Ground states and critical points for generalized frenkel-kontorova models in Z^d

GROUND STATES AND CRITICAL POINTS FOR GENERALIZED FRENKEL-KONTOROVA MODELS IN Zd RAFAEL DE LA LLAVE AND ENRICO VALDINOCI Abstract. We consider a multidimensional model of Frenkel-Kontorova type but we allow nonnearest neighbor interactions, which satisfy some weak version of ferromagnetism. For every possible frequency vector, we show that there are quasiperiodic ground states which enjoy further geometric properties. The ground states we produce are either bigger or smaller than their integer translates. They are at bounded distance from the plane wave with the given frequency. The comparison property above implies that the ground states and the translations are organized into laminations. If these leave a gap, we show that there are critical points inside the gap which also satisfy the comparison properties. In particular, given any frequency, we show that either there is a continuous parameter of ground states or there is a ground state and another critical point which is not a ground state. This is a higher dimensional analogue of the criterion of non-existence of invariant circles if and only if there is a positive Peierls-Nabarro barrier.

Introduction This paper deals with existence and multiplicity results for periodic and quasiperiodic equilibria of generalized Frenkel-Kontorova models. The equilibrium solutions we construct have further geometric properties, such as satisfying comparison properties and lying at a bounded distance from a linear function. The results presented here generalize some well known results in AubryMather theory. We allow the lattices to be of any dimension and interactions which are not nearest neighbor. Degenerate twists are also taken into account. Our Theorem 1 constructing ground states generalizes the main result in [A83, AD83, M82], whereas our main result, Theorem 2 constructing solutions which are not minimizers is an extension of the main result of [M86] to higher dimensions. Results similar to Theorem 1 for minimizers are in [B89, CLl98], but our proof is different from the ones presented in the literature and may be also applied to very general lattices and group actions (see [LlV06a, LlV06b] for a more general framework). The main difference with [B89] is that we consider first minimizers with periodic boundary conditions and then we pass to the limit. This allows us to ignore the problems given by the boundary terms. In [LlV06a, LlV06b] we fully eploit the advantage of this procedure, since we give there explicit applications to the Bethe lattice or other situations where the volume of a ball is comparable to its boundary. 2000 Mathematics Subject Classification. 82B20, 70H12, 37A60, 82C20, 82C22, 49J35. Key words and phrases. Minimal and nonminimal equilibrium solutions in statistical mechanics, variational methods, Aubry-Mather theory. The work of RdlL was supported by NSF grants. The work of EV was supported by MIUR Variational Methods and Nonlinear Differential Equations. 1

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RAFAEL DE LA LLAVE AND ENRICO VALDINOCI

One novelty of our variational approach is that we produce well ordered minimizers by considering an infimal minimizer. This method thus detects order properties from the comparison principle, but is independent of a maximum principle. We present some natural examples in Frenkel-Kontorova models where our theorem applies but, nevertheless, there are also non-well ordered minimizers (we present some conditions that exclude this behavior). The existence of non-well orderered minimizers is unavoidable also in models taking discrete values (see [CLl05]). The main result of this paper is the study nonminimal critical points of the energy in Theorem 2. We show that if the set of minimizers does not include a whole circle, then, there are critical points which are not minimizers. The technique of approximating by periodic problems does not work in this case. Indeed, problems of multiplicity are hard to approximate by other ones because, in passing to the limit, we need to ensure that the limit is indeed different. We use a gradient flow technique for a renormalized energy to overcome this difficulty. One technical advantage of the present approach is that we do not need to assume the full twist condition. We can allow the twist condition to degenerate. In the PDE case, this would correspond to degenerate ellipticity and indeed, we present in this paper a discrete analogue of the p-Laplace operator. For the theory of existence of minimizers, this is not a great advantage since simpler methods such as approximating by non-degenerate problems will also produce the result. On the other hand, for the multiplicity results, approximating by other problems has to consider the difficulty that the multiplicty could drop in the limit, so that the present approach has some advantages. Arguments along the same line as the ones presented here apply to PDEs too but, then, one has to pay attention to regularity, existence etc., which are easier issues in our case (see [LlV05]). In this other context, where uniqueness and maximum principles are sometimes more problematic, the present line of argument has some advantages. The Aubry-Mather theory had its origin in [A83, AD83, M82] and important precedents may be considered the works of [M24, H32, P74, P79]. A deep connection with the theory of viscosity solutions in PDEs has been developed too (see, e.g., [F97]). The techniques we use here will be a combination of the dynamical system and the PDE approaches. In this paper, we will not consider the most general case possible, but rather we will consider in detail some models, which illustrate the difficulties (including long range interactions, degenerate twist, higher dimensionality). We call attention to the fact that only a few properties of the model are used. More general theories can be found in [LlV06a, LlV06b]. The situation we consider deals with the lattice Zd . At every site, we have a particle whose state can be described by one real parameter. A configuration is an assignation of a state to each of the sites. That is, a configuration is ui ∈ R, with i ∈ Zd . Given a configuration, we consider the formal variational principle (1)

S(u) :=

X

i,j∈Zd

aij |ui − uj − cij |pij +

X

V (ui ) .

i∈Zd

We assume that pij = pji = p¯(i − j), cij = cji = c¯(i − j), aij = aji := a ¯(i − j), so that the system is invariant under translations in the lattice. We also assume

GENERALIZED FRENKEL-KONTOROVA MODELS

3

that 2 ≤ p¯ ≤ Λ, −Λ ≤ c¯ ≤ Λ, 0 ≤ a ¯ ≤ Λ and a ¯(ξ) = 0 if |ξ| ∈ {0} ∪ [R, +∞), for suitable Λ, R > 0. We also suppose that V is smooth and that V (r + 1) = V (r) for any r ∈ R. The assumption that a ¯(ξ) vanishes for large |ξ| means that the interaction is finite range. The assumption of periodicity of V is well justified if the order parameter at each of the sites is an angle. This happens, for example, if it is a spin, or if it is the displacement along a certain direction from a substratum, or if it is an internal phase. In these situations, the Physical meaning of S is an energy. In the one dimensional case, problems of the same form appear in twist mappings of a cylinder. The meaning of S is an energy in this case too and the periodicity condition is a reflection of the periodicity of the angle coordinate. A discussion of how one can extend our results by considering infinite range interactions as the limit of finite range ones is in [LlV06a]. Again, we note that the passage to the limits of minimizers is still a minimizer, but that the multiplicty results do not pass automatically to the limit. We also refer to those papers for some more general models. The assumptions on a ¯ and p¯ may be interpreted as a twist condition (in the d = 1 dynamical system framework) or as an ellipticity condition (in the PDE continuous analogue). To see this, one may look at the concrete case in which a ¯(1) = a ¯(−1) := 1/(2ℓ), a ¯(r) = 0 if r 6= ±1, p¯ := 2ℓ, with ℓ ∈ N, ℓ ≥ 1, cij := 0, d := 1 and R := 3/2, which, via the transformations θi := ui and Ii := ui − ui−1 reduces the formal critical points of S to the following generalization of the Standard Map:   (I, θ) 7→ I(I, θ), Θ(I, θ) , with

I(I, θ) :=

q I 2ℓ−1 + V ′ (θ) ,

2ℓ−1

Θ(I, θ) = θ + I(I, θ) .

The twist condition is then dynamically interpreted as ∂I I ≥ 0 (where defined). On the other hand, such system may also be seen as a discretization of the functional Z 1 |∇u|2ℓ + V (u) dx , 2ℓ which is of course related to elliptic PDEs of 2ℓ-Laplacian-type (see [PV05] for related results in this direction). The analogy between the dynamical system and the PDE case reflects into the fact that for ℓ = 1 the twist condition does not degenerate, namely ∂I I > 0, and the PDE is driven by the standard Laplacian. The degeneration, which corresponds to the case pij > 2, then reflects into the flatness of the twist condition in the dynamical system framework and in the degeneracy of the elliptic operator at the PDE level. Note that the dynamical systems analogy complicates when R ≫ 1, that is, when more and more nonnearest-neighbor interactions are allowed in a ¯, since more degrees of freedom arise, and it breaks when d > 1, since no map is then associated to the critical points of the system in a natural way. From the physical point of view, such a model may be seen as a system of (possibly nonlinear and nonhomogeneus) interacting springs in a background periodic potential. Another possible physical interpretation is spin waves. It is a generalization of the so called Frenkel-Kontorova model and the classical Standard Map of [C79] is included as a particular case. Notice that we allow lattices of higher dimensions and the interactions to be nonnearest neighbor, which is very natural

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for spin-waves. The form of the interaction – chosen here to be (possibly nonlinear) harmonic oscillators – is not too relevant. The only properties that matter are some convexity properties (configurations tend to be aligned), and the fact that the interaction is coercive (it penalizes close particles for becoming very different, see [LlV06a]). The techniques we expose here are very general (for instance, the interaction does not need to be power law, the finite range condition may be weakened, the lattice may have a nonEuclidean geometry, etc.): full details of such extensions are given in [LlV06a, LlV06b], while we focus here on a simpler, concrete model. The following definitions are standard in the calculus of variations. We say that a configuration ui is a ground state if X X V (ui ) aij |ui − uj − cij |pij + i∈Zd |i|≤K

i,j∈Zd min{|i|,|j|}≤K

(2)

X

≤

X

aij |vi − vj − cij |pij +

V (vi ) ,

i∈Zd |i|≤K

i,j∈Zd min{|i|,|j|}≤K

for any configuration vi such that vj = uj as long as |j| > K, for any given K > 0. It is easily seen that ui is a ground state if and only if X X aij |ui − uj − cij |pij + V (ui ) i∈Zd ∩Q

i,j∈Zd {i,j}∩Q6=∅

≤

X

i,j∈Zd {i,j}∩Q6=∅

aij |vi − vj − cij |pij +

X

V (vi )

i∈Zd ∩Q

for any vi such that vj = uj outside any bounded set Q ⊂ Rd . We say that ui is a critical point if i h X aij pij (ui − uj ) |ui − uj − cij |pij −2 + |uj − ui − cij |pij −2 + V ′ (ui ) j∈Zd

(3)

=

X

j∈Zd

i h aij pij cij |ui − uj − cij |pij −2 − |uj − ui − cij |pij −2

for any i ∈ Zd (of course, the case of ideal springs pij := 2 and cij := 0 is the easiest to visualize). Note that the series in (2) and (3) are just finite sums. Roughly, (2) says that, for a ground state, any change of a finite number of sites increases the energy (note that we sum the change of energy in the terms which are affected). Equation (3) is just the Euler-Lagrange equation of the functional S. In general, the energy functional S is only a formal sum and may well not converge, but the definitions of ground state and critical point make sense since they involve only finite sums. Ground states are critical points, as it is easily checked. Also, in case a ¯ := 0, i.e., when there is no spring interaction, critical points of the system are determined by critical points of the potential. For instance, given σ ∈ Rd , one can choose ui to be a critical point of V lying in [σ · i, σ · i + 1), with ui − κ ∈ Z, for some κ ∈ R. Such configurations satisfy |ui − σ · i| ≤ 1 and ui+j ≥ ui if σ · j ≥ 0. We will see that these are the basic estimates that we need to preserve even in presence of a nontrivial spring interaction a ¯. The first result we present deals with the existence of ground states which lie at a bounded distance from any given linear function. These ground states have an

GENERALIZED FRENKEL-KONTOROVA MODELS

5

interesting geometric property (known in the literature under the name of Birkhoff or selfconforming see (5) below). We will interpret geometrically this property by saying that the ground states and their translations are organized in a lamination. Theorem 1. Given any vector σ ∈ Rd , there exists a ground state ui such that (4)

|ui − σ · i| ≤ 1 , d

for any i ∈ Z . Moreover, (5)

if j ∈ Zd and s ∈ Z are such that σ · j ≥ s, we have that ui+j ≥ ui + s for any i ∈ Zd .

Theorem 1 has many features in common with analogous results of [B89], though it is valid under weaker, possibly degenerate, assumptions on the twist condition (for instance, powers greater than 2 do not satisfy the first condition in (2.1) of [B89]). Moreover, the proofs presented here are different, somewhat simpler, flexible to the construction of nonminimal critical points (as in Theorem 2 below) and feasible to the more general cases of [LlV06a]. We observe that (5) is an “order property” and it implies that ui+j = ui for any j ∈ Zd ∩ σ ⊥ , and thus, for σ ∈ Qd , it is just a periodicity condition. When σ ∈ Rd \ Qd , we obtain that the solutions are “quasiperiodic” since they are the limit of periodic solutions. Here we follow the notation common in variational method of calling quasiperiodic the limits in whatever sense of periodic solutions. In other areas of dynamics, one uses a different name depending on the different meanings of limit (e.g. pointwise, uniform etc.). So, our solutions could be called quasiautomorphic in the language of topological dynamics. It is worth to notice that the estimate in (4) is independent of σ and V . This is a consequence of (5) and of the discrete structure of Zd . In a similar continuous case dealt with in [M89] the situation with this respect is quite different, since the constants there depend on both the nonlinearity and the frequency, because of the use of the C α -Regularity Theory for PDEs. For more general group actions, estimates similar to (4) still hold, but the constants seem to become more involved (see [LlV06a] for explicit bounds). The property in (5) says that the integer translations of the graph are all comparable functions. Furthermore, it is easy to see that the pointwise limit of these functions is also a ground state. We will call lamination a set of comparable functions whose graphs in Zd × (R/Z) are a closed set under pointwise limits. These definition is very similar to the usual definition of laminations in topology. The graphs of each of the functions is a sheet. Nevertheless, in our case, we will not assume that different sheets are disjoint. We allow that they can touch, even if they cannot cross. As we will see later, with an extra assumption on the interaction we will prove a maximum principle that shows that the graphs of comparable equilibria have to be disjoint. The ground states satisfying (5) are organized in laminations, hence it is natural to formulate the results for laminations. It can happen that the lamination covers Zd × (R/Z) – this is analogue of the case of existence of invariant circles for twist maps – or that it contains gaps, which is an analogue of the minimizers being a Cantor set. Our next result shows that, in the case that a lamination of minimizers contains gaps, we can find a further critical point in the gap.

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RAFAEL DE LA LLAVE AND ENRICO VALDINOCI (0)

(1)

Theorem 2. Assume that the graphs of ui and ui belong to a lamination of (0) (1) ordered ground states as produced in Theorem 1. Suppose that ui ≥ ui for (0) (1) any i ∈ Zd and that uio > uio for some io ∈ Zd . (0)

Then, there exists a critical point ui satisfying (5), different from ui (1) (0) such that ui ≥ ui ≥ ui for any i ∈ Zd .

(1)

and ui ,

Under a further assumption, the result of Theorem 2 may be sharpened by obtaining a strict order property. For this, we say that the interaction is transitive if given any i, j ∈ Zd there exist i1 , . . . , iℓ ∈ Zd in such a way that (6)

aii1 , ai1 i2 , . . . , aiℓ j 6= 0 .

Physically, this means that two sites are always connected by a certain system of springs. Then, Theorem 2 takes the following stronger form: Corollary 1. Suppose that the interaction is transitive. Assume that the graphs (1) (0) of ui and ui belong to a lamination of ordered ground states as produced in (0) (1) (0) (1) Theorem 1. Suppose that ui ≥ ui for any i ∈ Zd and that uio > uio for some io ∈ Zd . (0) (1) Then, there exists a critical point ui satisfying (5) and such that ui > ui > ui d for any i ∈ Z . One of the tool of the proofs of Theorem 2 and Corollary 1 will be a comparison property (see Lemmata 1 and 2 below). The proofs of Theorems 1 and 2 that we present here are remarkably simple and they depend on very few elements of the problem. Thus, the ideas we expose may be applied to quite general variational problems which satisfy some sort of “ferromagnetism”, “coercivity” and “invariance under translations” (see [LlV06a, LlV06b]). Comparison Principles We will deal with the following differential equation: h X  t t t  a p (Φ (u) − Φ (u) ) |Φt (u)i − Φt (u)j − cij |pij −2 −∂ Φ (u) = ij ij i j t i     j∈Zd  i    +|Φt (u)j − Φt (u)i − cij |pij −2 + V ′ (Φt (u)i )    h X  |Φt (u)j − Φt (u)i − cij |pij −2 + a p c ij ij ij (7)   j∈Zd i    t t pij −2  for t > 0, −|Φ (u) − Φ (u) − c |  i j ij       Φ0 (u)i = ui .

Here, we will take t ∈ [0, T ⋆ ], for some T ⋆ > 0. The above Cauchy problem is the gradient flow of S in (1). As we will see, even if S is only a formal sum, the RHS of (7) involves only finite sums and so it is well defined. If we fix a frequency ω, we can consider the RHS defined on the affine space consisting of functions u such that |ui − ω · i| is bounded uniformly in i. We denote this space as Xω . It is an affine space modelled on ℓ∞ (Zd ). We note that when p ≥ 2, the RHS is a C 1 mapping from Xω to ℓ∞ . Furthermore, the Lipschitz constant is

GENERALIZED FRENKEL-KONTOROVA MODELS

7

uniform on bounded sets. Therefore, applying the standard theory of existence and regularity of solutions of equations on Banach spaces (or Banach manifolds) (see e.g. [H80]) we can conclude that, when the initial data u range over a bounded set, the flow is uniquely defined for a time T > 0 and, moreover, that the flow depends C 1 on the initial conditions. Furthermore, we can compute the derivative of the flow integrating the variational equations. Our next goal is to use the variational equations to show that there are Comparison Principles for the equation. We will prove a weak one, Lemma 1 and a strong one, Lemma 2 which requires an extra hypothesis. As we will see later, as a consequence of the Comparison Principles, we will obtain that, in the situations we are interested, the flow can be extended for all times. The Comparison Principles we obtain are the following: Lemma 1 (Weak Comparison Principle). Let (7) holds. Suppose that ui ≥ vi for all i ∈ Zd . Then Φt (u)i ≥ Φt (v)i for any i ∈ Zd and any t ∈ [0, T ⋆ ]. Lemma 2 (Strong Comparison Principle). Let the interaction be transitive, as defined in (6). Let the assumptions of Lemma 1 hold. If also u 6≡ v, then Φt (u)i > Φt (v)i for any i ∈ Zd and any t ∈ [0, T ⋆ ]. As we will now see in the course of the proof, Lemma 1 uses only that the nondiagonal elements of the variational equation are nonnegative, while Lemma 2 exploits the fact that the variational equations have a positive kernel. In order to prove Lemma 1 we argue as follows. For λ ∈ [0, 1], let uλi := λui + (1 − λ)vi and differentiate (7) to obtain that X µλij (t) ∂λ Φt (uλ )j , (8) −∂t (∂λ Φt (uλ )i ) = δiλ (t) ∂λ Φt (uλ )i − j∈Zd 0 0,

(13)

we gather from (12) that ∂λ Φto (uλ )io = 0, thence Ciλo (to ) = 0

(14)

for any λ ∈ (0, 1). Since Ciλo (0) ≥ 0 and the map t 7→ Ciλo (t) is nondecreasing due to (10), we conclude that Ciλo (t) = 0 for any t ∈ [0, to ] and any λ ∈ (0, 1).

(15) In particular,

0 = Ciλo (0) = uio − vio .

(16) Now, we prove that (17)

if (13) holds, τ

then φ (u)j = φτ (v)j for any τ ∈ [0, to ] and any j ∈ Zd such that aio j 6= 0.

To confirm this, we first note that, up to replacing ui and vi with Φτ (u)i and Φτ (v)i , we may and do assume that τ = 0 in (17). We also observe that, given j in such a way that uj 6= vj , there exist λj ∈ (0, 1) in such a way that λ

λ

uioj 6= uj j .

(18)

Indeed, if uλio = uλj for any λ, we would have that 0 = ∂λ (uλio − uλj ) = (uio − vio ) − (uj − vj ) = 0 − uj + vj , due to (16), which is a contradiction. Moreover, from (15) and (10), νiλo j (t)Cjλ (t) = 0 for any j ∈ Zd , any t ∈ [0, to ] and any λ ∈ (0, 1). In particular, if, by contradiction, uj 6= vj for some j with aio j 6= 0, we would have that λ

λ

λ

0 = νiojj (0) Cj j (0) = µiojj (0) (uj − vj ) λ

λ

λ

and so µiojj (0) = 0. By (9), this would give that uj j = ui0j , in contradiction with (18). This says that uj = vj for any j as long as aio j 6= 0, yielding (17). The proof of Lemma 2 is now ended by recursion, thanks to the fact that the interaction is transitive. For this, let to and io be as in (13), take any j ∈ Zd and suppose that aio i1 , ai1 i2 , . . . , aiℓ j 6= 0 , for suitable i1 , . . . , iℓ . Then, by (13) and (17), we have that Φτ (u)i1 = Φτ (v)i1 for any τ ∈ [0, to ]. Then, by applying (17) again, we get that Φτ (u)i2 = Φt (v)i2 for any τ ∈ [0, to ]. By repeatedly applying (17), we thus conclude that Φτ (u)ik = Φτ (v)ik for k = 1, . . . , ℓ and so Φτ (u)j = Φτ (v)j for any τ ∈ [0, to ]. In particular, uj = vj . The proof of Lemma 2 is thus concluded.  As a straightforward consequence of Lemma 2, we obtain: Corollary 2. Let the interaction be transitive. Let ui and vi be two critical points in such a way that ui ≥ vi for any i ∈ Zd and uio > vio for some io ∈ Zd . Then, ui > vi for any i ∈ Zd .

GENERALIZED FRENKEL-KONTOROVA MODELS

9

Therefore, when the interaction is transitive, we see that when we consider a lamination of critical points, the sheets are disjoint, in more agreement with the common usage in topology. If the interaction is not transitive, the conclusions of Corollary 2 could be false. For example, if we consider a layered material in which different layers do not interact, it is easy to have minimizers so that they agree on one layer but are different and comparable on the others. An explicit counterexample in our case is the following: take d ≥ 2, cij := 0, pij := 2, V (r) := sin(2πr) and  1 if ξ ∈ {e1 , −e1 }, a ¯(ξ) := 0 otherwise. Take also ui := 1/4 for any i ∈ Zd and  1/4 vi := −1/4

if i2 = 0, if i2 = 6 0.

Then, ui and vi are critical points, ui ≥ vi for any i ∈ Zd and they agree on {i2 = 0}. Proof of Theorem 1 The proof we present uses the rearrangement method of [CLl00] and it is based the construction of a “minimal minimizer” which possesses additional geometric properties. Further details and generalizations are given in [LlV06a]. We will first consider the case σ ∈ Qd and prove good uniform estimates. The case σ ∈ Rd \ Qd is then obtained by a limit process. We fix some finite index subgroup K of K(σ) := {k ∈ Zd s.t. σ · k ∈ Z}

(19)

and we consider the periodic configurations in n o (20) CK := u s.t. sup |ui − σ · i| ≤ 10 and ui+j = ui + σ · j for any j ∈ K . i∈Zd

In what follows, we identify, with a slight abuse of notation, the quotient Zd /K with an integer polytope centered at the origin. We can assume without loss of generality that the sides are much larger than the range of the interaction. We now quotient our variational problem with respect to K. That is, given i ∈ Qd , we consider the equivalence class [i] := {i′ ∈ Qd s.t. i − i′ ∈ K} . Analogously, given i, j ∈ Qd , we set [(i, j)] := {(i′ , j ′ ) ∈ Qd × Qd s.t. i − i′ = j − j ′ ∈ K} . Given any i ∈ Qd (resp., i, j ∈ Qd ), we then choose one element iK ∈ [i] (resp., (iK , jK ) ∈ [(i, j)]). We also consider the quotient energy functional X X V (uiK ) . aiK jK |uiK − ujK − cij |pij + (21) SK (u) := (iK ,jK )

iK

The functional SK is well defined for u ∈ CK and it does not depend on the choice of iK , jK . We also note that the series in (21) are just finite sums. Indeed, the set Zd /K is finite, since σ ∈ Qd and K has finite index in K(σ), thus we may always think that the contribution of the above series is given by some iK in such a finite

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RAFAEL DE LA LLAVE AND ENRICO VALDINOCI

set and by jK satisfying |iK − jK | < R, which, again, is a finite set. Note that the value of SK on CK ∋ u is determined once we know ui for i ∈ Zd /K, which is a finite set. Therefore, minimizing SK on CK is a finite dimensional problem. It is easy to check that if the distance between two sites goes to infinity, the energy goes to infinity. The problem is also invariant by addition of an integer to the state. Consequently, the set MK of minimizers of SK on CK is non empty. Moreover, if u ∈ MK , then so does v := u + k for any k ∈ Z and, if k≥

max |ui+ℓ | + |σ| max |j| ,

i,ℓ∈Zd /K

j∈Zd /K

d

we have that, given any i ∈ Z /K and j ∈ Zd , with j = j1 + j2 , j1 ∈ Zd /K and j2 ∈ K, vi+j = ui+j1 +j2 + k = ui+j1 + σ · j2 + k = ui+j1 + σ · j − σ · j1 + k ≥ σ · j . Consequently, the set M⋆K := MK ∩ {uj ≥ σ · j for any j ∈ Zd }

(22)

is non empty. Given two configurations ui and vi we now consider the configurations min{u, v}i := min{ui , vi }

and

max{u, v}i := max{ui , vi }.

The following inequality will be fundamental for our argument: (23)

SK (min{u, v}) + SK (max{u, v}) ≤ SK (u) + SK (v) .

To prove (23), we set αi := max{u, v}i −ui and βi := ui −min{u, v}i . Then αi βi ≥ 0, αi βi = 0 and vi = ui + αi − βi . Thus, if we consider the function [−1, 1]2 ∋ (s, t) 7−→ F (s, t) := |(u + tα + sβ)i − (u + tα + sβ)j − cij |pij , we have that X 2 pij (pij − 1)|(u + tα + sβ)i − (u + tα + sβ)j − cij |pij −2 (αi βj + αj βi ) ≤ 0 . ∂st F =− i,j

Accordingly, Z 1Z 0 ≥ 0

0

−1

2 ∂st F ds dt

= | max{u, v}i − max{u, v}j − cij |pij + | min{u, v}i − min{u, v}j − cij |pij −|ui − uj − cij |pij − |vi − vj − cij |pij , from which (23) plainly follows. The above considerations imply that uK i := min⋆ ui

(24)

u∈MK

M⋆K

is also in and it thus will be called the minimal minimizer. One of the main feature of uK is that it satisfies the no symmetry breaking condition, namely, that (25)

uK = uK(σ)

for any finite index subgroup K of K(σ). To confirm (25), we argue as follows. Given k ∈ K(σ), we observe that u ˜ ∈ M⋆K , K K ˜i and so where u ˜i := ui+k − σ · k, and therefore ui ≤ u K uK i ≤ min ui+k − σ · k . k∈K(σ)

GENERALIZED FRENKEL-KONTOROVA MODELS

11

Since the other inequality is obvious, we have that K uK i = min ui+k − σ · k k∈K(σ)

and so that K uK i = ui+k − σ · k

for any k ∈ K(σ). In particular, uK belongs to CK(σ) (and not only to CK ). This and the fact that SK = ♯(K(σ)/K) SK(σ) K

on CK(σ) yield that u ∈ MK(σ) and uK(σ) ∈ MK . More precisely, uK ∈ M⋆K(σ) and uK(σ) ∈ M⋆K , and so (25) follows. K(σ) Also, if j and s are as in (5), we have that vi := ui+j − s ∈ M⋆K(σ) and K(σ)

therefore vi ≥ ui , yielding the order property in (5). Let now j ∈ Zd . Take s such that σ · j ∈ [s, s + 1]. Then, the order property in (5) implies that K(σ)

K(σ)

σ · j − 1 ≤ s ≤ ui+j − ui

≤s+1≤ σ·j +1

thence K(σ)

K(σ)

sup |ui+j − ui

(26)

− σ · j| ≤ 1 .

i,j∈Zd

K(σ)

− σ · jo ∈ [0, 1].

K(σ) uj

− σ · j ≥ 1 for

We also observe that there must be some jo ∈ Zd such that ujo Indeed, if not, the fact that u

K(σ)

∈

M⋆K

would give that

K(σ) ui

d

− 1 would also lie in any j ∈ Z , and so vi := definition of minimal minimizer given in (24). Then, by (26), K(σ)

|ui

K(σ)

− σ · i| ≤ |ui

K(σ)

− ujo

M⋆K ,

in contradiction with the

− σ · (i − jo )| + 1 ≤ 2 ,

d

for any i ∈ Z . This and (22) give that K(σ)

0 ≤ ui

−σ ·i ≤ 2,

hence (4) follows by translating one unit down. From this, we also see that uK(σ) does not touch the boundary of CK as defined in (20). Also, uK(σ) is a ground state. Indeed, let K > 0 and vi be a configuration K(σ) for |i| > K. We take ℓ ∈ N so large that the set such that vi = ui {j ∈ Zd s.t. |i − j| ≤ R for some |i| ≤ K} is contained in the fundamental polytope Zd /(ℓK(σ)). We now introduce the “periodic extension” of v. Namely, given i ∈ Zd with i = i1 +i2 , i1 ∈ Zd /(ℓK(σ)) and i2 ∈ ℓK(σ), we define wi := vi1 +σ·i2 . Then, w ∈ CℓK(σ) and so SℓK(σ) (w) ≥ SℓK(σ) (uℓK(σ) ) = SℓK(σ) (uK(σ) ) . This says that X X aij |wi − wj − cij |pij + V (wi ) i∈Zd /(ℓK(σ)), |j−i|≤R

≥

X

i∈Zd /(ℓK(σ)), |j−i|≤R

i∈Zd /(ℓK(σ))

K(σ) aij |ui

−

K(σ) uj

− cij |p+ij +

X

i∈Zd /(ℓK(σ))

K(σ)

V (ui

).

12

RAFAEL DE LA LLAVE AND ENRICO VALDINOCI

Note that X

V (wi ) =

i∈Zd /(ℓK(σ))

X

V (vi ) +

K(σ)

X

V (ui

).

i∈Zd /(ℓK(σ)) |i|>K

|i|≤K

Also, it is easily seen that if i ∈ Zd /(ℓK(σ)), |i − j| ≤ R and |j| > K, we have that |j + h| > K for any h ∈ (ℓK(σ)), since ℓ is large, and so, for such a j, if j + h = j ′ ∈ Zd /(ℓK(σ)), we have |j ′ | > K and then K(σ)

wj = wj ′ − σ · h = uj ′ As a consequence, X

K(σ)

− σ · h = uj

.

aij |wi − wj − cij |pij

i∈Zd /(ℓK(σ)), |j−i|≤R

=

X

aij |vi − vj − cij |pij +

aij |vi − vj − cij |pij

i∈Zd /(ℓK(σ)), |j−i|≤R |i|>K≥|j|

|i|≤K |j−i|≤R

X

+

X

aij |vi − wj − cij |pij

i∈Zd /(ℓK(σ)), |j−i|≤R |i|,|j|>K

=

X

min{|i|,|j|}≤K

aij |vi − vj − cij |pij +

X

K(σ)

aij |ui

K(σ)

− uj

− cij |pij .

i∈Zd /(ℓK(σ)) |i|,|j|>K

The above considerations yields that uK(σ) is a ground state. The lamination property is a consequence of (5). This ends the proof of Theorem 1 in the case σ ∈ Qd . If σ ∈ Rd \ Qd , one takes a sequence σj ∈ Qd and uses Theorem 1 at the jth step, (j) (j) obtaining a ground state ui . Since (4) is uniform in σ, we can pass ui to the limit: (j) namely, ui − σj · i is compact (because the product of any collection of compact (j) topological spaces is compact by Tychonoff Theorem), thence ui approaches, up to subsequence, a suitable ui . By passing to the limit, we see that ui satisfies (4) and (5), and that it is a ground state.  Proof of Theorem 2 The proof we provide is based on the gradient flow method of [A90, KLlR97, CLl98, LlV05]. Further details and generalizations are given in [LlV06b]. We observe that, differently from what we did in the proof of Theorem 1, we need here to consider both the rational and irrational cases at the same time. Indeed, the irrational case cannot be obtained by taking limits of rational critical points, (1) (0) since ui might collapse onto either ui or ui during such a procedure, making the multiplicity result void. Thus, for the proof of Theorem 2, we fix σ ∈ Rd (not necessarily rational) and we consider the “resonant set” K(σ) given in (19). For ℓ ∈ N, with a slight abuse of notation, we identify the quotient spaces Zd /(ℓK(σ)) with a sequence of nested fundamental domains, that is, each Zd /(ℓK(σ)) is a polytope centered at the origin. Note that, when σ ∈ Rd \ Qd , some sides of such a polytope may have infinite length and, in fact, if K(σ) = {0} then Zd /(ℓK(σ)) = Zd . In any case, Zd /(ℓK(σ)) ⊂ Zd /((ℓ + 1)K(σ)) ⊂ . . . is a sequence of polytopes invading the

GENERALIZED FRENKEL-KONTOROVA MODELS

13

whole Zd . We also remark that, as a consequence of the periodicity and lamination properties,   X (0) (1) (27) |ui − ui | ≤ ♯ K(σ)/(ℓK(σ)) . i∈Zd /(ℓK(σ))

We define the space (0)

C := {ui satisfying (5) s.t. ui

(1)

≤ u ≤ ui

for any i ∈ N}

and, for u, v ∈ C, we consider the renormalized energy   X aij |ui − uj − cij |pij − |vi − vj − cij |pij Sℓ (u, v) := i∈Zd /(ℓK(σ)) j∈Zd

X

+

(28)

i∈Zd /(ℓK(σ))



 V (ui ) − V (vi ) .

Note that (1)

(29)

ui − uj ± cij ≤ ui (1)

≤ |ui

(0)

− uj + Λ (0)

− σ · i| + |σ · j − uj | + |σ| |i − j| + Λ ≤ 4 + |σ| R + Λ ,

if u ∈ C and |i − j| ≤ R, because of (4). Then, X aij (|ui − uj − cij |pij − |vi − vj − cij |pij ) i∈Zd /(ℓK(σ)) |i−j|≤R

X

≤ C

(|ui − vi | + |uj − vj |)

i∈Zd /(ℓK(σ)) |i−j|≤R

≤ CRd

X

|uj − vj | ,

j∈Zd /(CℓK(σ)

and so the series in (28) are convergent, thanks to (27). Notice also that if v (n) ∈ C is so that (n) lim v n→+∞ i

= vi for any i ∈ Zd /(ℓK(σ))

it follows from (27) and the Dominated Convergence Theorem that X (n) (30) lim |vi − vi | = 0 , n→+∞

i∈Zd /(ℓK(σ))

thence the pointwise and uniform convergence agree in C. We now show that the renormalized energy is good in detecting the ground states. Namely, we prove that, if u ∈ C, (31)

Sℓ (u, u(0) ) ≥ 0 and equality holds if and only if ui is a ground state.

The proof of such a statement will be accomplished in several steps. First, we show that (32)

if vi is a ground state, Sℓ (u, v) ≥ 0.

We may reduce to prove (32) under the additional assumption that (33) for some K > 0.

ui = vi for any i ∈ Zd /(ℓK(σ)) with |i| ≥ K,

14

RAFAEL DE LA LLAVE AND ENRICO VALDINOCI

Indeed, once (32) is proven under the assumption in (33), we may define  ui if i ∈ Zd /(ℓK(σ)) with |i| < K, (K) := ui vi if i ∈ Zd /(ℓK(σ)) with |i| ≥ K (K)

and then extend it periodically outside Zd /(ℓK(σ)). This way, ui satisfies (33) (K) and it approaches ui as k → +∞. Then, if (32) holds for ui , we gather from (30) that Sℓ (u, v) = lim Sℓ (u(K) , v) ≥ 0 , K→+∞

thence (32) holds for ui . In the light of these considerations, we now focus on the proof of (32) under condition (33). To this extent, we take a large n ∈ N and we consider the configuration  ui if i ∈ Zd /(nℓK(σ)), (n) vi := vi if i 6∈ Zd /(nℓK(σ)). (n)

By (33), vi = vi if i is outside a suitable bounded set Q ⊂ Zd /(nℓK(σ)). Then, since vi is a ground state, X X (n) (n) (n) (V (vi ) − V (vi )) ≥ 0 . aij (|vi − vj − cij |pij − |vi − vj − cij |pij ) + i∈Q

{i,j}∩Q6=∅

Consequently, Snℓ (v (n) , v) ≥

(n)

X

aij (|vi

+

X

(n)

− vj

− cij |pij − |vi − vj − cij |pij )

i∈Zd /(nℓK(σ)) j∈Zd i,j6∈Q

(n)

(V (vi ) − V (vi )) = 0 .

i∈(Zd /(nℓK(σ)))\Q

Thus, if r := dim (K(σ)), recalling (29) and (27), we conclude that   ℓK(σ) Sℓ (u, v) = Snℓ (u, v) Cnr Sℓ (u, v) ≥ ♯ nℓK(σ) X (n) (n) ≥ Snℓ (v (n) , v) − C (|ui − vi | + |uj − vj |) i∈Zd /(nℓK(σ)) j∈Zd |i−j|≤R

≥0−C

X

(n)

|uj − vj | = o(nr ) ,

dist (j,Zd /(nℓK(σ)))≤R j6∈Zd /(nℓK(σ))

where the above C’s are independent of n. A division by nr and a limit as n → +∞ yields that Sℓ (u, v) ≥ 0, thus proving (32). By exchanging the rˆoles of u and v, we also deduce from (32) that (34)

if ui and vi are ground states, Sℓ (u, v) = 0.

We now show that (35)

if vi is a ground state and Sℓ (u, v) = 0 then ui is a ground state too.

To prove this, let w be such that wi = ui if |i| > K. Let no so large that (36)

{j ∈ Zd s.t. |i − j| ≤ R for some i ∈ Zd , |i| ≤ K} ⊆ Zd /(no ℓK(σ)) .

We then use the fact that Sℓ (u, v) = 0 and (32) to conclude that Sno ℓ (u, w) = Sno ℓ (u, v) + Sno ℓ (v, w) = 0 − Sno ℓ (w, v) ≤ 0 ,

GENERALIZED FRENKEL-KONTOROVA MODELS

15

which implies that ui is a ground state, thanks to (36). By collecting the results of (32), (34) and (35), the claim in (31) follows. We also introduce the following gradient flow. Given ui ∈ C, we consider the configuration Φt (u)i , with i ∈ N and t ≥ 0 to be a solution of the Cauchy Problem (7), which turns out to be the gradient flow of (28). The right hand side of (7) is locally Lipschitz in ℓ∞ so the flow satisfies existence and uniqueness for short times. The Comparison Principle in Lemma 1 implies that if ui ≥ vi for all i ∈ Zd , then Φt (u)i ≥ Φt (v)i for all i ∈ Zd . Hence we conclude that Φt (C) ⊆ C. Moreover, since, for functions in C the generator of the flow is uniformly Lipschitz, we conclude that Φt extends for all time for initial data in C, due to the Prolongation Lemma (see [H80]). Furthermore, 2  X (37) −∂t Sℓ (Φt (u), u(0) ) = ∂t Φt (u)i ≥ 0 . i∈Zd /(ℓK(σ))

Note that the vanishing of the quantity in (37) is the same as being a critical point in (3). We now look at the evolution of the linear interpolation between u(0) and u(1) , that is, for s ∈ [0, 1], we define u(s) := su(1) + (1 − s)u(0) and we conclude by the above construction and Tychonoff Theorem (i.e., the product of any collection of compact topological spaces is compact), that for any s ∈ [0, 1] there is a sequence of times tn (s) and a critical point u(s,⋆) such that lim tn (s) = +∞ and

n→+∞

(s,⋆)

lim Φtn (s) (u(s) )i = ui

n→+∞

.

The proof of Theorem 2 will be completed once we are able to show that there exists s ∈ [0, 1] in such a way that u(s,⋆) does not coincide with u(0) nor with u(1) . For this, given r ≥ 0, we consider the energy ball Br⋆ := {u ∈ C s.t. Sℓ (u, u(0) ) ≤ r} . Then, in the light of (31), we may suppose that (38)

C ∩ B0⋆ = {u(0) , u(1) } ,

otherwise another ground state (and so, a fortiori a critical point) would exist, thus ending the proof. We now argue by contradiction and assume that no other critical point but u(0) and u(1) lies in C. We first argue that, under the assumption that there are no other critical points, for small r > 0, C ∩ Br⋆ has a connected component C (0) containing u(0) and another connected component C (1) containing u(1) . The reason is that, if not, for every r > 0 we could find a path contained in C ∩ Br⋆ joining u(0) and u(1) . In particular, for every r > 0, there would be a configuration u[r] ∈ C ∩ Br⋆ so that ||u[r] − u(0) ||, ||u[r] − u(1) || > 1/10||u(0) − u(1) ||. Passing to the limit r → 0, and using that C is compact, we can extract a convergent subsequence of u[r] and then construct a configuration whose renormalized energy is zero and which is not either u(0) nor u(1) .

16

RAFAEL DE LA LLAVE AND ENRICO VALDINOCI

Once we have the existence of the connected components, we conclude that once Φt (u(s) ) has entered C (0) (resp., C (1) ) it cannot leave it and it thus approaches u(0) (resp., u(1) ). We define the basin of attraction β (0) (resp., β (1) ) as all the s ∈ [0, 1] such that Φt (u(s) ) approaches u(0) (resp., u(1) ) as t → +∞. The continuous dependence of Φt on initial data implies that β (0) and β (1) are open. But then, β (0) and β (1) are open and disjoint and they cover [0, 1], which is a contradiction. This completes the proof of Theorem 2.  Proof of Corollary 1 The desired result plainly follows from Theorem 2 and Corollary 2.



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Department of Mathematics, University of Texas at Austin, Austin TX 78712-0257 E-mail address: [email protected] ` di Roma Tor Vergata Dipartimento di Matematica Roma, Italia Universita E-mail address: [email protected]