HUSCAP Journals

Instructions for use A uthor(s ) Garcke,Harald; K ohsaka,Y oshihito; S evcovic,D aniel

C itation Hokkaido University Preprint S eries in Mathematics, 900: 1-35

Is s ue D ate 2008

D O I 10.14943/84050

D oc UR L http://hdl.handle.net/2115/69708

T ype bulletin (article)

F ile Information pre900.pdf

for curvature flow with triple junction

Harald Garcke, Yoshihito Kohsaka, and Daniel ˇ

Sevˇcoviˇc

Abstract. In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida [13] is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove exponential stabilization of an evolving network starting from the vicinity of a linearly stable stationary network.

1

Introduction

The motion of curves under the curvature flow has been widely studied in the past [6, 11, 3]. Less is known about the evolution of networks under the curvature flow [4, 13, 20]. In this case the arcs in the network evolve in the normal direction with a speed proportional to the curvature of the arcs. At intersections with an outer boundary and at triple junctions boundary conditions have to hold. At the outer boundary one can prescribe the position (see [16, 18]), or the angle with the outer boundary [4, 13]. At the triple junction Young’s law, a force balance, leads to angle conditions. In this paper we are interested in the stability of stationary solutions to the curvature flow with a triple junction when we prescribe the natural angle condition of 90◦ _{at the outer boundary. For this case a linear}

stability criterion has been derived by Ikota and Yanagida [13] (see also [14]). We will demonstrate here that this criterion also leads to nonlinear stability.

We now specify the problem in detail. Let Ω be a bounded domain in R2 _{with C}3 -boundary ∂Ω. We introduce a C3-functionψ :R2 _→R _{with ∇ψ(x)6= 0 ifψ(x) = 0 such} that

Ω ={x∈R2 _{|ψ(x)<0}, ∂Ω = {x∈}R2 _{|ψ(x) = 0}.} We search families of curves Γ1

t, Γ2t, and Γ3t which are parameterized by timet and which

are contained in Ω. The three curves are supposed to meet at a triple junction p(t)∈ Ω at their one end point and at the other end point they are required to intersect with∂Ω, see Figure 1. We require for i= 1,2,3

βiVi =γiκi on Γit, (1.1)

3

X

i=1

γiTi = 0 at p(t), (1.2)

Γi_t⊥∂Ω at Γi_t∩∂Ω. (1.3)

Figure 1: The curvature driven flow Γt of a network with a triple junction at p(t) (left)

and a steady state Γ∗ (right).

Here Vi _{and κ}i _{are the normal velocity and curvature of Γ}i

t, respectively. The constants βi _{and γ}i _{are given physical parameters and T}i _{are unit tangents to the curve which are}

chosen such that they point away from the triple junction.

Equation (1.2) is a force balance and one can solve for the Ti’s if the condition

γi+γj ≤γk for all {i, j, k} mutually different,

is fulfilled. In the following we assume strict inequalities and an argument as in Bronsard and Reitich [4] gives that the angles θi _{between the tangents T}

j and Tk fulfill

sinθ1

γ1 =

sinθ2

γ2 =

sinθ3

γ3

with 0< θi _{< π(i= 1,2,3) andθ}1_+θ2_+θ3 _{= 2π. Existence of solutions to the evolution} problem (1.1)-(1.3) has been shown by Bronsard and Reitich [4]. We will show later that the energy functional

E[Γt] =

3

X

i=1

γiL[Γi_t]

where Γt = S3_i=1Γti and L[Γit] is the length of Γit, is a Ljapunov functional. The

con-stants γi _{can be interpreted as surface free energy densities (surface tensions) and the}

functional E is the total free energy of the systems. Sternberg and Zeimer [21] showed the existence of isolated local minimizers to E, which can be interpreted as solutions to a partitioning problem of two dimensional domains into three subdomains having (locally) least interfacial area.

Figure 2: Description of the local parametrization of the curve Γi_.

proceed by deriving useful geometric equations for the curvature and other geometric quantities in Section 5. Using the linearized stability criterion we show how to derive a priori estimates for Sobolev norms of the solution. These geometric equations are then used in order to prove usefull bounds for a solution. With the help of these energy type estimates we prove global existence of a classical solution. In the final section 7 we prove exponential stability of the stationary solution.

2

Parameterization and local existence

We consider line segments Γ1

∗, Γ2∗ and Γ3∗ meeting the outer boundary with an angle of

90◦ _{at their one end point and having without loss of generality p}

∗ = (0,0)T as their

common other end point where we assume that (1.2) holds. Then we define an arc-length parameterization of Γi

∗ (i= 1,2,3) as

Γi∗ ={Φi∗(σ)|σ∈[0, li]}

with Φi

∗(0) =p∗ = (0,0)T, Φi∗(li)∈∂Ω. In particular, we obtain thatliis the length of Γi∗.

Then we will extend Φi

∗ as an arc-length parameterization of the full line which contains

Γi

∗. We will now introduce a certain stretched coordinate system in order to allow for

parameterizations of curves close to Γi

∗ (i= 1,2,3) over fixed intervals [0, li].

Let Ti

∗ be the unit tangent to Γi∗ pointing from the triple junction p∗ to the outer

boundary and let Ni

∗ = RT∗i be a unit normal where R is the anticlockwise rotation by

π/2. We then define

µi = (p, T∗i)R2, µi_∂Ω(q) = max{σ|Φi_∗(σ) +qN_∗i ∈Ω}.

We remark that the parameterµi _{allows for a tangential movement of the triple junction}

along Γi

∗. We now set

Ψi(σ, q, µi) = Φi∗(ξi(σ, q, µi)) +qN∗i,

where

ξi(σ, q, µi) =µi+σ

li(µ i

We now define the parameterization of curves Γ = (Γ1_,Γ2_,Γ3_{) close to Γ}

∗ = (Γ1∗,Γ2∗,Γ3∗)

having their triple junction at the point pwith the help of functions

ρi : [0, li]→R

which fulfill the conditions

ρi(0) = (p, N∗i)R2 (i= 1,2,3) (2.1)

(see Fig. 2). Set

Φi(σ) = Ψi(σ, ρi(σ), µi), σ∈[0, li]. (2.2) Then the functions Φi _{parameterize the curves Γ}i _{in the neighborhood of Γ}

∗ as Γi =

{Φi_{(σ)|σ ∈ [0, l}i_{]}. Since Φ}i

∗(µi) = µiT∗i, we have Φi(0) = µiT∗i +ρi(0)N∗i = p, which

implies that

µ1T∗1+ρ1(0)N∗1 =µ2T∗2+ρ2(0)N∗2 =µ3T∗3+ρ3(0)N∗3. (2.3)

By virtue of the definition of µi_{, equation (2.1), and Young’s law}

3

X

i=1

γiT∗i = 0,

3

X

i=1

γiN∗i = 0,

we are led to

3

X

i=1

γiµi = 0,

3

X

i=1

γiρi(0) = 0.

Furthermore, identities (2.3) and the angle conditions give the following lemma.

Lemma 2.1 Let us define the matrix

Q=− 1

1−c1_c2_c3





c3_c1_s2 _s3 _c3_s1

c1_s2 _c1_c2_s3 _s1

s2 _c2_s3 _c2_c3_s1



,

where ci _{= cosθ}i_{, s}i _{= sinθ}i_{. Then, for ρ = (ρ}1_{, ρ}2_{, ρ}3_{), and µ = (µ}1_{, µ}2_{, µ}3_{), it holds}

µT =QρT(0).

Proof. It follows from (2.3) that

µi =µj(T∗j, T∗i)R2 +ρj(0)(N_∗j, T_∗i)_R2

for (i, j) = (1,2), (2,3), (3,1). By the angle condition, we have

(T_∗i, T_∗j)_R2 = cosθk, (T_∗i, N_∗j)_R2 = cos(θk+π/2) =−sinθk

for (i, j, k) = (1,2,3), (2,3,1), (3,1,2). This implies µi_−ck_µj _=−sk_ρj_{(0), so that we are}

led to _



−c2 _{0 1} 1 −c3 ₀ 0 1 −c1

 

 

µ1

µ2

µ3



=

 

−s2 _{0 0} 0 −s3 ₀ 0 0 −s1

 

 

ρ1₍₀₎

ρ2₍₀₎

ρ3₍₀₎

Then we obtain





µ1

µ2

µ3



= −1

1−c1_c2_c3





c3_c1 _{1 c}3

c1 _c1_c2 ₁ 1 c2 _c2_c3



 



s2 _{0 0}

0 s3 ₀ 0 0 s1



 



ρ1₍₀₎

ρ2₍₀₎

ρ3₍₀₎



=Q





ρ1₍₀₎

ρ2₍₀₎

ρ3₍₀₎





which completes the proof.

We now consider evolving curves

Γi(t) ={Φi(σ, t)|σ∈[0, li]},

where Φi_{(·, t) are defined as in (2.2) such that ρ}i_{(·, t) (i = 1,2,3) satisfy (2.1). We}

formulate the curvature flow for a network with the help of these parameterizations. For that purpose, the following quantities are needed

Ti = 1

|Φi σ|

Φi_σ, Ni = 1

|Φi σ|

RΦi_σ,

Vi = (Φi_t, Ni)R2, κi = 1

|Φi σ|

T_σi, Ni_R2, i= 1,2,3. (2.4)

Then we obtain the following formulation for the curvature flow of a network:

βi(Φi_t, Ni)R2 =γi 1

|Φi σ|

T_σi, Ni_R2, i= 1,2,3, (2.5)

3

X

i=1

γiρi = 0 at σ= 0, (2.6)

(T1, T2)R2 = cosθ3, (T3, T1)_R2 = cosθ2 at σ = 0, (2.7)

(Ni,∇ψ(Φi))_R2 = 0 at σ =li, i = 1,2,3. (2.8)

Note that the conditionsψ(Φi_(li_{, t)) = 0 (i= 1,2,3) and Φ}1_{(0, t) = Φ}2_{(0, t) = Φ}3_{(0, t) are} always fulfilled with our choice of the parameterizations. We can formulate the problem in terms of (ρ1_{, ρ}2_{, ρ}3_{) and obtain a system of three second order parabolic equations where} each equation is defined on a different spatial interval [0, li_{]. We obtain one boundary}

condition at σ =li _{for the i-th equation and the three equations are coupled through the}

three boundary conditions at the triple junction.

Let us derive the form of the nonlinear system for ρi_{(σ, t) (i= 1,2,3). Set I}i _{= [0, l}i_]

and Qi

t0,t1 =Ii×(t0, t1]. Equations (2.5) give

ρi_t =Li(ρi, ρi_σ, µi)κi(ρi, ρi_σ, ρ_σσi , µi) + Λi(ρi, ρi_σ, µi)µi_t for (σ, t)∈ Qi_0,T, (2.9)

where Li_(ρi_{, ρ}i

σ, µi) and Λi(ρi, ρiσ, µi) are

Li(ρi, ρi_σ, µi) = γ

i

βi_(Ψi

q, RΨiσ)R2

Ji(ρi, ρi_σ, µi),

Λi(ρi, ρi_σ, µi) =− 1

(Ψi

q, RΨiσ)R2

with the notationJi _=|Φi

σ|, and the curvatureκi =κi(ρi, ρiσ, ρiσσ, µi) is represented as κi(ρi, ρi_σ, ρi_σσ, µi)

= 1

{Ji_(ρi_{, ρ}i σ, µi)}3

h

(Ψiq, RΨiσ)R2ρi_σσ +2(Ψi_σq, RΨi_σ)_R2 + (Ψi_σσ, RΨi_q)_R2 ρi_σ

+(Ψi_qq, RΨi_σ)R2 + 2(Ψi_σq, RΨi_q)_R2 + (Ψi_qq, RΨi_q)_R2ρi_σ (ρ_σi)2+ (Ψi_σσ, RΨi_σ)_R2

i

.

By virtue of Lemma 2.1 and (2.9), we have

 

µ1

t µ2

t µ3

t



=QT0

 

ρ1

t ρ2

t ρ3

t



=QhT0M(ρ,ρ_σ,µ)i−1T0

 

L1_(u1_)κ1_(u1₎

L2_(u2_)κ2_(u2₎

L3_(u3_)κ3_(u3₎



,

whereQis the matrix as in Lemma 2.1,T0_{is the trace operator toσ = 0, i.e. T}0_{f =f}

σ=0, and M(ρ,ρ_σ,µ) is the matrix

M(ρ,ρ_σ,µ) = Id−diag(Λ1(u1),Λ2(u2),Λ3(u3))Q

with the notationui = (ρi_{, ρ}i σ, µi).

Remark 2.2 The matrix M(ρ,ρ_σ,µ) is invertible provided that

3

X

i=1

sup

t∈[0,T]

kρi(·, t)kC1_(Ii₎+ sup t∈[0,T]

|µi(t)|< δ0 (2.10)

for some δ0 > 0. Indeed, we have detM(ρ,ρσ,µ) = d

−1 + (c2 _{− Λ}1_(u1_)s2_)(c3 ₋ Λ2(u2)s3)(c1 −Λ3(u3)s1) , where d = −1/(1−c1c2c3). Then, Λi(0) = 0 (i = 1,2,3) imply that detM(0,0,0) = d(−1 +c1_c2_c3_{) = 16= 0. Since detM(ρ,ρ}

σ,µ) is continuous

with respect to ρ, ρ_σ, and µ, we can conclude that for ε < 1 there exists a δ0 > 0 such that detM(ρ,ρ_σ,µ)>1−ε >0 provided that (2.10) holds.

As a consequence, we are led to the following nonlinear nonlocal partial differential equa-tions for ρi_{(σ, t) (i= 1,2,3):}

ρi_t = ai(ρi, ρi_σ, µi)ρi_σσ+ Λi(ρi, ρi_σ, µi) 3

X

j=1

aij₁(T0ρ,T0ρ_σ,µ)T0ρj_σσ

+fi(ρi, ∂σρi,T0ρ,T0ρσ,µ) for (σ, t)∈ Qi0,T

where ai_(ρi_{, ρ}i

σ, µi) = γi/βi{Ji(ρi, ρiσ, µi)}2 and a ij

1(T0ρ,T0ρσ,µ) is the (i, j)-component

of the matrix

a1(T0_ρ,T0_ρ

σ,µ) =Q

h

T0_M(ρ,ρ

σ,µ)

i−1

Furthermore,fi _{is a smooth function inRwhich is evaluated at lower order terms. Then,}

recalling the boundary conditions (2.6)-(2.8) and Lemma 2.1, we have the following non-linear system:

                              

ρi

t =ai(ρi, ρiσ, µi)ρiσσ+ Λi(ρi, ρiσ, µi)

3

X

j=1

aij₁(T0ρ,T0ρ_σ,µ)T0ρj_σσ

+fi_(ρi_{, ρ}i

σ,T0ρ,T0ρσ,µ) for (σ, t)∈ Qi0,T,

3

X

i=1

γiρi = 0, g12(u12) = 0, g13(u13) = 0 at σ = 0,

bi

∂Ω(ρi, µi)ρiσ +g∂iΩ(ρi, µi) = 0 at σ =li (i= 1,2,3),

µT =QT0_ρT _{for t∈(0, T]}

(2.11)

where u1j = (ρ1_{, ρ}j_{, ρ}1

σ, ρjσ, µ1, µj) (j = 2,3) and g12_(u12_{) = (Ψ}1

σ,Ψ2σ)R2 + (Ψ1_σ,Ψ2_q)_R2ρ2_σ + (Ψ1_q,Ψ2_σ)_R2ρ1_σ + (Ψ1_q,Ψ2_q)_R2ρ1_σρ2_σ

−J1(ρ1, ρ1_σ, µ1)J2(ρ2, ρ2_σ, µ2) cosθ3,

g13(u13) = (Ψ3_σ,Ψ1_σ)R2 + (Ψ3_σ,Ψ1_q)_R2ρ1_σ + (Ψ3_q,Ψ1_σ)_R2ρ3_σ + (Ψ_q3,Ψ1_q)_R2ρ3_σρ1_σ

−J3_(ρ3_{, ρ}3

σ, µ3)J1(ρ1, ρ1σ, µ1) cosθ2,

bi_∂Ω(ρi, µi) = (RΨ_qi,∇ψ(Ψi))R2, g_∂i_Ω(ρi, µi) =−(RΨi_σ,∇ψ(Ψi))_R2.

Now we are ready to state a local existence result.

Theorem 2.3 (Local existence) Let α ∈ (0,1) and let us assume that ρi

0 ∈ C2+α(Ii) and µi

0 (i= 1,2,3) with sufficiently small norms kρi0kC1+α_(Ii₎ and |µi₀| fulfill the compati-bility conditions

    

3

X

i=1

γiρi₀ = 0, g12(u12₀ ) = 0, g13(u13₀ ) = 0 at σ = 0,

bi

∂Ω(ρi0, µi0)ρi0,σ+g∂iΩ(ρi0, µi0) = 0 at σ =li (i= 1,2,3), where u1₀j = (ρ1

0, ρ

j

0, ρ10,σ, ρ j

0,σ, µ10, µ

j

0) (j = 2,3). Then there exists a

T0 =T0

1/

3

X

i=1

kρi₀kC2+α_(Ii₎

>0,

T0 being an increasing function of its argument and such that the problem (2.11) with (ρi_{(·,0), µ}i_{(0)) = (ρ}i

0, µi0) (i= 1,2,3) has a unique solution (ρ1_{, ρ}2_{, ρ}3_{, µ}1_{, µ}2_{, µ}3_)∈C2+α,1_(Q1

0,T0)×C2+α,1(Q20,T0)×C2+α,1(Q30,T0)×

C1_{[0, T} 0]

3

In order to prove Theorem 2.3 by using a contraction principle, we need some prepara-tions which consist of three steps: 1) the linearization of (2.11) around the initial data; 2) the verification of the complementary conditions for the linearized system; 3) the deriva-tion of suitable a priori estimate for soluderiva-tions of the linearized system.

Step 1. Let us derive the linearization of (2.11) around the initial data ρi

0 ∈C2+α(Ii) and µi

0 (i= 1,2,3). First we define differential operators as

A0 = diag(a1(u10), a2(u20), a3(u30))∂σ2,

A1 = diag(Λ1(u10),Λ2(u20),Λ3(u30))a1(T0ρ0,T0ρ0,σ,µ0)T0∂σ2,

and also define, for given functions (ρi_{, µ}i_)∈C2+α,1_(Qi

0,T)×C1[0, T] (i= 1,2,3), Fi(σ, t) = ai(ui)−ai(ui₀) ρi_σσ

+ 3

X

j=1

Λi(ui)aij₁(T0ρ,T0ρ_σ,µ)−Λi(ui₀)aij₁(T0ρ₀,T0ρ_0,σ,µ₀) T0ρj_σσ

+fi(ρi, ρi_σ,T0ρ,T0ρ_σ,µ),

where ui₀ = (ρi

0, ρi0,σ, µi0), ρ0 = (ρ10, ρ20, ρ30), µ0 = (µ10, µ20, µ30), ui = (ρi, ρiσ, µi), ρ =

(ρ1_{, ρ}2_{, ρ}3_{), andµ= (µ}1_{, µ}2_{, µ}3_{). Then, settingF = (F}1_{, F}2_{, F}3_{), we have the linearization} of the differential equation given as

∂tρT =A0ρT +A1ρT +FT(σ, t).

Let us derive the linearization of the boundary conditions. For ρi

0 ∈ C2+α(Ii) and µi0 (i= 1,2,3), we define differential operators as

(Bki(T0ρ₀,T0ρ_0,σ,µ₀)∂σ)i=1,2,3 =

    

(γ1_{, γ}2_{, γ}3_{) for k = 1,} (b21_(T0_u12

0 )∂σ, b22(T0u120 )∂σ,0) for k = 2,

(b31_(T0_u13

0 )∂σ,0, b33(T0u130 )∂σ) for k = 3.

Here the components are represented as follows:

b21(u12) = (Ψ1_q,Ψ2_σ)R2 + (Ψ1_q,Ψ2_q)_R2ρ2_σ−(Ψ1_σ,Ψ1_q)_R2 +|Ψ1_q|2ρ1_σ

J2_(u2₎

J1_(u1₎cosθ 3_,

b22(u12) = (Ψ1_σ,Ψ2_q)R2 + (Ψ1_q,Ψ2_q)_R2ρ1_σ−

(Ψ2_σ,Ψ2_q)R2 +|Ψ2_q|2ρ2_σ

J1_(u1₎

J2_(u2₎cosθ 3_,

b31(u13) = (Ψ3_σ,Ψ1_q)R2 + (Ψ3_q,Ψ1_q)_R2ρ3_σ−(Ψ1_σ,Ψ1_q)_R2 +|Ψ1_q|2ρ1_σ J

3_(u3₎

J1_(u1₎cosθ 2_,

b33(u13) = (Ψ3_q,Ψ1_σ)_R2 + (Ψ3_q,Ψ1_q)_R2ρ1_σ−

(Ψ3_σ,Ψ3_q)_R2 +|Ψ3_q|2ρ3_σ

J1_(u1₎

J3_(u3₎cosθ 2_.

Also, we define differential operators as

(B_∂ki_Ω(Tli

ρi₀, µi₀)∂σ)i=1,2,3 =

    

(b1

∂Ω(Tl

1 ρ1

0, µ10)∂σ,0,0) for k= 1,

(0, b2

∂Ω(Tl

2 ρ2

0, µ20)∂σ,0) for k= 2,

(0,0, b3

∂Ω(Tl

3 ρ3

where Tli

(i= 1,2,3) is the trace operator ontoσ =li_{, i.e. T}li

f =f_σ=li. Then we set

B0(0 ;∂σ) = (Bki(T0ρ0,T0ρ0,σ,µ0)∂σ)k,i=1,2,3,

B0(li;∂σ) = (B∂kiΩ(Tl

i

ρi₀, µi₀)∂σ)k,i=1,2,3,

and also set, for (ρi_{, µ}i_)∈C2+α,1_(Qi

0,T)×C1[0, T] (i= 1,2,3), G1(t) = 0,

Gj(t) =T0h_bj1_(u1j

0 )ρ10,σ+bjj(u

1j

0 )ρ

j

0,σ− ∂g1j

∂ρ1 (u 1j

0 )(ρ1−ρ10)−

∂g1j ∂ρj (u

1j

0 )(ρj−ρ

j

0)

−∂g

1j ∂µ1 (u

1j

0 )(µ1−µ10)−

∂g1j ∂µj (u

1j

0 )(µj −µ

j 0) −1 2 Z 1 0

hD2g1j(ηu1j + (1−η)u1₀j)(u1j −u₀1j),u1j −u₀1jidηi (j = 2,3),

Gi_∂Ω(t) =Tlih−bi_∂Ω(ρi, µi)−bi_∂Ω(ρi₀, µi₀) ρi_σ+gi_∂Ω(ρi, µi)i (i= 1,2,3),

where Dg1j _{is the Fr´echet derivative of g}1j _{and the bracket h·,·i is the respective inner}

product. Then we have the linearization of the boundary conditions:

B0(0 ;∂σ)ρT =GT(t), B0(li;∂σ)ρT =GT∂Ω(t) for G(t) = (G1_{(t), G}2_{(t), G}3_{(t)) and G}

∂Ω(t) = (G1∂Ω(t), G2∂Ω(t), G3∂Ω(t)).

Step 2. Let us verify that the complementary conditions hold for the linearized system. We refer to Lunardi [17] for more information on the role of the complementary conditions. For that purpose, we make some preparations. Let L0(r,iζ) = (a0ij)i,j=1,2,3 where

aii₀ = 1

{Ji_(ui

0)}2

ζ2+r, aij₀ = 0 for i6=j.

Then we have

detL0 = 3

Y

i=1

1

{Ji_(ui

0)}2

ζ2+r

.

Setting ˆL0 = (ˆaij0) = (detL0)(L0)−1, we are led to

ˆ

aii0 = 3

Y

k=1,i6=k

1

{Ji_(ui

0)}2

ζ2+r

, aˆij0 = 0 for i6=j.

The matrix of the boundary conditions at σ= 0 is denoted by

(B₀ki(0 ;iζ))i=1,2,3 =

    

(γ1_{, γ}2_{, γ}3_{) for k = 1,} (ib21_(T0_u12

0 )ζ,ib22(T0u120 )ζ,0) for k = 2, (ib31_(T0_u13

0 )ζ,0,ib33(T0u130 )ζ) for k = 3, and at σ=li _by

(B₀ki(li;iζ))i=1,2,3 =

    

(ib1

∂Ω(Tl

1 ρ1

0, µ10)ζ,0,0) for k= 1, (0,ib2

∂Ω(Tl

2 ρ2

0, µ20)ζ,0) for k= 2, (0,0,ib3

∂Ω(Tl

3 ρ3

To verify the complementary condition we will show that the rows of the matrixB0Lˆ0 are linearly independent for all r∈C_{, Rer >0, modulo the polynomial}

P(r, ζ) = 3

Y

i=1

{ζ−ζ₀i(r)}

whereζi

0(r) = Ji(ui0)|r|

1 2ei(

Θ 2+

π

2) (i= 1,2,3). Here we note that ζi

0(r) are the roots of the polynomial detL0(r,iζ) which have a positive imaginary part.

First let us verify the complementary condition at σ = 0. To determine whether or not the complementary condition is satisfied, we have to verify that the system

3

X

k=1

ωkB0ki(0 ;iζ)ˆaii0(ζ)≡0 mod P(r, ζ) = 3

Y

i=1

(ζ−ζ₀i) (i= 1,2,3)

has the unique solution (ω1, ω2, ω3)T = 0 or equivalently that (ω1, ω2, ω3)T = 0 is the only vector satisfying P3_k=1ωkBki0 (0 ;iζ) ≡ 0 mod ζ −ζ0i (i = 1,2,3). That is, we may investigate that (ω1, ω2, ω3)T =0 is the only vector satisfying P_k3₌₁ωkB0ki(0 ;iζ0i) = 0 (i= 1,2,3). Thus it suffices to show that

det

  

γ1 _γ2 _γ3

b21_(T0_u12

0 ) b22(T0u120 ) 0

b31_(T0_u13

0 ) 0 b33(T0u130 )

 

6= 0. (2.12)

Indeed, in the case (ρi

0, µi0)≡(0,0) (i= 1,2,3), we have

det

  

γ1 _γ2 _γ3

b21_{(0) b}22_{(0) 0}

b31_{(0) 0 b}33₍₀₎

 

 = det

  

γ1 _γ2 _γ3

sinθ3 _−sinθ3 ₀

−sinθ2 _{0 sinθ}2

  

= −(γ1+γ2+γ3) sinθ2sinθ3 6= 0.

Since the determinant in (2.12) is continuous with respect to ρi

0 and µi0 (i = 1,2,3), we are led to (2.12) provided that kρi

0kC1_(Ii₎ and |µi₀| are small enough.

Next let us verify the complementary conditions at σ = li_{. Similarly as in the case} σ = 0, it suffices to show that

diag(b1

∂Ω(Tl

1 ρ1

0, µ10), b2∂Ω(Tl

2 ρ1

0, µ20), b3∂Ω(Tl

3 ρ1

0, µ30))6= 0. Indeed, in the case (ρi

0, µi0)≡(0,0) (i= 1,2,3), we have

detdiag(b1_∂Ω(0,0), b_∂2_Ω(0,0), b3_∂Ω(0,0)) = detdiag(|∇ψ(Φ1∗)|,|∇ψ(Φ2∗)|,|∇ψ(Φ3∗)|)

= −

3

Y

i=1

|∇ψ(Φi_∗)| 6= 0.

Since the determinant is also continuous with respect toρi

0andµi0(i= 1,2,3), we conclude detdiag(b1

∂Ω(Tl

1 ρ1

0, µ10), b2∂Ω(Tl

2 ρ1

0, µ20), b3∂Ω(Tl

3 ρ1

0, µ30)) 6= 0 provided that kρi0kC1_(Ii₎ and

|µi

Step 3. Let us analyze the linearized system. Set X = C(I1_)×C(I2_)×C(I3_{) and}

Y = C2_(I1_)×C2_(I2_{)× C}2_(I3_{). Define the realization of A}

0 in X with homogeneous boundary conditions as follows

D(A0) ={ϕ∈Y |ϕ, A0ϕ∈X, B0(0 ;∂σ)ϕ=0, B0(li;∂σ)ϕ=0}, A0ϕ=A0ϕ.

Then we have the following lemma which, in particular, characterizes the interpolation spaces DA0(β,∞). For a definition ofDA0(β,∞) we refer to Lunardi [17].

Lemma 2.4

(i) The linear operator A0 : D(A0)→X is sectorial.

(ii) The characterization of the interpolation spaces DA0(α,∞) is given as

DA0(β,∞) =

(

{ϕ∈Xβ |P3_i=1γiϕi(0) = 0} if β ∈(0,1/2),

{ϕ∈Xβ| B0(0 ;∂σ)ϕ=0, B0(li;∂σ)ϕ=0} if β ∈(1/2,1),

(2.13)

where Xβ :=C2β(I1)×C2β(I2)×C2β(I3).

Proof. See [22, Section 2] or adapt the argument in [17, Section 3.1.5] for the case of systems with the estimates in [2, Theorem 12.2].

Set A1ϕ=A1ϕ for ϕ∈D(A0). Then we obtain the following lemma.

Lemma 2.5 Let A=A0+A1. Then A : D(A0)→X is a sectorial operator.

Proof. According to [17, Proposition 2.4.1(ii)], A is a sectorial operator in X if A1 is a bounded linear operator fromD(A0) toDA0(α/2,∞). Indeed, by means of (2.13) and the

definition of a₁, we have, for ϕ ∈D(A0),

kA1ϕkDA0(α/2,∞) ≤ C0

3

X

i=1

kΛi(ui₀)kCα_(Ii₎

!

ka1(T0ρ₀,T0ρ_0,σ,µ₀)kXα/2kT

0_∂2

σϕkX

≤ Cˆ0kϕkD(A0)

where α ∈ (0,1) and C0, ˆC0 are constants which depend on kρi0kC1+α_(Ii₎ and |µi₀|. This

completes the proof.

Using an estimate as in the proof of [17, Proposition 2.4.1(ii)], we see that A : D(A) =

D(A0)→X is a sectorial operator such thatc1kϕkD(A0) ≤ kϕkD(A) ≤c2kϕkD(A0) forϕ∈

D(A0) and some constants c1, c2 >0. HenceDA(α,∞) =DA0(α,∞) with equivalence of

the respective norms.

By virtue of Lemma 2.5, we find that A generates the analytic semigroup etA_{. Then}

Proposition 2.6 Let us assume that ρi

0 ∈C2+α(Ii) (i= 1,2,3) satisfy the compatibility conditions

B0(0 ;∂σ)ρT0 =GT(0), B0(li;∂σ)ρT0 =GT∂Ω(0).

For (ρi_{, µ}i_)∈C2+α,1_(Qi

0,T)×C1[0, T] (i= 1,2,3), the linearized system

    

∂tρT =AρT +FT(σ, t),

B0(0 ;∂σ)ρT =GT(t), B0(li;∂σ)ρT =GT∂Ω(t),

ρi

0(·,0) = ρi0 (i= 1,2,3)

(2.14)

with the notation A=A0+A1 has a unique solution such that

3

X

i=1

kρik_C2+α,1_(Qi 0,T)

≤C

3

X

i=1

kρi0kC2+α_(Ii₎+kFik Cα,0_(Qi

0,T)+kG i_k

C(1+α)/2_[0,T]+kGi_∂Ωk_C(1+α)/2_[0,T]

. (2.15)

Proof. Adapt the argument in the proof of [17, Theorem 5.1.19] to our linearized system (2.14).

Now we are ready to prove Theorem 2.3 by using the contraction principle.

Proof of Theorem 2.3. Set

D =(ρ,µ)∈C2+α,1_(Q1

0,T)×C2+α,1(Q20,T)×C2+α,1(Q30,T)×

C1_{[0, T]}3 (ρi_{(·,0), µ}i_{(0)) = (ρ}i

0, µi0) (i= 1,2,3), kρkC_T2+α,1 +kµkC1

T ≤K

for some bounded positive parameters K and T where

kρk_C2+α,1

T =

3

X

i=1

kρik_C2+α,1_(Qi

0,T), k

µkC1

T =

3

X

i=1

kµikC1_[0,T].

Then, for (ρ,µ)∈ D, we define the mapping

F : D ∋(ρ,µ)7→(ρ,µ)

whereρis the solution of (2.14) andµis given by µT =QT0_ρT _{for such solutionρ. Once}

we prove that the mapping F is a contraction on D for suitable K and T, the mapping

F has a unique fixed point inD which implies that the nonlinear problem (2.11) admits a unique solution in [0, T].

Let us first prove that F maps D into itself. Note that the lower order terms in Fi

and Gi

∂Ω can be rewritten as

fi(u) =fi(u0) +

Z 1

0

hDfi(ηu+ (1−η)u0),u−u₀idη,

g_∂i_Ω(ui₁) =g_∂i_Ω(ui_1,0) +

Z 1

0

Hereu= (ρi_{, ρ}i

σ,T0ρ,T0ρσ,µ),u0 = (ρi0, ρi0,σ,T0ρ0,T0ρ0,σ,µ0),u1i = (ρi, µi), and ui1,0 = (ρi

0, µi0). Moreover, Dfi and Dg∂iΩ are the Fr´echet derivative of fi and gi∂Ω, respectively, and the bracket h·,·i is the respective inner product. Then, by means of (2.15) and

µT =QT0_ρT_{, we have}

kρk_C2+α,1

T +kµkC

1

T ≤

ˆ

C

_X3

i=1

kρi₀kC2+α_(Ii₎+kfi(u0)k_Cα_(Ii₎+|gi_∂Ω(Tl i

ui₀)|

+ 3

X

j=2

|bj1(T0_u1j

0 )||T0ρ10,σ|+|bjj(T0u

1j

0 )||T0ρ

j

0,σ|

+CKTν

for ν = min{α/2, (1−α)/2}. Thus, choosing

K = 2 ˆC

_X3

i=1

kρi₀kC2+α_(Ii₎+kfi(u0)k_Cα_(Ii₎+|g_∂i_Ω(Tl i

ui₀)|

+ 3

X

j=2

|bj1(T0u₀1j)||T0ρ1_0,σ|+|bjj(T0u₀1j)||T0ρj_0,σ|

, (2.16)

we conclude that there exists a time T1 >0 such that

kρk_C2+α,1

T +kµkC

1

T ≤K for T ≤T1. (2.17)

That is,F maps D into itself.

Let us prove that the mapping F is a contraction. For (ρ₁,µ₁), (ρ₂,µ₂) ∈ D with

T ≤T1, let

(ρ₁,µ₁) =F(ρ₁,µ₁), (ρ₂,µ₂) =F(ρ₂,µ₂)

be the solutions associated with the linearized problem (2.14). Then, applying a similar argument to [4, pp. 373-375] with µT =QT0_ρT_{, we are led to}

kρ₁−ρ₂k_C2+α,1

T +kµ1−µ2kC 1

T ≤

ˆ

CKTν(kρ1−ρ2kC_T2+α,1 +kµ1−µ2kC1 T)

for ν = min{α/2,(1 −α)/2}. Thus, F is a contraction for T ≤ T2, which satisfies ˆ

CKT2ν ≤ 1/2. Consequently, choosing T∗ = min{T1, T2}, we find that F has a unique fixed point inDforT ≤T∗, so that the nonlinear problem (2.11) has a unique solution in

[0, T] with (2.17) for T ≤T∗. Further, checking the details of the estimate for the linear

system, we obtain fort ∈[0, T]

3

X

i=1

kρi(·, t)kC1+α_(Ii₎≤m₀+C_KTν,

where m0 depends on kρi0kC1+α_(Ii₎ and |µi₀|. Then, there exists a time T₃ > 0 such

that m0 + CKTν ≤ 2m0 for T ≤ T3. Thus, choosing T0 = min{T∗, T3}, we have

P3

i=1kρi(·, t)kC1+α_(Ii₎ ≤ 2m₀ for t ∈ [0, T] with T ≤ T0. It is possible to

guaran-tee 2m0 ≤ δ0 for sufficiently small kρi0kC1+α_(Ii₎ and |µi₀|, where δ₀ is as in (2.10). By

Lemma 2.1,|µi_{(t)|is estimated by}P3

i=1kρi(·, t)kC0_(Ii₎, so that|µi(t)|can be smaller than δ0 if kρi0kC1+α_(Ii₎ and |µi₀| are small enough. This completes the proof of Theorem 2.3.

3

Linearization

In order to linearize the nonlinear system (2.11) around the stationary solution Γ∗ = S3

i=1Γi∗, we need to establish the following properties of Ψ at (q, µi) = (0,0).

Lemma 3.1 For the parameterizations Ψi_{, i = 1,2,3, in Section 2, the following}

prop-erties hold:

(i) Ψi_{(σ,0,0) = Φ}i

∗(σ) and Ψi(σ, q,0) = Φi∗(σµi∂Ω(q)/li) +qN∗i(σµ∂iΩ(q)/li). (ii) Ψi

σ(σ,0,0) =T∗i, Ψiq(σ,0,0) =N∗i, andΨiµ(σ,0,0) = (1−σ/li)T∗i.

(iii) Ψi

σσ(σ,0,0) = (0,0)T, Ψiσq(σ,0,0) = (0,0)T, and Ψiσµ(σ,0,0) = (−1/li)T∗i.

(iv) Ψi

σσq(σ,0,0) = (0,0)T and Ψiσσµ(σ,0,0) = (0,0)T.

Proof. By the definition of Ψi_{, (i) is obvious. Let us prove (ii). Differentiating Ψ}i_{(σ,0,0) =}

Φi

∗(σ) with respect to σ, we readily derive Ψiσ(σ,0,0) =T∗i(σ). Applying a similar

argu-ment to [7], we obtain {µi

∂Ω(q)}′|q=0 = 0. Thus (i) gives Ψqi(σ,0,0) = N∗i(σ). Further, by

the definition ofξi_{, we have}

ξ_µi(σ,0,0) = 1−σ/li.

It follows from the definition of Ψi _{and the Frenet-Serret formulae that}

Ψi_µ(σ, q, µi) =ξµ(σ, q, µi)(1−qκi∗)T∗i(ξ(σ, q, µi)) =ξµ(σ, q, µi)T∗i(ξ(σ, q, µi)).

Putting (q, µi_{) = (0,0), the third property of (ii) is derived. Finally, by using (ii), we}

have (iii)-(iv).

By virtue of Lemma 3.1, we are led to the linearization of (2.11) around the stationary solution Γ∗ =

S3

i=1Γi∗.

Proposition 3.2 The linearization of (2.11) is given by

            

βi_ρi

t =γiρiσσ for σ∈(0, li),

3

X

i=1

γiρi = 0 at σ= 0,

ρ1

σ =ρ2σ =ρ3σ at σ= 0, ρσ +hi∗ρi = 0 at σ=li,

(3.1)

for i= 1,2,3, where hi

∗ is the curvature of ∂Ω at the point Φi∗(li)∈Γi∗∩∂Ω.

We remark that (3.1) corresponds to the linearized problem which was derived in a formal way by Ikota and Yanagida [13].

Proof of Proposition 3.2. Applying the same argument as in [7, Section 3], using Lemma 3.1 and κi

∗ = 0 (i = 1,2,3) we obtain from equations (2.9) and the boundary conditions

atσ =li_{, the first and fourth equations in (3.1). Thus we only derive the third equation}

of (3.1). To simplify the notation, we set

e

Then it is easy to obtain

e

Ji(ρi, µi) = 1, ∂Jei(0,0)[ρi, µi] = d

dεJe

i_(ερi_{, εµ}i₎

ε=0 =−1

liµ i_,

where ∂Jei_{(0,0) is the Fr´echet derivative of Je}i _{at (0,0). Recalling the definition of}

e

g1j

(j = 2,3) and using Lemma 3.1, we have

∂eg1j(0,0,0,0)[ρ1, ρj, µ1, µj] = −1

l1(T 1

∗, T∗j)R2µ1− 1 lj(T

1

∗, T∗j)R2µj + (T_∗1, N_∗j)_R2ρj_σ

+(N_∗1, T_∗j)_R2ρ1_σ −

−1

l1µ 1₋ 1

ljµ j

cosθk

for (j, k) = (2,3) or (3,2), where∂g_e1j_{(0,0,0,0) is the Fr´echet derivative of}

e

g1j_{at (0,0,0,0).}

Since the angle conditions atσ = 0 give

(T∗1, T∗2)R2 = cosθ3, (T_∗1, N_∗2)_R2 =−sinθ3, (N_∗1, T_∗2)_R2 = sinθ3,

(T∗1, T∗3)R2 = cosθ2, (T_∗1, N_∗3)_R2 = sinθ2, (N_∗1, T_∗3)_R2 =−sinθ2,

it follows that

∂_eg12(0,0,0,0)[ρ1, ρ2, µ1, µ2] = (−ρ2_σ +ρ1_σ) sinθ3, ∂eg13(0,0,0,0)[ρ1, ρ3, µ1, µ3] = (ρ3_σ −ρ1_σ) sinθ2

atσ = 0. Hence the linearization of _eg1j_(ρ1_{, ρ}j_{, µ}1_{, µ}j_{) = 0 (j = 2,3) is}

(−ρ2_σ +ρ1_σ) sinθ3 = 0, (ρ3_σ−ρ1_σ) sinθ2 = 0 at σ = 0,

so that, by virtue of θj _{∈(0, π) (j = 2,3), we have}

ρ1_σ =ρ2_σ =ρ3_σ at σ = 0.

This completes the proof.

In [13] Ikota and Yanagida investigated linearized stability for the curvature flow with a triple junction (3.1). They derived a criterion according to which one can determine whether the stationary solution is linearly stable or unstable. In what follows, we recall their linearized stability criterion. The main result of [13, Theorem 1.1] is concerned with the analysis of the self-adjoint eigenvalue problem associated to the linearized system of equations (3.1). We now recall their linearized stability result.

Theorem 3.3 The maximal eigenvalue of the linearized problem (3.1), i.e. βi _{= 0,} i = 1,2,3, is negative and the stationary solution is linearly stable if and only if one of the following conditions is fulfilled:

(a) either all h1

∗, h2∗, andh3∗ are positive,

(b) or, at most one of them is non-positive, and they satisfy

We will also need a variational characterization of the linearized stability property. To this end, let us introduce the bilinear form

I∗[ϕ,ϕ] =

3

X

i=1

γi

Z li

0

(ϕi_s)2ds+h∗i(ϕi)2|s=li

(3.2)

for all ϕ ∈ E(Γ∗), where

E(Γ∗) =

(ϕ1, ϕ2, ϕ3)∈H1(0, l1)×H1(0, l2)×H1(0, l3) 3

X

i=1

γiϕi(0) = 0 .

This bilinear form was also considered in [13]. The following lemma is a simple conse-quence of the variational characterizations of the largest eigenvalue.

Lemma 3.4 Let λ be the maximal eigenvalue of the time independent linearized system (3.1), i.e. we set βi _{= 0, i= 1,2,3. Then}

I∗[ϕ,ϕ]>(−λ)

3

X

i=1

γikϕik2

L2_(Γi_∗) for ϕ∈ E(Γ∗).

Remark 3.5 In order to simplify the presentation, we will henceforth consider only the case βi _{= γ}i_{, for i = 1,2,3. It is worth to note that the linearized stability criterion}

is invariant with respect to the positive constants βi _{> 0 for i = 1,2,3. As it should}

be obvious from all the energy type estimates to follow, the full nonlinear stability of the stationary solution will not be affected by a different choice of positive mobility constants βi _{>0 for i= 1,2,3.}

4

Uniqueness of the stationary solution

In this section we prove the uniqueness of the stationary solution in a smallH2_{-neighborhood.} The inverse mapping theorem also gives a bound on the H2_{-norm of ρ = (ρ}1_{, ρ}2_{, ρ}3_{) in} terms of the L2_{-norm of the curvature κ= (κ}1_{, κ}2_{, κ}3_).

To this end, let us introduce the function space

M=(ρ1, ρ2, ρ3)∈H2(0, l1)×H2(0, l2)×H2(0, l3) 3

X

i=1

γiρi(0) = 0 .

Then ρ via parametrization (2.2) defines a neighboring triple junction configuration such that the end points lie on ∂Ω.

Theorem 4.1 Let I∗ be positive. Then there exists a H2-neighborhood of ρ ≡ 0 in M,

such that ρ≡0 is the only solution of the problem

κi = 0, ∢_(∂Ω,Γi

t) =π/2, (4.1)

Proof. The idea of the proof is to use the local inverse mapping theorem for the cur-vature operator with appropriate boundary conditions. The positivity of I∗ will ensure

invertibility of the linearization.

Using the notation (2.4) of Section 2 we obtain

κi(ρ) = ai(ρi, ρi_σ,µ(ρ(0)))ρi_σσ+fi(ρi, ρi_σ,µ(ρ(0)))

with ai_(ρi_{, ρ}i

σ,µ(ρ(0))) = 1/{Ji(ρi, ρiσ,µ(ρ(0)))}2, a smooth function fi, and a linear

mapping µfrom R3 _to R3_{. The boundary conditions in (4.1) and (4.2) can be written as}

gi_(ρi_(li_{), ρ}i

σ(li),µ(ρ(0))) = 0 for i= 1,2,3, gi_(ρ(0),ρ

σ(0),µ(ρ(0))) = 0 for i= 4,5

whereg1_{, . . . , g}5 _{are smooth functions. We define N =L}2_{(0, l}1_)×L2_{(0, l}2_)×L2_{(0, l}3_{) and} observe that solving (4.1) and (4.2) is equivalent to finding a zero of the mapping

F :Bε(0)→ N ×R5, F(ρ) = (κ1(ρ), κ2(ρ), κ3(ρ), g1, . . . , g5)

whereBε(0) is a ball of a radiusε >0 around zero in the spaceM. Since H2 is embedded

in C1 _{the mapping is well defined. Arguing similarly as in Section 3 we obtain}

DF(0) : M → N ×R5

ρ 7→ (ρ1

σσ, ρ2σσ, ρ3σσ, ρ1σ+h1ρ1, ρ2σ +h2ρ2, ρ3σ +h3ρ3, ρ1σ−ρ2σ, ρ1σ −ρ3σ).

Similarly as in [13], since I∗ is positive we can conclude thatDF(0) is injective and hence

the Fredholm alternative gives that DF(0) is invertible. Now the local inverse mapping theorem (see e.g. [24]) gives that there is a neighborhood around 0 such that onlyρ≡0 solves (4.1) and (4.2).

It is worthwhile noting that the mapping F analyzed in the proof of the above theo-rem is in fact a local diffeomorphism. Therefore its inverse mapping is locally Lipschitz continuous. Hence we have the following corollary.

Corollary 4.2 There exist constants C, δ1 >0 such that

kρkH2 ≤Ckκk_L2

provided that kκk2

L2 < δ1 and ρ ∈ M fulfills F(ρ) = (κ1, κ2, κ3,0,0,0,0,0).

5

Governing equations for the curvature and other

geometric quantities

In order to show the global existence and the nonlinear stability of solutions for which the bilinear form of [7] is positive, we apply an energy method similar to the one used in [5] and [15]. For such a method it is important to derive evolution equations for the curvature.

Letsbe the arc-length parameter along the phase boundary Γtand letX be a smooth

map such that X(·, t) is an arc-length parameterization of Γt with

Γi_t={Xi(s, t)|s∈[0, ri(t)]}

where ri _{is smooth such that L[Γ}i

t] = ri(t) which is the length of Γit. Let Ni(=Ni(s, t))

be the unit normal vector of Γi

t. It can be written as

Ni(s, t) =

cosωi_{(s, t)}

sinωi_{(s, t)}

.

Then we have

Ni

s =−κiTi, Tsi =κiNi, Ni

t =−ωitTi, Tti =ωtiNi

(5.1)

where Ti _{is the unit tangent vector of Γ}i

t and κi is the curvature of Γit. In addition, we

define

Vi = (X_ti, Ni)R2, vi = (X_ti, Ti)_R2

and hence

X_ti =ViNi+viTi. (5.2)

Differentiating (5.2) with respect tos and using (5.1), we have

Xi

ts = VsiNi+ViNsi+vsiTi+viTsi (5.3)

= (Vsi+κivi)Ni+ (−κiVi+vis)Ti. (5.4)

Lemma 5.1 Let Xi _{be a smooth arc-length parameterization as above. Then}

ω_ti =V_si+κivi, v_si =κiVi.

Proof. Since Xi

ts =Xsti and Xsi =Ti, it follows from (5.1) and (5.2) that

ωi_tNi = (V_si+κivi)Ni+ (−κiVi +vi_s)Ti.

Thus we obtain the desired results.

As a consequence of Lemma 5.1 we have the following lemma.

Lemma 5.2 Let Xi _{be a smooth arc-length parameterization of Γ}i

t as above. Then the

curvature κi _{(i= 1,2,3) satisfies the evolution equation:}

Proof. By ωi

s=κi and Lemma 5.1, we obtain

κi_t=ω_sti =ωi_ts = (V_si+κivi)s =Vssi +κivis+κisvi =Vssi + (κi)2Vi+κisvi.

This completes the proof.

By the assumption that Γi

t meets ∂Ω at the one end point with the angleπ/2, we have

ψ(Xi(ri(t), t)) = 0, (∇ψ(Xi), Ni)R2 = 0 at s=ri(t).

Differentiating the identity ψ(Xi_(ri_{(t), t)) = 0 with respect to t and taking into account}

the transversality condition (∇ψ(Xi_{), N}i₎

R2 = 0 and the governing equationX_ti =ViNi+ vi_Ti _{we can derive the following lemma.}

Lemma 5.3 At the point where Γi

t (i= 1,2,3) meets the outer boundary ∂Ω we have

vi+ (ri)′ = 0 for s=ri(t).

Next we derive corresponding boundary conditions at the triple junction point p(t). It is assumed that phase boundaries Γ1_{, Γ}2_{, and Γ}3 _{meet at the triple junction. Let}

i, j, k ∈ {1,2,3} be mutually different. Let Γk _{and γ}k _{be the interface and the surface}

energy density between phases i,j. Following the arguments in Bronsard and Reitich [4] the anglesθi _{(i= 1,2,3) of the phases at the triple junction pointp(t) fulfill Young’s law}

sinθ1

γ1 =

sinθ2

γ2 =

sinθ3

γ3 (5.5)

(see [23]). Young’s law can be expressed as a force balance in the following form

3

X

i=1

γiTi = 3

X

i=1

γiNi = 0. (5.6)

Let p(t)∈R2 _{denote a triple junction. At the triple junction the following boundary} conditions hold:

X1_{(0, t) =X}2_{(0, t) = X}3_{(0, t) (=p(t)), (5.7)} (X_si, X_sj)R2 = cosθk at p(t) (5.8)

for (i, j, k) = (1,2,3),(2,3,1), (3,1,2). Then we obtain the following lemma.

Lemma 5.4 At the triple junction p(t) we have the following equality:

3

X

i=1

Proof. Differentiating (5.7) with respect tot, we obtain

dp

dt =X

1

t =Xt2 =Xt3. (5.9)

For i= 1,2,3, it holds

dp dt, T

i

R2 = (X i

t, Ti)R2 =vi.

This fact and Young’s law imply that

3

X

i=1

γivi = 3

X

i=1

γi dp dt, T

i

R2 = dp dt,

3

X

i=1

γiTi_R2 = 0.

Hence the proof is complete.

In the next lemma we derive evolution equations and boundary conditions for the curvature.

Lemma 5.5 A smooth solution of the curvature flow equations

Vi =κi, i= 1,2,3, (5.10)

with the boundary conditions

    

∢_(Γi_(t),Γj_{(t)) = cosθ}k _{for i, j, k ∈ {1,2,3} mutually different,} ∢_(∂Ω,Γi

t) =π/2 at Γit∩∂Ω, ∂Γi

t ⊂∂Ω

(5.11)

fulfills when expressed in the above arc-length parameterization the evolution equations

κi_t =κi_ss+ (κi)3+κi_svi, i= 1,2,3.

Furthermore, at the triple junction p(t) we have

3

X

i=1

γiκi = 0, (5.12)

κ1_s+κ1v1 =κ2_s+κ2v2 =κ3_s+κ3v3 (5.13)

and at Γi

t∩∂Ω the identity

κi_s+hiκi = 0 (5.14)

holds. Here hi _{is the curvature of ∂Ω at the point X}i_(ri_{(t), t)∈Γ}i t∩∂Ω.

Proof. From (5.9) we deduce

dp dt, N

i

R2 = (X i

t, Ni)R2 =Vi.

This fact and Young’s law imply that

3

X

i=1

γiVi = 3

X

i=1

γi dp dt, N

i

R2 = dp dt,

3

X

i=1

Since Vi _=κi_{, we are led to (5.12).}

Differentiating (5.8) with respect to t, we obtain

(X_sti, X_sj)R2 + (X_si, X_stj)_R2 = 0.

It follows from (5.3) and (5.11) that

(V_si+κivi) sinθk+ (V_sj +κjvj)(−sinθk) = 0.

By 0< θk< π we have sinθk >0, so that

V_si+κivi =V_sj +κjvj.

Finally, Lemma 5.2 and the fact that Vi _=κi _{imply (5.13).}

6

A priori estimates and global existence of a smooth

solution

The purpose of this section is to derive a priori estimates guaranteeing global existence of a smooth solution and its convergence to a steady state. First, we derive a priori estimates for the L2_{-norm of the curvature. Next we proceed with higher order energy estimates} yielding a priori estimates for the H2_{-norm of the curvature. As a consequence of these} estimates we will be able to prove exponential decay of theH2_{-norm of the curvature. We} remark that due to the parabolic regularization property the solution of Theorem 2.3 will become smoother for positive time such that all derivatives in the following computations exist.

6.1

First order a priori estimates

Let us define the energy functional

E[Γ] = 3

X

i=1

γiL[Γi]

where Γ = S3_i=1Γi _{and L[Γ}i_{] is the length of Γ}i_{. Further, throughout Section 6, we use}

the following notations:

kϕikLp =

Z

Γi t

|ϕi|pds

!1 p

(1≤p <∞), kϕikL∞ = ess sup Γi

t

|ϕi|, (6.1)

kϕkLp =

3

X

i=1

γikϕikp_Lp

!1 p

, kϕkL∞ = max

i kϕ i_k

L∞ (6.2)

for a vector function ϕ= (ϕ1_{, ϕ}2_{, ϕ}3_).

(i) d

dtE[Γt] +kκk

2

L2 = 0,

(ii) d dtkκk

2

L2 =−2

3 X i=1 γi Z Γi t

(V_si)2ds+hi(Vi)2_s=ri

+kκk4_L4 +

3

X

i=1

γi(κi)2vi_s=0

where hi _{is evaluated at X}i_(ri_{(t), t).}

Proof. By means of the identity L[Γi

t] =ri(t), Lemmata 5.1, 5.3 and 5.4, we have

d

dtE[Γt] =

3

X

i=1

γi(ri)′ ₌₋

3

X

i=1

γi{vi(ri, t)−vi(0, t)}

= − 3 X i=1 γi Z Γi t

v_sids =−

3 X i=1 γi Z Γi t

κiVids =−

3 X i=1 γi Z Γi t

(κi)2ds.

In order to prove (ii), we compute

2

Z

Γi t

κiκi_tds= 2

Z

Γi t

κi{κi_ss+ (κi)3+κi_svi}ds. (6.3)

The identity d dt Z Γi t

(κi)2ds= 2

Z

Γi t

κiκi_tds+ (κi)2_s=ri(r i₎′

,

and Lemma 5.3 imply

2

Z

Γi t

κiκi_tds= d

dt

Z

Γi t

(κi)2ds+ (κi)2vi_s=ri. (6.4)

For the right hand side of equation (6.3) we can use the boundary condition (5.14). Integration by parts yields

2

Z

Γi t

κi{κi_ss+ (κi)3+κi_svi}ds

= 2

κiκs

s=ri s=0 −

Z

Γi t

(κi_s)2

ds+ 2

Z

Γi t

(κi)4ds+ 2

Z

Γi t

κiκi_svids

=−2

Z

Γi t

(κi_s)2ds+hi(κi)2_s=ri

−2κiκi_ss=0+ 2

Z

Γi t

(κi)4ds+ 2

Z

Γi t

κiκi_svids. (6.5)

Using the identities vi

s=κiVi = (κi)2 and integrating by parts we obtain

Z

Γi t

κiκi_svids = (κi)2vis_s=₌₀ri −

Z

Γi t

κi(κi_svi+κivi_s)ds

= (κi)2vis_s=₌₀ri −

Z

Γi t

κiκi_svids−

Z

Γi t

(κi)4ds.

Thus we have _Z

Γi t

κiκisvids =

1 2

(κi)2vis_s=₌₀ri −

Z

Γi t

(κi)4ds

It follows from (6.4), (6.5), and (6.6) that d dt Z Γi t

(κi)2ds = −2

Z

Γi t

(κis)2ds+hi(κi)2

s=ri

+

Z

Γi t

(κi)4ds

−2κiκi_ss=0−(κi)2vi_s=0.

By (5.12) and (5.13) we have

3

X

i=1

γin_2κi_κi s

s=0+ (κ

i₎2_vi s=0 o = 2 3 X i=1

γi_κi_(κi

s+κivi)

s=0− 3

X

i=1

γi_(κi₎2_vi s=0

= −

3

X

i=1

γi(κi)2vi_s=0.

Thus we are led to the identity

d dt 3 X i=1 γi Z Γi t

(κi)2ds = −2 3 X i=1 γi Z Γi t

(κi_s)2ds+hi(κi)2_s=ri

+ 3 X i=1 γi Z Γi t

(κi)4ds

+ 3

X

i=1

γi_(κi₎2_vi s=0.

Since Vi _=κi _{the proof of the lemma follows.}

Let us define a bilinear form I as

I[ϕ,ϕ] = 3

X

i=1

γi

Z ri

0

(ϕi_s)2ds+hi(ϕi)2|s=ri

for ϕ∈ E(Γ) where

E(Γ) =(ϕ1, ϕ2, ϕ3)∈H1(0, r1)×H1(0, r2)×H1(0, r3) 3

X

i=1

γiϕi(0) = 0 .

Since V ∈ E(Γ) we can rewrite the statement ii) of Lemma 6.1 as

d dtkκk

2

L2 + 2I[V,V] =kκk4_L4 +

3

X

i=1

γi(κi)2vi_s=0. (6.7)

The following lemmata are crucial in the derivation of a priori estimates.

6.2

Higher order estimates for the curvature.

We define the averaged curvature along the curve Γi

t as κiav = L[Γ1i t]

R

Γi tκ

i_ds.

Lemma 6.2 The following estimates for a C3_{-curve Γ}i

(i) kκi−κi_avkL∞ ≤(L[Γi

t]) 1 2kκi

skL2, |κi_av| ≤

1 (L[Γi

t]) 1 2

kκi_k L2,

(ii)

Z

Γi t

(κi)4ds≤2nL[Γit]kκisk2L2 +

1

L[Γi t]

kκik2

L2

o

kκik2

L2,

(iii) There are m1 =m1(L[Γit]) and m2 =m2(1/L[Γit]) such that

(κi_|

s=0)3

≤m1kκikL2kκi_sk2

L2 +m2kκik3_L2.

Proof. The estimates in (i) are established in a standard way and we do not present details here. By the estimates in (i), we have

kκikL∞ ≤ kκi−κi

avkL∞+|κi

av| ≤(L[Γit]) 1 2kκi

skL2 +

1 (L[Γi

t]) 1 2

kκikL2.

It implies that

Z Γi t

(κi)4ds

≤ kκik2L∞kκ

i_k2

L2

≤

(L[Γi_t])12kκi

skL2 + 1

(L[Γi t])

1 2

kκikL2

2

kκik2

L2.

The statement (ii) now follows from the elementary inequality (a+b)2 _≤2(a2_+b2_). Let ri

0 be such thatκi(r0i, t)−κiav(t) = 0. We then obtain

(κi_−κi

av)3|s=0 =−3

Z ri 0

0

κi

s(κi−κiav)2ds.

This implies that

(κi−κi_av)3|s=0

≤ 3

Z

Γi t

|κi_s||κi−κi_av|2_ds

≤ 3kκi−κi_avkL∞kκi

skL2kκi−κi_avk_L2

≤ 3(L[Γi_t])12kκi sk2L2

kκikL2 + (L[Γi_t]) 1 2|κi

av|

≤ 6(L[Γi_t])12kκik_L2kκi_sk2_L2.

Note that (κi₎3 _{= (κ}i_−κi

av)3+ 3κiκiav(κi−κiav) + (κiav)3. Then it follows that

kκiκi_av(κi−κi_av)kL∞ ≤ kκi

av(κi−κiav)2kL∞ +k(κi

av)2(κi−κiav)kL∞

≤ L[Γi_t]|κi_av|kκi_sk2

L2 + (L[Γi_t]) 1 2|κi

av|2kκiskL2

≤ (L[Γi_t])12kκik_L2kκi_sk2_L2 + (L[Γi_t]) 1 2|κi

av| ·

1 2 |κ

i

av|2+kκisk2L2

≤ (L[Γi_t])12kκik

L2kκi_sk2_L2 +

1 2kκ

i_k L2

1

L[Γi t]

kκik2_L2 +kκi_sk2_L2

=

(L[Γi_t])12 + 1

2

kκikL2kκi_sk2_L2 +

1 2L[Γi

t]

Thus we have

(κi|s=0)3

≤ (κi−κi_av)3|s=0

+ 3kκiκi_av(κi−κi_av)kL∞+|κ_av|3

≤

9(L[Γi_t])12 +3

2

kκikL2kκi_sk2_L2 +

3 2L[Γi

t]

+ 1 (L[Γi

t])3/2

kκik3_L2.

Hence the proof is complete.

Lemma 6.3 For smooth solutions of (5.10), (5.11) we have

(v1_{, v}2_{, v}3₎T _=Q(V1_{, V}2_{, V}3₎T_,

for any t∈(0, T], where the matrix Q was defined in Lemma 2.1.

Proof. At the triple junction p(t), we have, for all i, j ∈ {1,2,3}, dp

dt =v

i_Ti_+Vi_Ni _=vj_Tj_+Vj_Nj_.

Taking the inner product withTi _{we obtain}

vi =vj(Ti, Tj) +Vj(Ti, Nj).

By (5.8), (Ti_{, T}j_{) = cosθ}k _{and (T}i_{, N}j_{) =−sinθ}k_{. Thus we derive}

vi−ckvj =−skVj.

If we solve this with respect to (v1_{, v}2_{, v}3₎T_{, we are led to the desired result.}

By Lemma 6.3 and Vi _=κi_{, we have}

3

X

i=1

γi(κi)2vi_s=0

≤C

_X3

i=1

γi(κi)2_s=0

_X3

i=1

|κi|_s=0

≤Cˆ

3

X

i=1

γi(κi)3_s=0. (6.8)

6.3

Structural stability of the bilinear form

The aim of this subsection is to show that positivity of the bilinear form I∗ is invariant

with respect to small perturbations of the curve parametrization ρi_{. More precisely, we}

will show that the bilinear form I is positive definite provided thatI∗ is positive definite

and ρ = (ρ1_{, ρ}2_{, ρ}3_{) is small in the C}1_{-norm. Taking into account Corollary 4.2 and} the continuity of embedding H2 _{֒→ C}1 _{the positive definiteness of the bilinear form is} preserved if the L2_{- norm of the curvature κ= (κ}1_{, κ}2_{, κ}3_{) is small.}

According to Lemma 3.4, I∗ is positive provided the maximal eigenvalue for the

linearized problem is negative. The following lemma is a direct consequence of [13, Lemma 3.1 and Prop. 3.3].

Lemma 6.4

(i) Let λ be the maximal eigenvalue of the linearized problem. For ε >0 there exists a δ >0such that, for any perturbation|hi_−hi

∗|< δand|L[Γi]−L[Γi∗]|< δ, i= 1,2,3,

we have

I[ϕ,ϕ]>(−λ−ε)kϕk2

(ii) There exists a c > 0 such that

ckϕ_sk2

L2 ≤I[ϕ,ϕ] +kϕk2_L2 for ϕ∈ E(Γ).

Using Lemma 6.4, we obtain the existence of constants δ∗ >0 and c∗ >0 such that

I[ϕ,ϕ]>−λ

2kϕk 2

L2 +c∗kϕsk2L2 for ϕ ∈ E(Γt) (6.9)

provided that, for i= 1,2,3,

|hi−hi∗|< δ∗, |L[Γi]−L[Γi∗]|< δ∗. (6.10)

Lemma 6.5 We have the following estimates.

(i) There exist constants δ2, C > 0 such that |hi −h∗i| ≤ CkρikC0_(Ii₎ provided that

kρi_k

C0_(Ii₎< δ₂.

(ii) There exist constants δ3, C >0 such that

|L[Γi]−L[Γi∗]| ≤CkρkC1 and |p| ≤Ckρk_C0

provided that kρkC1 < δ₃, where p is the triple junction of Γ =S3 i=1Γi. Here we have denoted kρk_Ck+α =P3_i=1kρik_Ck+α_(Ii₎ for k ∈N∪ {0}, α ∈[0,1). Proof. To prove (i), we recall that κ∂Ω(Xi) is represented by

κ∂Ω(Xi) = − 1

|∇ψ(Xi_)|([D

2_ψ(Xi_)]T

∂Ω(Xi), T∂Ω(Xi))R2. (6.11)

Since the right hand side does not depend on derivatives of ρi_{, the mean value theorem}

implies the second inequality of (i).

In order to prove (ii), we have to analyze properties of the function µi

∂Ω introduced in Section 2. From the definition it follows that µi

∂Ω(0) =li and p∗+µi∂Ω(q)T∗i+qN∗i ∈∂Ω.

Thereforeψ(p∗+µi_∂Ω(q)T∗i+qN∗i) = 0 and so we can compute the derivative ofµi∂Ω(q) as

d dqµ

i

∂Ω(q) = −

(∇ψ(p∗+µi_∂Ω(q)T∗i+qN∗i), N∗i)_R2

(∇ψ(p∗+µi_∂Ω(q)T∗i+qN∗i), T∗i)_R2 .

Since (∇ψ, Ni

∗)R2 = 0 on∂Ω we obtain d dqµ

i

∂Ω(0) = 0. Now, by taking the second derivative of µi

∂Ω and taking into account the expression for the curvature hi∗ at the intersection of

∂Ω and Γ′

∗ we obtain d

2

dq2µi_∂Ω(0) =hi∗. Thus

µi_∂Ω(q) =li+h∗iq2+o(q2) as q→0.

We recall that the parameterization Φi _{of the curve Γ}i _{is given by}

Φi(σ) =p∗+ h

µi+σ

li{µ i

∂Ω(ρi(σ))−µi}

i

T_∗i+ρi(σ)N_∗i.

Using the above property of the function µi

∂Ω and the fact µT =Qρ(0)T we obtain

|L[Γi]−L[Γi∗]|=

Z li

0

(|Φiσ(σ)| −1)dσ

≤CkρkC1.

Similarly, asp=µi_Ti

∗ +ρi(0)N∗i we obtain|p| ≤CkρkC0. With this all statements of the