数理解析研究所講究録 1356 巻 2004 年 100-107

100

Stability of Stationary Interfaces of Binary-Tree Type: Overview

九州大学・大学院数理学研究院井古田亮 (IKOTA)
Faculty of Mathematics,$\mathrm{R}\mathrm{y}o$Kyushu University
東北大学・大学院理学研究科柳田英二 (E 小 YANAGIDA)
Mathematical Institute,Tohoku University

1Intoroduction

A curvature-driven motion with a triple junction has been introduced by Mullns [10]as a model of grain boundary motion in two dimensions. Later, the motion was derived
formally by Bronsard and Reitich [1] as the singular limit of a vector-valued Allen-Cahn
equation. Bronsard and Reitich [1] also showed short-time existence of the motion. Let
represent curves at time $t>0$ contained in a twO-dimensional bounded
Suppose $\Gamma_{i}(t)(i=1,2,3)$ meet at one point $m(t)$ .
region with smooth boundary
The evolving interface that we consider is subject to the following laws:$\Gamma_{\dot{1}}(t)(i=1,2,3)$

an.

$\Omega$(M1) The normal velocity of the interface is given by its curvature.

is givenand(M2) At the triple junction $m(t)$ , the contact angle between1),
positive(3,2).
is,
3,
1,, (2,forThat
by Young’s law, whereconstants , , ,$\mathrm{p}_{:}(t)$$\theta_{k}$$\Gamma_{j}(t)$$(\mathrm{i}\mathrm{j}, k)=(1,2,3)$$\sigma_{1}$$\sigma_{2}$$\sigma_{3}$$\frac{\sin\theta_{1}}{\sigma_{1}}=\frac{\sin\theta_{2}}{\sigma_{2}}=\frac{\sin\theta_{3}}{\sigma_{3}}$where $0<\theta_{k}<\pi$ and,$\theta_{1}+\theta_{2}+\theta_{3}=2\pi.$(M3) At the other end of each curve,$\Gamma_{i}(t)$touches

an at the right angle.

The interfaces have Energy $E(t)$ , which decreases as time goes:,$E(t)=\sigma_{1}|\Gamma_{1}(t)|+\sigma_{2}|\Gamma_{2}(t)|+\sigma_{3}|\Gamma_{3}(t)|$mean the lengths of curves Ti(t). Stationary interfaces of thewhere
motion can be viewed as critical points of the energy. In this connection Sternberg and$|\mathrm{Y}\mathrm{i}(\mathrm{t})|(i=1,2,3)$

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Ziemer [12] have proved the existence of local minimizers of the energy in clover-likeregions. Here we remark that stationary interfaces consist of straight line segments.
On the other hand, Ikota and Yanagida [8] have studied stabilities of stationary
interfaces of the motion
by linearizing corresponding equations around thestationary interfaces. They linearized the equations formally and analyzed the resulting
elliptic operator rigorously to obtain astability criterion which determines the unstabledimension in terms of the parameters. They also checked by numerical experimentsthat the linearized stability criterion suggests a nonlinear stability.
Recently Ikota and Yanagida have extended their results to stationary interfacesof binary-tree type with more than one triple junctions. In this article we give anoverview of their results.
$(\mathrm{M}1)-(\mathrm{M}3)$2

Formulation of the Problem

Consider a network of curves with triple junctions in . We assume the network
consists of curves denoted by ), $i=1,2$ ,, , and contacts withat endpoints (see Figure 1). We regard as the set of the curves, and denote$\Omega$$\Gamma=\Gamma(t)$

$n$

an

$=\gamma_{\dot{l}}(t)$$\ldots$$\mathrm{j}$$n$$\Gamma$$\{\gamma_{i}\}$$\Omega$

Figure 1: An interface$\Gamma$with triple junctions.

an.

by $B$ the subset of that consists of curves touching
Let : be positive constants
representing surface energy of . per unit length. We denote by $L(t)$ the lengths ofand by $V=\{x_{l}\}$ the set of triple junctions.
Every curve is driven to the center of curvature at the normal speed V. that isequal to the curvature of at each point. At each triple junction , three curves, say$\Gamma$$\sigma$$\gamma_{1}$$\gamma_{\dot{l}}$$\gamma\dot{.}$$\cdot$$\gamma_{i}$$x\iota$

102$\gamma_{i}$,$\gamma_{j}$

,

$Yk$

, meet with prescribed angles and satisfy Young’s law:$\frac{\sin\theta_{l,i}}{\sigma_{i}}=\frac{\sin\theta_{l,j}}{\sigma_{j}}=\frac{\sin\theta_{l,k}}{\sigma_{k}}$,,are alike. Eachat
andandis the angle between
where
at the right angle.
contacts with
We consider perturbations that can be represented as graphs of functions on , anddescribe the motion of nearby interfaces by using some nonlinear partial differential
equations with moving boundaries.
as themeet at a triple junction $x_{l}\in V$ . We take
Suppose that , ,origin of
coordinate system. For , the -axis is taken along , and the 77-axis
radian counter-clockwise. In this coordinate
is taken by rotating the -axis bysystem we consider a perturbation which can be represented as a graph of $\eta=w:(\xi)$ .
Approximating the time evolution of , we obtain a linear operator at . We takecoordinate systems for and in the same way, and describe perturbations by using. For details of this procedure, we refer to ourandsome functions
previous paper [8].
There are two ways of introducing a coordinate system on . because both end
points can be the origin. We will choose one of these coordinate systems according tosituations in order to make the presentation simple.
$(u_{1,12}, \ldots,u_{n})$ ,Now let us describe the linear operator more precisely. Put
be the length of . Then is written as
where : is defined on . Let$\theta_{l,i}$$\gamma_{\dot{*}}\in B$$\theta_{l,k}$$\theta_{l,j}$$x_{t}$$\gamma_{k}$$\gamma_{j}$an$\Gamma$$7\mathrm{i}$$\gamma_{j}$$x_{l}$$\gamma_{k}$$\xi$$\xi-\eta$$\gamma_{i}$$\mathrm{Y}\mathrm{i}$$\pi/2$$\xi$$\mathcal{L}$$tJ_{i}$$Yj$$w_{j}(t, \xi)$$\gamma_{\dot{1}}$$Ytt$$w_{k}(t,\xi)$$\gamma_{1}$$\mathcal{L}$$u$$\gamma.\cdot$$\mathrm{u}=$$\mathcal{L}$$L_{:}$(1)$7\mathrm{i}$$\mathcal{L}[\mathrm{u}]$$= \frac{\partial^{2}\mathrm{u}}{\partial\xi^{2}}$.

The associated boundary conditions are given as follows.

1. For . $\in B,$$\gamma_{1}$

(2)

2. If

$\frac{\partial u_{i}}{\partial\xi}(L_{i})+h:u:(L.\cdot)=0.$$\gamma.\cdot$,$\gamma_{j}$

,

$Yk$

meet at $x_{l}\in V,$$\sigma_{\dot{1}}u_{*}.(0)+$$\mathrm{v}_{j}u_{j}(0)+\sigma_{k}u_{k}(0)=0,$$!_{(\mathrm{O})}$$= \frac{\partial u_{j}}{\partial\xi}(0)=\frac{\partial u_{k}}{\partial\xi}(0)$.

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We set’$H:=\oplus L^{2}(0, L_{i})\gamma_{\dot{*}}\in\Gamma$

and treat$\mathcal{L}$as an operator from $H$ to $H$ with a domain of definition$\mathrm{D}(\mathcal{L})$$D(\mathcal{L})=\{\mathrm{u}\in\gamma_{j}\in\Gamma\oplus$=\{\mathrm{u}\in\gamma_{j}\in\Gamma\oplus H^{2}(0,H^{2}(0, L_{i})|\mathrm{u}$L_{i})|\mathrm{u}$The inner product ,$(\cdot, \cdot)_{H}$$\cdot)_{H}$$(\cdot$

satisfies conditions (3) and (4).

of $H$ is given by.v:dl .

$\{\sigma:\int_{0}^{L:}u_{1}$$( \mathrm{u}, \mathrm{v})_{\mathrm{H}}:=.\sum_{\gamma.\in}\mathrm{r}$

3$\}$

$\}$

Results

Our results are stated as follows.
Theorem 3.1. Let
be a stationary interface that is homeomorphic to a binarytree. Define a characteristic index by$\Gamma=\{\gamma_{\dot{1}}\}$$D$,$D=. \sum_{\gamma_{*}\in\Gamma}\sigma:L:\mathrm{x}.\prod_{\gamma.\in B}h_{\dot{l}}+.\sum_{\gamma.\in B}\{\sigma_{\dot{l}}\prod_{\gamma_{\mathrm{j}}\in B\backslash \{\gamma:\mathrm{J}}h_{j}\}$

an

where
denotes the curvature ofat the point of contact withis taken to be nonpositive if is convex.)$h_{i}$$h\dot{.}$$\gamma.\cdot\in B$. (Note that$\Omega$(i) The unstable dimension$N_{\mathrm{U}}$is given by$m-$$1$for$(-1)^{m}D\leq 0,$for$(-1)^{m}D>0,$$N_{\mathrm{U}}=\{$$m$

where $m=\#$ {A $<0$ }.
(ii) The stationary interface is degenerate ($i.e.$ , there exists a zero eigenvalue) andif
only if $D=0.$

We remark that the index$D$is independent of the topology of .
$\Gamma$104

Variational Methods

4

The operator$\mathcal{L}$naturally leads to a bilinear form.

Definition 4.1. A bilinear for$m$ $J:V\mathrm{x}Varrow$ R is defined by,$J( \mathrm{u},\mathrm{v}):=.\sum_{\gamma_{*}\in B}h_{i}u_{t}(L_{i})v:(L_{i})+.\sum_{\gamma.\in\Gamma}\sigma:\int_{0}^{L}:\partial_{\xi}u_{i}(\xi)\partial_{\xi}v_{i}(\xi)d\xi$

where
$V:=$

The inner product{$\mathrm{u}\in\gamma_{*}.\in\Gamma\oplus H^{1}$$(\cdot, \cdot)_{V}$(0,$L_{j}$)satisfies the condition (3)}.

$|\mathrm{u}$is given by$( \mathrm{u},\mathrm{v})_{\mathrm{V}}:=\sum_{\gamma.\in\Gamma}.\{\sigma:\int_{0}^{L}.\cdot(u:v_{\dot{1}} +\partial_{\xi}u_{i}\partial_{\xi}v:)d\xi\}$

In addition we introduce a functional$I$:

We can characterize the eigenvalues ofthe following result:$\mathcal{L}$R defined by$V\backslash \{\mathrm{O}\}arrow$$I( \mathrm{u}):=\frac{J(\mathrm{u},\mathrm{u})}{(\mathrm{u},\mathrm{u})_{\mathrm{R}}}$.

.

in terms of . Discussions similar to [8] yield$I$

Proposition 4.1. There eist positive numbers$c$andsuch that$d$for all$||\mathrm{u}||_{\mathrm{V}}^{2}\leq \mathrm{c}(\mathrm{u}, \mathrm{u})_{\mathrm{H}}+dJ(\mathrm{u}, \mathrm{u})$$\mathrm{u}\in V.$

Prom this we deduce that the operator is self-adjoint.
Let j) be the family of all finite dimensional subspaces of $H$ . Denote by the th. The eigenvalues a characterized by theeigenvalue of C. Then we have-inf principle:$\mathcal{L}$$\lambda_{j}$$\mathrm{X}_{j}\geq \mathrm{X}_{j+1}$$)_{\mathrm{j}}$$j$$\mathrm{e}$$\mathrm{s}\mathrm{u}\triangleright$(5)$-\mathrm{A}_{j}=$$\sup$$\inf$$I(\mathrm{v})$,$\dim K\leq-1\kappa\epsilon \mathrm{r}_{\dot{J}}v\in K^{[perp]}\backslash \{0\}$where$K^{[perp]}:=${$\mathrm{u}\in V|(\mathrm{u},\mathrm{v})_{\mathrm{H}}=0$

for aU$\mathrm{v}\in K$}.

For the proof, see Section 1, Chapter 13 of [11].
as parameters, each eigenvalue is a continuous and monotone deIf we takecreasing function of . See Theorems 6 and 9 in Chapter 6 of [2].
In view of the above observations we can prove the proposition below.
$\{h_{9}\}$$h_{i}$

Proposition 4.2. Put $m=\#\{h_{*}|<0| 7* \in B\}$ . Then

$N_{\mathrm{U}}\geq m-1$

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5Characteristic Functions

As in [8] we make characteristic functions.
Proposition 5.1. For any stationary interface
there exists a complex-valued characteristic function $F=F(\mu)$ of a complex variable with parameters aif ,andsatisfying the following properties:$\Gamma_{i}$$L_{*}.$$\mu$$(\gamma_{\dot{1}} \in\Gamma)$$h_{i}(\gamma_{i}\in B)$(i) For$F(\mu)=0$$\mu\neq 0,$(ii) $F(\mu)$ is analytic in
to real numbers.

if and only if$\mu_{l}\sigma_{\dot{\iota}}$,$L_{\dot{l}}$(iii) $F(\mu)$ is odd with respect to, and$\mu$$\lambda=\mu^{2}$$h_{\dot{l}}$is an eigenvalue of Z.

. FuMer,

is real-valued if$\mathrm{F}(\mathrm{f}\mathrm{i})$. In particular, $F(0)=0$ for any$\sigma_{\dot{1}}$

,

$\mu$

$L_{:}$,

is restricted$h_{\dot{l}}$.

(iv) Any zero of $F(\mu)$ lies on the real axis or imaginary axis, and it depends oncontinuously.

$\sigma_{\dot{1}}$,$L_{i}$,$h_{i}$(v)

as$F(\mu)arrow+\mathrm{o}\mathrm{o}$(vi) For eachpendent of$\muarrow+\mathrm{o}\mathrm{o}$$\gamma_{i}\in B_{f}F$$h_{i}$.

.

is notice as $F=P(\mu)h:+Q(\mu)$ , where $P$ and$Q$

are inde-We prove this by induction. First notice that the assertion was proved in [8] if
has only one triple junction.
Next, let be a stationary interface with two or more triple junctions. Take anedge
We divide into two parts
andby introducing a virtual
boundary which intersects
orthogonally (See Fig. 2). Denote by the curvatureof at the intersection point with . Then the curvature of $C$ is $-h$ forat thesame intersection point. Suppose that the assertion is true for,andand denoteby
andthe characteristic functions forand , respectively, satisfying theproperties. By (vi), we can write them as$\Gamma$$\Gamma$$\gamma_{k}\in\Gamma\backslash B.$$\Gamma$$C$$\Gamma^{\beta}$$\Gamma^{\alpha}$

$h$

$Ytt$

$C$

$\Gamma^{\alpha}$$\Gamma^{\beta}$$\Gamma^{\beta}$$\Gamma^{\alpha}$$F^{\beta}$$F^{\alpha}$$\Gamma^{\alpha}$$\Gamma^{\beta}$$(\mathrm{i})\sim(\mathrm{v}\mathrm{i})$$F^{\alpha}(\mu)=P^{a}(\mu)h+Q^{\alpha}(\mu)_{1}$(6)$F^{\beta}(\mu)=-P^{\beta}(\mathrm{p})h$ $+Q^{\beta}(\mu)$whereeliminate, ,are independent of . From $\mathrm{F}(0)=0$ andto define a function
by$P^{\alpha},Q^{\alpha}$

(7)

$h$,

$P^{\beta}$$Q^{\beta}$$h$$F^{\beta}(\mu)$$F^{\alpha+\beta}(\mu)$$F^{\alpha+\beta}( \mu):=\frac{P^{\alpha}(\mu)Q^{\beta}(\mu)+Q^{\alpha}(\mu)P^{\beta}(\mu)}{\sigma_{\dot{*}}\mu}$for$\mu\neq 0$$=0,$

we can108

Figure 2: The interface divided into two parts. The dotted line stands for the virtualboundary.

We can prove this function satisfies the desired properties.
andMoreover, from (7), we obtain an explicit expression of $(dF/d\mu)|_{\mu=0}$ .
$F^{\alpha+\beta}(0)=0.$

Proposition 5.2. Let be the characteristic function constructed as above. Then thederivative of $F(\mu)$ at $\mu=0$ is given by$F$.

$D:=–|_{\mu=0}dF. \cdot.:\overline{d}\overline{\mu}=\sum_{\gamma.\in\Gamma}\sigma.L:\mathrm{x}\prod_{\gamma.\in B}h+.\sum_{\gamma.\in B}\{\sigma_{i}\prod_{:\gamma_{j}\in B\backslash \{\gamma\}}h_{j}\}$6

Outline of the Proof

We have the following lemma on the nondegeneracy of zero eigenvalues.
Lemma 6.1. If at most one of$h_{i}(\gamma:\in B)$

is zero, then any zero eigenvalue is simple.

First notice that we can deform 0 without changing the shape of a given stationary
as variable parameters. Without loss ofinterface. Hence we may regard$(\gamma_{i}\in B)$$h_{i}$

generality, we put$B=\{\gamma_{1}, \ldots,\gamma_{k}\}$.

We count the number of positive eigenvalues as follows. Assume first that , ,Next, we decrease the values of , , . . . ,..., $h_{k}>0.$ Then (5) implies
one by one to negative values. By this procedure, the index can change its sign
On the other hand, Proposition 4.2 showsat most $m$ times and henceis even if, , $h_{k}>0$ ,Hence $N_{\mathrm{U}}=m-1$ or . Since $D>0$ if$D>0$ and is odd if $D<0.$ Thus (i) is proved.
$\mathrm{h}\mathrm{i}$$h_{1}$$N_{\mathrm{U}}=0.$$h_{2}$$D$$N_{\mathrm{U}}\leq m.$$N_{\mathrm{U}}\geq m-1.$$m$$h_{1},h_{2}$$N_{\mathrm{U}}$$\ldots$$h_{2}$$h_{m}$

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:$124\mathrm{A}$