have
(2.4)$\{\begin{array}{ll}\epsilon^{2}(u_{n,\epsilon})_{xx}(x)+\overline{g}_{n,\epsilon}(x)u_{n,\epsilon}(x)=0, in (0,1) ,(u_{n,\epsilon})_{x}(0)=(u_{n,\epsilon})_{x}(1)=0, \end{array}$where$\overline{g}_{n,\epsilon}(x):=\frac{f(u_{n,\epsilon}(x))}{u_{n,\epsilon}(x)}.$By comparing these two problems with the Sturm-Liouville theory and (A2), (A3), it is(see e.g., [1]). This fact concludes
obtained thatthatis unstable.
We now summarize some fundamental properties of:$\lambda_{0}^{n,\epsilon}<\cdots<\lambda_{n-1}^{n,\epsilon}<0<\lambda_{n}^{n,\epsilon}<\cdots<+\infty$$u_{n,\epsilon}$$g_{n,\epsilon}(x)$(G1)$g_{n,\epsilon}$

is $1/n$ -periodic,(G2) for each $\ell=1$ , . . . , $2n-1,$(G3)$g_{n,\epsilon}$

is even with respect to$x= \frac{\ell}{2n},$

is monotone increasing for $x\in[O, 1/2n],$(G4) for each $\ell=1$ , . . . , $n,$(G5)$g_{n,\epsilon}$$\lim_{\epsilonarrow 0}g_{n,\epsilon}(\epsilon z)=f_{u}(u_{\pm})$$\lim_{\epsilonarrow 0}g_{n,\epsilon}(x_{\ell}^{n}+\epsilon z)=f_{u}(U(z))$, in the topology of$C_{loc}^{2}(R)$(G6) for arbitrality fixed $\delta\in(0,1/(4n))$ , there exists$\lim_{\epsilonarrow 0}\overline{\beta}_{n,\epsilon}=f_{u}(u_{\pm})$, and for suﬃciently small, in the topology of,,$\beta_{n,\epsilon},$$\overline{\beta}_{n,\epsilon}\in R$

such that$\beta_{n,\epsilon}\leq g_{n,\epsilon}(x)\leq\overline{\beta}_{n,\epsilon}$$\epsilon,$$C_{loc}^{2}(R)$for$\lim_{\epsilonarrow 0}\beta_{n,\epsilon}=$$x\in[0, \delta]$

85
Roughly speaking, $(G1)-(G2)$ , (G3) and $(G4)-(G6)$ provide the symmetic, monotone andasymptotic properties of
In particular, we see from (2.5) that$\varphi_{0}^{n,\epsilon}$(2.5)$\{\begin{array}{ll}(U_{z})_{zz}(z)+f_{u}(U(z))U_{z}(z)=0 in R,U_{z}\in H^{1}(R) , \end{array}$and hence, we can expect that$\lambda_{0}^{n,\epsilon}\sim 0$

as$\epsilonarrow 0$

and,$\varphi_{0}^{n,\epsilon}(x)\sim U_{z}(\frac{x-z_{\ell}^{n}}{\epsilon})$in a neighborhood of
However, we would need more precise estimates onthan (G6) in order tojustify the above observation. Under the assumptions $(A1)-(A3)$ one can derive from
Proposition 2.1 that as
and
with
some
. They lead us to the asymptotic estimates of critical eigenvalues, which is
obtained in Carr-Pego [2] already stated as in the introduction.
Here we would like to refer the result of Carr and Pego, and prepare the following
proposition, which is an easy consequence of the result by [2].
$z_{\ell}^{n}.$$g_{n,\epsilon}(x)$$\epsilonarrow 0,$$d,$$\beta_{n,\epsilon}=f_{u}(u_{\pm})+O(e^{-d/n\epsilon})$

Proposition 2.4. Assume $(A1)-(A3)$ . Let $n\in N$ and(i)$\epsilon\in(0, \sqrt{f_{u}(0)}/n\pi)$. Then,$\lim_{\epsilonarrow 0}\lambda_{0}^{n,\epsilon}=0,$(ii) for $0<j<n,$2.3$\overline{\beta}=f_{u}(u_{\pm})+O(e^{-\overline{d}/n\epsilon})$$\overline{d}>0$$\lim_{\epsilonarrow 0}\frac{\lambda_{j}^{n,\epsilon}-\lambda_{0}^{n,\epsilon}}{\epsilon^{2}}=0.$

Properties of principal eigenfunction

With use of (i) of Proposition 2.4, one can easily obtain some properties of the principal
eigenfunctions.

Proposition 2.5. Assume $(A1)-(A3)$ . Let $n\in N$ and(i)$\varphi_{0}^{n,\epsilon}$

is $1/n$ -periodic,(ii)$\varphi_{0}^{n,\epsilon}$

is even $w.r.t.$(iii)$\varphi_{0}^{n,\epsilon}$$\epsilon\in(0, \sqrt{f_{u}(0)}/n\pi)$. Then,$x=z_{\ell}^{n},$is monotone increasing for $x\in[0, 1/2n].$Proof. The claims can be proved by using the properties $(G1)-(G3)$ . We omit a detail.
Proposition 2.6. Assume $(A1)-(A3)$ . Let $n\in N$ anda suitable normalizations on$\epsilon\in(0, \sqrt{f_{u}(0)}/n\pi)$$\square$. Then, Under$\varphi_{0}^{n,\epsilon},$$\lim_{\epsilonarrow 0}\varphi_{0}^{n,\epsilon}(z_{\ell}^{n}+\epsilon z)=(-1)^{\ell j}U_{z}(z)$(2.6)$\lim_{\epsilonarrow 0}\frac{\varphi_{0}^{n,\epsilon}(x_{\ell}^{n}+\epsilon z)}{\varphi_{0}^{n,\epsilon}(0)}=\cosh(\sqrt{-f_{u}(u_{\pm})}z)$

(2.7)

and

in the topology of$C_{loc}^{2}(R)$.

86
Proof. The claims can be proved by using the properties (G4), (G5) and (i) of Proposition

2.4. We omit a detail.

$\square$Proposition 2.7. Assume $(A1)-(A3)$ . Let $n\in N$ and$\delta\in(0,1/(4n))$$\epsilon\in(0, \sqrt{f_{u}(0)}/n\pi)$. Then for any,$\lim_{\epsilonarrow 0}\frac{1}{\epsilon}\int_{0}^{x}\frac{\varphi_{0}^{n,\epsilon}(0)^{2}}{\varphi_{0}^{n,\epsilon}(\xi)^{2}}d\xi=\frac{1}{\sqrt{-f_{u}(u_{\pm})}}$,

uniformly in$[\delta, z_{1}^{n}].$Proof. The comparison argument with (G6) and (i) of Proposition 2.4 proves the propo-

sition. We omit a detail.

3$\square$Symmetric Properties on eigenfunctions

Fix $n\in N,$$\epsilon\in(0, \sqrt{f_{u}(0)}/n\pi)$

and $0<j<n$ . Let us consider the auxially problem:(3.1)$\{\begin{array}{ll}\epsilon^{2}\tilde{\varphi}_{xx}(x)+(g_{n,\epsilon}(x)+\lambda)\tilde{\varphi}(x)=0 in (0,1) ,\tilde{\varphi}(0)=0, \end{array}$where
is to be chosen in later. Whenthe solution of (3.1) by$\tilde{\varphi}_{x}(0)\neq 0$$\lambda=\lambda_{j}^{n,\epsilon}$

for some $0<j<n$ , we denote$\tilde{\varphi}_{j}^{n,\epsilon}$Set$M_{j}^{n,\epsilon}(x):=[(\varphi_{j}^{n,\epsilon})_{x}(x)\varphi_{j}^{n,\epsilon}(x) (\tilde{\varphi}_{j}^{n,\epsilon})_{x}(x)\tilde{\varphi}_{j}^{n,\epsilon}(x)]$The following proposition on the monodromy martix is proved by Floquet theory with
(G1) and (G2), and by a standard linear algebra.

Proposition 3.1. Let $n\in N,$(3.1), there exists a unique$\epsilon\in(0, \sqrt{f_{u}(0)}/n\pi)$$\gamma_{j}^{n,\epsilon}>0$

and a uniqueand. For anysuch that$\varphi_{j}^{n,\epsilon}$$\tilde{\varphi}_{j}^{n,\epsilon}$of (2.3) and$\kappa_{j}^{n,\epsilon}>0$$M_{j}^{n,\epsilon}( \frac{1}{n})M_{j}^{n,\epsilon}(0)^{-1}=[_{-\gamma_{j}^{n,\epsilon}\sin\frac{j\pi}{n}}\cos\frac{j\pi}{n} \frac{1}{\gamma_{j}^{n_{)}\epsilon}}\sin\frac{j\pi}{n}\cos\frac{j\pi}{n}],$$M_{j}^{n,\epsilon}( \frac{1}{2n})M_{j}^{n,\epsilon}(0)^{-1}=[_{-\frac{\gamma_{j}^{n,\epsilon}}{\kappa_{j}^{n,\epsilon}}\sin\frac{j\pi}{2n}}\kappa_{j}^{n,\epsilon}\cos\frac{j\pi}{2n} \frac{}{}\cos\frac{j\pi 2nj\pi}{2n}\frac{\kappa_{j}^{n,\epsilon}}{\kappa_{j}^{n,\epsilon}\gamma_{j1}^{n,\epsilon}}\sin\frac{}{}]$

Remark 3.1. In the proof of Proposition 3.1, (G2) plays an essential role (see [6]). In
seems to be complicated.
absense of (G2), the representation of$M_{j}^{n,\epsilon}$

Moreover, under a suitable normalizing condition onproperties are obtained.

$\varphi_{j}^{n,\epsilon}$and$\tilde{\varphi}_{j}^{n,\epsilon}$, fine symmetric

87
Corollary 1. Suppose that
andUnder the normalizing condition$\gamma_{j}^{n,\epsilon}$$\kappa_{j}^{n,\epsilon}$be positive numbers given in Proposition 3.1.

$(\tilde{\varphi}_{j}^{n,\epsilon})_{x}(0)=\gamma_{j}^{n,\epsilon}\varphi_{j}^{n,\epsilon}(0)$the following $(i)-(iv)$ hold:

(i)

(ii)

$M_{j}^{n,\epsilon}( \frac{1}{n}-x)=\{\begin{array}{ll}1 00-1 \end{array}\}M_{j}^{n,\epsilon}(x)[ \cos\frac{j\pi}{J^{\pi}nn}\sin\frac{}{}$$- \cos\frac{j\pi}{n}\sin\frac{j\pi}{n}],$$M_{j}^{n,\epsilon}(x+ \frac{1}{n})=M_{j}^{n,\epsilon}(x)\{\begin{array}{ll}cos\frac{j\pi}{n} sin\frac{j\pi}{n}-sin\frac{J^{\pi}}{n} cos\frac{J^{\pi}}{n}\end{array}\},$(iii) for $\ell=0$ , 1, . . . , $n,$$M_{j}^{n,\epsilon}(x_{\ell}^{n})=\varphi_{j}^{n,\epsilon}(0)\{\begin{array}{lll}cos j\pi x_{\ell}^{n} sinj\pi x_{\ell}^{n}-\gamma_{j}^{n,\epsilon} sinj\pi x_{\ell}^{n} \gamma_{j}^{n,\epsilon}cosj\pix_{\ell}^{n}\end{array}\}$Let$\varphi_{j}^{n,\epsilon}$and$\tilde{\varphi}_{j}^{n,\epsilon}$be as in Proposition 3.2. Then,$\frac{d}{dx}(\frac{\tilde{\varphi}_{j}^{n,\epsilon}}{\varphi_{j}^{n,\in}})=\frac{1}{(\varphi_{j}^{n,\epsilon})^{2}}W(\varphi_{j}^{n,\epsilon},\tilde{\varphi}_{j}^{n,\epsilon})=\frac{\gamma_{j}^{n,\epsilon}\varphi_{j}^{n,\epsilon}(0)^{2}}{\varphi_{j}^{n,\epsilon}(x)^{2}},$where $W$ denotes the Wronskian. It implies that$\frac{\tilde{\varphi}_{j}^{n,\epsilon}(x)}{\varphi_{j}^{n,\epsilon}(x)}=\int_{0}^{x}\frac{\gamma_{j}^{n,\epsilon}\varphi_{j}^{n,\epsilon}(0)^{2}}{\varphi_{j}^{n,\epsilon}(\xi)^{2}}d\xi$

for suﬃciently small$x\in[0, z_{1}^{n}]$and putting $x=1/(2n)$ we see$\tan\frac{j\pi}{2n}=\int_{0}^{\frac{1}{2n}}\frac{\gamma_{j}^{n,\epsilon}\varphi_{j}^{n,\epsilon}(0)^{2}}{\varphi_{j}^{n,\epsilon}(\xi)^{2}}d\xi$

4Outline of proofs

4.1

Quotient of eigenfunctions

Fix $n\in N,$
and the eigenfunction. Suppose $0<j<n$ . For the eigenfunctionof the auxially problem (3.1), set$\epsilon\in(0, \sqrt{f_{u}(0)}/(n\pi))$$\tilde{\varphi}_{j}^{n,\epsilon}$$\psi_{j}^{n,\epsilon j}(x)$$:= \frac{\varphi_{j}^{n,\epsilon}(x)}{\varphi_{0}^{n,\epsilon}(x)}$and$\tilde{\psi}_{j}^{n,\epsilon}(x)$

(3.2)

.

$:= \frac{\tilde{\varphi}_{j}^{n,\epsilon}(x)}{\varphi_{0}^{n,\in}(x)}$$\varphi_{j}^{n,\epsilon}$of (2.3)

(4.1)

88
respectively, A direct calculation shows us thatlinear diﬀerential equation$\psi_{j}^{n,\epsilon}$

and also$\tilde{\psi}_{j}^{n,\epsilon}$satisfy the following(4.2)$( \psi_{j}^{n,\epsilon})_{xx}+2\frac{(\varphi_{0}^{n,\epsilon}(x))_{x}}{\varphi_{0}^{n,\epsilon}(x)}(\psi_{j}^{n,\epsilon})_{x}+\frac{\lambda_{j}^{n,\epsilon}-\lambda_{0}^{n,\epsilon}}{\epsilon^{2}}\psi_{j}^{n_{\rangle}\epsilon}=0$$(in a$subinterval $in [0,1])$ withand$(\psi_{j}^{n,\epsilon})_{x}(0)=(\psi_{j}^{n_{\}}\epsilon})_{x}(1)=0$$\tilde{\psi}_{j}^{n,\epsilon}(0)=\tilde{\psi}_{j}^{n,\epsilon}(1)=0$

Without loss of generality, we can always assume that$\psi_{j}^{n,\epsilon}(0)=\cos j\pi z_{\ell}^{n}.$In addition, for each$\varphi_{j}^{n,\epsilon},$$\tilde{\varphi}_{j}^{n,\epsilon}$should be chosen to satisfy the normalized condition$(\tilde{\varphi}_{j}^{n,\epsilon})_{x}(0)=\gamma_{j}^{n,\epsilon}\varphi_{j}^{n,\epsilon}(0)$as in Proposition 3.1; so that$( \tilde{\psi}_{j}^{n,\epsilon})_{x}(0)=\frac{(\tilde{\varphi}_{j}^{n,\epsilon})_{x}(0)}{\varphi_{j^{*}}^{n,\epsilon}(0)}=\gamma_{j}^{n,\epsilon}\psi_{j}^{n,\epsilon}(0)$.

Proposition 4.1. Let $n\in N$ , and let $0<j<n$ be fixed. Suppose thatthe functions defined in (4.1). Then,

uniformly in$\lim_{\epsilonarrow 0}\psi_{j}^{n,\epsilon}(x)=\cos j\pi z_{1}^{n}$

Proof. We first remark that$\varphi_{n}^{n,\epsilon}(x)>0$

from Sturm comparison theorem andbounded:for$x\in[0, z_{1}^{n}$$\psi_{j}^{n,\epsilon}$and$\tilde{\psi}_{j}^{n,\epsilon}$

are$[0, z_{1}^{n}].$) because of (G2). Hence it follows
thatis positive and$0<\lambda_{j}^{n,\epsilon}-\lambda_{0}^{n,\epsilon}<\lambda_{n}^{n,\epsilon}-\lambda_{0}^{n,\epsilon}$$\cos j\pi z_{1}^{n}\frac{\varphi_{n}^{n,\epsilon}(x)}{\varphi_{0}^{n,\epsilon}(x)}\leq\psi_{j}^{n,\epsilon}(x)\leq\cos j\pi z_{1}^{n}$for$\psi_{j}^{n,\epsilon}$$x\in[O, z_{1}^{n}].$Now we rewrite (4.2) as follows:$[\varphi_{0}^{n,\epsilon}(x)^{2}(\psi_{j}^{n,\epsilon})_{x}]_{x}+\underline{\lambda\lambda_{0_{\varphi_{0}^{n,\epsilon}(x)^{2}\psi_{j}^{n,\epsilon}}}^{n,\epsilon}}=0.$

By integrating the above equation from

$0$

to

$x$

we have$( \psi_{j}^{n,\epsilon})_{x}(x)+\underline{\lambda\lambda_{0}^{n,\epsilon}}\int_{0}^{x}(\frac{\varphi_{0}^{n,\epsilon}(\xi)}{\varphi_{0}^{n,\epsilon}(x)})^{2}\psi_{j}^{n,\epsilon}(\xi)d\xi=0$

.

(4.3)

By using (iii) of Theorem 2, we obtain$|(\psi_{j}^{n,\epsilon})_{x}(x)|\leq^{\underline{\lambda\lambda_{0}^{n,\epsilon}}}\cdot z_{1}^{n}\cos j\pi z_{1}^{n}.$Therefore, by (i) of Theorems 2$\lim_{\epsilonarrow 0}(\psi_{j}^{n,\epsilon})_{x}(x)=0$

uniformly in$[0, z_{1}^{n}],$and we obtain$\lim_{\epsilonarrow 0}\psi_{j}^{n,\epsilon}(x)=\cos j\pi z_{1}^{n}$

uniformly in$[0, z_{1}^{n}].$$\square$89
Corollary 2. Let $n\in N$ , and let $0<j<n$ be fixed. Suppose
Proposition 3.2. Then,be the number as in$\gamma_{j}^{n,\epsilon}$$\lim_{\epsilonarrow 0}\epsilon\gamma_{j}^{n,\epsilon}=\sqrt{-f_{u}(u_{\pm})}\tan\frac{j\pi}{2n}.$Proof. We see that for each$x\in[0, z_{1}^{n}],$$\frac{\tilde{\psi}_{j}^{n,\epsilon}(x)}{\psi_{j}^{n,\epsilon}(x)}=\epsilon\gamma_{j}^{n,\epsilon}\int_{0}^{x}\frac{\cos^{2}j\pi z_{1}^{n}}{\psi_{j}^{n,\epsilon}(\xi)^{2}}\cdot\frac{1}{\epsilon}\frac{\varphi_{0}^{n,\epsilon}(0)^{2}}{\varphi_{0}^{n,\epsilon}(\xi)^{2}}d\xi$

(4.4)

.

Set $x=z_{1}^{n}=1/(2n)$ in (4.4). It follows from (3.2) with the mean value theorem that$\tan\frac{j\pi}{2n}=\epsilon\gamma_{j}^{n,\epsilon}\cdot\frac{\cos^{2}j\pi z_{1}^{n}}{\psi_{j}^{n,\epsilon}(\xi_{1})^{2}}\cdot\frac{1}{\epsilon}\int_{0}^{1/2n}\frac{\varphi_{0}^{n,\epsilon}(0)^{2}}{\varphi_{0}^{n,\epsilon}(\xi)^{2}}d\xi,$where
By letting$\xi_{1}\in(0, z_{1}^{n})$.

$\epsilonarrow 0$

with Propositions 4.1 and 5.1, we obtain a desired result.

Proposition 4.2. Let $n\in N,$ $0<j<n,$any with $0<\delta<1/(4n)$ ,$\psi_{j}^{n,\epsilon}$and$\tilde{\psi}_{j}^{n,\epsilon}$$\square$as in Proposition 5.1. Then, for$\delta$$\lim_{\epsilonarrow 0}\tilde{\psi}_{j}^{n,\epsilon}(x)=\sin j\pi z_{1}^{n}$

uniformly in$[\delta, z_{1}^{n}].$Proof. The proof is a modification of the proof of Corollary 2. We omit a detail.
4.2$\square$

Proof of Theorem 1

Proof of Theorem 1. Now we extend the convergence result in Proposition 5.1 to$[0, x_{1}^{n}-\delta]$. Suppose$x\in[z_{1}^{n}, x_{1}^{n}-\delta]$$x\in$. By (i) of Proposition 3.2 we have$\psi_{j}^{n,\epsilon}(x)+i\tilde{\psi}_{j}^{n,\epsilon}(x)=(\psi_{j}^{n,\epsilon}(\frac{1}{n}-x)-i\tilde{\psi}_{j}^{n,\epsilon}(\frac{1}{n}-x))e^{ij\pi x_{1}^{n}}.$Hence it follows from Proposition 4.1 that$\lim_{\epsilonarrow 0}(\psi_{j}^{n,\epsilon}(x)+i\tilde{\psi}_{j}^{n,\epsilon}(x))=e^{-ij\pi/(2n)}e^{ij\pi/n}=e^{ij\pi z_{1}^{n}}$uniformly in$[z_{1}^{n}, x_{1}^{n}-\delta]$

.

(4.5)

. By combining Proposition 4.1 with (4.5), we obtain$\lim_{\epsilonarrow 0}\psi_{j}^{n,\epsilon}(x;\epsilon)=\cos j\pi z_{1}^{n}$

uniformly in$[0, x_{1}^{n}-\delta]$

(4.6)

.

(4.7)

and
$\lim_{\epsilonarrow 0}\tilde{\psi}_{j}^{n,\epsilon}(x;\epsilon)=\sin j\pi z_{1}^{n}$

uniformly in$[\delta, x_{1}^{n}-\delta]$

Therefore (i) of Theorem 1 follows from (4.6).
We note tha (iii) of Theorem 1 can be proved in similarly.
Finally, we suppose
for some $\ell=2$ , . . . , . It follows from (ii) of
Proposition 3.2 that$x\in[x_{\ell-1}^{n}+\delta, x_{p}^{n}-\delta]$$n$$\psi_{j}^{n,\epsilon}(x)+i\tilde{\psi}_{j}^{n,\epsilon}(x)=(\psi_{j}^{n,\epsilon}(x-\frac{\ell-1}{n})+i\tilde{\psi}_{j}^{n,\epsilon}(x-\frac{\ell-1}{n}))e^{ij\pi(\ell-1)/n}.$90
Therefore (4.6) and (4.7) leads us to$\lim_{\epsilonarrow 0}(\psi_{j}^{n,\epsilon}(x;\epsilon)+i\tilde{\psi}_{j}^{n,\epsilon}(x;\epsilon))=e^{ij\pi/(2n)}e^{ij\pi(\ell-1)/n}=e^{ij\pi z_{\ell}^{n}}.$uniformly in$[x_{\ell-1}^{n}+\delta, x_{\ell}^{n}-\delta]$. In particular, we have$\lim_{\epsilonarrow 0}\psi_{j}^{n,\epsilon}(x;\epsilon)=\cos j\pi z_{\ell}^{n}$

uniformly inand it proves (ii) of Theorem 1.
Thus it completes the proof of Theorem 1.
$[x_{\ell-1}^{n}+\delta, x_{\ell}^{n}-\delta]$

4.3$\square$

Proofs of Theorem 2

Proof of Theorem 2. For each $j=1$ , . . . , $n-1$ and$\ell=1$, . . . , $n-1$ and$\Psi_{j}^{n,\epsilon}(z;x_{\ell}^{n}):=\psi_{j}^{n,\epsilon}(x_{\ell}^{n}+\epsilon z)=\frac{\varphi_{j}^{n_{\rangle}\epsilon}(x_{p}^{n}+\epsilon z)}{\varphi_{0}^{n,\epsilon}(x_{p}^{n}+\epsilon z)}.$Similarly as (4.2) we can see that$\Psi_{j}^{n,\epsilon}(\cdot;x_{\ell}^{n})$satisfy the diﬀerential equation$( \Psi_{j}^{n,\epsilon})_{zz}+2\frac{\epsilon(\varphi_{0}^{n_{)}\epsilon})_{x}(x_{\ell}^{n}+\epsilon z)}{\varphi_{0}^{n,\epsilon}(x_{\ell}^{n}+\epsilon z)}(\Psi_{j}^{n,\epsilon})_{z}+(\lambda_{j}^{n,\epsilon}-\lambda_{0}^{n,\epsilon})\Psi_{j}^{n,\epsilon}=0$

By (iii) of Proposition 3.2 and Theorem 1 the initial condition on
given by.

$\Phi_{j}^{n,\epsilon}(\cdot;x_{\ell}^{n})$(4.8)

at $z=0$ is$\Phi_{j}^{n,\epsilon}(0;x_{\ell}^{n})=\cos j\pi z_{1}^{n}\cos j\pi x_{\ell}^{n}=\frac{1}{2}(\cos j\pi z_{\ell+1}^{n}+\cos j\pi z_{\ell}^{n})$and$( \Phi_{j,\ell}^{n,\epsilon})_{z}(0;x_{\ell}^{n})=\frac{\epsilon\gamma_{j}^{n,\epsilon}}{\tan j\pi z_{1}^{n}}\cdot\frac{1}{2}(\cos j\pi z_{\ell+1}^{n}-\cos j\pi z_{\ell}^{n})$.

Hence it follows from (ii) of Proposition 2.4, and Corollary 2 that$\lim_{\epsilonarrow 0}\Psi_{j}^{n,\epsilon}(z;x_{\ell}^{n})=\Psi_{j}^{n,0}(z;x_{\ell}^{n})$uniformly in any compact subset in $R$ , where$\Psi_{j}^{n,0}(z;x_{\ell}^{n})$is the solution of$(\Psi_{j}^{n,0})_{zz}+2\sqrt{-f_{u}(u_{\pm})}\tanh\sqrt{-f_{u}(u_{\pm})}z(\Psi_{j}^{n,0})_{z}=0$(4.9)

with the initial condition$\Psi_{j}^{n,0}(0;x_{\ell}^{n})=\frac{1}{2}(\cos j\pi z_{\ell+1}^{n}+\cos j\pi z_{\ell}^{n})$,$( \Psi_{j}^{n,0})_{z}(0;x_{\ell}^{n})=\frac{\sqrt{-f_{u}(u_{\pm})}}{2}(\cos j\pi z_{\ell+1}^{n}-\cos j\pi z_{\ell}^{n})$.

By solving (4.9) with the initial condition above, we are led to$\Psi_{j,\ell}^{n,0}(z)=\frac{\cos z_{\ell+1}^{n}+\cos z_{\ell}^{n}}{2}+\frac{\cos z_{\ell+1}^{n}-\cos z_{\ell}^{n}}{2}\tanh\sqrt{-f_{u}(u_{\pm})}z.$

Thus it completes the proof.

$\square$91

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