Stationary solutions to the Falk system on shape memory alloys

Research Article

Received 15 April 2009 Published online 18 August 2009 in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/mma.1229MOS subject classification: 35 J 60; 35 Q 72; 74 N 30

Stationary solutions to the Falk systemon shape memory alloys

Takashi Suzuki and Souhei Tasaki∗†

Communicated by H.-D. Alber

We study the Falk model system describing martensitic phase transitions in shape memory alloys. Its physicallyclosed stationary state is formulated as a nonlinear eigenvalue problem with a non-local term. Then, some results onexistence, stability, and bifurcation of the solution are proven. In particular, we prove the existence of dynamicallystable nontrivial stationary solutions. Copyright © 2009 John Wiley & Sons, Ltd.

Keywords: shape memory; Falk model; non-local elliptic problem; stability

1. Introduction

The Falk model [1, 2] is a system of partial differential equations describing austenite–martensite phase transitions occurring on arod made by shape memory alloys, which is written as⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

�utt +��4x u= (�F′

1(ux)+F′2(ux))x in (0, l)×(0, T)

cv�t −k�xx =�F′1(ux)utx in (0, l)×(0, T)

u|t=0 =u0, ut|t=0 =u1, �|t=0 =�0 in (0, l)

ux =�2x ux =�x =0 on {0, l}×(0, T)

(1)

with the length l>0 of the rod. Unknown functions u=u(x, t) and �=�(x, t) are real-valued, denoting the displacement and theabsolute temperature, respectively. Physical constants �, �, cv , and k are positive, and stand for the mass density, capillarity, specificcaloric heat, and heat conductivity, respectively. In this paper, we always assume that the nonlinear terms F1, F2 ∈C4 satisfy

F′1(0)=F′

2(0)=0, F′′1 (0)>0, F1, F2�−c for a constant c>0 (2)

This nonlinearity can stand for the Falk model

F1(ux)=�1u2x , F2(ux)=�3u6

x −�2u4x −�1�cu2

x (3)

for positive physical constants �1, �2, �3 and the critical temperature �c. It accounts for the experimental observed behavior, seeFalk [3]. For the physical background, we refer to the monograph by Brokate–Sprekels [4]. Without loss of generality under theassumption (2), we put

F1(0)=F2(0)=0, F′′1 (0)=2�1, F(4)

1 (0)=24�1, F′′2 (0)=−2�1�c, F(4)

2 (0)=−24�2

for constants �1>0, �2, �1, �c ∈R. Furthermore, the following assumptions will sometimes be imposed:

F1 and F2 are even functions (4)

F′1(v)>0 for any v>0, F′

1(v)<0 for any v<0 (5)

F1(v)→+∞ or F2(v)→+∞ as |v|→+∞ (6)

Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan∗Correspondence to: Souhei Tasaki, Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka

University, Osaka 560-8531, Japan.†E-mail: [email protected]

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Copyright © 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2010, 33 994–1011

T. SUZUKI AND S. TASAKI

From the physical point of view, these conditions are quite natural. In particular, (4) is derived from the symmetric property of thefree energy, caused by the equivalence of shearing directions. Note that the Falk nonlinearity (3) also satisfies these assumptions.

On the global in time existence of the solution to the Falk model system, there are several results, such as Aiki [5], Yoshikawa [6]concerning the boundary condition

u=�2x u=�x =0 on {0, l}×(0, T)

and for more general nonlinearities. In this paper, however, the class of the solution is considered as in the following theorem.

Theorem A (Suzuki and Yoshikawa [7])Assume that �0�0. For any (u0, u1,�0)∈H4 ×H2 ×H2 problem (1) has a unique global in time solution (u,�) satisfying

u ∈ C([0,∞); H4), ut ∈C([0,∞); H2)

� ∈ C([0,∞); H2), �t ∈L2loc(0,∞; H1)

Moreover, if we assume that �∗ =minx∈[0,l] �0(x)>0, then for any T>0 there exists �>0 determined by �∗, ‖‖utx‖∞‖L∞(0,T), and‖‖ux‖∞‖L∞(0,T) such that

�(t)��∗ exp(−�t)

for t ∈ [0, T].

Here and henceforth, for 1�p�∞, ‖·‖p denotes the standard Lp-norm on (0, l),

‖v‖LpT X =‖‖v‖X‖Lp(0,T) =

(∫ T

0‖v(·, t)‖X dt

)1/p

and the norm for L∞T X is defined similarly. This theorem was proven for more general nonlinearity, that is, under the assumption

that the nonlinear terms F1, F2 ∈C4 satisfy

F′1(0)=0, F2�−c for a constant c>0

instead of (2).One of the purposes of this paper is to prove the existence of stable nontrivial stationary solutions. Thus, we turn to the stationary

problem to (1). First, system (1) satisfies the momentum conservation law

d

dtM(ut)=0

and the energy conservation law

d

dtE(u, ut,�)=0

where the momentum and the energy are defined by

M(ut) :=∫ l

0ut dx

E(u, ut,�) := �

2‖ut‖2

2 + �

2‖�xux‖2

2 +∫ l

0F2(ux)+cv�dx

The momentum and the energy are thus determined by the initial data (u0, u1,�0), that is,

a :=M(u1)=M(ut), � :=E(u0, u1,�0)=E(u, ut,�)

In the stationary state (u,�)= (u,�) of (1) such that ut =�t =0, it follows from the second equation of (1) that � is a constant:

�xx =0 in (0, l), �x =0 on {0, l}

Regarding the energy conservation, this constant � is to be associated with u as

�= 1

cvl

(�− �

2‖�xux‖2

2 −∫ l

0F2(ux)dx

)


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where �=E(u0, u1,�0)=E(u, 0,�). Therefore, it holds that⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

−��4x u=−(�F′

1(ux)+F′2(ux))x in (0, l)

ux(0)=ux(l)=�2x ux(0)=�2

x ux(l)=0

�= 1

cvl

(�− �

2‖�xux‖2

2 −∫ l

0F2(ux)dx

)>0

(7)

In other words, given constants l, �, cv>0, and the initial data (u0, u1,�0)∈H4 ×H2 ×H2 satisfying �0>0, the stationary problem maybe formulated by ⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

−�′′ =−�F′1()−F′

2() in (0, l)

(0)=(l)=0

�=�(�,)= 1

cvl

(�− �

2‖′‖2

2 −∫ l

0F2()dx

)>0

(8)

Here =(x)=ux(x) is the strain function usually denoted by =(x) which acts as an order parameter, and �=E(u0, u1,�0)∈R. Anysolution to (8) satisfies ′′(0)=′′(l)=0, so that (8) can stand for (7) completely.

Concerning the stationary problem to the Falk system, there are very few mathematical studies. For the case that the Falknonlinearity (3) is adopted and the problem does not have any non-local term due to giving the absolute temperature �, seeFriedman–Sprekels [8]. In [8], some results on the existence, nonexistence, and bifurcation of the solution are proven under the

boundary condition u(0)=u(l)=�2x u(0)=�2

x u(l)=0, where the temperature � is regarded as the bifurcation parameter. On the otherhand, there is no result on (8) except for Suzuki–Yoshikawa [7] concerning the stability.

Problem (8) has the variational functional

J�()=− log(�−J2())+ 1

cvlJ1()

defined for ∈U� :={∈H10(0, l) | J2()<�}, where

J1() :=∫ l

0F1()dx, J2() := �

2‖′‖2

2 +∫ l

0F2()dx

This assertion means that (8) is equivalent to �J�()=0, that is, ∈U� and

d

dsJ�(+s�)

∣∣∣∣s=0

=0

for any �∈H10(0, l). In fact, (8) reads

�J2()=��J1()= 1

cvl(�−J2())�J1()

which is equivalent to �J�()=0.From the above-described variational structure, several stabilities of critical points of J� can be defined. A critical point ∈U�

of J� is said to be infinitesimally stable if there exists 0>0 such that any 1 ∈ (0,0 / 2] admits �0>0 such that ‖−‖H1(0,l)<0and J�()−J�()<�0 imply ‖−‖H1(0,l)<1. On the other hand, a critical point ∈U� of J� is said to be linearized stable if thequadratic form

Q(�,�)= d2

ds2J�(+s�)

∣∣∣∣∣s=0

is positive definite, that is, Q(�,�)>0 for any �∈H10(0, l)\{0}. It is clear that any linearized stable critical point is infinitesimally

stable. The following theorem assures the dynamical stability of such stationary solutions. For more information of this theory, werefer to the monograph [9].

Theorem B (Suzuki and Yoshikawa [7])Let the initial data (u0, u1,�0)∈H4 ×H2 ×H2 satisfy minx∈[0,l] �0(x)>0 and

∫ l0 u1 dx =0. Assume that =ux ∈U� is an infinitesimally

stable critical point of J� and that

�=�(�,)= 1

cvl

(�− �

2‖′‖2

2 −∫ l

0F2()dx

)>0

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Then (u,�) is dynamically stable in the sense that for any >0, there exists �>0 such that

‖(u0 −u)x‖H1(0,l)<�,

∣∣∣∣∣1

l

∫ l

0log�0 dx− log�

∣∣∣∣∣<�

implies

supt�0

‖(u(·, t)−u)x‖H1(0,l)<, supt�0

∣∣∣∣∣1

l

∫ l

0log�(·, t)dx− log�

∣∣∣∣∣<

Motivated by the above-described property, our concern now turns to the structure of the set of stationary solutions and theirlinearized stability. Here and henceforth, when we consider the stationary problem, is said to be a solution if it solves the problemand ∈C2(0, l)∩C[0, l], that is, it is a classical solution.

For the moment, we drop the condition �>0 and consider the following problem:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−�′′ =−�F′1()−F′

2() in (0, l)

(0)=(l)=0

�=�(�,)= 1

cvl

(�− �

2‖′‖2

2 −∫ l

0F2()dx

) (9)

where l, �, cv>0, and �∈R are given constants. Defining the operator H :R×X →Y by

H(�,) := −�′′+ 1

cvl

(�− �

2‖′‖2

2 −∫ l

0F2()dx

)F′

1()+F′2()

X = {∈C2[0, l] | (0)=(l)=0}, Y =C[0, l]

problem (9) is formulated by H(�,)=0. Here, we choose the energy � as the bifurcation parameter which is conserved in (1). If(�,) is a solution, then the linearized operator L=L�, ∈L(X, Y) is realized as

L(�) = −��′′+ 1

cvl

(�− �

2‖′‖2

2 −∫ l

0F2()dx

)F′′

1 ()�+F′′2 ()�

− 1

cvl

(�∫ l

0′�′ dx+

∫ l

0F′

2()�dx

)F′

1() with �(0)=�(l)=0 (10)

The linearized stability of the solution to (8) is equivalent to the positive definiteness of L whenever �=�(�,�)>0. When =0, L=L�,0is written as

L(�)=−��′′+2�1

(�

cvl−�c

)� with �(0)=�(l)=0

Thus, the eigenvalues { n}∞n=1 of L=L�,0 and the corresponding eigenfunctions {�n}∞n=1 are

n = �n2�2

l2+2�1

(�

cvl−�c

), �n(x)=sin

n�x

l

The sign of 1 controls the linearized stability of =0.The first result of this paper provides with the fundamental property concerning the bifurcation of the solution. It guarantees,

for example, that secondary bifurcations never occur. Then, the linearized stability of the solution changes at the turning pointwhenever �=�(�,)>0.

Theorem 1.1Assume (4) and (5). Let (�∗,∗)∈R×X be any nontrivial solution to (9). Then in any sufficiently small neighborhood of (�∗,∗),solutions to (9) generate a unique branch (one-dimensional manifold) in R×X .

We also consider the bifurcation from the branch of the trivial solutions. The following theorem classifies the bifurcation pointsand the directions of the bifurcated branch emerged from the branch of the trivial solutions.

Theorem 1.2The bifurcation points from the branch of trivial solutions {(�, 0) |�∈R} to (9) are �=�n, where

�n =cvl

(�c − �n2�2

2�1l2

), n=1, 2, . . . (11)


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In a neighborhood of the bifurcation point (�n, 0), the bifurcated branch consisting of nontrivial solutions to (9) can be describedas follows:

Cn ={(�(s),(s))∈R×X | s∈S}, (s)=s�n +z(s)

where S is an open interval containing 0, �(0)=�n, z(0)= z(0)=0,

� :S→R, z :S→Z, ˙= ��s

, �n(x)=sinn�x

l

and Z is a complement of span{�n} in X . Moreover, �(0)=0 and

�(0) =(

1+ 3cv�1

�21

)(�n2�2

2l−�1�cl

)+ 3cvl�2

�1

= −�1

cv

(1+ 3cv�1

�21

)�n + 3cvl�2

�1

From �(0)=0 and the representation of �(0), we can see that both super- and sub-critical bifurcations can occur for some parametersetting. For example, if �1 is sufficiently large and �c>0, then �n>0 and �(0)<0. In this case, �=�(�,)>0 holds near the bifurcationpoint (�n, 0). Thus, taking n=1, the bifurcated branch consists of linearized stable nontrivial solutions to (8) in a neighborhood of(�n, 0) from the local theory [10], and consequently, these nontrivial solutions are dynamically stable from Theorem B. When �(0)>0,on the other hand, there arises a hysteresis concerning the change of stable stationary states as � decreases and increases.

Next, we shall more precisely investigate the structure of the set of the stationary solutions and their stability by using thevariational structure.

To begin with, =0 is a solution to (8) whenever �>0. Moreover, by calculating the linearized eigenvalues, we can easily seethat the trivial solution =0 is linearized stable if and only if �>�1. Then we shall study nontrivial stationary solutions.

If is a nontrivial solution to (8), then any zero x =x0 of =(x), (x0)=0, is simple, that is, ′(x0) =0. In fact, if this property isnot the case, then it follows from the assumption (2) and the uniqueness of solutions to ODEs that ≡0, a contradiction. Henceforth,a nontrivial solution is said to be (n−1)-nodal if =(x) has exactly (n−1) nodal zeros in (0, l), where n=1, 2, . . . . We notethat if 0<x1<x2< · · ·<xn−1<l denote the nodal points of an (n−1)-nodal solution =(x), then =(x) has symmetric reflectionsat x =x1 / 2, (x1 +x2) / 2, . . . , (xn−1 + l) / 2. A nontrivial solution is said to be n-mode if it is (n−1)-nodal and has antisymmetricreflections at x =x1, x2, . . . , xn−1. Then, under the assumption (4), any nontrivial solution is n-mode for some n=1, 2, . . . .Furthermore, xj = jl / n for j=1, 2, . . . , n−1, see Figure 1.

To begin with, we are concerned with to give a necessary condition for the stationary solution to be stable. The next theorempresents a necessary condition imposed on the local minimizer of J�.

Theorem 1.3Let ∈U� be any local minimizer of J�. Then the following facts hold:

(i) =0, : 0-nodal, or 1-nodal.(ii) Under (4), =0 or : 0-nodal.

The following theorem is concerning the existence of the nontrivial stationary solution, recall (11).

Theorem 1.4There exists a solution to (8). If �<�1, then there exists a nontrivial solution to (8). Moreover, under (4), if �<�n for some n=1, 2, . . .,then there exists an n-mode nontrivial solution to (8).

Figure 1. Solution and the assumption (4).

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The solution obtained by Theorem 1.4 is a global minimizer. Then our interest turns to its stability. The following theorem assuresthe stability of local minimizers.

Theorem 1.5Assume (4)–(6), and F1, F2 ∈C�. Then any local minimizer ∈U� of J� is infinitesimally stable.

Theorems 1.4, 1.5, and B result in the following corollary.

Corollary 1.1Assume (4)–(6), and F1, F2 ∈C�. Then there exists a stationary solution (u,�) which is dynamically stable in the sense of Theorem B.Moreover, if �<�1, then there exists a stationary solution (u,�) which is dynamically stable in the sense of Theorem B and =ux isnon-constant.

As the solution obtained by Theorem 1.4 and Corollary 1.1 is a global minimizer, it satisfies the condition in Theorem 1.3. Inparticular, =ux is 0-nodal for the dynamically stable nontrivial solution (u,�) obtained by Corollary 1.1 in the low-energy case,�<�1. Then the rod consists of a single martensitic phase and shears in the same direction everywhere. On the other hand, in thehigh-energy case, that is, when �>�1, the trivial solution (u,�)= (�,� / (cvl)) is dynamically stable, where � is a constant. Actually, weexpect from the physical point of view that the trivial solution =ux =0, that is, the stationary state uniformly consisting of anaustenitic phase, is stable when the temperature is high, owing to the shape memory effect or the pseudoelasticity of the rod. Werecall that the free energy consists of the temperature-dependent term �F1(ux) and the potential F2(ux) which is usually doublewell. From the dynamical point of view, however, we consider the variational functional J�. It is double well in the low-energy case.Then it would become triple well as the energy increases when the nontrivial solutions bifurcate sub-critically, and finally singlewell in the high-energy case.

This paper is composed of seven sections. Theorems 1.1–1.5 are proven in Sections 2–6, respectively. Section 7 is devoted tosome results on constraints with respect to the temperature for the existence of the nontrivial solutions.

2. Proof of Theorem 1.1

In this section, we show some properties of the linearized operator L defined by (10) and prove Theorem 1.1. First, in any smallneighborhood of an (n−1)-nodal nontrivial solution (�∗,∗) which does not include the trivial solution =0, the number of thezeros of any solution is (n−1). In fact, if not, there is a solution =(x) having a degenerate zero. Then by the uniqueness ofsolutions to ODEs, should be zero. Therefore, under (4), it is sufficient for the proof of Theorem 1.1 to consider the 1-modeproblem because any n-mode solution may be considered as a 1-mode solution to (9) in (0, l / n).

Lemma 2.1Assume (5) and let (�,) be a nontrivial 0-nodal solution to (9). If L=L�, has the eigenvalue 0, then the eigenfunction �=�(x) ofL=L�, which corresponds to the eigenvalue 0 satisfies ∫ l

0F′

1()�dx =0

ProofBy using (9), the linearized operator (10) can be rewritten as

L(�)=−��′′+(�F′′1 ()+F′′

2 ())�+ �

c2v l2

(∫ l

0F′

1�dx

)F′

1() with �(0)=�(l)=0

Suppose ∫ l

0F′

1()�dx =0

Then we have

−��′′+(�F′′1 ()+F′′

2 ())�=0, �(0)=�(l)=0

On the other hand, differentiating (9) with respect to x, it holds that

−��′′1 +(�F′′

1 ()+F′′2 ())�1 =0

for �1 =′. We note that �′1(0)=�′

1(l)=0 and that �1 has exactly one nodal zero x = l / 2 in (0, l). Then it follows from Sturm’scomparison theorem (see [11] for instance) that �=�(x) is 0-nodal in (0, l). Without loss of generality, we assume that �>0 and>0 in (0, l). Then F′

1()>0 in (0, l) from (5). Hence we have∫ l

0F′

1()�dx>0

which is a contradiction. This completes the proof. �


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Lemma 2.2Assume (5) and let (�,) be a nontrivial 0-nodal solution to (9). If L=L�, has the eigenvalue 0, then it is simple.

ProofLet �1 and �2 be eigenfunctions of L=L�, which correspond to the eigenvalue 0 : L(�1)=L(�2)=0. It follows from Lemma 2.1that ∫ l

0F′

1()�1 dx =0,

∫ l

0F′

1()�2 dx =0

Then there exist nonzero constants c1 and c2 such that �3 =c1�1 +c2�2 satisfies∫ l

0F′

1()�3 dx =0, L(�3)=0

Hence from Lemma 2.1, it holds that �3 ≡0, that is, �1 and �2 are linearly dependent. �

Proof of Theorem 1.1If L=L�∗ ,∗ does not have the eigenvalue 0, then the assertion is obvious by using the implicit function theorem. Thus supposethat L has the eigenvalue 0. From Lemma 2.2, it is simple. Let � be the corresponding eigenfunction: L(�)=0. Then it holds that

Ker(L)={�� | �∈R}Define the operator � :R×R× X →Y by

�(s,�, v) :=H(�∗+�,∗+s�+v)

where

X ={

v ∈C2[0, l]

∣∣∣∣∣ v(0)=v(l)=0,

∫ l

0v�dx =0

}

Zeros of � have one-to-one correspondence with that of H and �(0, 0, 0)=H(�∗,∗). Considering the linearized operator

�(�,v)(0, 0, 0)=(

1

cvlF′

1() L

):R× X →Y

it is an isomorphism because of Lemma 2.1. Therefore, it follows from the implicit function theorem that for |s|�1, there exists aunique C3 mapping

s �→ (�(s), v(s))∈R× X

such that

�(0)=0, v(0)=0, �(s,�(s), v(s))=0

Consequently,

C∗ ={(�(s),(s))∈R×X | |s|�1}is the unique branch in the assertion, where �(s)=�∗+�(s), (s)=∗+s�+v(s). �


The bifurcation points from the branch of the trivial solutions {(�, 0) |�∈R} are obtained by the bifurcation from simple eigenvalues,see Crandall–Rabinowitz [12]. In fact, we have

Ker(Ln) = {��n | �∈R}

Ran(Ln) ={

v ∈C[0, l]

∣∣∣∣∣∫ l

0v�n dx =0

}

H�(�n, 0)�n = 2�1

cvl�n ∈Ran(Ln)

�n(x) = sinn�x

l

where Ln =L�n,0.

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Next, consider the bifurcated branch Cn. For (�(s),(s))∈Cn, it holds that

H(�(s),(s))=−��2x(s)+A(s)F′

1((s))+F′2((s))=0

where

A(s) = 1

cvl

{�(s)− �

2‖�x(s)‖2

2 −∫ l

0F2((s))dx

}

A(0) = 1

cvl�n =�c − �n2�2

2�1l2

Differentiating this with respect to s, we have

0=−��2x (�n + z(s))+A(s)F′

1((s))+A(s)F′′1 ((s))(�n + z(s))+F′′

2 (�n + z(s))

where

A(s) = 1

cvl

{�(s)−�

∫ l

0�x(s)�x(�n + z(s))dx−

∫ l

0F′

2((s))(�n + z(s))dx

}

A(0) = 1

cvl�(0)

By differentiating this once more, we have

0 = −��2x z(s)+A(s)F′

1((s))+2A(s)F′′1 ((s))(�n + z(s))

+A(s){F(3)1 ((s))(�n + z(s))2 +F′′

1 ((s))z(s)}+F(3)2 ((s))(�n + z(s))2 +F′′

2 (�n + z(s)) (12)

where

A(s) = 1

cvl

{�(s)−�

∫ l

0|�x(�n + z(s))|2 dx−�

∫ l

0�x(s)�xz(s)dx

−∫ l

0F′′

2 ((s))(�n + z(s))2 dx−∫ l

0F′

2((s))z(s)dx

}

A(0) = 1

cvl

{�(0)−�

∫ l

0|�x�n|2 dx+2�1�c

∫ l

0�2

n dx

}

= 1

cvl

(�(0)+ �1

cv�n

)= 1

cvl

(�(0)− �n2�2

2l+�1�cl

)

because

‖�n‖22 = l

2, ‖�x�n‖2

2 = n2�2

l2‖�n‖2

2 = n2�2

2l

Taking s=0 in this equation, we obtain

Lnz(0)+ 4�1

cvl�(0)�n =0

and therefore,

4�1

cvl�(0)‖�n‖2

2 =0

Hence, �(0)=0 and z(0)=0.By differentiating (12) once more, we have

0 = −��2x z(3)(s)+A(3)(s)F′′

1 ((s))+3A(s)F′1((s))(�n + z(s))+3A(s){F(3)

1 ((s))(�n + z(s))2 +F′′1 ((s))z(s)}

+A(s){F(4)1 ((s))(�n + z(s))3 +3F(3)

1 ((s))(�n + z(s))z(s)+F′′1 ((s))z(3)(s)}

+F(4)2 ((s))(�n + z(s))3 +3F(3)

2 ((s))(�n + z(s))z(s)+F′′2 ((s))z(3)(s)


10

01


where

A(3)(s) = 1

cvl

{�(3)(s)−3�

∫ l

0�x(�n + z(s))�xz(s)dx−�

∫ l

0�x(s)�xz(3)(s)dx

−∫ l

0F(3)

2 ((s))(�n + z(s))3 dx−3

∫ l

0F′′

2 ((s))(�n + z(s))z(s)dx−∫ l

0F′

2((s))z(3)(s)dx

}

A(3)(0) = 1

cvl�(3)(0)

Taking s=0 in this equation, we obtain

Lnz(3)(0)+ 6�1

cvl

(�(0)− �n2�2

2l+�1�cl

)�n +

{12�1

�1

(2�1�c − �n2�2

l2

)−24�2

}�3

n =0

and therefore,

6�1

cvl�(0)‖�n‖2

2 = 6�1

cvl

(�n2�2

2l−�1�cl

)‖�n‖2

2 −{

12�1

�1

(2�1�c − �n2�2

l2

)−24�2

}‖�n‖4

4

Hence, we have

�(0) =(

1+ 3cv�1

�21

)(�n2�2

2l−�1�cl

)+ 3cvl�2

�1

= −�1

cv

(1+ 3cv�1

�21

)�n + 3cvl�2

�1

This completes the proof. �


Let be an (n−1)-nodal nontrivial solution to (8). Differentiating (8) with respect to x, it holds that

−��′′1 +(�F′′

1 ()+F′′2 ())�1 =0 in (0, l) (13)

for �1 =′. On the other hand, it holds that

Q(�,�) = d2

ds2J�(+s�)

∣∣∣∣∣s=0

= 1

�−J2()

∫ l

0��′2 +(�F′′

1 ()+F′′2 ())�2 dx+ 1

c2v l2

(∫ l

0F′

1()�dx

)2

thanks to (8). Let 0<x1<x2< · · ·<xn−1<xn = l be the zeros of =(x).

(i) Suppose n�3. Then, =(x) has at least two nodal zeros in (0, l). Define the following function:

H10(0, l) �1 = �1(x) :=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0(

x ∈[

0,x1

2

])

�1(x)

(x ∈[

x1

2,

x2 +x3

2

])

0

(x ∈[

x2 +x3

2, l

])

�1 = �1(x) is antisymmetric and =(x) is symmetric with respect to x = (x1 +x2) / 2=x3 / 2 in [0, x3], see Figure 2.Therefore, ∫ l

0F′

1()�1 dx =∫ x3

0F′

1()�1 dx =0 (14)

It follows from (13) and (14) that

Q(�1, �1)=0

10

02



Figure 2. Graph �1 (i).

Figure 3. Graph �1 (ii).

Recalling that is a local minimizer of J�, it holds that Q(�,�)�0 for any �∈H10(0, l). Hence, �= �1 is a global minimizer

of Q =Q(�,�) and it satisfies

−��′′1 +(�F′′

1 ()+F′′2 ())�1 =0 in (0, l), �1(0)= �

′1(0)=0

This implies �1 ≡0, which is a contradiction.(ii) Suppose (4) and n�2. Then, =(x) has at least one nodal zero in (0, l). Define the following function:

H10(0, l) �1 = �1(x) :=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0(

x ∈[

0,x1

2

])

�1(x)

(x ∈[

x1

2,

x1 +x2

2

])

0

(x ∈[

x1 +x2

2, l

])

�1 = �1(x) is symmetric and =(x) is antisymmetric with respect to x =x1 =x2 / 2 in [0, x2]. It follows from theassumption (4) that F′

1()=F′1((x)) is antisymmetric with respect to x =x1 =x2 / 2 in [0, x2], see Figure 3.

Therefore,

∫ l

0F′

1()�1 dx =∫ x2

0F′

1()�1 dx =0 (15)

It follows from (13) and (15) that Q(�1, �1)=0. Then from a similar argument, we obtain �1 ≡0, which is a contradiction.


In this section, we prove Theorem 1.4 by using a direct method in the calculus of variations.


10

03


Lemma 5.1There exists a global minimizer ∈U� of J�, that is, J�()= inf∈U�

J�().

ProofU� =∅. In fact, for any initial data (u0, u1,�0)∈H4 ×H2 ×H2 satisfying �0>0, the solution (u,�) stated in Theorem A satisfies

� = E(u0, u1,�0)=E(u(·, t), ut(·, t),�(·, t))

>�

2‖�xux(·, t)‖2

2 +∫ l

0F2(ux(·, t))dx = J2(ux(·, t))

for any t>0.Next, it follows from the assumption (2) that

J�()�− log(�+ lc)− c

cv

for any ∈U�, that is, J� is bounded from below. Thus we put

J� := inf∈U�

J�()>−∞

Defining

U�(�) :={∈H10(0, l) | J2()��−�}

for �>0, we have

inf∈U�\U�(�)

J�()>J�, U�(�) =∅

with sufficiently small �>0 chosen adequately. Then we obtain

J� = inf∈U�

J�()= inf∈U�(�)

J�()

and therefore, there exists a sequence {j}⊂ U�(�) such that J�(j)→J�. As {j}⊂H10(0, l) is a bounded sequence and U�(�)⊂H1

0(0, l)

is closed, it weakly converges to ∈ U�(�) passing to a subsequence. Moreover, by using Sobolev’s imbedding theorem, {j} is asequence in {∈C[0, l] | (0)=(l)=0} and strongly converges passing to a subsequence. Then from the assumption F1, F2�−cand Fatou’s lemma, we have

J1()� lim infj→∞

J1(j), J2()� lim infj→∞

J2(j)

Thus, we observe the lower semi-continuity of J�:

J�()� lim infj→∞

J�(j)=J�

Consequently, we have

J�()=J�, ∈ U�(�)⊂U�

where is an interior point, and therefore, it is a critical point. This completes the proof. �

Proof of Theorem 1.4From Lemma 5.1, we have obtained a solution to (8). Assume �<�1. Then, even if =0 is a solution to (8), that is, �>0, this trivialsolution =0 is not a local minimizer of J�, so that there exists a nontrivial solution.

Next, assume (4) and �<�n for some n=1, 2, . . . . Any n-mode solution to (8) satisfies⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−�′′ =−�F′1()−F′

2() in

(0,

l

n

)

(0)=

(l

n

)=0

�=�(�,)= 1

cvl

(�− n�

2‖′‖2

L2(0, ln )

−n

∫ l/n

0F2()dx

)>0

(16)

10

04



Any solution =(x) to (16) is realized by

(x)=

⎧⎪⎪⎨⎪⎪⎩

N

(l

2n−x

) (x ∈[

0,l

2n

])

−N

(x− l

2n

) (x ∈[

l

2n,

l

n

]) (17)

where N =N(x) is a solution to the corresponding Neumann problem⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−�′′N =−�F′

1(N)−F′2(N) in

(0,

l

n

)

′N(0)=′

N

(l

n

)=0

�=�(�,N)= 1

cvl

(�− n�

2‖′

N‖22 −n

∫ l/n

0F2(N)dx

)>0

(18)

Problem (18) has the variational functional

J�,n(N)=− log(�−nJ2,n(N))+ n

cvlJ1,n(N)

defined for N ∈U�,n :={N ∈H1(0, l) | nJ2,n(N)<�}, where

J1,n(N) :=∫ l/n

0F1(N)dx, J2,n(N) := �

2‖′

N‖2L2(0, l

n )+∫ l/n

0F2(N)dx

This means that (18) is equivalent to �J�,n(N)=0. Recalling U� =∅ in the proof of Lemma 5.1, we also obtain U�,n =∅. Thus, theassertion of Lemma 5.1 is also valid to J�,n on U�,n. From the assumption �<�n, the trivial solution =0 is not a global minimizer,and we have a nontrivial solution N =N(x) to (18). Moreover, it is monotone in (0, l / n). Actually, if this property is not the case,then we may assume that N =N(x) is m-times symmetric for m=2, 3, . . .. Defining = (x) :=N(x / m), we have J1,n()= J1,n(N)and

J2,n() = �

2‖′‖2

L2(0, ln )

+∫ l/n

0F2()dx

= �

2m2‖′

N‖2L2(0, l

n )+∫ l/n

0F2(N)dx<J2,n(N)

Therefore, ∈U�,n and

J�,n()<J�,n(N)

that is, N is not a global minimizer, which is a contradiction. Consequently, we obtain an n-mode nontrivial solution to (16)by (17). �


Throughout this section, we omit the subscript of the L2-inner product: (·, ·)= (·, ·)L2(0,l).

It holds for �∈H10(0, l) that

J�(+�)−J�()= 1

2

∞∑i=0

i|(�,�i)|2 +o(‖�x‖22), ‖�x‖2 �1 (19)

where 0� 1� · · · and �0, �1, . . . stand for the eigenvalues and the corresponding L2-normalized eigenfunctions of L=L�,,respectively. Suppose that the critical point ∈U� is non-degenerate. Then 0>0 because is a local minimizer of J�. Hence thereexists 0<0 �1 such that given >0, we have �1>0 such that ‖�x‖2<20 and J�(+�)−J�()<�1 imply ‖�x‖2< / 2, whichmeans that is infinitesimally stable.

Suppose that the critical point ∈U� is degenerate. Then 0 =0 and it is simple by Lemma 2.2:

0= 0< 1� 2� · · ·Similar to the non-degenerate case, it follows from (19) that there exists 0<0 �1 such that given >0, we have �1>0 such that‖�x‖2<20 and J�(+�)−J�()<�1 imply

‖�x‖2<

2(20)


10

05


where �=∑∞i=1(�,�i)�i =�−(�,�0)�0 =�−s�0, and we have

J�(+�)−J�(+s�0)= 1

2

∞∑i=1

i|(�,�i)|2 +o(‖�x‖22), ‖�x‖2 �1 (21)

by (19).Setting

J(s) :=J�(+s�0)

J is real analytic at s=0. To prove this, set

J1(s) := J1(+s�)=∫ l

0F1(+s�0)dx

Then J1 ∈C∞ and

J(�)1 (s)=

∫ l

0F(�)

1 (+s�0)��0 dx

for any �=1, 2, . . . . Take s∗ ∈R and >0 arbitrarily. Putting

m= maxx∈[0,l]

|(x)|+(|s∗|+) maxx∈[0,l]

|�0(x)|

it follows from F1 ∈C� that there exist constants C1, C2>0 such that

|F(�)1 (v)|�C1C�

2�! for any v ∈ (−m, m)

This implies

supx∈[0,l]

|F(�)1 (+s�0)|�C1C�

2�! for any s∈ (s∗−, s∗+)

Therefore, it holds for any s∈ (s∗−, s∗+) that

|J(�)1 (s)|�

∫ l

0|F(�)

1 (+s�0)||�0|� dx�C1C�2�! max

x∈[0,l]|�0|�

which means J1 ∈C�. Similarly, set

J2(s) := J2(+s�0)=∫ l

0

�

2|′+s�′

0|2 +F2(+s�0)dx

Then J2 ∈C∞ and

J(�)2 (s)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫ l

0�(′+s�′

0)�′

0 +F′2(+s�0)�0 dx (�=1)

∫ l

0��′2

0 +F′′2 (+s�0)�2

0 dx (�=2)

∫ l

0F(�)

2 (+s�0)��0 dx (�=3, 4, . . .)

Hence, J2 ∈C� can be shown similarly to J1. In view of

J(s)=− log(�− J2(s))+ 1

cvlJ1(s)

J is real analytic at s=0. Therefore, it holds that

J�(+s�0)−J�() = J(s)−J(0)

=∞∑

j=1

1

j!J

(j)(0)sj, |s|�1 (22)

10

06



Assume that

1

j!J

(j)(0)=0 for any j=1, 2,. . . (23)

By the uniqueness theorem, we have

J(s)=J(0) for any s∈R (24)

On the other hand, we also obtain

J(s) � − log(�− J2(s))− c

cv

J(s) � − log(�+cl)+ 1

cvJ1(s) (25)

The eigenfunction �0 is real analytic on (0, l) because it is a solution to the analytic elliptic equation

−��′′0 +(�F′′

1 ()+F′′2 ())�0 +�F′

1()=0

provided some constants �, �∈R and derived from L�,(�0)=0. Hence, any zero of �0 is isolated and it holds for almost everyx ∈ (0, l) that

|(x)+s�0(x)|→+∞ as |s|→+∞

uniformly on a set of positive measure of x. This fact, (6), and (25) contradict (24). Thus, Equation (23) does not hold, that is, thereexists j=1, 2, . . . such that

j,0 :=J(j)

(0) =0

Therefore, as is a degenerate local minimizer, there exists j=4, 6, 8, . . . such that

j,0

>0 and j,0 =0 for any j<j (26)

Here, for any r =1, 2, . . . , it holds that

J�(+�)−J�()=r∑

j=0

1

j!djJ�()[�,. . . ,�]+o(‖�‖r), ‖�‖�1 (27)

where

djJ�()[�,. . . ,�]= dj

dsjJ�(+s�)

∣∣∣∣∣s=0

, j =1, 2,. . .

is a j-linear form. It follows from (26) and (27) that

J�(+s�0)−J�() = 1

j!djJ�()[�0,. . . ,�0]|s|j +o(|s|j)

= 1

j!

j,0|s|j +o(|s|j), |s|�1 (28)

Hence, there is �2>0 such that |s|<20 and J�(+�)−J�()<�2 imply

|s|<

2(29)

We note that

J�(+�)−J�()= 1

2

∞∑i=1

i|(�,�i)|2 + 1

j!

j,0|s|j +o(‖�x‖2

2)+o(|s|j), ‖�2‖2 �1

by (21) and (28). Thus inequalities (20) and (29) signify that is infinitesimally stable.We finally note that if we take the Falk nonlinearity (3), then j=4 or j=6 in (26). In fact, if F1, F2 ∈C6 and a degenerate local

minimizer ∈U� satisfies

4,0 = d4J�()[�0,. . . ,�0]=0


10

07


then it follows from a simple calculation that

6,0 = d6J�()[�0,. . . ,�0]

= 403

(cvl)6

(∫ l

0F′

1()�0 dx

)6

+ 158

(cvl)5

(∫ l

0F′

1()�0 dx

)4 ∫ l

0F′′

1 ()�20 dx

+ 117

(cvl)4

(∫ l

0F′

1()�0 dx

)2(∫ l

0F′′

1 ()�20 dx

)2

+ 62

(cvl)4

(∫ l

0F′

1()�0 dx

)3 ∫ l

0F(3)

1 ()�30 dx

+ 74

(cvl)3

∫ l

0F′

1()�0 dx

∫ l

0F′′

1 ()�20 dx

∫ l

0F(3)

1 ()�30 dx

+ 19

(cvl)3

(∫ l

0F′

1()�0 dx

)2 ∫ l

0F(4)

1 ()�40 dx+ 18

(cvl)3

(∫ l

0F′′

1 ()�20 dx

)3

+ 15

(cvl)2

∫ l

0F′′

1 ()�20 dx

∫ l

0F(4)

1 ()�40 dx+ 10

(cvl)2

(∫ l

0F(3)

1 ()�30 dx

)2

+ 6

(cvl)2

∫ l

0F′

1()�0 dx

∫ l

0F(5)

1 ()�50 dx+ 1

cvl

∫ l

0F(6)

1 ()�60 dx+ 1

cvl�

∫ l

0F(6)

2 ()�60 dx

Hence, we can see 6,0>0 by (3) and Lemma 2.1.

7. Remarks on the temperature for the nontrivial stationary solution

The next theorem provides with necessary conditions of the temperature � for the existence of the nontrivial solution to (9). Roughlyspeaking, there is no nontrivial solution to (9) for high temperatures.

Theorem 7.1Let (�,) be any nontrivial solution to (9). Then the following facts hold:

(i) (Necessary conditions of the temperature �)

(i-1) Assume (5). Then there exists a constant �= �(F1, F2)>0 such that

�=�(�,)<�

(i-2) Assume (3). Then

�=�(�,)<�c + �22

4�1�3

(ii) (Necessary conditions of the temperature � and the length l>0)

(ii-1) Assume (5) and put

c1 := infv∈R

F′1(v)

v�0

Assume furthermore that there exists c2 ∈R such that

F′2(v)

v�−c2

for any v ∈R. Then

c1�=c1�(�,)�c2 − ��2

l2

(ii-2) Assume (3). Then

�=�(�,)<�2

23�1�3

+�c − ��2

2�1l2

10

08



Such kind of constraint, that is, �(�,)<� for a constant � implies

�

2‖′‖2

2 +∫ l

0F2()dx>�−cvl�

for any nontrivial solution (�,) to (9). Concerning (8), in particular, it is easy to see the following facts:

• Assume that there exists a constant �>0 such that �(�,)<� for any nontrivial solution (�,) to (9) as in Theorem 7.1. Then itholds that

�−cvl�<�

2‖′‖2

2 +∫ l

0F2()dx<�

for any nontrivial solution (�,) to (8). In particular, from the latter inequality we obtain the a priori estimate

‖′‖22<

2

�(�+cl)

• By Theorem 7.1 we obtain the following facts:

◦ Assume (5) and that there exists c2 ∈R such that

F′2(v)

v�−c2

for any v ∈R. If

c2 − ��2

l2<0

then nontrivial solutions to (8) do not exist.◦ Assume (3). If

�22

3�1�3+�c − ��2

2�1l2�0

then nontrivial solutions to (8) do not exist.

Hence, if l>0 is sufficiently small, then nontrivial solutions to (8) do not exist. In other words, when l>0 is sufficiently small, anynontrivial solution (�,) to (9) satisfies �(�,)�0.

Proof of Theorem 7.1Although (i-2) and (ii-2) of Theorem 7.1 are essentially proven in Friedman–Sprekels [8], we prove all the assertions of this theoremfor completeness. Let (�,) be any nontrivial solution to (9).

(i-1) It follows from (9) that

′(x)2 = 2

�

(�F1((x))+F2((x))−�F1

(( x1

2

))−F2

(( x1

2

)))where x =x1 / 2 is the first zero of ′ =′(x), in other words, x =x1 ∈ (0, l] is the first zero of =(x). Taking x =0 in this equation,we have

′(0)2 = 2

�

(−�F1

(( x1

2

))−F2

(( x1

2

)))>0 (30)

Next, it holds that

F1(v) = F1(0)+F′1(0)v+ 1

2 F′′1 (0)v2 +o(v2)=�1v2 +o(v2) (31)

F2(v) = F2(0)+F′2(0)v+ 1

2 F′′2 (0)v2 +o(v2)=−�1�cv2 +o(v2) (32)

for |v|�1, where �1>0 and �c ∈R are constants determined by F1 and F2. Suppose the assertion does not hold. Then there existsa nontrivial solution (�1,1) satisfying �(�1,1)>�c. It follows from (31) and (32) that

−�F1(v)−F2(v)=−�1(�−�c)v2 +o(v2) for |v|�1 (33)

Hence, there exists a constant �1 =�1(F1, F2)>0 such that

F1(v) > 12 �1�2 for |v|��1 (34)

−�F1(v)−F2(v) � 0 for 0�|v|��1 (35)


10

09


by (5), (31), and (33). Moreover, if ��0, then it follows from (2) and (34) that

−�F1(v)−F2(v)�−�F1(v)+c�− 12 �1�2

1�+c for |v|��1 (36)

Here, as the assertion does not hold, there exists a nontrivial solution (�2,2) satisfying �(�2,2)>max(2c / (�1�21),�c). We can

furthermore assume �(�2,2)��(�1,1). Then inequalities (35) and (36) still remain true for (�2,2) because of F1�0, and therefore,we obtain

−�F1(v)−F2(v)�0 for any v ∈R

by (35) and (36). This contradicts (30).(i-2) From (30), it holds for some v ∈R that

−�F1(v)−F2(v)=−�3v2

{(v2 − �2

2�3

)2− �2

2

4�23

− �1

�3

(�c −�

)}>0

Hence, we obtain

− �22

4�23

− �1

�3(�c −�)<0

that is,

�<�c + �22

4�1�3

which completes the proof.(ii-1) Multiplying the first equation of (9) by and integrating it over (0, l), we have

�∫ l

0′2 dx =−

∫ l

0(�F′

1()+F2())dx

By using Poincaré’s inequality

∫ l

0′2 dx��2

l2

∫ l

02 dx

we obtain

��2

l2

∫ l

02 dx�−

∫ l

0(�F′

1()+F′2())dx

that is,

∫ l

02

(�F′

1()+F′2()

+ ��2

l2

)dx�0 (37)

If the assertion does not hold, then

�F′1()+F′

2()

+ ��2

l2�c1�−c2 + ��2

l2>0

which contradicts (37).(ii-2) If the assertion does not hold, then we have

�F′1()+F′

2()

+ ��2

l2=6�3

(2 − �2

3�3

)2+2�2

(�− �2

23�1�3

−�c + ��2

2�1l2

)�0

It follows from (37) that

=0 or 2 = �2

3�3for almost every x ∈ (0, l)

a contradiction. �

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References1. Falk F. Ginzburg–Landau theory of static domain walls in shape-memory alloys. Zeitschrift für Physik B Condensed Matter 1983; 51:177--185.2. Falk F. Ginzburg–Landau theory and solitary waves in shape-memory alloys. Zeitschrift für Physik B Condensed Matter 1984; 54:159--167.3. Falk F. Elastic phase transitions and nonconvex energy functions. Free Boundary Problems, Theory and Applications, vol. I. Pitman Research Notes

in Mathematical Series, vol. 185. Longman: Harlow, 1990; 45--59.4. Brokate M, Sprekels J. Hysteresis and Phase Transitions. Applied Mathematical Sciences, vol. 121. Springer: New York, 1996.5. Aiki T. Weak solutions for Falk’s model of shape memory alloys. Mathematical Methods in the Applied Sciences 2000; 23:299--319.6. Yoshikawa S. Weak solutions for the Falk model system of shape memory alloys in energy class. Mathematical Methods in the Applied Sciences

2005; 28:1423--1443. DOI: 10.1002/mma.621.7. Suzuki T, Yoshikawa S. Stability of the steady state for the Falk model system of shape memory alloys. Mathematical Methods in the Applied

Sciences 2007; 30:2233--2245. DOI: 10.1002/mma.889.8. Friedman A, Sprekels J. Steady states of austenitic–martensitic domains in the Ginzburg–Landau theory of shape memory alloys. Continuum

Mechanics and Thermodynamics 1990; 2:199--213.9. Suzuki T. Mean Field Theories and Dual Variation. Atlantis Studies in Mathematics for Engineering and Science, vol. 2. Atlantis Press: Amsterdam,

Paris, 2008.10. Crandall MG, Rabinowitz PH. Bifurcation, perturbation of simple eigenvalues, and linearized stability. Archive for Rational Mechanics and Analysis

1973; 52:161--180.11. Coddington EA, Levinson N. Theory of Ordinary Differential Equations. McGraw-Hill: New York, 1955.12. Crandall MG, Rabinowitz PH. Bifurcation from simple eigenvalues. Journal of Functional Analysis 1971; 8:321--340.


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Stationary solutions to the Falk system on shape memory alloys