Ripples in Graphene: A Variational Approach

Suspended graphene samples are observed to be gently rippled rather than being flat. In Friedrich et al. (Z Angew Math Phys 69:70, 2018), we have checked that this nonplanarity can be rigorously described within the classical molecular-mechanical frame of configurational-energy minimization. There, we have identified all ground-state configurations with graphene topology with respect to classes of next-to-nearest neighbor interaction energies and classified their fine nonflat geometries. In this second paper on graphene nonflatness, we refine the analysis further and prove the emergence of wave patterning. Moving within the frame of Friedrich et al. (2018), rippling formation in graphene is reduced to a two-dimensional problem for one-dimensional chains. Specifically, we show that almost minimizers of the configurational energy develop waves with specific wavelength, independently of the size of the sample. This corresponds remarkably to experiments and simulations.


Introduction
Carbon forms a variety of different allotropic nanostructures. Among these a prominent role is played by graphene, a pure-carbon structure consisting of a one-atom thick layer of atoms arranged in a hexagonal lattice. Its serendipitous discovery in 2005 has been awarded the 2010 Nobel Prize in Physics to Geim and Novoselov and has sparkled an exponentially growing research activity. The fascinating electronic properties of graphene are believed to offer unprecedented opportunities for innovative applications, ranging from next-generation electronics to pharmacology, and including batteries and solar cells. A new branch of Materials Science dedicated to lower-dimensional systems has developed, cutting across Physics and Chemistry and extending from fundamental science to production [8].
Despite the progressive growth of experimental, computational, and theoretical understanding of graphene, the accurate description of its fine geometry remains to date still elusive. Indeed, suspended graphene samples are not exactly flat but gently rippled [1,22] and waves of approximately one hundred atom spacings have been predicted computationally [7]. Such departure from planarity seems to be necessary in order to achieve stability at finite temperatures, in accordance with the limitations imposed by the classical Mermin-Wagner Theorem [15,20,21]. Even in the zero-temperature limit, recent computations [12] suggest that nonplanarity is still be expected due to quantum fluctuations. Note that, beside the academic interest, the fine geometry of graphene sheets is of a great applicative importance, for it is considered to be the relevant scattering mechanism limiting electronic mobility [13,26].
The aim of this paper is to prove that the emergence of waves with a specific, samplesize independent wavelength can be rigorously predicted. We move within the frame of Molecular Mechanics, which consists in describing the carbon atoms as classical particles and in investigating minimality with respect to a corresponding configurational energy. This energy is given in terms of classical potentials and takes into account both attractive-repulsive two-body interactions, minimized at some given bond length, and three-body terms favoring specific angles between bonds [2,3,24,25]. With respect to quantum-mechanical models, Molecular Mechanics has the advantage of being simpler and parametrizable, although at the expense of a certain degree of approximation. Remarkably, it delivers the only computationally amenable option as the size of the system scales up. In addition, it often allows for a rigorous mathematical analysis. In particular, crystallization results for graphene in two dimensions have been proved both in the thermodynamic limit setting [5,6] and in the case of a finite number of atoms [4,19]. The fine geometry of other carbon nanostructures has also been investigated [9,10,[16][17][18]23].
A first step toward the understanding of rippling in graphene is detailed in the companion paper [11] where we investigate ground-state deformations of the regular hexagonal lattice with respect to configurational energies including next-to-nearest-neighbor interactions. (Note that pure nearest-neighbor interactions predict flat minimizers.) In such setting, optimal hexagonal cells are not planar, see Fig. 1 left. The main result of [11] is a classification of all graphene ground states into two distinct families: rolled-up and rippled configurations. Rolled-up structures ideally correspond to carbon nanotubes. Their optimality recalls remarkably the experimental evidence that free-standing graphene samples tend to roll up [14]. Rippled configurations, see Fig. 1, would instead correspond to suspended graphene patches, where the rolling-up is prevented by the adhesion to a probing frame.
Our focus is here on rippled configurations. These are not planar and feature a specific direction in three-dimensional space along which they are periodic. The full threedimensional description of rippled configurations is hence completely determined by orthogonal sections to such specific periodicity direction ( see the free edge at the bottom of the samples in Fig. 1, for instance). The aim of this paper is to address the geometry of such orthogonal sections (and hence of the whole rippled configuration) from a variational viewpoint. In fact, such sections are nothing but one-dimensional chains in two dimensions.
We introduce an effective energy for such sections by considering cell centers as particles and favoring a specific distanceb between cell centers and a specific angle π −ψ between segments connecting neighboring cell centers. Figure 2 illustrates this setting in the case of the rippled configuration on the left of Fig. 1.
Specific wave patterns of the rippled structure will then correspond to waves in the chain of cell centers, as in the case of the rippled configuration on the right of Fig. 1. By slightly abusing terminology, we shall hence call particles such cell centers and bonds the segments between two neighboring cell centers.
In the following, two choices for the effective energy are considered. At first, we analyze the reduced energy (3) taking into account nearest-and next-to-nearest-neighbor interactions and favoring nonaligned consecutive bonds. This leads to a large variety of energy minimizers with many different geometries, see Fig. 3. We then specialize the description via the (general) energy (7) taking additionally longer-range interactions into account. This second choice leads to a finer characterization of energy minimizers since the energy accounts also for curvature changes of the chain.
Our main result (Theorem 2.4) states the possibility of finding an optimal wavelength for energy minimizers. More precisely, for all prescribed overall lengths of the chain one finds an optimal wavelength λ such that all almost minimizers of the energy with that specific length can be viewed as compositions of λ waves, up to lower-order terms. Note that by fixing a given length of the chain one actually imposes a boundary condition, which corresponds to suspending the sample. Without such a boundary condition, no optimal wavelength is to be expected, for the sample would be rolling up, an instance which is indeed captured by our description. The crucial point of our result is that the optimal wavelength λ is independent of the size of the system. This corresponds to experimental and computational findings [7,22]. It is worth at this point to emphasize that the model features no ad-hoc addition of a mesoscopic lengthscale and that the optimal wavelength exclusively arises from minimality.
All results of the paper are presented in Sect. 2. The corresponding proofs are based on elementary arguments but are technically very involved and are detailed in Sects. 3-6. A first step is achieved in Sect. 3 where we consider a cell energy depending just on three consecutive particles. Here, convexity allows to check that minimizers are configurations where the two bonds between the particles are not aligned.
In Sect. 4 we consider the single-period problem of a chain which changes its curvature only once. In particular, we identify the optimal wavelength λ l max depending on the number of bonds l (later referred to as discrete-wave period). To this aim, it is instrumental to check for the concavity of the mappings l → λ l max and l → λ l max /l (see Lemma 4.3 and Lemma 4.5) where λ l max /l represents the normalized wavelength. Eventually, by convexity arguments we are able to control the deviation of the length of the chain from the optimal wavelength λ l max in terms of the energy excess, see Lemma 4.9. The strategy is then to identify candidate minimizers by composing more single-period chains, see Fig. 8 for an illustration. This turns out to be properly doable for even discrete-wave periods only. The treatment of odd discrete-wave periods is surprisingly much more intricate, see e.g. Lemma 4.8. One resorts there in showing that the combination of two single-period waves with odd discrete-wave periods is unfavored with respect to the combination of two single-period waves with even discrete-wave periods having the same overall length.
Once the single-period problem is settled, we tackle in Sect. 5 the multiple-period problem, by allowing the chain to change curvature more than once. We show here that the energy of the chain depends on the number of particles where the chain changes its curvature, see Lemma 5.3. We also quantify the length of the chain in terms of the number of different discrete-wave periods composing it (Lemma 5.1) and we show that the length excess can be controlled in terms of the energy excess (Lemma 5.2). Section 6 finally contains the proof of the main result. We firstly address the characterization of the minimal energy (Theorem 2.2). The upper bound for the minimal energy is obtained via an explicit construction composing single-period chains. The proof of the matching lower bound is more subtle and relies on the fine geometry of almost minimizers. In particular, we show that a chain with almost minimal energy essentially consists exclusively of single-period chains with a specific discrete-wave period, which only depends on the choice of the boundary conditions. The main underlying observation is made in terms of normalized wavelengths (wavelength divided by discrete-wave period): (1) the normalized wavelength of chains with larger discrete-wave periods is too short to accommodate the boundary conditions and (2) chains with smaller discrete-wave period, although having sufficiently large normalized wavelength, cost too much energy due to a large number of curvature changes. The arguments rely on a fine interplay of the longer-range contributions and the wavelength λ l max for different discrete-wave periods l.

Admissible configuration and configurational energy.
We consider chains consisting of n ∈ N particles and corresponding deformations y : {1, . . . , n} → R 2 . We write y i = y(i) for i = 1, . . . , n and introduce the set of admissible configurations by (1) The conditions above ensure that only consecutive points in the chain are bonded. In particular, apart from i = 1 and i = n, each atom is bonded to exactly two other particles.
Here, the value 1.5 is chosen for definiteness only. For two vectors a 1 , a 2 ∈ R 2 we let (a 1 , a 2 ) ∈ [0, 2π) be the angle between a 1 and a 2 , measured counterclockwisely. We define the bond lengths and angles of the chain by In the following we introduce the configurational energy E n of a chain, and we detail the hypotheses which we assume on E n throughout the paper. The energy is given by the sum of two contributions, respectively accounting for two-body and three-body interactions among particles that are respectively modulated by the potentials v 2 and v 3 , see (3) and (7).
We assume that the two-body potential v 2 : (0, ∞) → [−1, ∞) is smooth and attains its minimum value only at 1 with v 2 (1) = −1 and v 2 (1) > 0. Moreover, we suppose that v 2 is strictly increasing right of 1. Referring to the modeling of graphene sheets [11], this potential models the effective interaction between different graphene-lattice cells, favoring a specific distance of cell centers, here normalized to 1.
The three-body potential v 3 : [0, 2π ] → [0, ∞) is assumed to be smooth and symmetric around π , namely v 3 (π − ϕ) = v 3 (π + ϕ). Moreover, we suppose that the minimum is attained only at π with v 3 (π ) = v 3 (π ) = v 3 (π ) = v 3 (π ) = 0, and v 3 (π ) > 0. With reference to the modeling of graphene sheets, the latter potential describes the energy associated with the flatness of adjacent graphene-lattice cells [11]. In particular, v 3 is not related to angles between bonded carbon atoms but contributes an effective descriptor of flatness of cells. The reader is referred to [11,Section 5] and in particular to formula [11, (5.2)] for a discussion of this term.
We introduce a configurational energy by where the cell energy is defined as for y 1 , y 2 , y 3 ∈ R 2 . The constant ρ > 0 will be chosen to be suitably small later on. More precisely, one could reformulate the whole theory by prescribing a single two-body potentialṽ 2 and letting

Fig. 3. Energy minimizers of (3) with different geometries
In this setting, the dimensionless constant ρ > 0 would measure the ratio between the energetic contributions of first and second neighbors. (Specifically,ṽ 2 (1) = −1 and |ṽ 2 (2)| ≤ ρ in a neighborhood of 2.) Since our analysis is largely based on the smallness of such ratio, we prefer to highlight this in the notation and stick to the equivalent form in (4).
Since in the sequel we will consider also a more general energy, the configurational energy (3) is called the reduced energy. Let us mention that due to the fact that E red n is written as a sum over cell energies, the two-body contributions at the left and right end of the chain are counted only once and not twice. However, since we focus on the case of large numbers of particles n and we are not interested in describing the fine geometry close to the ends of the chain, this effect will be negligible for our analysis.
Our first result addresses the configurations with minimal reduced energy. In particular, we check that all configurations minimizing the reduced energy have bonds of equal length and show exactly two possible bond angles. Theorem 2.1 (Minimizers for the reduced energy). Let ρ > 0 be small depending only on v 2 and v 3 . Then there exist e cell ∈ R, 0 <b < 1, andψ ∈ (0, π/8) such that min y∈A n E red n (y) = (n − 2)e cell and each configuration y ∈ A n with minimal energy satisfies b i =b for i = 1, . . . , n −1 and ϕ i = π +ψ or ϕ i = π −ψ for i = 2, . . . , n − 1.
The result relies on the properties of the cell energy (4) and is proved in Sect. 3. We observe that there are many minimizers of the energy with very different geometries, see Fig. 3. In particular, to exclude certain geometries, in the following we will take given boundary conditions into account. This is realized by specifying the length of the chain in direction e 1 . Indeed, let us fix the straining parameter μ in the set of admissible values M, with M ⊂ (2/3, 1) being a closed interval, and define A n (μ) = {y ∈ A n | (y n − y 1 ) · e 1 = (n − 1)μ}.
θ Fig. 4. Almost minimizers of (3) consisting of single-period waves with different wavelengths (or in other words: different discrete-wave periods), represented by smooth waves for illustration purposes. Observe that the second and third configuration have different global geometries in spite of accommodating the same boundary conditions. The last configuration is only an almost minimizer since the angle θ is not π ±ψ Note that the length |y n − y 1 | of a minimizer of the reduced energy is necessarily strictly smaller than n − 1, forb < 1 andψ > 0. This implies that the choice of values of μ close to 1 in A n (μ) actually corresponds to straining the chain. Even by restricting to the special subclass A n (μ), (almost) minimizers of (3) may have very different geometries, see Fig. 4.
To investigate the qualitative differences and different geometries of various configurations with (almost) minimal reduced energy in more detail, we now introduce a general, refined energy. For y ∈ A n we let The term on the right accounts for longer-range interactions. The constantρ > 0 will be chosen suitably small with respect to ρ later on, again reflecting the different relevance of the different contributions. We note that we could take more general interactions into account, but the contributions of third neighbors are already sufficient for our subsequent analysis and here we prefer simplicity rather than generality. Let us also mention that a reformulation of (7) in terms of a single two-body potentialṽ 2 , similar to (5), is possible.

Characterization of minimal energy.
We will now identify the minimal energy E n for given μ ∈ M. We set The energy has a zero order term e gen cell which is constant for all values μ ∈ M and is a small perturbation of e cell given in Theorem 2.1, i.e., |e gen cell − e cell | ≤ cρ. Differences in the minimal energy in terms of μ appear only in the first order termρe range which is associated to the longer-range interactions. For the exact definitions of e gen cell and e range we refer to (54) and (58) below, respectively. For an illustration of the graph of the function e range we refer to Fig. 10.
In Theorem 2.3 below we will see that almost minimizers of the minimization problem (8) can be interpreted as waves (in a discrete sense). Then,ρe range is essentially related to the wavenumber of the minimizer. In particular, smaller values of μ correspond to a smaller wavenumber or, respectively, to a larger wavelength. Compare also the first and the second configuration in Fig. 4. Roughly speaking, this effect corresponds to the waves having 'constant curvature', induced by the angleψ from Theorem 2.1. In this context, the finite set M res = {μ ∈ M| e range is not differentiable in μ} (10) of resonant lengths plays a pivotal role since for μ ∈ M res minimizers of (8) are (almost) periodic waves, cf. Theorem 2.3 below. We remark that the minimal energy can be characterized only up to small error terms of the form 1/n andρ 2 . The term 1/n accounts for boundary effects at the left and right end of the chain, induced by the longer-range interactions. The termρ 2 on the right-hand side of (9) reflects the fact that periodic waves with different wavelengths lead to a different longer-range interaction. This effect will be discussed in more detail in Lemma 5.3.

Characterization of almost minimizers.
We now proceed with the characterization of almost minimizers. Recalling (2) we define which can be interpreted as particles where the chain 'changes its curvature'. For convenience, we write for a strictly increasing sequence of integers, where N (y) ∈ N depends on y. We will interpret |y i k+1 − y i k |, k = 1, . . . , N (y) − 1, as the wavelength of a wave. In the following, we say y ∈ A n (y) is an almost minimizer of (8) if where c is the constant from Theorem 2.2. We now present two results on the characterization of almost minimizers, starting from the resonant case μ ∈ M res .
Theorem 2.3 states that, despite of nonuniqueness, the minimizers can be characterized in terms of the wavelength λ(μ). We remark that the parts of the chain satisfying (13) correspond to a fixed number of bonds, also referred to discrete-wave period in the following, i.e., l μ := i k+1 −i k is constant for all i k ∈ K. More precisely, we will show below in Lemma 4.3 that the connection between μ, the wavelength, and the discrete-wave period is given by the formula with the bond lengthb and the angleψ from Theorem 2.1. Notice that the fact that the sequence λ(μ) is decreasing in μ (or equivalently, l μ is decreasing in μ) is in accordance with the above remark that smaller values of μ correspond to larger wavelengths, see again Fig. 4. Let us remark that the assumption n ≥ρ −2 can be dropped at the expense of a more complicated estimate (14). We however prefer to keep this assumption for simplicity since we are indeed interested in the case of a large number of particles.
In Corollary 6.1, we will explicitly provide an example of a chain involving waves of different discrete-wave periods in order to show that in general it is energetically favorable that #(C(y) \ K) is positive. In particular, minimizers are not expected to be periodic, but only periodic 'outside of a small set', controlled in terms ofρ. In particular, Corollary 6.1 will show that (a) the minimal energy in Theorem 2.2 can be characterized only up to a higher order error term of the formρ 2 and that (b) the characterization given in Theorem 2.3, see (14), is sharp.
Let us now drop the resonance assumption and present a characterization result for almost minimizers for general μ.
where σ only depends on μ, but not on y. In particular, in accordance with Theorem This result states that, for μ between two resonant lengths μ and μ , the almost minimizer shows essentially the two wavelengths λ(μ ) and λ(μ ) in proportion σ and 1 − σ , respectively, where σ depends just on μ.
The proofs of Theorems 2.3-2.4 are contained in Sects. 4-6. We start with the analysis of a single-period problem in Sect. 4, move on to the problem of multiple periods in Sect. 5, and finally give the proof of the main results in Sect. 6. We warn the Reader that in the following all generic constants may depend on the potentials v 2 and v 3 without explicit mentioning. Dependencies on other constants such as ρ,ρ, or ε, will always be indicated in brackets after the constant. Moreover, we will often use the notation x = max{z ∈ Z : z ≤ x} and x = min{z ∈ Z : x ≤ z} for x ∈ R. This continuous, simplified setting is still capable of illustrating some of the main features of the general model. In particular, it allows to identify an optimal wavelength, independently of the sample size. On the other hand, it avoids many technicalities and, correspondingly, it is much less detailed.
As said, configurations correspond to curves consisting of a finite number of arcs of a circle, whose radius is normalized to 1, and having non-overlapping secants on some given axis. The configuration is hence identified by the lengths {θ 1 , . . . , θ k } ∈ [0, π) k of the corresponding arcs. The total length of the curve is given by On the other hand, the projection of the curve on the axis has length Note that, for all k ∈ N given, the maximal length of the projection is achieved by the configuration made of k equal arcs with length /k. In fact, the concavity of sin on [0, π] entails that ≤ 2k sin( /(2k)), where equality holds iff θ i = /k for all i.
We now reformulate the variational problems by restricting to those curves of fixed length > 0 fulfilling the boundary condition = μ , where the given straining parameter μ ∈ (0, 1) has the exact same meaning as in (6). As all arcs have the same curvature, to minimize the energy in this case corresponds to minimize the number of curvature changes, i.e., k − 1. Let f : [2/π, 1] → [0, π/2] be the inverse function of τ → sin(τ )/τ , which is concave and strictly decreasing. The minimal value k min can be computed in terms of μ as In case μ is such that /(2 f (μ)) ∈ N, we have that the configuration with minimal energy is the juxtaposition of k min arcs of equal length θ * := /k min . For all μ which do not belong to such discrete set, the optimal curve consists of k min arcs, which necessarily cannot be all of equal length. Note that the optimal arc length θ * is invariant with respect to the length of the curve: given μ with /(2 f (μ)) ∈ N, among curves with length := m/k min for m ∈ N, the optimal configuration is the juxtaposition of m arcs of the same optimal length /m = /k min = θ * . This in particular illustrates in this simplified setting the onset of a specific, sample-size independent optimal wavelength.

The Cell Problem
In this short section we focus on the cell energy (4) and prove Theorem 2.2. Let us firstly note that the cell energy can be written equivalently in terms of bond lengths and angles. More precisely, we introducẽ ). Owing to this notation, we can now state the following.
(ii) The cell energyẼ cell is smooth in a neighborhood of the minimizers and there exists c conv = c conv (ρ) > 0 such that its Hessian at the minimizers satisfies where I ∈ R 3×3 denotes the identity matrix.
Proof. Ad (i). Fix ε > 0 small. Since for ρ = 0 the energy is uniquely minimized by (1, 1, π), for ρ small (depending on ε) the minimizers ofẼ cell lie in The second derivative of f reads as Consequently, f (π ) < 0 since v 2 is strictly increasing right of 1 and v 3 (π ) = 0. Moreover, as v 3 is symmetric around π , f is symmetric around π as well. Thus, it suffices to identify a unique minimizer of (b 1 . After a transformation, this is equivalent to show that has a unique minimizer on D ε : and g 2 = G − g 1 /ρ. Let the functions λ 1 and λ 2 denote the smallest eigenvalues of D 2 g 1 and D 2 g 2 , respectively. Using a Taylor expansion for v 3 around π , we compute D 2 g 1 (1, /12). Thus, for ε small enough, λ 1 is positive on D ε by the assumptions on v 2 and v 3 . Consequently, for ρ small enough, depending only on v 2 and v 3 , we find a constant c G > 0 such that This implies that the minimizer of G is uniquely determined and, by the symmetry of G in the variables b 1 and b 2 , it has the form (b,b,θ). We conclude thatẼ cell is minimized exactly at (b,b, π ±ψ) withψ = √θ . The first order optimality condition for ε > 0 small, we getb < 1 by the assumptions on v 2 . Similarly, possibly taking ε small enough, we findψ ∈ (0, π/8).
Ad (ii). The smoothness of the cell energyẼ cell in a neighborhood of the minimizers follows directly from the assumptions on v 2 and v 3 . For brevity we set Since DG(T (d 0 )) = 0 by the first order optimality conditions, we obtain . This together with (20) and the fact that DT (d 0 ) = diag(1, 1, 2(ϕ − π)) yields (18) and concludes the proof.

Remark 3.2. (Smallness ofψ)
The proof shows thatψ → 0 as ρ → 0. In the following sections, we will frequently assume thatψ is small with respect to constants depending on v 2 , v 3 , and the closed interval M introduced before (6). This will amount to choosing ρ sufficiently small.
We conclude this section with the proof of Theorem 2.1.
Proof of Theorem 2.1. The statement follows immediately from Lemma 3.1 and (3) with the constant e cell =Ẽ cell (b,b, π +ψ).

The Single-Period Problem
The goal of this section is to consider chains y ∈ A n , n fixed and small, so that we expect minimizers to be represented by a wave consisting of one single period. In this section, we will only consider the reduced energy introduced in (3). We will first investigate the geometry and the length of configurations with minimal energy. Here, it will turn out that the analysis is considerably different for even and odd numbers of bonds. Afterwards, we study small perturbations of energy minimizers and show that the length excess can be controlled by the energy excess. y i0 Fig. 6. A single-period chain y ∈ U 6 4.1. Geometry and length of energy minimizers. We investigate the geometry and the length of configurations y ∈ A n with minimal energy, i.e., E red n (y) = (n − 2)e cell , see Theorem 2.1. Let n = l + 1, where l will stand for the discrete-wave period. Recall the definition of the bond lengths b i and the angles ϕ i in (2). Moreover, letb andψ be the values found in Lemma 3.1. By U l we denote the family of configurations y ∈ A l+1 such that the bond lengths coincide with that of minimizers of the cell energy, namely and such that there Note that, in particular, all configurations in U l are minimizers of E red l+1 . Moreover, given the index i 0 , the position of the points y ∈ U l is determined uniquely up to a rotation and a translation. In particular, the length of the chain, denoted by |y l+1 − y 1 |, is completely determined by the choice of i 0 .
To identify the length of the chain, we will frequently use the formulas for θ ∈ [0, 2π) which can be derived by using a geometric series argument and the representations cos We recall that the angle between two vectors a 1 , a 2 ∈ R 2 , measured counterclockwisely, is denoted by (a 1 , a 2 ). We define the maximal possible discrete-wave period by and show that configurations U l for l ≥ l max are not admissible.
Proof. Consider y ∈ U l . We first show that y / ∈ A l+1 if i 0 ≥ 2π/ψ − 1. Let j = 2π/ψ and θ = (e 1 , y 2 − y 1 ). We observe that j − 1 ≤ i 0 . Then we compute by (21), (22), and (23) where the last step follows fromb ≤ 1 (see Theorem 2.1) and Thus, the assumption in (1) is violated and therefore y / ∈ A l+1 . Likewise, we argue to Combining the two conditions on the choice of i 0 , we find that l Recall that the length of the chain |y l+1 − y 1 | is completely determined by the choice of i 0 from (22). Thus, we can interpret |y l+1 − y 1 | as a function of i 0 . More precisely, recalling also Lemma 4.1 we introduce where y ∈ U l ⊂ A l+1 is a configuration satisfying (22) for i 0 . The maximum of the function will be denoted by λ l max . Since the length is invariant under inversion of the order of the labels of the particles, we get λ l (i) = λ l (l − i + 1) for i ≤ l/2 .
After a rotation we may suppose that (y l+1 − y 1 ) · e 2 = 0 and (y l+1 − y 1 ) · e 1 > 0. In this case, letting for i = 1, . . . , l, we note that We now determine the maximizer of λ l . Proof. We argue by contradiction. Suppose that the maximum is attained by a configuration y ∈ U l with i 0 = l/2 , (l + 1)/2 . After a rotation we may assume that (y l+1 − y 1 ) · e 2 = 0 and observe that (28) holds. In view of (22), a short computation yields with the angles φ i defined in (27). Recall the symmetry λ l (i) = λ l (l − i + 1) for i ≤ l/2 , see right after (26).
We define a configurationȳ ∈ U l with index i 0 = i 0 − 1 (see (22)) andφ 1 = φ 2 , where we indicate the angles in (27) corresponding toȳ byφ i . Note that the configuration is characterized uniquely up to a translation. More precisely, we obtain By (28) and (29) this gives Consequently, the length |y l+1 − y 1 | is not maximal among all configurations in U l . This contradicts the assumption and shows that the maximum is attained The previous result shows that for even l ∈ 2N ∩ [2, l max ] the maximum of λ l is attained at i 0 = l/2, i 0 = l/2 + 1 and we call λ l max = λ l (l/2) the wavelength of a wave with discrete-wave period l. The following lemma provides the relation between wavelength and even discrete-wave periods. Odd discrete-wave periods have to be treated differently, cf. Lemma 4.8 below.
With the help of (23), we then indeed get λ l max = 2b sin(ψl/4)/ tan(ψ/2). Remark 4.4. The proof shows that a configuration y ∈ U l as in (27) and (28) which realizes the maximal length λ l max necessarily satisfies φ 1 , φ l ∈ {(l/4 − 1)ψ, lψ/4}. Let l mid = 6/ψ for brevity. In the following a distinguished role will be played by the normalized wavelength (normalized with respect to the number of bonds) : [2, l max ] → R, being the function which satisfies  Fig. 7. The fact that the function is affine on the intervals between two even discretewave periods will be crucial (a) to identify the function e range in Theorem 2.2 and (b) to give the characterization (17) in Theorem 2.4. Indeed, it will turn out that is the set of resonant lengths M res introduced in (10). We now study the properties of the normalized wavelength .  (30), and the fact that l midψ /4 ≤ 3/2, we obtain that is strictly decreasing and concave. Moreover, one can check that From this we deduce that (l) ≥ λ l max /l for all l ∈ 2N ∩ (l mid , l max ]. Moreover, note that (l) = λ l max /l for all l ∈ 2N ∩ [2, l mid ] by definition. Finally, by Lemma 4.3 we compute (2) =b cos(ψ/2) and ( 6/ψ ) = 2/3 sin(3/2)b + O(ψ), which shows that ([2, l mid ]) ⊃ (2/3,b cos(ψ/2)) for ρ (and thusψ, cf. Remark 3.2) sufficiently small.

Remark 4.6.
[Strict concavity of ] Clearly, as piecewise affine function, the normalized wavelength is not strictly concave. However, the strict concavity of x → sin(x)/x implies (νl 1 When we speak of strict concavity of in the following, we refer exactly to this property. Before we proceed with the case of odd discrete-wave periods, we briefly note that configurations U l can be connected to longer chains.  22), we find that all bonds and angles of y (see (2)) satisfy b i =b and ϕ i = π ±ψ. Thus, E red 2l+1 (y) = (2l − 1)e cell with e cell from Theorem 2.1. We now investigate in more detail the case of odd discrete-wave periods l ∈ 2N + 1. From Lemma 4.2 we get that the maximum of λ l is attained exactly for i 0 = (l + 1)/2. Without going into details, we remark that one can calculate forψ sufficiently small that This in particular shows that the normalized wavelength does not capture correctly the wavelength for odd l. We hence proceed here by remarking that, under suitable conditions, the length for two consecutive waves with odd atomic period can be controlled in terms of the lengths of waves with even atomic period. This will eventually allow us to control the wavelength in terms of the normalized wavelength also for odd l. More precisely, for odd l 1 , l 2 ∈ (2N+1)∩[2, l max ] we let y : {1, . . . , l 1 +l 2 +1} → R 2 be a configuration with (y 1 , . . . , y l 1 +1 ) ∈ U l 1 , (y l 1 +1 . . . , y l 1 +l 2 +1 ) ∈ U l 2 , and the junction angle ϕ l 1 +1 − π =ψ (see (2)). In view of (22), we find (y 1 , . . . , y l 1 +2 ) ∈ U l 1 +1 , (y l 1 +2 . . . , y l 1 +l 2 +1 ) ∈ U l 2 −1 . Consequently, by the definition of the function λ l in (26) and the triangle inequality we obtain This estimate can be obtained also for more general junction angles as the following lemma shows.
Note that the right-hand side of (33) is well defined in the sense that l 1 +t, l 2 −t ≤ l max for t ∈ {−1, 1} since l 1 , l 2 ≤ l max and l max is even (see (24)). Notice that in contrast with the discussion before (32), the chains are connected at point y l 1 +2 .
Proof. Let y be given as in the assumption. After a rotation we may suppose that (y l 1 +l 2 +1 − y 1 ) · e 2 = 0. Similarly to (27), we define the angles φ i , where the sum now runs from 1 to l 1 + l 2 . As ϕ l 1 +2 − π ∈ (1 + 2Z)ψ, we get As (y 1 , . . . , y l 1 +2 ) ∈ U l 1 +1 and (y l 1 +2 . . . , y l 1 +l 2 +1 ) ∈ U l 2 −1 , we derive similarly to (32) This shows (33) for l 2 > l 1 . From now on we suppose l 1 ≥ l 2 . In order to conclude the proof, it suffices to show the strict inequality Indeed, since the number of different admissible configurations (up to rigid motions) and the number of different l 1 , l 2 is bounded by a number only depending on l max , we obtain the statement for a positive constant c mix , which only depends on l max (and thus only on ρ). It remains to show (35). First, suppose that l 1 − l 2 ≥ 2. Then we use Lemma 4.3, (24), and the strict concavity of sin on [0, π] to get If now l 1 = l 2 , we assume by contradiction that the inequality in (35) was not strict. Equality would imply (y l 1 +2 − y 1 )·e 2 = (y l 1 +l 2 +1 − y l 1 +2 )·e 2 = 0, i.e., the two parts of the chain, lying in U l 1 +1 and U l 2 −1 , respectively, satisfy (27) and (28). But then Remark 4.4 Since l 1 = l 2 , we obtain a contradiction to (34). This establishes (35) and concludes the proof.

Small perturbations of energy minimizers.
In this section, we investigate the length of single periods for configurations being small perturbations of energy minimizers. To this end, we introduce the set of small-perturbed chains where, as before, the angles ϕ i andφ i corresponding to y andȳ, respectively, are defined in (2). Likewise, the bond lengths will again be denoted by b i . (For the angles the sum runs only from 2 to l.) In the following, we use the notation (a) 2 + = (max{a, 0}) 2 for a ∈ R and the quantity e cell from Theorem 2.1. Recall also l max defined in (24). We first treat the case of even discrete-wave periods. whereȳ ∈ U l is a configuration corresponding to y as given in the definition of U l δ .
Proof. Let y ∈ U l δ andȳ ∈ U l be given. By Lemma 3.1 and a Taylor expansion we get for some c > 0 for all i = 2, . . . , l, where the last step follows with the definition of U l δ and the choice cδ 0 ≤ c conv /4. By (3) and Jensen's inequality we get For i = 1, . . . , l we let φ i andφ i be the angles defined in (27), associated to y andȳ, respectively. Possibly after rotations, it is not restrictive to suppose that (y l+1 −y 1 )·e 2 = 0 and that φ 1 =φ 1 . Clearly, we get Since, cos is Lipschitz with constant 1, we then derive for each i = 1, . . . , l where we also used the fact thatb < 1. We now get which together with (38) and the choice C = c conv /(4(2l −1)l 2 ) gives the first inequality of the statement. The second inequality follows from Lemma 4.5.
We close this section with the observation that also for configurations in U l δ the maximal discrete-wave period is given by l max .

The Multiple-Period Problem
In this section, we study the relation between length and energy for a chain consisting of more than one single discrete-wave period. More precisely, we will investigate configurations belonging to for δ > 0 to be specified below, where the bond lengths b i and angles ϕ i are defined in (2). (As before, for the angles indices run only from 2 to n − 1.) For later purpose, we note that by Lemma 3.1,(ii) we have for c conv = c conv (ρ) > 0 and δ ≤ δ 0 with δ 0 from Lemma 4.9, cf. (37) for the exact computation. We split our considerations into two parts concerning the reduced and the general energy, respectively.

The multiple-period problem for the reduced energy. We introduce the index set
The index set is denoted by 'sgn' to highlight that at the points y i , i ∈ I sgn , the sign of ϕ i − π changes from plus to minus. For the application in Sect. 6 it is convenient to also take the index i = 1 into account. Sometimes we will also consider the 'shifted' index set We also define a decomposition of I sgn by for l ∈ N, l ≥ 2. For a minimizer y of E red n , the length of the waves corresponding to even discretewave periods (I l sgn ) l∈2N can be estimated by where we used Lemma 4.5. In the previous section, see particularly Lemma 4.8, we have also seen that the length of waves with odd discrete-wave period can be controlled in terms of waves with even discrete-wave period. For later purpose, we introduce the maximal length of odd discrete-wave periods L : 2N + 1 → (0, ∞) by Recall the definition of the maximal discrete-wave period l max in (24). For convenience, we introduce also a relabeling I sgn ∪ {n} = {i 1 , . . . , i J } for an increasing sequence of integers (i j ) J j=1 . The following lemma controls (up to some boundary effects) the length of the chain in terms of the contributions of waves with even and odd discrete-wave periods.
Lemma 5.1 (Length of chain with minimal energy). Let y : {1, . . . , n} → R 2 with y ∈ A n be a minimizer of E red n . Then where c mix > 0 is the constant from Lemma 4.8.
Proof. Consider the labeling I sgn ∪ {n} = {i 1 , . . . , i J }. Moreover, we choose indices j 1 < j 2 < . . . < j K such that l∈2N+1 I l sgn = {i j 1 , . . . , i j K }. Note that K = l∈2N+1 #I l sgn . In the following, we will consider pairs of indices i j k , i j k+1 for odd k with corresponding lengths l k 1 = i j k +1 − i j k and l k 2 = i j k+1 +1 − i j k+1 . We will suppose that The case #K < K /2 is very similar by considering the chain in reverse order. We indicate the necessary adaptions at the end of the proof. Consider a pair of indices i j k , i j k+1 for odd k. Recall that l k 1 = i j k +1 − i j k , l k 2 = i j k+1 +1 − i j k+1 are odd and l m,k := i j k +m+1 − i j k +m are even for all m ∈ {1, . . . , M k − 1}, where M k := j k+1 − j k . We can decompose the chain (y i j k , . . . , y i j k+1 +1 ) into the partŝ Here, we have used Theorem 2.1 and the fact that i j k +m + 1 ∈ I sgn (cf. (41)-(42)) to see that the chains have the form introduced in (21)- (22). (We refer to Fig. 8 for an illustration of composed single-period waves.) We also define the configurationỹ k : By the definition of the function λ l in (26) and the triangle inequality we get From Theorem 2.1 we recall that each angle ϕ i (see (2)) enclosed by two bonds is π +ψ or π −ψ. Due to the fact that the discrete-wave periods l m,k for m ∈ {1, . . . , M k − 1} are even, we find (i j k+1 + 1) − (i j k +1 + 1) ∈ 2N. Thus, i.e., the junction angleφ l k 1 +2 atỹ l k 1 +2 satisfiesφ l k 1 +2 − π ∈ (1 + 2Z)ψ. Consequently, we can apply Lemma 4.8 and find together with (48) Here, we have also used that the discrete-wave periods l k 1 + t, l k 2 − t, and l m,k are even and have applied Lemma 4.5.
We proceed in this way for all k ∈ {1, 3, . . . , 2 K /2 − 1} and then we derive by (45), (49), Lemma 4.5, and the triangle inequality where here and in the following the sum in k always runs over {1, 3, . . . , 2 K /2 − 1}. Note that the last three terms appear since Lemma 4.5 and the estimate (49) are possibly not applicable. (The very last term is only necessary for odd K .) However, in view of y ∈ A n , Lemma 4.1, and the fact thatb ≤ 1 (see Theorem 2.1), their contribution can be bounded by 3l max . Moreover, note that K /2 ≥ l∈2N+1 #I l sgn /2−1 and c mix ∈ (0, 1). To conclude, it therefore remains to show For each k choose t k ∈ {−1, 1} such that the maximum is attained. If K is even, we set r l = #{k | l k We then find l∈2N r l = K = l∈2N+1 #I l sgn and l∈2N lr l = 2 l∈2N+1 l#I l sgn /2 . Then (50) follows from (44). Finally, we briefly indicate the necessary changes if K − #K ≥ K /2 (see (45)). In this case, we consider the chainŷ = (y n , . . . , y 1 ) in reverse order and note that the index set introduced in (41) corresponding toŷ is given by I sgn ∪ {n} (as defined in (42) for the configuration y). The above reasoning is then applied on the pairs of indices i j k+1 + 1 and i j k + 1 for k ∈ {1, 3, . . . , We now investigate the length of general configurations in A δ n . Recall the notation (a) 2 + = (max{a, 0}) 2 for a ∈ R. Lemma 5.2 (Energy excess controls length excess) There exist δ 0 > 0, c odd > 0, and c el > 0 only depending on ρ such that for all 0 ≤ δ ≤ δ 0 and each y ∈ A δ n we have where I l sgn as in (43) and n odd = l∈2N+1 #I l sgn l.
Proof. Let y ∈ A δ n be given and define I sgn and I l sgn as in (41) and (43), respectively. Choose a configurationȳ : {1, . . . , n} → R 2 minimizing the energy E red n and satisfying sgn(ϕ i −π) = sgn(φ i −π) for i = 2, . . . , n −1, where sgn denotes the sign function and ϕ i are the angles defined in (2) corresponding toȳ. Note thatȳ is determined uniquely by y up to a rigid motion.
We will follow the lines of the previous proof by taking additionally the deviation from energy minimizers into account, where we will employ Lemma 4.9 and Lemma 4.10. Similarly to the previous proof, we consider the labeling I sgn ∪ {n} = {i 1 , . . . , i J } as well as l∈2N+1 I l sgn = {i j 1 , . . . , i j K }. For odd k we also define l k 1 = i j k +1 − i j k and l k 2 = i j k+1 +1 − i j k+1 . Moreover, let K be defined as in (45). Without loss of generality we can reduce ourselves to the case #K ≥ K /2 since otherwise one may consider the chain in reverse order, as commented at the end of the previous proof.
We consider the parts of the chain having odd discrete-wave period. For odd k, we define the configurationỹ k : {1, . . . , l k 1 + l k 2 + 1} → R 2 as in (47). Accordingly, we define the configurationỹ k corresponding toȳ. By the triangle inequality (cf. (48)) we get that whereŷ m,k is defined in (46). We now estimate the various terms in the above right-hand side by starting with the terms includingŷ m,k . By Lemma 4.9, Hölder's inequality, and the fact that with C > 0 from Lemma 4.9, where in the last step we have used Theorem 2.1. We now consider the first term in the right-hand side of (51). The difference of the junction anglesφ k , respectively, can be estimated by Consequently, applying Lemma 4.10 and summing over all k ∈ {1, 3, .
Repeating the arguments in (48)-(50), in particular using Lemma 4.8 for |ỹ k For brevity we set E = E red n (y) − (n − 2)e cell . Combining the previous estimate with (51) and (52), and using again Hölder's inequality together with (40), we get k odd For the remaining parts with even period we repeat the argument in (52). All in all we get where the last three terms appear since Lemma 4.9 is possibly not applicable on these parts of the chain. (The very last term is only necessary for odd K .) Similarly to the proof of Lemma 5.1, by Lemma 4.11 we can show that |y i 2 − y 1 | + |y n − y i J −1 | + |y i j K +1 − y i j K | ≤ 3(b + δ)l max ≤ 3l max for δ 0 sufficiently small. Therefore, using also K /2 ≥ l∈2N+1 #I l sgn /2 − 1 and c mix ∈ (0, 1) we get where c el := (2/ √ C + 4l max / √ c conv ) −2 . Now choose δ 0 so small that 2l 2 max δ ≤ c mix /4 and set c odd = c mix /(4l max ). This implies (c mix /2−2l 2 max δ)K ≥ c odd l max K ≥ c odd n odd , where the last step follows from n odd /l max ≤ l∈2N+1 #I l sgn = K . From this, the statement of the lemma follows.

The multiple-period problem for the general energy.
Let y ∈ A n and observe that the general energy (7) including the longer-range interaction can be written as where E gen cell denotes the generalized cell energy defined by Let y ∈ A n be a minimizer of E red n . Choose i ∈ {1, . . . , n − 3} with sgn(ϕ i+1 − π) = sgn(ϕ i+2 − π), where sgn denotes the sign function, and define e gen cell := E gen cell (y i , y i+1 , y i+2 , y i+3 ).
Clearly, the value is independent of the particular choice of the configuration y and the index i. Recalling Theorem 2.1, we also see |e As before, the value is independent of y and the choice of i. We find e per > 0, which follows from the geometry of the four points y i , y i+1 , y i+2 , y i+3 determined by the condition sgn(ϕ i+1 − π) = sgn(ϕ i+2 − π) and sgn(ϕ i+1 − π) = sgn(ϕ i+2 − π), respectively, and the fact that v 2 is strictly increasing right of 1. (We refer to Fig. 9 for an illustration.) Recall the definition of I sgn in (41). The general energy (53) for a configuration y ∈ A n with E red n (y) = (n − 2)e cell can now be estimated by where we used that #{i = 1, We now formulate the main result of this section about the relation between the reduced and the general energy. Notice that the higher order error term nc rangeρ 2 appears since due to the longerrange interactions, the energy can be slightly decreased by small rearrangement of the particles. Note that Lemma 5.3 together with Lemma 5.2 allows to control the length excess in terms of the energy excess for the general energy.
Proof. Let y ∈ A δ n be given. Exactly as in the proof of Lemma 5.2, we choose a configurationȳ : {1, . . . , n} → R 2 minimizing the energy E red n and satisfying sgn(ϕ i − π) = sgn(φ i − π) for i = 2, . . . , n − 1, where ϕ i ,φ i are the angles defined in (2) corresponding to y andȳ, respectively. Denote the bonds introduced in (2) again by b i andb i . Recall that the energy E n (ȳ) can be estimated by (56). Using (40) and Theorem 2.1 we find For each i ∈ {1, . . . , n − 3} an elementary computation shows for some c > 0. (The argument is very similar to the one in (39) and we therefore omit the details.) Consequently, we find a constantC > 0 only depending on v 2 such that Minimizing the last expression amounts to choosing each |b i −b i | and |ϕ i −φ i |, equal to 4Cρ/c conv . Thus, we deduce This together with (56) yields the claim for c range = 4C 2 /c conv .

Proof of the Main Result
In this section we give the proofs of our main results Theorems 2.
As announced right after Theorem 2.3, a chain involving waves with different discretewave periods (and wavelengths) can be energetically more favorable, even for μ ∈ M res . Consequently, almost minimizers cannot be expected to be periodic, but only essentially periodic, i.e., periodic up to a small set of points, see (14). We close this section with an example in that direction and show that the upper bound can be improved in terms of the higher order error nρ 2 . Recall (11) and (12).

Lower bound and characterization of minimizers.
We first derive the lower bound for the minimal energy (8). Afterwards, based on the lower bound estimates, we will provide the characterization of configuration with (almost) minimal energy.
Outline of the proof: In Step 1 we identify the set of defects consisting of particles where the cell energy deviates too much from the minimum. We will see that on the complement of the defect set the results from Sect. 5 are applicable. In this context, we partition the chain into various regions associated to even and odd discrete-wave periods, where the periods l and l will play a pivotal role. In Step 2 we estimate the length of the various parts, particularly using the concavity of the mapping (see (30)). In Step 3 we provide estimates for the energy of the chain and based on Lemma 5.2, Lemma 5.3, we derive relations between length and energy. Finally, in Step 4 we show that it is energetically convenient if the chain consists (almost) exclusively of waves with discrete-wave period l and l , from which we can deduce the statement.
Step 1: Partition of the chain. Choose 0 < δ ≤ δ 0 with δ 0 being the minimum of the constants given in Lemma 4.9, Lemma 4.11, Lemma 5.2, and Lemma 5.3. We note that δ 0 only depends on the choice of v 2 , v 3 , and ρ, but is independent ofρ. Below in (84) and (93) we will eventually chooseρ sufficiently small in terms of δ 0 whose choice then only depends on v 2 , v 3 , and ρ.
Define the index set of defects by withb andψ from Theorem 2.1. We introduce the set and the labeling Notice that in the parts of the chain between indices I * def we will be in the position to apply our results from Sect. 5. In particular, denotes the indices of the first particles of these parts of the chain. Similarly to (41), we let We also introduce a decomposition of I sgn by (compare to (43)) for l ∈ N, l ≥ 2. We will also use the notation for i j ∈ I wave . As i j − 1 / ∈ I sgn for i j ∈ I * def and i j / ∈ I sgn for i j ∈ I * def \ I wave , we get i j ∈I wave I sgn, j = I sgn and i j ∈I wave I l sgn, j = I l sgn . Moreover, we introduce the number of particles related to different discrete-wave periods by n good = #I l sgn l , n good = #I l sgn l , n good = n good + n good , N good = #(I l sgn ∪ I l sgn ) We indicate the waves with discrete-wave period l and l as good since they are expected to appear in a configuration minimizing the energy (7), cf. Theorem 2.4. On the other hand, the other even discrete-wave periods are called bad. We also recall that in Sects. 4 and 5 we have seen that waves with odd discrete-wave period have to be treated in a different way. Below we will show that the numbers n odd , n bad , and n def are negligible with respect to n good . From the definitions in (64), (65), and (67) we also get Finally, we introduce the mean discrete-wave periods associated to the different parts. First, for the even discrete-wave periods we set l good * = n −1 good n good l + n good l , On the other hand, for the odd discrete-wave periods we define where (r l ) l∈2N is some admissible sequence in (44) with l∈2N r l l (l) = L (#I l sgn ) l∈2N+1 .
Step 2: Length of the chain. We consider the indices I sgn and estimate the length of the various contributions related to 'good', 'bad', and 'odd' discrete-wave periods. We start with the bad discrete-wave periods. Using the fact that is concave (see Lemma 4.5) and applying Jensen's inequality, we deduce by (71) We now address odd discrete-wave periods. Recalling the definition of the maximal length of odd discrete-wave periods in (44) and using (68), we derive Recall that is concave. Then from (72), the fact that l∈2N lr l ≤ n odd (see (44) and (69b)), and Jensen's inequality we get Now combining (73)-(75) and using (68) as well as n good + n bad + n odd ≤ n − 1 (see (70)) we derive We close this step with an estimate about the contribution of I def . Recall the definitions of I * def and I wave in (64)-(65). For each j ∈ {1, . . . , J − 1}, we define In view of the boundary conditions (y n − y 1 ) · e 1 = (n − 1)μ (see (6)) and the fact that the length of each bond is bounded by 3/2 (see (1)), we find by (70) Step 3: Energy estimates. First, recalling (7), (64)-(65) and defining n j = i j+1 − i j + 1 ≥ 3 for i j ∈ I wave , we get where we used that, by the decomposition at each defect two longer-range contributions are neglected and v 2 ≥ −1. We consider the first sum. In view of the fact that the cell energyẼ cell is minimized exactly for (b,b, π +ψ) and (b,b, π −ψ) by Lemma 3.1, (63) implies for j ∈ {2, . . . , J − 1} for a constant c def = c def (δ) > 0. As δ depends only on ρ, also c def depends only on ρ.
On the other hand, if i j ∈ I wave , we can apply Lemma 5.3 and get where for brevity we have set E red j = E red n j (y i j , . . . , y i j+1 ) − (n j − 2)e cell . Here, we have also used that the set I sgn, j coincides with the one considered in Sect. 5, see (41) and (68).
Our goal is to estimate the sum in (79). As a preparation, we recall that |e cell −e gen cell | ≤ cρ, as observed below (54), and we calculate Recalling n j = i j+1 −i j + 1, by an elementary computation, using (65) and (70), we find that i j ∈I wave (n j − 2) = (n − 1) − #{i ∈ I * def | i + 1 ∈ I * def } − #I wave = n − 2 − n def . Thus, we obtain i j ∈I wave (n j − 3)e gen cell + e cell + n def e cell ≥ (n − 2)e gen cell − (2n def + 1)cρ, where we used that #I wave ≤ n def + 1, see (64) As c def = c def (ρ) and c range = c range (ρ) are independent ofρ, we can selectρ so small that the last term in the first line can be bounded from below by n def c def /2 + n defρ e per . Thus, we derive for c rest = 2c rangeρ + 3e per + c. The next steps will be to derive suitable lower bounds for 2#I sgnρ e per and i j ∈I wave E red j /2 . We first estimate the latter. Recalling (68), (77), and the definition of E red j in (81), we find by Lemma 5.2 where n j odd = l∈2N+1 #I l sgn, j l. Here, we have again used that the sets I sgn, j and I l sgn, j coincide with the ones considered in Sect. 5. By a computation similar to the one before (82), using #I wave ≤ n def +1, we get i j ∈I wave n j = n −2−n def +2#I wave ≤ n +n def ≤ 2n. Then, taking the sum over all i j ∈ I wave and using Jensen's inequality, we derive Consequently, in view of (76), (78), and #I wave ≤ n def + 1, we find i j ∈I wave E red j ≥ c el 2n (n − 1)μ − 3/2(n def + 1) − L + n bad c + n odd c odd − 4(n def + 1)l max where we have also used i j ∈I wave n j odd = n odd , see (68) and (69b). We now consider the term 2#I sgnρ e per . Recall that ϒ, defined in (57), is convex. By Jensen's inequality we compute with (69c) and (71) where ϒ (l * ) denotes the right derivative of ϒ at l * . Likewise, for the good discrete-wave periods using (69a), (71), and the fact that that ϒ is affine on [l , l ] we obtain 2N good = 2#(I l sgn ∪ I l sgn ) = n good ϒ(l ) + n good ϒ(l ) = n good ϒ n −1 good (n good l + n good l ) = n good ϒ(l good * ) ≥ n good ϒ(l * ) + ϒ (l * )(l good * − l * ) .
We close with the characterization of almost minimizers.