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Phase transitions with a minimal number of jumps in the singular limits of higher order theories
P.I. Plotnikov
a, J.F. Toland
b,∗aLavryentyev Institute of Hydrodynamics, Russian Academy of Sciences, Lavryentyev pr. 15, Novosibirsk 630090, Russia bDepartment of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
Received 1 March 2009; received in revised form 5 October 2009; accepted 5 October 2009 Available online 10 November 2009
Abstract
For a smoothW:(0,∞)×Rd→Rand a family ofL-periodicW1,2-functionsϑ:R→Rdwithϑ ϑ, the basic problem is to understand the weak* limit as→0 ofL-periodic minimizers of
L
0
2ϕ2+W (ϕ, ϑ)
ds. (†)
It is assumed thatW (φ, θ )→ ∞asφ→0,∞, and thatW (·, θ ), which has no more than three critical points counting multiplicity depending onθ∈Rd, is of a type that arises in the Cahn–Hilliard theory of phase separations whered=1. The limiting problem with=0 is to minimize, over boundedL-periodic measurable functionsϕ,
L
0 W
ϕ(s), ϑ (s)
ds. (‡)
Minimizers of (‡) need not be unique (there may be uncountably many), they may be discontinuous and minimizers with only simple jumps may coexist with minimizers with much more complicated discontinuities. Weak* limits of minimizers of (†) as →0 are minimizers of the relaxation of (‡). However it is shown that if, for a sequence of minimizers of (†),
lim sup k→∞
√k L
0 ϕ
k(s)2ds <∞, k→0,
then the weak* limit of any subsequence of{ϕk}is an actual minimizer of (‡) which is continuous except at a finite number of simple jumps. Moreover, for sequencesk→0 from a set of positive Lebesgue density, it is shown that the weak* limit of L-periodic minimizers of (†) is a minimizer of (‡) with a finite number of simple jumps. Under additional hypotheses it is shown that, for sequences from a set of full Lebesgue density, the weak* limits ofL-periodic minimizers of (†) are minimizers of (‡) with aminimal number of simple jumps.
©2009 Elsevier Masson SAS. All rights reserved.
* Corresponding author.
E-mail addresses:[email protected] (P.I. Plotnikov), [email protected] (J.F. Toland).
0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2009.11.002
MSC:82B26; 49S05; 49J45; 34E15
Keywords:Phase transitions; Cahn–Hilliard; Singular limits; Relaxed minimizers; Regularized minimizers; Minimal jump principle; Gamma limits
1. Introduction
A crucial feature in models of phase transitions is the non-convexity of bulk energy functions with respect to the phase variableϕ. For instance, in one-dimensional Cahn–Hilliard theory the Helmoltz free energy density is given by
W (ϕ, ϑ )=1 4
ϕ2−12
−ϑ ϕ, (1.1)
whereϑis the chemical potential. IfϕandϑareL-periodic, the total mesoscopic energy per period is J(ϕ):=
L 0
2ϕ2+1
4
ϕ2−12
−ϑ ϕ
dx, >0, (1.2)
whereϕ2/2 corresponds to the energy of phase interactions,√
characterizes the width of the interfaces between phases and in practiceis very small. Critical points ofJsatisfy
−ϕ(x)+ϕ(x)3−ϕ(x)=ϑ (x), x∈R. (1.3)
Alternatively, (1.3) occurs as a quasi-steady, phase field model in the theory of solidification withϑrepresenting tem- perature [1,2]. One-dimensional phase transition modelling can also give rise to higher order non-convex variational problems [10,12], see [9] for a general discussion. However we confine ourselves to studying generalizations of (1.2) and the weak* limits inL∞perof its minimizers as→0 and two questions arise.
(1) How does one describe the weak* limits of minimizers of (1.2) and its generalizations?
(2) Is there is a “macroscopic” variational problem with minimizers that coincide with these weak* limits?
A common belief is that both issues can be resolved usingΓ-convergence theory, but this is not always the case.
Recall that a sequence of functionalsF:X→ [α,∞],α >−∞, defined on a metric spaceX, hasΓ-limitF:X→ [α,∞]if, for everyϕ0andϕ→ϕ0,
F (ϕ0)lim inf
→0 F(ϕ)
and there exists a sequenceϕ→ϕ0so that F (ϕ0)=lim
→0F(ϕ).
LetMFandMF be the sets of minimizers ofFandF respectively, and letLF be set of limit points of sequences x∈MFas→0. It is clear thatLF ⊂MF, but they are not equal in general. In the mesoscopic theory of phase transitions F would represent the total free energy and ϕ∈MF the corresponding stable equilibrium states. If a macroscopic theory is to be regarded as a limit of mesoscopic theory, then macroscopic stable equilibria should belong to LF and the Γ-limit F interpreted as an approximation to the macroscopic free energy. The validity of such an approach depends on the size ofMF\LF. If it is not empty, an additional selection principle is needed to identify the solutions of the macroscopic problem that are relevant to the mesoscopic theory (particularly ifLF is small inMF).
The following two examples related to our problem illustrate different relations betweenMF andLF. In both, ϑ is a given, continuousL-periodic function. ForL >0 fixed, letLpper orL∞perbe the space ofL-periodic functions with restrictions that arepth-power integrable or essentially bounded on(0, L).
The first problem is to minimize the Ginzburg–Landau energy functional (1.2):
J
ϕ∗
= min
ϕ∈L4perJ(ϕ), J(ϕ)= L 0
ϕ2+1 4
ϕ2−12
−ϕϑ
ds. (1.4)
The second is to minimize the scaled free energy functional J
ψ∗
= min
ψ∈L4per
J(ψ ), J(ψ )= L 0
√
ψ2+ 1 4√
ψ2−12
−ψ ϑ
ds. (1.5)
Note that ifϑ in (1.4) is replaced by √
ϑ, then (1.4) is transformed into (1.5), but there is an essential difference between the two as they stand. Let X=L4per with the weak topology in which bounded sets are metrizable. The Γ-limitJ ofJis a particular case of general theory, see, for example, [6],
J(ϕ)=:Γ-limJ(ϕ)= L 0
W∗∗(ϕ, ϑ ) ds, (1.6)
where, for fixedθ∈R,W∗∗(·, θ )denotes the convex envelope ofW (·, θ )in (1.1). SinceW is bounded below, the set of minimizers of (1.6) is non-empty and there are various possibilities. For example, (1.6) may have a unique minimizer. This happens if the set of zeros of ϑ is discrete, a case that was thoroughly investigated in [1,2] (see also [10] for generalizations and further discussion). Alternatively, (1.6) may have an infinite set of minimizers, including Young measure solutions, as happens whenϑ in (1.1) vanishes on some interval. Moreover, a minimizer may be discontinuous at every point of such an interval.
On the other hand, see [4, Chapter 6], [11], we have that
Γ-limJ(ψ )=J(ψ ):=
⎧⎪
⎪⎨
⎪⎪
⎩
℘0N(ψ )−L
0 ψ ϑ ds if|ψ| =1 almost everywhere on[0, L) andψis piecewise constant, +∞ otherwise,
(1.7)
whereN(ψ )is the number of discontinuities ofψin[0, L)and
℘0=2 1
−1
1 4
φ2−12
dφ=4
3. (1.8)
Clearly elements ofMJare piecewise constant functionsψwith a finite number of jumps at pointssi ∈ [0, L), and N(ψ )℘0is a weighted count of jumps per period. Roughly speaking the first term in (1.7) strives to minimize the number of jumps, but this process is controlled byϑ.
A difference betweenMJandMJis that elements ofMJhave a regularity property independent ofbecause the set
Φ(ψ): ψ∈MJ, ∈(0,1)
, whereΦ(φ)= φ
0
s2−1ds,
is bounded in the Sobolev space Wper1,1. By contrast, the minimizers of the Ginzburg–Landau energy J have no regularity independent of . An analysis of the relation betweenMJ andLJ is consequently more difficult and is our concern here. A special case of our results concernsLJ under the assumption that the limiting problem, (1.4) with =0, has at least one piecewise continuous minimizer, which is true when{ϑ=0} ∩ [0, L)has a finite number of connected components. The conclusion is that there exists a setE⊂(0,1)which is Lebesgue dense at 0 (see (5.1)) and the elements ofLJ which arise from sequences inEare piecewise continuous functions with the minimal possible weighted number of jumps (see (2.3)).
Similar non-convex variational problems arise in the classical nonlinear theory of elastic rods [7] (also [5]) as follows. Suppose thatx∈Rrepresents the positions of material points in an elastic rod with uniform densitywhen it is in equilibrium in the absence of external forces on a straight line. Suppose thatu(x, t ) denotes the position of the same point when the rod is deformed in the same straight line by a force fieldf (x, t ), and thatf and the stretch fielduxareL-periodic inx. If the elastic energy density is given in terms of the stretch and stretch gradient by the formula
2uxx2+W (ux),
whereWis convex (a one-phase material), thenusatisfies the wave equation ut t= −uxxxx+
W(ux)
x+f, ux(x+L, t )=ux(x, t ).
In the corresponding travelling-wave problem,f (x, t )=f (x−ct ),u=u(x−ct ), and the stretch variableϕ(s)= u(s), withϑ(s)=f (s), satisfies
−ϕ+∂φW
ϕ(s), ϑ (s), γ
=0, ϕ(s+L)=ϕ(s), for some constantγ, where
W (φ, θ, γ )=W (φ)−c2φ2/2+(θ+γ )φ.
Note that, for largec,Wis non-convex inφeven thoughW is convex. The external functions areϑandγ. If instead of remaining straight, a compressible rod with bending stiffness is prescribed to lie on a given periodic curve in a vertical plane, the stored energy is again non-convex in the stretch, and its curvature and height are two additional external functions in the travelling-wave equation. This observation in [13] explains our interest in periodic boundary conditions and multiple external functions.
Main results. LetHper1 be the Hilbert space ofL-periodic continuous functions onRwith restrictions inW1,2(0, L).
In what follows, d∈N,ϕ:R→(0,∞),ϑ:R→Rd andA:R→RareL-periodic functions, while(φ, θ, A, )∈ Rd+3. We study minimization problems for generalizations of (1.2) of the following type. LetB:R→Rbe a smooth increasing function and, for a setE⊂(0,1]with a limit point at 0, let{ϑ: ∈E}be bounded in(Hper1 )d. Then the problem is to findϕ∈Hper1 such that
J(ϕ)= inf
ϕ∈Hper1
J(ϕ)=:E(), >0, (1.9)
where J(ϕ)=
L 0
2
B(ϕ)2
+W (ϕ, ϑ)
ds, ϕ∈Hper1 . (1.10)
For the Ginzburg–Landau functional (1.2),B(φ)=φandW is given by (1.1), whereas the scaled Ginzburg–Landau functional (1.5) corresponds toW in (1.1) withB(φ)=√
2φ andϑ =√
ϑ. A solutionϕ to problem (1.9), for >0, satisfies the Euler–Lagrange equation
B ϕ(s)
B(ϕ)(s)−∂φW
ϕ(s), ϑ(s)
=0 onR. (1.11)
It follows from the maximum principle and (H1) below that solutionsϕof (1.11) haveϕL∞per+ ϕ−1L∞perbounded by a constant independent ofwhenϑ(Hper1 )d ME,∈E. Our goal is to give an explicit upper bound for the number of jumps ofϕ, a weak* limit inL∞per of minimizersϕk, for a “almost all sequencesk tending to zero” in a sense that will be made precise. Before summarizing our result we discuss the essential properties of the potentialsW and exterior functionsϑunder consideration.
Conditions (H1–3), which are formulated precisely and illustrated by a figure in Section 2, describe a function W (·, θ )which is either a single or double well potential depending onθ∈Rd. (H1) is a coercivity condition that W (φ, θ )→ ∞asφ→α, βwhereα, β∈ [−∞,∞]. To avoid repetition, we consider only the mixed caseα >−∞, β= ∞and takeα=0. (H2) says that, forθ∈Rd, the functionW (·, θ )has no more than three critical points counting multiplicity. (H3) implies that the distance between local or global minimizers φ±(θ )ofW (·, θ )is bounded below by a positive constant. Therefore the only possible bifurcations of critical points ofW (·, θ )are from inflection points to local-minimum–local-maximum pairs. Thus W is of a type that arises in first order phase transition problems.
Hypotheses (H1–3) induce a decomposition ofRd into disjoint sets corresponding to the number and type of critical points of W. The most important is G03, which consists of thoseθ∈Rd for whichW (·, θ )has two distinct global minima corresponding to two distinct coexisting stable phases (see Fig. 1).
(H4) in Section 4 means, roughly speaking, that, for someϑ∈(Hper1 )dandΘ0, Θ1<∞, lim sup
→0 ϑ−ϑ(L1per)d/√
=Θ0 and lim sup
→0
√∂ϑ(L1per)d =Θ1.
Fig. 1. Graphs ofW (·, θ )depending onθ∈Rd.
(H5) is the requirement that there exists a piecewise regular solutions to the primary variational problem:
L 0
W ϕ∗, ϑ
ds= inf
ϕ∈L∞per
L 0
W (ϕ, ϑ ) ds=:E(0). (1.12)
It holds if the critical setG30(ϑ )= {s: ϑ (s)∈G03}has a finite number of connected components.
By definition, a piecewise regular minimizer of (1.12) has finitely many jumps per period at pointssiwithϑ (si)∈ G03and to each such minimizerϕ∗we can assign a weighted number of jumpsW(ϕ∗), which is commensurate with, but not necessarily equal to, the actual number of jumps per period,
W ϕ∗
=
si∈[0,L)
℘ ϑ (si)
.
Here℘:G03→Ris given by formula (2.3a), which is similar to (1.8) in that special case. LetWminbe the infimum of the weighted number of jumps of all piecewise regular minimizers. We will see thatWminis attained. LetN∗<∞ be the maximum of the actual number of jumps of all piecewise regular minimizers with minimal weighted number of jumps.
(H6) in Section 5 is the requirement that Edefined by (1.9) is locally absolutely continuous on (0,1). (In Ap- pendix A we discuss (H6) which is often satisfied trivially; when ϑ is independent of it follows becauseE is concave.)
The following, Theorem 5.6, is one of the main results of the paper.
Minimal number of jumps principle.Suppose that(H1–6)hold with Θ0=Θ1=0 in(H4). Then, for anyδ >0 there is a setEδ⊂(0,1]which is Lebesgue dense at0, see(5.1), with the following property. If a sequence{ϕn:Eδ n→0}of minimizers converges weak* inL∞pertoϕ, thenϕis a piecewise regular minimizer of (1.12)with weighted number of jumpsW(ϕ)Wmin+δand actual number of jumpsN(ϕ)N∗.
Method. As mentioned above, the passage to the limit in (1.9) is difficult because its solutions lack regular- ity properties independent of . To cope, we use the fact that the Euler–Lagrange equation (1.11) implies that A(s)=∂θW (ϕ(s), ϑ(s))ϑ(s)where the adiabatic variableA(see [3]) is defined by
A(s):=W
ϕ(s), ϑ(s)
− 2
B(ϕ)(s)2
.
Combining this observation with the remark following (1.11), we conclude that solutions of (1.11) satisfy the estimates M−1ϕ(s), ϕ(s)M fors∈R, AHper1 M,
in which the constantMdepends only onW and supϑ(Hper1 )d. In particular, if periodic solutions of (1.11) converge weak* inL∞pertoϕ, then, after passing to a subsequence, we can assume that(A, ϑ)converges weakly in(Hper1 )d+1 to some(A, ϑ ).The main idea is to obtain a representation for weak* limits of solutions to(1.11)in terms ofAandϑ.
To this end we introduce the averaging operator
Ψ[ϕ](s)= 1 h()√
h()√
/2
−h()√ /2
ϕ(s+t ) dt≡ 1 h()
h()/2
−h()/2
v(y, s) dy,
whereh()=ln|ln|for∈(0,1/e]andv(y, s)=ϕ(s+√
y). By Lemma 3.7, the functionvcan be approximated with an accuracy ofp,p∈(0,1/4), by a solution to the autonomous ordinary differential equation
1 2
B(u)(y)2
−W
u(y, s), ϑ(s)
+A(s)=0, y∈
−h(), h() .
This choice ofhis more or less essential for the existence of such an approximation and leads to the representation Ψ[ϕ](s)= 1
h()
h()/2
−h()/2
u(y, s) dy+O p
.
In Section 2.3 we show thatΨ (s)=lim→0Ψ[ϕ](s)exists for everys∈R, and that the weak* limit is continuous on[0, L)except at points of the critical set
s∈F(A, ϑ ):=
s∈ [0, L): ϑ (s)∈G03andA(s)=inf
φ W
φ, ϑ (s) .
Since the weak* limitϕ ofϕ equalsΨ almost everywhere, we can identifyϕwithΨ and thus replace the analysis of solutions to (1.11) with an analysis of the corresponding sequenceΨ[ϕ]. The advantage gained is thatΨ[ϕ] converges pointwise to ϕ and the limiting behaviour of Ψ[ϕ],∈E, nears∈F(A, ϑ ) can be characterized by what we call the oscillation defect. This oscillation defect, osc-defE(s), is defined by (3.3) in terms of the family {ϕ: ∈E}and not in terms of its weak* limit points. Neverthelessϕis continuous atswhen osc-defE(s)=0.
The relation between the oscillation defect and the limiting behaviour of the energy functional is established in Theorem 3.3 which is the first significant result of the paper. If a sequence of periodic solutionsϕ,∈E, of Eq. (1.11) converges weak* inL∞perto a functionϕ, it follows that
si
℘ ϑ (si)
lim inf
E→0
√ 2
L 0
B
ϕ(s)2
ds, (1.13)
where the sum is taken over the set of pointssi∈ [0, L)∩F(A, ϑ )with non-zero oscillation defect. This theorem holds for solutions of the Euler–Lagrange equation, not only for minimizers ofJ. However, when applied to a sequence of minimizers, Theorem 3.3 implies that if the limit on the right in (1.13) is finite, thenϕis a piecewise regular minimizer and
W(ϕ)lim inf
E→0
√ 2
L 0
B
ϕ(s)2
ds.
The second significant result of the paper is Theorem 4.2 which can be regarded as the “inverse” of Theorem 3.3.
In the simplest case, when Θ0=0 in (H4), if the primary variational problem has at least one piecewise regular minimizerϕ, then Theorem 4.2 says that
lim sup
→0
E()−E(0)
√ 2W(ϕ), whereE()=J(ϕ).
Theorems 3.3 and 4.2 give estimates, from below and above, of the weighted number of jumps of a weak* limit of minimizers, in terms of the total energyE()=J(ϕ)and the interfacial energy
B(ϕ)= 2 L 0
B
ϕ(s)2
ds.
Notice that whileE()depends only on, the interfacial energy depends on the minimizerϕ, since (1.9) may have many solutions. The relation between total and interfacial energies is important in the general theory of singularly perturbed variational problems. For example, for minimizers of the scaled Ginzburg–Landau functional (1.5), the in- terfacial and bulk energy contributions to the total energy are approximately equal whenis small. This is an example of the principle of equipartition of energy which plays a crucial role in the analysis of (1.5) and its multidimensional generalizations. However it is special, and for more general problems we need a different technique.
Our approach is based on Struwe’s monotonicity method [14, Chapter II, Section 9]. Application of this method to problem (1.9), in the simplest case whenΘ1=0 in (H4), leads to the inequality, see Lemma 5.4,
B()E()+Λ1(), √
Λ1()→0 as→0, (1.14)
in whichB()=supϕ−1B(ϕ). In its turn, this lead to a criterion for estimating the number of jumps of a weak*
limit of minimizers whenΘ0=Θ1=0 in (H4), namely, forδ >0, lim sup
k→∞
√kE(k)Wmin+δ, k→0 ⇒ W(ϕ)Wmin+δ.
That this criterion is valid for sequences in a set which is Lebesgue dense at zero is the content of Lemma 5.5. Its proof, which depends on the energy estimate (1.14), might be regarded as the main insight in this paper.
2. Minimization problems
Theprimary problemis
ϕ∈infL∞per
L 0
W
ϕ(s), ϑ (s)
ds, (2.1)
whereϑ∈(Hper1 )dis given andW satisfies the following hypotheses which are illustrated in Fig. 1.
(H1) W∈C3((0,∞)×Rd),
W (φ, θ )→ ∞, ∂φW (φ, θ )→ ∞ asφ→0,∞, uniformly forθin compact subsets ofRd.
(H2) For every θ∈Rd, the function W (·, θ )has no more than three, counting multiplicity, critical points each of which is one of the following types: a stable global minimizerφs(θ )which is non-degenerate, an unstable local maximizerφu(θ )which is non-degenerate, a metastable local minimizerφm(θ )which is non-degenerate, or a critical inflection pointφum(θ ). With this hypothesis we can writeRd as the union of three disjoint sets,G1, G2,G3as follows.
G1is the set ofθ∈Rdsuch thatW (·, θ )has only one critical point – the non-degenerate global minimizerφs(θ );
G2is the set ofθ∈Rdsuch thatW (·, θ )has two critical points – the non-degenerate global minimizerφs(θ ) and a critical inflection pointφum(θ )withW (φs(θ ), θ ) < W (φum(θ ), θ ).
G3is the set ofθ∈Rdsuch thatW (·, θ )has three critical points – a non-degenerate global minimizerφs(θ ), a non-degenerate local maximizerφu(θ ), and a non-degenerate local minimizerφm(θ )with
W
φs(θ ), θ
W
φm(θ ), θ
< W
φu(θ ), θ .
In its turn,G3contains the setG03ofθsuch thatW (·, θ )has two non-degenerate global minimizers,φs−(θ ) <
φs+(θ ), and one non-degenerate local maximizerφu(θ ). In other words, a metastable state has become stable andφmcoincides with one of the pointsφs±.
Let F⊂Rd+1be defined by F =
(A, θ )∈Rd+1: A=W
φs±(θ ), θ
whereθ∈G03 . (H3) Forθin a compact set there existsc >0 such that
φu(θ )−φs(θ )c, φm(θ )−φs(θ )c, φs+(θ )−φ−s (θ )c, W
φu(θ ), θ
−W
φs(θ ), θ
c forθ∈G3\G03, W
φu(θ ), θ
−W
φ±s (θ ), θ
c forθ∈G03, W
φum(θ ), θ
−W
φs(θ ), θ
c forθ∈G2,
∂φ2W
φs(θ ), θ
c forθ∈R\G03,
∂φ2W
φs±(θ ), θ
c forθ∈G03,
∂φ2W
φu(θ ), θ
<0 forθ∈G3,
∂φ2W
φm(θ ), θ
>0 forθ∈G3\G03.
Remark 2.1.IfW is a real-analytic function, thenG03is a real-analytic variety. More generally, except in degenerate situations, possibly caused by symmetries, the setG03is typically the closure of a union of manifolds with dimensions d−1, or smaller. In particular, whend=1,G03is often a discrete set of points.
To proceed we need some additional notation. Let(A, ϑ )∈(Hper1 )d+1be given. WithF,Gi,i=1,2,3, andG03 defined in terms ofWby (H2), let
Gi(ϑ )=
s∈R: ϑ (s)∈Gi
, i=1,2,3, G30(ϑ )=
s∈R: ϑ (s)∈G03 , F(A, ϑ )=
s∈R:
A(s), ϑ (s)
∈F
⊂G30(ϑ ).
Since the prescribed functionsϑare continuous, the setsG1(ϑ ),G3(ϑ )are open andG2(ϑ ),G30(ϑ )are closed; so the setsR\G30(ϑ )andG3(ϑ )\G30(ϑ )are open. SinceG30(ϑ )= {s: ϑ (s)∈G03}, it follows from Remark 2.1 that ifϑ is real-analytic, thenG30is a one-dimensional real-analytic variety. If it is not the whole spaceRit is a discrete set of points.
2.1. Minimizers of primary problem
For anyϑ∈(Hper1 )d,ϕis a minimizer of the primary problem (2.1) if ϕ(t )∈
{φs(ϑ (t ))}, t∈R\G30(ϑ ) {φ+s (ϑ (t )), φ−s (ϑ (t ))}, t∈G30(ϑ )
almost everywhere. Minimizers always exist but they may not be unique and often cannot be continuous. By (H3), the size,φ+s (θ )−φs−(θ ),θ∈G03, of possible jumps of minimizers is bounded below by a positive constant. We are interested in minimizers that are piecewise regular in the following sense. Say that{Sn}is adiscrete periodicsequence ifSn+1−Sn∈ [a, b]for somea, b >0 and alln, and{Sn: n∈Z} = {Sn+L: n∈Z}.
Piecewise regular minimizers. Forϑ∈(Hper1 )d, a minimizerϕ∈L∞per of the primary problem (2.1) is said to be piecewise regularif there exists a discrete periodic sequence{Sn} ⊂G30(ϑ )with the following properties. For everyn, the restrictionϕ|(Sn,Sn+1)∈W1,2(Sn, Sn+1),
s→limSn±0ϕ(s)∈ φs−
ϑ (Sn) , φs+
ϑ (Sn)
and the functionϕhas a jump atSnwith magnitudeφs+(ϑ (Sn))−φs−(ϑ (Sn)).
For a piecewise regular minimizer theactual number of jumps per periodis
N(ϕ):=cardQ(ϕ) <∞ whereQ(ϕ)= [0, L)∩ {Sn: n∈N}. (2.2) It will be convenient to estimate the number of jumps of a piecewise regular minimizer by assigning to each a number other than unity. LetB:(0,∞)→Rbe the smooth strictly increasing function in (1.10) and, forθ∈G03, let
℘ (θ )= 1
√2
φs+(θ ) φs−(θ )
B(φ)
W (φ, θ )−A dφ whereA=W
φs±(θ ), θ
. (2.3a)
Then theweighted number of jumps ofϕper periodis W(ϕ)=
s∈Q(ϕ)
℘ ϑ (s)
<∞. (2.3b)
A piecewise regular minimizerϕhas aminimal weighted number of jumpsif W(ϕ)=inf
˜
ϕ W(ϕ)˜ =:Wmin, (2.3c)
where the infimum is taken over all piecewise regular minimizersϕ. Lemma 2.3 asserts that˜ Wminis attained.
Remark 2.2.Note that, forθ∈G03,
℘ (θ )=1 2 T
0
B(u)2
dt
ifusatisfies A=W
u(t ), θ
−1 2
B(u)2
, u(0)=φs−(θ ), u(T )=φs+(θ ),
whereA=W (φs±(θ ), θ ). Note also that, for a givenB,℘ (ϑ (s)),s∈G30(ϑ ), is bounded below by a positive con- stant k(M), for any ϑ with ϑ(Hper1 )d M. To see this, suppose (A, θ )∈F andθcLM, where ϑ (s) cLϑ(Hper1 )d cLM. Then, since{φ−(θ ), φ+(θ ): θcLM}is bounded and sinceW∈C3, by Taylor’s theorem,
W (φ, θ )−A−1 2∂φ2W
φ−(θ ), θ
φ−φ−(θ )2
Cφ−φ−(θ )3
for φ−(θ )φφ+(θ ), where C depends on M andW. Therefore there exists φ1>0, independent of θ with θcLM, such thatφ−(θ )+φ1φ+(θ )and
W (φ, θ )−A
√c 2
φ−φ−(θ )
forφ∈(φ−(θ ), φ−(θ )+φ1), wherecis as in (H3). Therefore
φ+(θ ) φ−(θ )
B(φ)
W (φ, θ )−A dφ
φ−(θ ) +φ1
φ−(θ )
B(φ)
W (φ, θ )−A dφC,
whereCis independent ofθwithθcLM.
We now record an observation that will be useful later.
Lemma 2.3.Suppose that(H1–3)hold and that there exists at least one piecewise regular minimizer. Then there exists a piecewise regular minimizer with a minimal weighted number of jumps. Let
N∗=max
ϕminN(ϕmin),
where the maximum of the actual number of jumps is taken over all piecewise regular minimizers with minimal weighted number of jumps. Then there existsδ >0such thatN(ϕ)N∗ifW(ϕ)Wmin+δ.
Proof. LetW(ϕk)Wmin+1/kfor a sequence of piecewise regular minimizers. Denote bySn(k)∈G30(ϑ )the points of discontinuities ofϕk. Since, by Remark 2.2, the quantities℘ (ϑ (S(k)n ))are uniformly bounded away from 0, the numbersN(ϕk)of discontinuities ofϕkform a finite set of integers. Therefore, after passing to a subsequence, we can assume thatN(ϕk)=N for allk. Since the setSn(k)is invariant with respect to the shifts→s+L, we can assume that, for everyn, the sequence{Sn(k)}converges ask→ ∞, toSnsay, where{Sn}is invariant with respect to the shift andSn∈G30(ϑ ).
The piecewise regular minimizersϕk are periodic and continuous ats∈(Sn(k)−1, Sn(k))and hence coincide with one ofφs(ϑ (s)),φs±(ϑ (s))there. However, from (H3), each of the critical pointsφs(ϑ (s0)),φs±(ϑ (s0))is non-degenerate, for eachs0∈(Sn(k)−1, Sn(k)). Hence there is a unique continuous branch of critical pointsϕ(t )˜ defined in a vicinity ofs0 by the system
∂φW
˜
ϕ(t ), ϑ (t )
=0, ϕ(s˜ 0)=ϕk(s0).
Therefore ϕk coincides with ϕ˜ in a neighbourhood of each point s0∈(Sn(k)−1, Sn(k)). Moreover the non-degeneracy condition (H3) implies that
ϕk(s0)=ϕ˜(s0)=
∂φθ2 W (ϕk(s0), ϑ (s0))·ϑ(s0)
∂φ2W (ϕk(s0), ϑ (s0))
c(M)ϑ(s0).
Thus, sinceϑis square-integrable on(0, L),
n∈Z
(0,L)∩(Sn(k)−1,Sn(k))
ϕk(s)2ds
=:
L 0
ϕk2
dsc.
We can therefore assume thatϕkconverges uniformly to some continuous functionϕ on each compact subset of the interval(Sn−1, Sn)and that
n∈Z
(0,L)∩(Sn−1,Sn)
ϕ(s)2ds
=:
L 0
ϕ2
dsc,
whereϕk converges weak* inL∞per toϕ. Since the set of relaxed minimizers (Section 2.2) is weakly closed,ϕ is a relaxed minimizer. Hence it is a piecewise regular minimizer of (2.1).
Now chooseη >0 so that the pointsηandL+ηdo not lie in the sequence{Sn}. Then W(ϕ)=
Sn∈(η,L+η)
℘ ϑ (Sn)
,
and for all sufficiently largek, W(ϕk)=
Sn(k)∈(η,L+η)
℘ ϑ
Sn(k) .
In the limiting process some points S(k)n can be lost: more precisely, for each Sn there is a finite set P (n) of cardinalityp(n)1 so that
Si(k)→Sn ask→ ∞for alli∈P (n).
In other words we can split the set of sequences{Sn(k)} ∩(η, L+η)into disjoint clusters of cardinalityp(n)such that all sequences in a cluster converge to one pointSn. Obviously
Sn∈(η,L+η)
p(n)=N. Since, fori∈P (n),
℘ ϑ
Si(k)
→℘ ϑ (Sn)
ask→ ∞for alli∈P (n), we obtain that
Sn∈(η,L+η)
p(n)℘
ϑ (Sn)
= lim
k→∞
Sn(k)∈(η,L+η)
℘ ϑ
S(k)n
=Wmin
Sn∈(η,L+η)
℘ ϑ (Sn)
.
We conclude thatϕ is a piecewise regular minimizer with a minimal weighted number of jumps, thatp(n)=1 for alln, and thatN∗is finite.
Suppose that for noδ >0 doesW(ϕ)Wmin+δ imply thatN(ϕ)N∗.Then there is a sequence of piecewise regular minimizers{ϕk}with
W(ϕk)Wmin+1/k, and N(ϕk)N∗+1.
Now we repeat the preceding argument to obtain, in the weak* limit ofϕk, a piecewise regular minimizerϕ with minimal weighted number of jumps andp(n)=1 for alln. Sincep(n)=1 for alln,N(ϕ)N∗+1. ButN(ϕ)N∗, becauseϕ is a piecewise regular minimizer with minimal weighted number of jumps. This contradiction proves the result. 2
Generalized solutions. For givenϑ∈(Hper1 )d, a pair(A, ϕ)∈Hper1 ×L∞peris called ageneralized solutionof
∂φW (ϕ, ϑ )=0 (2.4)
if
(GS)(i): for allt∈R\G30(ϑ ), ϕ(t )∈
φι ϑ (t )
, ι=s, m, u, um
and A(t )=W
ϕ(t ), ϑ (t )
; (GS)(ii): for allt∈G30(ϑ ),
A(t )∈ W
φ±s ϑ (t )
, ϑ (t ) , W
φu
ϑ (t ) , ϑ (t )
;
(GS)(iii):
ϕ(t )=φu ϑ (t )
fort∈G30(ϑ )\F(A, ϑ ), ϕ(t )=k(t )φs−
ϑ (t ) +
1−k(t ) φs+
ϑ (t )
fort∈F(A, ϑ ), wherek:F(A, ϑ )→ [0,1]is measurable.
The continuity ofArestricts the behaviour ofϕas follows.
Lemma 2.4.Forϑ∈(Hper1 )d, let(A, ϕ)be a generalized solution of(2.4). Thenϕis continuous onR\F(A, ϑ ).
Proof. To simplify notation, letϕι(t )=φι(ϑ (t )),ι∈ {s, u, m, um},t∈R. Let s0∈G1(ϑ ), which is open sinceG1 is open andϑ is continuous. Thenϕs(t )is the unique critical point ofW (·, ϑ (t )), for t in a neighbourhood of s0. Hence, by (GS)(i), ϕ(t )=ϕs(t )in this neighbourhood ofs0. However, by the implicit function theorem,ϕs(t )is a continuous function oftin a neighbourhood ofs0, sinceϕs(s0)is a non-degenerate critical point ofW (·, ϑ (s0)). Thus ϕ is continuous onG1(ϑ ).
Lets0∈G3(ϑ )\G30(ϑ ), an open set. It follows from (GS)(i) that, fortin a neighbourhood ofs0,ϕ(t )∈ {ϕι(t ): ι= s, u, m}andA(t )=W (ϕ(t ), ϑ (t )). Since, by (H3),
∂φW
ϕι(s0), ϑ (s0)
=0 and ∂φ2W
ϕι(s0), ϑ (s0)
=0,
if follows from the implicit function theorem that ϕι depends continuously on t in a neighbourhood of s0. Since s0∈G3(ϑ )\G30(ϑ ), there exists >0 such that, for all|t−s0|sufficiently small,
W
ϕs(t ), ϑ (t )
+W
ϕm(t ), ϑ (t )
W
ϕu(t ), ϑ (t )
−.
Sinceϕ(t )∈ {ϕι(t ): ι=s, u, m}andA(t )=W (ϕ(t ), ϑ (t )), and sinceAis continuous, it follows thatϕ(t )=ϕι(t ), t in a neighbourhood ofs0, forone choice of ι. Thusϕis continuous ats0for anys0∈G3(ϑ )\G30(ϑ ).
Lets0∈G30(ϑ )\F(A, ϑ ). Since, by (H3),
∂φW
ϕu(s0), ϑ (s0)
=0 and ∂φ2W
ϕu(s0), ϑ (s0)
=0,
by the implicit function theorem,ϕu(t )depends continuously ontin a neighbourhood ofs0. Also, by (H3),
∂φW
ϕs±(s0), ϑ (s0)
=0 and ∂φ2W
ϕs±(s0), ϑ (s0)
=0.
It follows that there are continuous functionsϕ˜±, defined fort in a neighbourhood ofs0in the open setG3(ϑ )with, for some >0,
˜
ϕ±(s0)=ϕs±(s0), ∂φW
˜
ϕ±(t ), ϑ (t )
=0,
˜
ϕ−(t )+ϕu(t )ϕ˜+(t )−, W
˜
ϕ±(t ), ϑ (t )
+W
ϕu(t ), ϑ (t ) .
Therefore, fortin this neighbourhood ofs0inG3(ϑ ),{ϕu(t ),ϕ˜±(t )}are the three critical points ofW (·, ϑ (t ))and, for all sucht,
A(t )∈ W
ϕu(t ), ϑ (t ) , W
˜
ϕ±(t ), ϑ (t ) . However, by (GS)(iii),
ϕ(s0)=ϕu(s0), A(s0)=W
ϕu(s0), ϑ (s0) ,
whereAis continuous ats0. It follows thatϕ(t )=ϕu(t )in a neighbourhood ofs0and the continuity ofϕat a point ofG30(ϑ )\F(A, ϑ )follows.
Next assume that s0∈G2(ϑ ). Then the functionW (·, ϑ (s0))has exactly two critical points ϕs(s0)andϕum(s0) with
W
ϕs(s0), ϑ (s0)
< W
ϕum(s0), ϑ (s0)
. (2.5)
MoreoverAis continuous and A(s0)∈
W
ϕum(s0), ϑ (s0) , W
ϕs(s0), ϑ (s0) .
