Exact periodic stripes for a minimizers of a local/non-local interaction functional in general dimension
EExact periodic stripes for minimizers of a local/nonlocal interactionfunctional in general dimension
Sara Daneri ∗ and Eris Runa † FAU Erlangen–N¨urnberg Max Planck Institut f¨ur Mathematik in den Naturwissenschaften, Leipzig
Dedicated to Prof. Stephan Luckhaus on the occasion of his 65th birthday.
Abstract
We study the functional considered in [25, 26, 28] and a continuous version of it, analogousto the one considered in [30]. The functionals consist of a perimeter term and a nonlocal termwhich are in competition. For both the continuous and discrete problem, we show that theglobal minimizers are exact periodic stripes. One striking feature of the functionals is that theminimizers are invariant under a smaller group of symmetries than the functional itself. In thecontinuous setting, to our knowledge this is the first example of a model with local/nonlocalterms in competition such that the functional is invariant under permutation of coordinatesand the minimizers display a pattern formation which is one-dimensional. Such behaviour for asmaller range of exponents in the discrete setting was already shown in [28].
In this paper we study a discrete local/nonlocal functional considered in a series of papers byGiuliani, Lebowitz, Lieb and Seiringer (cf. [25, 26, 28]) and a continuous version of it.The discrete functional is the following: given E ⊂ Z d , d ≥ L ∈ N ˜ F dsc J,L ( E ) := 1 L d (cid:16) J (cid:88) x ∈ [0 ,L ) d ∩ Z d (cid:88) y ∼ x | χ E ( x ) − χ E ( y ) | − (cid:88) x ∈ [0 ,L ) d ∩ Z d , y ∈ Z dx (cid:54) = y | χ E ( x ) − χ E ( y ) || x − y | p (cid:17) , (1.1)where J is a positive constant, p ≥ d + 2, y ∼ x if x and y are neighbouring points in the lattice,and χ E ( x ) := (cid:40) x ∈ E ,0 otherwise. ∗ [email protected] † [email protected] a r X i v : . [ m a t h . A P ] J un n the continuous setting we consider the following functional: for E ⊂ R d , d ≥ L > F J,L ( E ) = 1 L d (cid:16) J Per ( E, [0 , L ) d ) − ˆ [0 ,L ) d ˆ R d | χ E ( x ) − χ E ( y ) | K ( x − y ) d y d x (cid:17) , (1.2)where J is a positive constant,Per ( E, [0 , L ) d ) := ˆ ∂E ∩ [0 ,L ) d | ν E ( x ) | d H d − ( x ) , | z | = d (cid:88) i =1 | z i | , with ν E ( x ) exterior normal to E in x , is the 1-perimeter of E and K is a kernel. The generalassumptions on K will be specified in (2.3)-(2.6). One particular kernel in this class is K ( ζ ) = 1( | ζ | + 1) p . (1.3)Both of the discrete and the continuous models describe systems of particles in which there is ashort-range attracting force (the perimeter) and a repulsive long-range force (nonlocal term).The behaviour of the functional is very similar to the one considered in [30], namely the kernelhas the same scaling, C | ζ | +1) p ≤ K ( ζ ) ≤ C | ζ | +1) p , and retains the same symmetries as thefunctional considered in [30], but (1.3) has the advantage that, due to the positivity of the inverseLaplace transform of ζ d (cid:55)→ ´ R d − K ( ζ , . . . , ζ d ) d ζ · · · d ζ d − , the reflection positivity technique canbe applied (see Section 4).The aim of this paper is to study the structure of the minimizers of the discrete and continuousfunctionals.In order to make our problem well-posed we impose periodic boundary conditions, namely werestrict the functional to [0 , L ) d -periodic sets. However, as we will see from the statement ofTheorems 1.4 and 1.5, due to the fact that our result is independent of L , this will not be arestriction.For both the discrete and continuous problem, there exists a critical constant J dsc c ( J c respectively)such that if J > J dsc c (respectively J > J c ), then the global minimizers are trivial, namely eitherempty or the whole domain.The critical constants J dsc c and J c are (as proven in [25] for the discrete setting and in [30] for thecontinuous case) J dsc c := (cid:88) y > , ( y ,...,y d ) ∈ Z d − y ( y + · · · + y d ) p/ and J c := ˆ R d | ζ | K ( ζ ) d ζ. (1.4)When J = J dsc c − τ (resp. J = J c − τ ) with 0 < τ ≤ ¯ τ for some ¯ τ > { e i } di =1 . A unionof stripes in the continuous setting is a [0 , L ) d -periodic set which is, up to Lebesgue null sets, of theform V ⊥ i + ˆ Ee i for some i ∈ { , . . . , d } , where V ⊥ i is the ( d − e i and ˆ E ⊂ R with ˆ E ∩ [0 , L ) = ∪ Nk =1 ( s i , t i ). A union of stripes is periodic if ∃ h > ν ∈ R s.t.ˆ E ∩ [0 , L ) = ∪ Nk =0 (2 kh + ν, (2 k + 1) h + ν ). In the following, we will also sometimes call unions ofstripes simply stripes. 2he family of unions of stripes will be denoted by S , and the family of unions of stripes which are[0 , L ) d -periodic will be denoted by S per L .In the discrete setting the concept of union of stripes is the same, up to intersecting with thediscrete lattice and considering h, L ∈ N . As for periodic stripes, the width and distance of theintersection of the continuous stripes with the discrete lattice must be the same. In particular, ifthe period L is a prime number there can be no periodic stripes of period less than L , thus theonly [0 , L ) d -periodic stripes in the discrete setting are Z d and ∅ .We will show that the conjecture on the structure of minimizers holds both for the discrete andcontinuous setting. For the discrete setting, a different proof was already given in [28] for thesmaller range of exponents p > d .We would like to point out that our method applies if, instead of periodic boundary conditions, weimposed optimal periodic Dirichlet boundary conditions, namely asking that the sets E are, outside[0 , L ) d , periodic unions of stripes of optimal period. We prefer periodic boundary conditions sincethe problem is invariant under coordinate exchange, and thus we are not preselecting a particulardirection.If one optimizes among periodic unions of stripes, then unions of stripes with optimal energy for ˜ F dsc J,L (respectively ˜ F J,L ) have width and distance of order τ − / ( p − d − and energy of order τ ( p − d ) / ( p − d − .Thus it is natural to rescale the functional in such a way that the width and the energy of the stripesare of order O (1) for τ small. Letting β := p − d −
1, the rescaling is the following: x := τ − /β ˜ x, L := τ − /β ˜ L and ˜ F J,L ( E ) := τ ( p − d ) /β F τ, ˜ L ( ˜ E ) . (1.5)In particular, notice that since the minimizers of the rescaled functional F τ,L are unions of stripes,also the minima of ˜ F J,L with J = J c − τ are unions of stripes.Making the substitutions in (1.5), letting also ζ = τ − /β ˜ ζ and in the end dropping the tildes, onehas that the rescaled functional in the continuous setting is given by F τ,L ( E ) = 1 L d (cid:16) − Per ( E, [0 , L ) d ) + ˆ R d K τ ( ζ ) (cid:104) ˆ ∂E ∩ [0 ,L ) d d (cid:88) i =1 | ν Ei ( x ) || ζ i | d H d − ( x ) − ˆ [0 ,L ) d | χ E ( x ) − χ E ( x + ζ ) | d x (cid:105) d ζ (cid:17) , (1.6)where K τ ( ζ ) = τ − p/β K ( ζτ − /β ).In the discrete setting, for E ⊂ κ Z d [0 , L ) d -periodic, where κ = τ /β and L is a multiple of κ , let˜ E κ := (cid:91) i ∈ E (cid:16) i + [ − κ/ , κ/ d (cid:17) . Per ,κ ( E, [0 , L ) d ) := (cid:88) x ∈ [0 ,L ) d ∩ κ Z d (cid:88) y ∼ x | χ E ( x ) − χ E ( y ) | κ d − . (1.7)Then, the rescaled functional has the form F dsc τ,L ( E ) = 1 L d (cid:16) − Per ,κ ( E, [0 , L ) d ) + (cid:88) ζ ∈ κ Z d K dsc κ ( ζ ) (cid:104) ˆ ∂ ˜ E κ ∩ [0 ,L ) d d (cid:88) i =1 | ν ˜ E κ i ( x ) || ζ i | d H d − ( x ) − ˆ [0 ,L ) d | χ ˜ E κ ( x ) − χ ˜ E κ ( x + ζ ) | d x (cid:105)(cid:17) , (1.8)3here K dsc κ ( ζ ) = κ d | ζ | p .For fixed τ >
0, consider first for all
L > F τ,L on [0 , L ) d -periodicstripes (denoted above by S per L ) and then the minimal among these values as L varies in (0 , + ∞ ).We will denote this value by C ∗ τ , namely C ∗ τ := inf L> inf E ∈S per L F τ,L ( E )By the reflection positivity technique (see Section 4), this value is attained on periodic stripes ofwidth and distance h ∗ τ > C ∗ , dsc τ and h ∗ , dsc τ .In general, both for the discrete and the continuous setting, one does not expect h ∗ τ and h ∗ , dsc τ tobe unique (for the discrete setting see [28]).For the continuous setting, our main theorems are the following.In our first theorem, we show that for kernels satisfying assumptions (2.3)-(2.6), the period h ∗ τ isunique, provided 0 < τ ≤ ˆ τ with ˆ τ > Theorem 1.1.
Let d ≥ , p ≥ d + 2 and a family of kernels { K τ } satisfying (2.4) - (2.6) . Thenthere exists ˆ τ > s.t. whenever < τ < ˆ τ , h ∗ τ is unique. In the next two theorems, we deal with the occurrence of pattern formation for F τ,L . Theorem 1.2.
Let d ≥ , p ≥ d + 2 , L > . Then there exists ¯ τ > such that ∀ < τ ≤ ¯ τ thereexists h τ,L such that the minimizers of F τ,L are periodic stripes of width and distance h τ,L . The next theorem shows that h τ,L is close to h ∗ τ whenever L is large. Theorem 1.3.
There exists a constant C such that for every < τ ≤ ¯ τ , one has that the width h τ,L of a minimizer of F τ,L satisfies | h ∗ τ − h τ,L | ≤ CL . (1.9)In Theorem 1.2 the constant ¯ τ depends on L . One expects ¯ τ to be independent on L . In thisrespect, when L is of the form L = 2 kh ∗ τ , the independence is shown in Theorem 1.4, namely τ does not depend on L if L = 2 kh ∗ τ . Theorem 1.4.
Let d ≥ , p ≥ d + 2 and h ∗ τ be the optimal stripes’ width for fixed τ . Then thereexists τ , such that for every τ < τ , one has that for every k ∈ N and L = 2 kh ∗ τ , the minimizers E τ of F τ,L are optimal stripes of width h ∗ τ . Notice that the periodic boundary conditions were imposed in order to give sense to the functionalwhich is otherwise not well-defined. If one is interested to show that optimal periodic stripes ofwidth and distance h ∗ τ are ”optimal” if one varies also the periodicity, then it is not difficult to seethat Theorem 1.3 is sufficient. This corresponds to the ”thermodynamic limit” and is relevant inphysics.For the discrete setting, choosing now h ∗ , dsc τ equal to one of the admissible optimal periodic widthsfor τ > heorem 1.5. Let d ≥ , p ≥ d + 2 and h ∗ , dsc τ be an optimal width for the optimal periodic stripes.Then there exists τ > s.t. ∀ < τ < τ and L = 2 kh ∗ , dsc τ , k ∈ N , the minimizers of F dsc τ,L areperiodic stripes of width h ∗ , dsc τ . As already noticed, in the discrete for a union of stripes E to be [0 , L ) d -periodic of width anddistance h one has that L/h ∈ N . Given that h ∈ τ /β Z this is not always possible. Therefore,there can not be an analogous statement to Theorem 1.2 and Theorem 1.3. The competition between short-range and long-range forces is at the base of pattern formation inmany areas of physics and biology (see e.g. [18, 31, 35, 13, 41]).In dimension d = 1, there are many instances in which pattern formation is rigorously shown(see e.g. [39, 14, 21]).However, in dimension d ≥
2, showing pattern formation is a rather difficult problem which isrigorously solved in very few models: in the discrete case, to our knowledge, only in [28, 45, 9],and, in the continuous setting, in [27]. On the general issue of crystallization see [8].In the continuous setting, the closest and most famous model is the sharp interface version of theOhta-Kawasaki [40] model. This functional is well-studied (e.g. [15, 17, 33, 2, 41, 39, 16, 29, 34, 38]).Even though periodic pattern formation is expected due to physical experiments and numericalsimulations (e.g. [44, 15]), the problem is still open.One of the main difficulties is that the minimizers are invariant under a smaller group of symmetriesthan the functional itself. This is sometimes called breaking of symmetry. In the continuous setting,to our knowledge this is the first example of a model with local/nonlocal terms in competition suchthat the functional is invariant under permutation of coordinates and the minimizers display apattern formation which is one-dimensional. Such behaviour for a smaller range of exponents inthe discrete setting was already shown in [28].As already stated in the beginning of the introduction, in this paper we are considering questionsthat have already been studied in a series of papers in [21, 22, 25, 26, 27, 28]. For this reason wewould like discuss some of the similarities and differences to the most recent paper [28]. In [28],the discrete setting is considered. It can be shown that their setting is equivalent to ours.In the smaller range of exponents p > d , they prove a result similar to Theorem 1.5. We improvetheir result to the range of exponents p ≥ d + 2. Our results can be viewed as progress towards theaim of proving pattern formation for the more “physical” exponents. Among them we recall the caseof thin magnetic films ( p = d + 1 see e.g. [44, 21]), 3D micromagnetics ( p = d see e.g. [37, 43, 32])and diblock copolymers ( p = d − E ⊂ R d and not E ⊂ Z d . For this reason one needs to find a decomposition of the functionalinto terms that measure in a certain sense how much a minimizer deviates from being a union ofstripes. Such quantities have been introduced in [30] (see Section 2). The penalization for not beinga union of stripes is expressed through the rigidity estimate (see Proposition 3.2). The two-scaleapproach, although widely used in applied analysis, in this context appeared for the first timein [27].A second common point to [28] is the use of the technique of reflection positivity, which wasintroduced in the context of quantum field theory in [42] and applied for the first time to statisticalmechanics in [20]. For the first appearance of the technique in models with short-range and Coulombtype interactions see [19]. For further generalizations to one-dimensional models see [21, 23, 24]and for applications to two-dimensional models see [22, 27]. Such technique allows to show thatminimal stripes must be periodic.In [30], for a smaller range of exponents ( p > d instead of p ≥ d + 2) a rigidity estimate was shown,leading to prove that minimizers of F τ,L converge in L to periodic stripes as τ ↓
0. In the presentpaper, we show that pattern formation really appears not only for τ tending to 0 (as was done in[30]) but for a positive fixed τ , in the range p ≥ d + 2. For this we need a new rigidity argumentand a stability result (namely, Lemma 6.1). In the rigidity estimate we use a result of [10] (seeSection 3). Moreover, we show that in case L is an even multiple of the optimal period h ∗ τ , such τ does not depend on how big L is.M. Goldman, B. Merlet and V. Millot communicated to us that they have an alternative proof ofProposition 3.2. This paper is organized as follows: in Section 2, we explain the setting and some preliminary resultthat will be used in the following; in Section 3 we improve the key estimate of [30], namely therigidity estimate, to exponents p ≥ d + 2 (See Theorem 2.3 and Theorem 3.1); in Section 4 weshow Theorem 1.1 and that for τ > F dsc τ,L or F τ,L among sets which are union ofstripes then the minimizers are periodic stripes (this is done by the so-called reflection positivity); inSection 5, we show analogous results to [30] for the discrete setting using our technique; in Section 6,we prove Theorems 1.2 and 1.3; in Section 7 we prove Theorem 1.4 and Theorem 1.5, namely thatthe parameter τ > L ,provided L is an even multiple of h ∗ τ . In this section, we set the notation and recall some preliminary results on the functional (1.2)proven in [30] which will be used in the proof of our main theorems.6 .1 Notation and preliminary definitions
In the following, we let N = { , , . . . } , d ≥
1. On R d , we let (cid:104)· , ·(cid:105) be the Euclidean scalar productand | · | be the Euclidean norm. We let ( e , . . . , e d ) be the canonical basis in R d and for ζ ∈ R d welet ζ i = (cid:104) ζ, e i (cid:105) e i and ζ ⊥ i := ζ − ζ i . For z ∈ R d , let | z | = (cid:80) di =1 | z i | be its 1-norm and | z | ∞ = max i | z i | its ∞ -norm.Given a measurable set A ⊂ R d , let us denote by H d − ( A ) its ( d − | A | its Lebesgue measure.Given a measurable function f : R d → R , Df denotes its distributional derivative.For a measure µ on R d , we denote by | µ | its total variation.We are now ready to recall the definition of set of locally finite perimeter (see [3]). Such a propertyis fundamental because the functional (1.2) is finite only on [0 , L ) d -periodic sets of locally finiteperimeter. Definition 2.1.
A set E ⊂ R d is of (locally) finite perimeter if the distributional derivative of χ E is a (locally) finite measure. We let ∂E be the reduced boundary of E , namely the set of points x ∈ spt( Dχ E ) such that the limit ν E ( x ) := − lim r ↓ Dχ E ( B ( x, r )) | Dχ E | ( B ( x, r )) exists and satisfies | ν E ( x ) | = 1 . We call ν E the exterior normal to E . In particular, Dχ E = − ν E H d − (cid:120) ∂E . Notice that a [0 , L ) d -periodic set (as those considered in the paper) can not be of finite perimeter.For this reason we need to introduce the sets of locally finite perimeter.We define now (up to multiplying by a positive constant J ) the first term of the functional (1.2),namely Per ( E, [0 , L ) d ) := ˆ ∂E ∩ [0 ,L ) d | ν E ( x ) | d H d − ( x )and, for i ∈ { , . . . , d } Per i ( E, [0 , L ) d ) = ˆ ∂E ∩ [0 ,L ) d | ν Ei ( x ) | d H d − ( x ) , (2.1)thus Per ( E, [0 , L ) d ) = (cid:80) di =1 Per i ( E, [0 , L ) d ). Notice that in the definition of Per the norm appliedto the exterior normal ν E is not isotropic. For more general reference on anisotropic surface energiessee [36].Because of periodicity, w.l.o.g. we always assume that | Dχ E | ( ∂ [0 , L ) d ) = 0.Now, let us look at the definition and the assumptions on the second term of (1.2), namely thenonlocal term. Given a function K : R d → R , one defines it as − ˆ R d ˆ [0 ,L ) d K ( ζ ) | χ E ( x ) − χ E ( x + ζ ) | d x d ζ. Now we set our assumptions on K . They will be expressed in terms of the rescaled kernels K τ ( ζ ) = τ − p/β K ( ζτ − /β ) , with τ > τ := J c − J with J c defined in (1.4) and J < J c . For such rescalings the kernel of the nonlocalterm has the form K τ depending on K as above. Rescaling the functional in (1.2) will have theadvantage that the width and distance of the periodic optimal stripes as well as their energy willbe of order O (1).We assume that K is such that the rescaled kernels K τ satisfy the following ∃ C : 1 C | ζ | + τ /β ) p ≤ K τ ( ζ ) ≤ C | ζ | + τ /β ) p , p ≥ d + 2 , (2.3) K τ ( ζ ) converges monotonically increasing for τ ↓ | ζ | p or to 1 | ζ | p ,K τ is symmetric under exchange of coordinates, i.e. K τ ( P ζ ) = K τ ( ζ )for all permutations P on d indices , (2.5) (cid:98) K τ is the Laplace transform of a nonnegative function , (2.6)where (cid:98) K τ ( ζ i ) = ˆ R d − K τ ( ζ ⊥ i , ζ i ) d ζ ⊥ i , which is independent of i thanks to (2.5).An example of such family of kernels is given by the rescaling of K ( ζ ) = | ζ | +1) p , namely K τ ( ζ ) = 1( | ζ | + τ / ( p − d − ) p . (2.7)Indeed, the first two properties are trivial, and there exists a constant C q such that (cid:98) K τ ( z ) = C q | z | + τ /β ) q , q = p − d + 1 . (cid:98) K τ is the Laplace transform of a nonnegative function since, for s > s q = 1Γ( q ) ˆ + ∞ α q − e − αs d α where Γ( q ) is the Euler’s Gamma function. Thus1( s + τ /β ) q = 1Γ( q ) ˆ + ∞ α q − e − ατ /β e − αs d α. Property (2.6) will be used in the Section 4 and in particular in all one-dimensional optimizations.Property (2.3) will be the main source of inequalities in Section 6 and Section 7. Property (2.4) isused in order to obtain the Γ-limit.From now on, we fix a kernel satisfying properties (2.3)-(2.6).8 emark 2.2.
Given that all our analysis depends only on properties (2.3) - (2.6) , if one is notinterested in relating the structure of minimizers of F τ,L with the structure of minimizers of ˜ F J,L ,then it is not necessary to assume that K τ is obtained via rescaling K as in (2.2) . Sets of locally finite perimeter are defined up to Lebesgue null sets, therefore in statements re-garding a set of locally finite perimeter E we will always mean they hold up to null sets or for aprecise representative. Given that both the perimeter and the nonlocal quantities defining (1.2) areinvariant under modifications on null sets, we can always assume to have a precise representative.For example, the main theorems asserting that in some regimes minimizers of (1.2) are unions ofperiodic stripes hold neglecting a Lebesgue-null set.Slicing (used in different contexts e.g. in [3, 4, 5, 6, 7, 11, 12]) will be the main tool of our analysis.For i ∈ { , . . . , d } , let x ⊥ i be a point in the subspace orthogonal to e i . We define the one-dimensionalslices of E ⊂ R d by E x ⊥ i := (cid:8) t ∈ [0 , L ) : te i + x ⊥ i ∈ E (cid:9) . Notice that in the above definition there is an abuse of notation as the information on the directionof the slice is contained in the index x ⊥ i . As it would be always clear from the context which is thedirection of the slicing, we hope this will not cause confusion to the reader.Given a set of locally finite perimeter E , for a.e. x ⊥ i its slice E x ⊥ i is a set of locally finite perimeterin R and the following slicing formula (see [36]) holds for every i ∈ { , . . . , d } Per i ( E, [0 , L ) d ) = ˆ ∂E ∩ [0 ,L ) d | ν Ei ( x ) | d H d − ( x ) = ˆ [0 ,L ) d − Per ( E x ⊥ i , [0 , L )) d x ⊥ i . Whenever d = 1, a set E of locally finite perimeter is up to Lebesgue-null sets a locally finite unionof intervals (see [3, 36]). Therefore, w.l.o.g. we will write E = ∪ i ∈ Z ( s i , t i ) with t i < s i +1 . Moreover,one has that the reduced boundary ∂E coincides with the topological boundary of ∪ i ∈ Z ( s i , t i ).Thus, when d = 1 one can definePer ( E, [0 , L )) = Per( E, [0 , L )) = ∂E ∩ [0 , L )) , where ∂E is the reduced boundary of E .While writing slicing formulas, with a slight abuse of notation we will sometimes identify x i ∈ [0 , L ) d with its coordinate in R w.r.t. e i and { x ⊥ i : x ∈ [0 , L ) d } with [0 , L ) d − ⊂ R d − .In Sections 6 and 7 we will have to apply slicing on smaller cubes around a point. Therefore weneed to introduce the following notation. For r > x ⊥ i we let Q ⊥ r ( x ⊥ i ) = { z ⊥ i : | x ⊥ i − z ⊥ i | ∞ ≤ r } or we think of x ⊥ i ∈ [0 , L ) d − and Q ⊥ r ( x ⊥ i ) as a subset of R d − . Since the subscript i will be alwayspresent in the centre (namely x ⊥ i ) of such ( d − i of Q ⊥ r ( x ⊥ i ) should be clear. We denote also by Q ir ( t i ) ⊂ R the interval of length r centred in t i .In Section 7, instead of integrals on [0 , L ) d one will often consider integrals on smaller cubescentred at other points of [0 , L ) d . Therefore, for z ∈ [0 , L ) d and r >
0, we define Q r ( z ) = { x ∈ R d : | x − z | ∞ ≤ r } .While doing estimates on slices, we will consider E ⊂ R a set of locally finite perimeter and s ∈ ∂E a point in the relative boundary of E . We will denote by s + := inf { t (cid:48) ∈ ∂E, with t (cid:48) > s } s − := sup { t (cid:48) ∈ ∂E, with t (cid:48) < s } . (2.8)9stimates will be obtained (see the following subsection) through the function η : ∂E × R → R defined as η ( s, z ) := min( z + , s − s − ) + min( z − , s + − s ) , (2.9)where z + = max { z, } and z − = − min { z, } . In particular, given a [0 , L ) d -periodic set E of locallyfinite perimeter, the functions η x ⊥ i : ∂E x ⊥ i × R → R are defined as above for the slices E x ⊥ i .In the paper, we will denote constants which depend on L > d with thesymbol C d,L and the constants which depend only on the dimension with C d . In Section 7, wherethe constants do not depend on L , in order to simplify notation we will use A (cid:46) B , whenever thereexists a constant C d depending only on the dimension d such that A ≤ C d B . Notice that, since akernel has been fixed, then the constants depend also implicitly on the chosen kernel.In presence of multiple integrals, we use the convention ˆ A . . . ˆ A n f ( x , . . . x n ) d x n . . . d x = ˆ A (cid:16) . . . (cid:16) ˆ A n f ( x , . . . x n ) d x n (cid:17) . . . (cid:17) d x . Let E = E h be a periodic union of stripes of width and distance h in the direction e i . Up torelabeling coordinates we can assume that i = 1, thus E h = ˆ E h × [0 , L ) d − , and˜ F J,L ( E h ) = − τh + ˆ R d K ( ζ ) (cid:16) | ζ | h − L d ˆ [0 ,L ) d | χ E h ( x ) − χ E h ( x + ζ ) | d x (cid:17) d ζ. As in [30], it is possible to compute the energy ˜ F J,L ( E h ) to get˜ F J,L ( E h ) (cid:39) − τh + h − ( p − d ) . Optimizing in h , one finds that the optimal stripes have a width of order τ − / ( p − d − and energyof order − τ ( p − d ) / ( p − d − . Letting β := p − d −
1, this motivates the rescaling x := τ − /β ˜ x, L := τ − /β ˜ L and ˜ F J,L ( E ) := τ ( p − d ) /β F τ, ˜ L ( ˜ E ) . (2.10)In these variables, the optimal stripes have width of order O (1).Making the substitutions in (2.10) letting also ζ = τ − /β ˜ ζ and in the end dropping the tildes, onehas (see Lemma 3.6 in [30]) F τ,L ( E ) = 1 L d (cid:16) − Per ( E, [0 , L ) d ) + ˆ R d K τ ( ζ ) (cid:104) ˆ ∂E ∩ [0 ,L ) d d (cid:88) i =1 | ν Ei ( x ) || ζ i | d H d − ( x ) − ˆ [0 ,L ) d | χ E ( x ) − χ E ( x + ζ ) | d x (cid:105) d ζ (cid:17) , (2.11)where K τ ( ζ ) = τ − p/β K ( ζτ − /β ).Let us now state an important estimate from below for F τ,L (see [30, Lemma 3.2]).10 χ E ( x ) − χ E ( x + ζ ) | = | χ E ( x ) − χ E ( x + ζ i ) | + | χ E ( x + ζ i ) − χ E ( x + ζ ) |− | χ E ( x ) − χ E ( x + ζ i ) || χ E ( x + ζ i ) − χ E ( x + ζ ) | . (2.12)If the kernel K τ is symmetric (namely, K τ ( ζ , . . . , ζ i , . . . , ζ d ) = K τ ( ζ , . . . , − ζ i , . . . , ζ d ) for every i = 1 , . . . , d ), one has that ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ ) − χ E ( x + ζ ) || χ E ( x + ζ ) − χ E ( x ) | K τ ( ζ ) d ζ d x = ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ ⊥ ) − χ E ( x ) || χ E ( x + ζ ) − χ E ( x ) | K τ ( ζ ) d ζ d x. For the general case, by using property (2.3), we have that ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ ) − χ E ( x + ζ ) || χ E ( x + ζ ) − χ E ( x ) | K τ ( ζ ) d ζ d x ≤ C ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ ) − χ E ( x + ζ ) || χ E ( x + ζ ) − χ E ( x ) | ( | ζ | + τ /β ) p d ζ d x ≤ C ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ ) − χ E ( x ) || χ E ( x + ζ ⊥ ) − χ E ( x ) | ( | ζ | + τ /β ) p d ζ d x ≤ C ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ ⊥ ) − χ E ( x ) || χ E ( x + ζ ) − χ E ( x ) | K τ ( ζ ) d ζ d x, (2.13)where C is the constant appearing in (2.3).In the same way as in [30, Lemma 3.2], by using (2.12) and in addition (2.13), one has that ˆ [0 ,L ) d ˆ R d K τ ( ζ ) | χ E ( x ) − χ E ( x + ζ ) | d ζ d x ≤ ˆ [0 ,L ) d ˆ R d K τ ( ζ ) d (cid:88) i =1 | χ E ( x ) − χ E ( x + ζ i ) | d ζ d x − C d ˆ [0 ,L ) d ˆ R d K τ ( ζ ) d (cid:88) i =1 | χ E ( x ) − χ E ( x + ζ i ) || χ E ( x ) − χ E ( x + ζ ⊥ i ) | d ζ d x, . (2.14)As it will be clear from the proof, the result does not depend on the particular value of C , withoutloss of generality we may assume that C = 1.Notice also that (2.13) is an equality if and only if χ E represents unions of stripes.Define then, for i ∈ { , . . . , d } , G iτ,L ( E ) := ˆ R (cid:98) K τ ( ζ i ) (cid:104) ˆ ∂E ∩ [0 ,L ) d | ν Ei ( x ) || ζ i | d H d − ( x ) − ˆ [0 ,L ) d | χ E ( x ) − χ E ( x + ζ i ) | d x (cid:105) d ζ i , where (cid:98) K τ ( z ) = ´ R d − K τ ( z, ζ (cid:48) ) d ζ (cid:48) and I iτ,L ( E ) := 2 d ˆ [0 ,L ) d ˆ R d K τ ( ζ ) | χ E ( x ) − χ E ( x + ζ i ) || χ E ( x ) − χ E ( x + ζ ⊥ i ) | d ζ d x,I τ,L ( E ) := d (cid:88) i =1 I iτ,L ( E ) . (2.15)11stimate (2.14) implies (see Lemma 3.6 in [30]) F τ,L ( E ) ≥ L d (cid:16) − Per ( E, [0 , L ) d ) + d (cid:88) i =1 G iτ,L ( E ) + d (cid:88) i =1 I iτ,L ( E ) (cid:17) . (2.16)The estimate we need now is the following (see Lemma 3.4 of [30]): for every one-dimensional L -periodic set E ⊂ R of locally finite perimeter and every z ∈ R , ˆ L | χ E ( x ) − χ E ( x + z ) | d x ≤ (cid:88) x ∈ ∂E ∩ [0 ,L ) η ( x, z ) , (2.17)where η is the function defined in (2.9)For every τ ≥ ζ ∈ R d , [0 , L ) d -periodic set E ⊂ R d of locally finite perimeter and i ∈ { , . . . , d } ,by using (2.17) with a slicing argument (see Lemma 3.4 in [30]) one has that ˆ [0 ,L ) d ˆ R d K τ ( ζ ) | χ E ( x ) − χ E ( x + ζ i ) | d ζ d x ≤ ˆ ∂E ∩ [0 ,L ) d | ν Ei ( x ) | ˆ R d K τ ( ζ ) η x ⊥ i ( x i , ζ i ) d ζ d H d − ( x ) . (2.18)We will use the following slicing formula G iτ,L ( E ) = ˆ [0 ,L ) d − G dτ,L ( E x ⊥ i ) d x ⊥ i , (2.19)where G dτ,L ( E x ⊥ i ) := ˆ R (cid:98) K τ ( z ) (cid:16) Per( E x ⊥ i , [0 , L )) | z | − ˆ L | χ E x ⊥ i ( x ) − χ E x ⊥ i ( x + z ) | d x (cid:17) d z. (2.20)As a consequence of (2.18) and the fact that | z | ≥ η x ⊥ i ( x, z ), G iτ,L ( E ) ≥ G dτ,L (see [30, Lemma 3.7]): for everyone-dimensional L -periodic set E of locally finite perimeter (recall that β = p − d − G dτ,L ( E ) ≥ C d,L (cid:88) x ∈ ∂E ∩ [0 ,L ) min(( x + − x ) − β , τ − ) + min(( x − x − ) − β , τ − ) , (2.21)where x + and x − are defined as in (2.8).Moreover, for every δ ≥ τ /β , one has that (see [30, Lemma 3.7])Per( E, [0 , L )) − ≤ C d,L ( Lδ − + δ β G dτ,L ( E )) . (2.22)Although in [30] the estimates above hold for a slightly different kernel (namely, | ζ | p +1 ), theycontinue to hold for the type of kernels considered here since the only thing which is used in theproof is assumption (2.3).Optimizing in δ in (2.22), one has thatPer( E, [0 , L )) − ≤ C d,L max (cid:0) τ G dτ,L ( E ) , G dτ,L ( E ) /p − d (cid:1) . Integrating it for E x ⊥ i w.r.t. x ⊥ i ∈ [0 , L ) d − one obtains F τ,L ( E ) ≥ C d,L (cid:104) − L d (cid:16) d (cid:88) i =1 G iτ,L ( E ) + d (cid:88) i =1 I iτ,L ( E ) (cid:17)(cid:105) (2.23)12nd Per ( E, [0 , L ) d ) ≤ C d,L L d max(1 , F τ,L ( E )) . (2.24)For details see Lemma 3.9 in [30].The main result of [30] is the following. Theorem 2.3 ([30, Theorem 1.1]) . Let p > d and L > . Then one has that F τ,L Γ -converge inthe L -topology as τ → to a functional F ,L which is invariant under permutation of coordinatesand finite on sets which are (up to permutation of coordinates) of the form E = F × R d − where F ⊂ R is L -periodic with { ∂F ∩ [0 , L ) } < ∞ .Moreover, let { E τ } be a family of [0 , L ) d -periodic subsets of R d such that there exists M suchthat for every τ one has that F τ,L ( E τ ) < M . Then, up to a permutation of coordinates, one hasthat there is a subsequence which converges in L to some set of the form E = F × R d − with { ∂F ∩ [0 , L ) } < ∞ . p > d to p ≥ d + 2 In this section we will prove the following theorem:
Theorem 3.1 ([30, Theorem 1.1] improved) . Let p ≥ d + 2 and L > . Then one has that F τ,L Γ -converge in the L -topology as τ → to a functional F ,L which is invariant under permutationof coordinates and finite on sets (up to permutation of coordinates) of the form E = F × R d − ,where F ⊂ R is L -periodic with { ∂F ∩ [0 , L ) } < ∞ .On sets of the form E = F × R d − the functional is defined by F ,L ( E ) = 1 L (cid:16) − { ∂F ∩ [0 , L ) } + ˆ R d | ζ | p (cid:104) (cid:88) x ∈ ∂F ∩ [0 ,L ) | ζ | − ˆ L | χ F ( x ) − χ F ( x + ζ ) | d x (cid:105) d ζ (cid:17) . (3.1) Moreover, let { E τ } be a family of [0 , L ) d -periodic subsets of R d such that there exists M such thatfor every τ one has that F τ,L ( E τ ) < M , then, up to a permutation of coordinates, one has that thereis a subsequence which converges in L ([0 , L ) d ) to some set E = F × R d − with { ∂F ∩ [0 , L ) } < ∞ . The proof of Theorem 3.1 consists of two main parts: a part in which compactness of sets ofequibounded energy and lower semicontinuity of the functionals G iτ,L and I iτ,L is proved and a partin which a rigidity estimate is proved. The rigidity estimate roughly says that in the limit as τ ↓ L ([0 , L ) d ). This in turn says that thelimiting problem is one-dimensional.For the first part we refer to [30], where the quantities and the estimates defined in Section 2 areused.The second part is based on different arguments. Indeed, the rigidity result is the core of Theo-rem 3.1. Such result is contained in the following proposition, which substitutes Proposition 4.3 of[30]. Proposition 3.2 (Rigidity) . Let p ≥ d + 2 and let E be a [0 , L ) d -periodic set of locally finiteperimeter such that (cid:80) di =1 G i ,L ( E ) + I ,L ( E ) < + ∞ . Then, E is one-dimensional, i.e. up topermutation of the coordinates, E = (cid:98) E × R d − for some L − periodic set (cid:98) E .
13n the proof of Proposition 3.2, we will apply the following result (see [10, Proposition 1]). It willbe applied to the function r iλ ( u, · ) : R d − → R (for the definition see (3.6) below). Theorem 3.3.
Let Ω ⊂ R d − be an open and connected set in R d − and f : R d − → R be ameasurable function such that ˆ Ω × Ω | f ( x ) − f ( y ) || x − y | d < + ∞ . (3.2) Then f is constant almost everywhere, namely there is constant function ˜ f such that f = ˜ f up toa set of null Lebesgue measure. Notice that the Theorem 3.3 is trivial whenever f is smooth. In order to obtain the nonsmoothcase, a regularization step is needed (see [10] for the details).Let us also recall our notation: given t ∈ R d , we will denote by t i = (cid:104) t, e i (cid:105) e i , t ⊥ i = t − t i and we set f E ( t ⊥ i , t i , t (cid:48)⊥ i , t (cid:48) i ) := | χ E ( t ⊥ i + t i + t (cid:48) i ) − χ E ( t i + t ⊥ i ) || χ E ( t ⊥ i + t i + t (cid:48)⊥ i ) − χ E ( t i + t ⊥ i ) | . (3.3)In order to be able to use Theorem 3.3, we need to change variables and have K ( t (cid:48) − t ) instead of K ( t (cid:48) ) in our formulas. For this reason we will make the change of variables ˜ t = t (cid:48) + t . Thus wehave that d I i ,L ( E ) = ˆ [0 ,L ) d ˆ R d | χ E ( t + t (cid:48)⊥ i ) − χ E ( t ) || χ E ( t + t (cid:48) i ) − χ E ( t ) | K ( t (cid:48) ) d t (cid:48) d t = ˆ [0 ,L ) d ˆ R d | χ E ( t i + ˜ t ⊥ i ) − χ E ( t ) || χ E ( t ⊥ i + ˜ t i ) − χ E ( t ) | K (˜ t − t ) d˜ t d t = ˆ [0 ,L ) d ˆ R d | χ E ( t i + t (cid:48)⊥ i ) − χ E ( t ) || χ E ( t ⊥ i + t (cid:48) i ) − χ E ( t ) | K ( t (cid:48) − t ) d t (cid:48) d t ≥ ˆ [0 ,L ) d ˆ [0 ,L ) d | χ E ( t i + t (cid:48)⊥ i ) − χ E ( t ) || χ E ( t ⊥ i + t (cid:48) i ) − χ E ( t ) | K ( t (cid:48) − t ) d t (cid:48) d t Since we will make a slicing argument, we further rewrite the above as d I i ,L ( E ) ≥ ˆ [0 ,L ) d − ˆ [0 ,L ) d − Int( t ⊥ i , t (cid:48)⊥ i ) d t ⊥ i d t (cid:48)⊥ i , (3.4)where Int( t ⊥ i , t (cid:48)⊥ i ) := ˆ L ˆ L f E ( t ⊥ i , t i , t (cid:48)⊥ i − t ⊥ i , t (cid:48) i − t i ) K ( t − t (cid:48) ) d t i d t (cid:48) i . (3.5)It is useful to think of Int( t ⊥ i , t (cid:48)⊥ i ) as the interaction between two different slices.In this section we will use the following notation: given λ ∈ (0 , L ), u ∈ ( λ, L − λ ) and E a set oflocally finite perimeter and t ⊥ i ∈ [0 , L ) d − , we will denote by r iλ ( u, t ⊥ i ) := min (cid:8) inf {| u − s | : s ∈ ∂E t ⊥ i and s ∈ ( λ, L − λ ) } , | u − λ | , | L − λ − u | (cid:9) r io ( t ⊥ i ) := inf s ∈ ∂E t ⊥ i ∩ [0 ,L ] min( s + − s, s − s − ) , (3.6)where s + , s − are defined in (2.8).Notice that for a set of finite perimeter the followings hold:14 E t ⊥ i is a set of finite perimeter for almost every t ⊥ i , • for every set of finite perimeter in F ⊂ R , there exist a finite number of intervals { J i } Ni =1 such that F ∩ [0 , L ) = (cid:83) Ni =1 J i , where the equality is intended in the measure theoretic sense,namely up to a set of null measure (see [3]).Thus the above map is well-defined for almost every t ⊥ i and measurable. Remark 3.4.
When λ = 0 , term r iλ ( u, t ⊥ i ) measures the distance of u from jump points on ∂E t ⊥ i which are close. The role of λ > is technical: it is used to handle the situation in which the nextjump point s ∈ ∂E t ⊥ i is in a λ -neighborhood of { , L } . In Proposition 3.2, this technical point wouldbe unnecessary since E is [0 , L ) d -periodic. However, we introduce it because it will be needed in theproof of the local rigidity lemma, namely Lemma 7.6 (when instead of [0 , L ) d we will consider [0 , l ) d with l < L and therefore E is not [0 , l ) d -periodic). We prefer to give here already a more generalproof instead of repeating twice a similar argument. Suppose that, for every u , one has that r iλ ( u, · ) is constant almost everywhere: if this holds forevery i , then it is not difficult to see that E is (up to null sets) either a union of stripes or acheckerboards, where by checkerboards we mean any set whose boundary is the union of affinesubspace orthogonal to coordinate axes, and there are at least two of these directions.Via an energetic argument one can rule out checkerboards (see the comment at the end of the proofof Proposition 3.2).In order to obtain that r iλ ( u, · ) is constant almost everywhere, we will apply Theorem 3.3.The main use of the term r io ( t i ) is the following: since G i ,L ( E ) < + ∞ and inequality (2.21) holdsfor τ = 0, one has that ˆ [0 ,L ) d − ( r i ( t ⊥ i )) − β d t ⊥ i ≤ G i ,L ( E ) < + ∞ . (3.7)The next lemma gives a lower bound for the interaction term of close slices. Lemma 3.5.
Let λ ∈ (0 , L/ and let t (cid:48)⊥ i , t ⊥ i ∈ [0 , L ) d − , t ⊥ i (cid:54) = t (cid:48)⊥ i be such that min( r io ( t ⊥ i ) , r io ( t (cid:48)⊥ i )) > | t (cid:48)⊥ i − t ⊥ i | and | t (cid:48)⊥ i − t ⊥ i | ≤ λ . Then for every u ∈ ( λ, L − λ ) it holds Int( t (cid:48)⊥ i , t ⊥ i ) ≥ C d,L | r iλ ( u, t (cid:48)⊥ i ) − r iλ ( u, t ⊥ i ) || t (cid:48)⊥ i − t ⊥ i | d . (3.8) Proof.
W.l.o.g. let us assume that r iλ ( u, t ⊥ i ) < r iλ ( u, t (cid:48)⊥ i ). In particular this implies that r iλ ( u, t ⊥ i ) < min( | u − λ | , | L − λ − u | ), and hence there exists a point s o ∈ ( λ, L − λ ) such that | u − s o | = inf {| u − s | : s ∈ ∂E t ⊥ i , s ∈ ( λ, L − λ ) } . For simplicity of notation denote δ = | t (cid:48)⊥ i − t ⊥ i | and r = | r iλ ( u, t ⊥ i ) − r iλ ( u, t (cid:48)⊥ i ) | . Since r o ( t ⊥ i ) > δ ,one has that( s o − δ, s o + δ ) ∩ E t ⊥ i = ( s o , s o + δ ) or ( s o − δ, s o + δ ) ∩ E t ⊥ i = ( s o − δ, s o ) . λ ≥ δ , we have that ( s o − δ, s o + δ ) ⊂ [0 , L ). We will assume( s o − δ, s o + δ ) ∩ E t ⊥ i = ( s o , s o + δ ) (3.9)The other case is analogous.We will distinguish two subcases:(i) Suppose r > δ/
2. From the definition of δ and r , then for the slice in t (cid:48)⊥ i one has that( s o − δ/ , s o + δ/ ∩ E t (cid:48)⊥ i = ( s o − δ/ , s o + δ/
2) or ( s o − δ/ , s o + δ/ ∩ E t (cid:48)⊥ i = ∅ . Indeed on the slice E t (cid:48)⊥ i , the closest jump point to s o is at least r distant and r > δ/
2. Wewill assume the first alternative above. The other case is analogous.Then for every a ∈ ( s o − δ/ , s o ) and a (cid:48) ∈ ( s o , s o + δ/ f E ( t ⊥ i , a, t (cid:48)⊥ i − t ⊥ i , a (cid:48) − a ) = 1 . Thus given that r iλ ( u, t ⊥ i ) ≤ L , we have thatInt( t ⊥ i , t (cid:48)⊥ i ) = ˆ L ˆ L f E ( t ⊥ i , t i , t (cid:48)⊥ i − t ⊥ i , t (cid:48) i − t i ) K ( t (cid:48) − t ) d t i d t (cid:48) i ≥ ˆ s o s o − δ/ ˆ s o + δ/ s o f E ( t ⊥ i , t i , t (cid:48)⊥ i − t ⊥ i , t (cid:48) i − t i ) K ( t (cid:48) − t ) d t (cid:48) i d t i ≥ ˆ s o s o − δ/ ˆ s o + δ/ s o K ( t (cid:48) − t ) d t (cid:48) i d t i ≥ C d δ δ p ≥ C d | r iλ ( u, t ⊥ i ) − r iλ ( u, t (cid:48)⊥ i ) | L δ p − ≥ C d,L | r iλ ( u, t ⊥ i ) − r iλ ( u, t (cid:48)⊥ i ) || t (cid:48)⊥ i − t ⊥ i | d , where in the second last line above we have used that for every t i ∈ ( s o − δ/ , s o ) and t (cid:48) i ∈ ( s o , s o + δ/
2) one has that K ( t (cid:48) − t ) ≥ C d | t (cid:48)⊥ i − t ⊥ i | p . (3.10)(ii) Let us assume now that r ≤ δ/
2. Since r o ( t (cid:48)⊥ i ) , r o ( t ⊥ i ) > δ , one has that either( s o − r, s o + δ/ ∩ E t (cid:48)⊥ i = ( s o − r, s o + δ/
2) or ( s o − r, s o + δ/ ∩ E t (cid:48)⊥ i = ∅ or ( s o − δ/ , s o + r ) ∩ E t (cid:48)⊥ i = ( s o − δ/ , s o + r ) or ( s o − δ/ , s o + r ) ∩ E t (cid:48)⊥ i = ∅ . Indeed if none of the above were true we would have that ∂E t (cid:48)⊥ i ∩ ( s o − δ/ , s o + δ/ ≥ r o ( t (cid:48)⊥ i ) > δ .W.l.o.g. (see Figure 1) we will assume( s o − r, s o + δ/ ∩ E t (cid:48)⊥ i = ( s o − r, s o + δ/ . λs + δ/ s s − r ut ⊥ i t (cid:48)⊥ i ( a, t (cid:48)⊥ i )( a, t ⊥ i )( a (cid:48) , t ⊥ i )Figure 1: The points in the proof of Lemma 3.5 case (ii) are depicted. Recall that δ = | t ⊥ i − t (cid:48)⊥ i | ≤ λ , r = | r iλ ( u, t (cid:48)⊥ i ) − r iλ ( u, t ⊥ i ) | is less than or equal to δ/
2. In the estimate for the interaction termInt( t (cid:48)⊥ i , t ⊥ i ) one takes points ( a, t ⊥ i ), ( a, t (cid:48)⊥ i ) with a ∈ ( s o − r, s o ) and ( a (cid:48) , t ⊥ i ) with a (cid:48) ∈ ( s o , s o + δ ).The other cases are analogous.Then for every a ∈ ( s o − r, s o ) and a (cid:48) ∈ ( s o , s o + δ/ f E ( t ⊥ i , a, t (cid:48)⊥ i − t ⊥ i , a (cid:48) − a ) = 1.ThusInt( t ⊥ i , t (cid:48)⊥ i ) = ˆ L ˆ L f E ( t ⊥ i , t i , t (cid:48)⊥ i − t ⊥ i , t (cid:48) i − t i ) K ( t (cid:48) − t ) d t i d t (cid:48) i ≥ ˆ s o s o − r ˆ s o + δ/ s o f E ( t ⊥ i , t i , t (cid:48)⊥ i − t ⊥ i , t (cid:48) i − t i ) K ( t (cid:48) − t ) d t (cid:48) i d t i ≥ ˆ s o s o − r ˆ s o + δ/ s o K ( t (cid:48) − t ) d t (cid:48) i d t i ≥ C d δrδ p ≥ C d,L | r iλ ( u, t ⊥ i ) − r iλ ( u, t (cid:48)⊥ i ) || t (cid:48)⊥ i − t ⊥ i | d +1 , where in the last line we have used (3.10). Lemma 3.6.
Assume that E ⊂ R d is a set of locally finite perimeter such that ( r io ) − β ∈ L ([0 , L ) d ) , p ≥ d + 2 and I ,L ( E ) < + ∞ . Let r iλ ( u, · ) be as defined in (3.6) . Then, we have that r iλ ( u, · ) isconstant almost everywhere.Proof. We will apply Theorem 3.3 to the function r iλ ( u, · ). Notice that because [0 , L ) d − ⊃ (0 , L ) d − ,by showing (3.2) with [0 , L ) d − instead of (0 , L ) d − we have that r iλ ( u, · ) is constant almost every-where. 17 [0 ,L ) d − ˆ [0 ,L ) d − | r iλ ( u, t ⊥ i ) − r iλ ( u, t (cid:48)⊥ i ) || t (cid:48)⊥ i − t ⊥ i | d d t (cid:48)⊥ i d t ⊥ i ≤ C d,L ( G i ,L ( E ) + I i ,L ( E )) < + ∞ . (3.11)and since I i ,L ( E ) is finite we have that r iλ ( u, · ) is constant almost everywhere.We have that ˆ [0 ,L ) d − ˆ [0 ,L ) d − | r iλ ( u, t ⊥ i ) − r iλ ( u, t (cid:48)⊥ i ) || t (cid:48)⊥ i − t ⊥ i | d d t ⊥ i d t (cid:48)⊥ i ≤ ˆ [0 ,L ) d − ˆ [0 ,L ) d − χ A ( t ⊥ i , t (cid:48)⊥ i ) | r iλ ( u, t (cid:48)⊥ i ) − r iλ ( u, t ⊥ i ) || t ⊥ i − t (cid:48)⊥ i | d d t (cid:48)⊥ i d t ⊥ i + ˆ [0 ,L ) d − ˆ [0 ,L ) d − χ B ( t ⊥ i , t (cid:48)⊥ i ) | r iλ ( u, t (cid:48)⊥ i ) − r iλ ( u, t ⊥ i ) || t ⊥ i − t (cid:48)⊥ i | d d t (cid:48)⊥ i d t ⊥ i + ˆ [0 ,L ) d − ˆ [0 ,L ) d − χ C ( t ⊥ i , t (cid:48)⊥ i ) | r iλ ( u, t (cid:48)⊥ i ) − r iλ ( u, t ⊥ i ) || t ⊥ i − t (cid:48)⊥ i | d d t (cid:48)⊥ i d t ⊥ i (3.12)where A := (cid:110) ( t (cid:48)⊥ i , t ⊥ i ) ∈ [0 , L ) d − × [0 , L ) d − : min( r o ( t (cid:48)⊥ i ) , r o ( t ⊥ i )) ≤ | t ⊥ i − t (cid:48)⊥ i | , and | t (cid:48)⊥ i − t ⊥ i | ≤ λ (cid:111) B := (cid:110) ( t (cid:48)⊥ i , t ⊥ i ) ∈ [0 , L ) d − × [0 , L ) d − : min( r o ( t (cid:48)⊥ i ) , r o ( t ⊥ i )) > | t ⊥ i − t (cid:48)⊥ i | , and | t (cid:48)⊥ i − t ⊥ i | ≤ λ (cid:111) C := (cid:110) ( t (cid:48)⊥ i , t ⊥ i ) ∈ [0 , L ) d − × [0 , L ) d − : | t (cid:48)⊥ i − t ⊥ i | > λ (cid:111) . The last term on the r.h.s. of (3.12) is trivially bounded.Because of (3.8), we have that ˆ [0 ,L ) d − ˆ [0 ,L ) d − χ B ( t ⊥ i , t (cid:48)⊥ i ) | r iλ ( u, t (cid:48)⊥ i ) − r iλ ( u, t ⊥ i ) || t ⊥ i − t (cid:48)⊥ i | d d t (cid:48)⊥ i d t ⊥ i ≤ C d,L ˆ [0 ,L ) d − ˆ [0 ,L ) d − Int( t (cid:48)⊥ i , t ⊥ i ) ≤ C d,L d I i ,L ( E ) (3.13)Since A = { ( t ⊥ i , t (cid:48)⊥ i ) : r o ( t ⊥ i ) ≤ | t ⊥ i − t (cid:48)⊥ i |} ∪ { ( t ⊥ i , t (cid:48)⊥ i ) : r o ( t (cid:48)⊥ i ) ≤ | t ⊥ i − t (cid:48)⊥ i |} ,
18t is sufficient to estimate the contribution of { ( t ⊥ i , t (cid:48)⊥ i ) : r o ( t ⊥ i ) ≤ | t (cid:48)⊥ i − t ⊥ i |} . Thus, one has that ¨ [0 ,L ) d − × [0 ,L ) d − χ A ( t (cid:48)⊥ i , t ⊥ i ) | r iλ ( u, t ⊥ i ) − r iλ ( u, t (cid:48)⊥ i ) || t ⊥ i − t (cid:48)⊥ i | d d t (cid:48)⊥ i d t ⊥ i ≤ ˆ [0 ,L ) d − ˆ { r o ( t ⊥ i ) < | t (cid:48)⊥ i − t ⊥ i |} | r iλ ( u, t ⊥ i ) − r iλ ( u, t (cid:48)⊥ i ) || t ⊥ i − t (cid:48)⊥ i | d d t (cid:48)⊥ i d t ⊥ i ≤ ˆ [0 ,L ) d − ˆ { r o ( t ⊥ i ) < | t (cid:48)⊥ i − t ⊥ i |} L | t ⊥ i − t (cid:48)⊥ i | d d t (cid:48)⊥ i d t ⊥ i ≤ C d ˆ [0 ,L ) d − ˆ Lr o ( t ⊥ i ) Lρ d − ρ d d ρ d t ⊥ i ≤ C d,L ˆ [0 ,L ) d − r o ( t ⊥ i ) − d t ⊥ i ≤ C d,L ˆ [0 ,L ) d − r o ( t ⊥ i ) − β d t ⊥ i ≤ C d,L G i ,L ( E ) (3.14)where from the third to the fourth line we have used a change to polar coordinates, from the fifthto the sixth line the fact that β = p − d − ≥ Proof of Proposition 3.2.
From the Lemma 3.6 r iλ ( u, · ) is constant almost everywhere for every u and for every i . Fix u and λ sufficiently small such that r iλ ( u, · ) (cid:54) = min( | u − λ | , | L − λ − u | ). Ifthis is not possible, then E ∩ [0 , L ) d is either [0 , L ) d or ∅ . Therefore for every t ⊥ i , the minimizers of | s − u | for s ∈ ∂E t ⊥ i are either u + r iλ ( u, t ⊥ i ) or u − r iλ ( u, t ⊥ i ). The fact that r iλ ( u (cid:48) , t ⊥ i ) is also constantalmost everywhere for u (cid:48) ∈ ( u − ε, u + ε ) implies that one of the following three cases holds(a) u + r iλ ( u, t ⊥ i ) ∈ ∂E t ⊥ i for all t ⊥ i ∈ [0 , L ) d − (b) u − r iλ ( u, t ⊥ i ) ∈ ∂E t ⊥ i for all t ⊥ i ∈ [0 , L ) d − (c) u + r iλ ( u, t ⊥ i ) ∈ ∂E t ⊥ i for all t ⊥ i ∈ [0 , L ) d − and u − r iλ ( u, t ⊥ i ) ∈ ∂E t ⊥ i for all t ⊥ i ∈ [0 , L ) d − .Thus, since this holds for every i , we have that E must be a checkerboard or a union of stripes. Werecall that by a checkerboard we mean any set whose boundary is the union of affine hyperplanesorthogonal to coordinate axes, and there are at least two of these directions.However, one can rule out checkerboard. To see this we consider the contribution to the energygiven in a neighbourhood of an edge. W.l.o.g. we may assume that around this edge the set E isof the following form − ε ≤ x ≤ − ε ≤ x ≤ x i ∈ ( − ε, ε ) for i (cid:54) = 1 ,
2. Notice that forevery ζ such that ζ + x > ζ + x > ζ i ∈ ( − ε, ε ) for i (cid:54) = 1 ,
2, the integrand in I ,L ( E ) isequal to 1 / | ζ | p . Then by setting A x ,x := ( − x , − x + ε ) × ( − x , − x + ε ) × ( − ε, ε ) d − , one hasthat d I ,L ( E ) ≥ ˆ ( − ε, × ( − ε,ε ) d − ˆ A x ,x | ζ | p d ζ d x ≥ ˆ (0 ,ε ) d x + ζ + x + ζ + (cid:80) di =3 ζ i ) p d ζ d x ε d − ˆ (0 ,ε ) d +2 | t | p d t which, by making a change of variables to polar coordinates, diverges. A similar calculation wasmade also in [25]. Let us consider the following one-dimensional functional: on an L -periodic set E ⊂ R of locallyfinite perimeter F τ,L ( E ) = 1 L (cid:16) − Per( E, [0 , L )) + ˆ R (cid:98) K τ ( z ) (cid:104) Per( E, [0 , L )) | z | − ˆ L | χ E ( x ) − χ E ( x + z ) | d x (cid:105) d z (cid:17) , where (cid:98) K τ ( z ) = ˆ R d − K τ ( z, z (cid:48) ) d z (cid:48) . We will consider as before kernels which satisfy conditions (2.3)-(2.6). In particular, C q C ( τ /β + | z | ) q ≤ (cid:98) K τ ( z ) ≤ CC q ( τ /β + | z | ) q , q := p − d + 1 , (4.1)where C is the constant appearing in (2.3) and C q appears in ˆ R d − d ζ · · · d ζ d − ( τ /β + | ζ | + · · · + | ζ d | ) p = C q ( τ /β + | ζ | ) q . The above functional corresponds to F τ,L ( E ) when the set E is a union of stripes.The main tool of this section is the reflection positivity technique, introduced in the context ofquantum field theory [42] and then applied for the first time in statistical mechanics in [20]. Forthe first appearance of the technique in models with short-range and Coulomb type interactionssee [19]. See [21], [23], [24] for further generalizations to one-dimensional models and [22], [27] forapplications to two-dimensional models.Only for notational reasons, we follow [30]. Except for Theorem 1.1, the rest of this section can beindeed obtained by simple modifications of well-known results.The main statements that will be shown in this section are Theorem 1.1 and the following Theorem 4.1.
There exists
C > and ˆ τ > such that for every ≤ τ < ˆ τ , ∃ h ∗ τ > s.t., forevery L > , the minimizers of F τ,L are periodic stripes of period h τ for some h τ > satisfying | h τ − h ∗ τ | ≤ CL .h ∗ τ is the period of the stripes giving the optimal energy density among all [0 , L ) d -periodic sets, as L varies.
20n order to prove Theorem 1.1 and Theorem 4.1, we proceed using the method of reflection positivity.Where the proofs are the same as in [30], we refer directly to that paper for the details.For h >
0, recall that E h := ∪ k ∈ Z [(2 k ) h, (2 k + 1) h ]. Then, define e ∞ ,τ ( h ) := F τ, h ( E h ) = lim L → + ∞ F τ,L ( E h ) . Analogously to [30, Lemma 6.1], one can see that e ∞ ,τ ( h ) = − h + A τ ( h ) , (4.2)where A τ ( h ) := ˆ + ∞ h ( z − h ) (cid:98) K τ ( z ) d z + (cid:88) k ∈ N ˆ h ˆ (2 k +1) h kh (cid:98) K τ ( x − y ) d y d x. Thus one can define h ∗ τ equivalently, as the minimizer of e ∞ ,τ .For convenience of notation, let us denote by ¯ e ∞ ,τ , the corresponding e ∞ ,τ relative to the specifickernel defined in (2.7), namely K τ ( ζ ) = 1( | ζ | + τ / ( p − d − ) p . Proof of Theorem 1.1.
We will show that for τ small enough the function e ∞ ,τ has a unique mini-mizer.We will divide the proof of into several steps. Step 0:
It is not difficult to see that for the specific kernel defined in (2.7) the minimizer of ¯ e ∞ , is unique. Indeed, performing calculations analogous to those in Lemma 6.1 of [30], one has thatfor that kernel ¯ e ∞ , = − h + A ( h ) = − h + ¯ C q h q − , (4.3)which has the unique minimizer ¯ h ∗ = (( q −
1) ¯ C q ) / ( q − . Step 1: (cid:98) K τ is C ∞ and convex. This comes from the definition of reflection positivity. Indeed, if (cid:98) K τ ( z ) = ´ + ∞ e − λz d µ ( λ ), for a positive measure µ , then (cid:98) K (cid:48)(cid:48) τ ( z ) = ´ + ∞ λ e − λz d µ ( λ ) ≥ Step 2:
Let f : R → R be a C and convex function and let f n : R → R be a family of C and convex functions such that for every x ∈ R one has that f n ( x ) → f ( x ). One can show that f (cid:48) n ( x ) → f (cid:48) ( x ) for every x . Indeed, let x ∈ R and ε >
0. By definition, there exists h > f ( x + h ) − f ( x ) h < f (cid:48) ( x ) + ε . Moreover, since f n converges to f pointwise, there exists n ∈ N suchthat f k ( x + h ) − f k ( x ) h < f (cid:48) ( x ) + ε for all k ≥ n . By using the convexity and differentiability of f k ,one can observe that f (cid:48) k ( x ) ≤ f k ( x + h ) − f k ( x ) h , hence f (cid:48) k ( x ) < f (cid:48) ( x ) + ε for all k ≥ n . The inequality f (cid:48) ( x ) − ε ≤ f (cid:48) k ( x ) can be obtained via a similar reasoning. Step 3:
We now show that for every δ > τ > | (cid:98) K (cid:48) τ ( z ) | ≤ z q +1 for every z ≥ δ and τ ≤ ¯ τ . Indeed, let z ≥ δ . Then, by assumption, one has that (cid:98) K τ ( z ) ≥ C q C ( τ q/β + z q ) and (cid:98) K τ ( z/ ≤ CC q τ q/β +( z/ q . Thus, from the convexity of (cid:98) K τ proved in Step 1 one has that | (cid:98) K (cid:48) τ ( z ) | ≤ | (cid:98) K τ ( z ) − (cid:98) K τ ( z/ | z/ ≤ (cid:12)(cid:12)(cid:12) C q C ( τ q/β + z q ) − CC q τ q/β +( z/ q (cid:12)(cid:12)(cid:12) z/ . (4.4)21iven that τ (cid:28) δ , there exist ˆ c , ˆ C and ¯ C , such that | (cid:98) K (cid:48) τ ( z ) | ≤ (cid:12)(cid:12)(cid:12) C q C ( τ q/β + z q ) − C q τ q/β +( z/ q (cid:12)(cid:12)(cid:12) z/ ≤ (cid:12)(cid:12)(cid:12) ˆ cz q − ˆ C ( z/ q (cid:12)(cid:12)(cid:12) z/ ≤ ¯ Cz q +1 . Step 4:
From Steps 2 and Step 3, one has that the convergence of (cid:98) K τ ( z ) to z q in property (2.4) canbe upgraded to uniform convergence on compact intervals of (0 , + ∞ ). Indeed, let [ a, b ] ⊂ (0 , + ∞ )and let y ∈ [ a, b ]. Then (cid:98) K τ ( y ) = (cid:98) K τ ( a ) + ´ ya (cid:98) K (cid:48) τ ( t ) d t . On one side, (cid:98) K τ ( a ) converges to a q . Onthe other side, the pointwise convergence of (cid:98) K (cid:48) τ from Step 2 turns in a convergence of the integral ´ ya (cid:98) K (cid:48) τ ( t ) d t thanks to the bound in Step 3 for τ small enough depending on a and dominatedconvergence. Step 5:
From the definition of A τ ( h ) one has that A (cid:48) τ ( h ) = ˆ + ∞ h (cid:98) K τ ( z ) d z + (cid:88) k ∈ N ˆ (2 k +1) h kh (cid:98) K τ ( h − y ) d y + (cid:88) k ∈ N (2 k + 1) ˆ h (cid:98) K τ ( x − (2 k + 1) h ) d x − (cid:88) k ∈ N k ˆ h (cid:98) K τ ( x − kh ) d x and A (cid:48)(cid:48) τ ( h ) = − (cid:98) K τ ( h ) + (cid:88) k ∈ N (2 k + 1) (cid:98) K τ ( − kh ) − (cid:88) k ∈ N k (cid:98) K τ ((1 − k ) h ) + (cid:88) k ∈ N ˆ (2 k +1) h kh (cid:98) K (cid:48) τ ( h − y ) d y − (cid:88) k ∈ N (2 k + 1) ˆ h (cid:98) K (cid:48) τ ( x − (2 k + 1) h ) d x + (cid:88) k ∈ N (2 k + 1) (cid:98) K τ ( − kh )+ (cid:88) k ∈ N (2 k ) ˆ h (cid:98) K (cid:48) τ ( x − (2 k ) h ) d x − (cid:88) k ∈ N k (cid:98) K τ ((1 − k ) h ) . In particular, for δ >
0, on [ δ, + ∞ ) A (cid:48) τ and A (cid:48)(cid:48) τ converge uniformly to A (cid:48) respectively A (cid:48)(cid:48) , where A is defined in (4.3). This follows from Step 4, the decay property (2.3), pointwise convergence of (cid:98) K (cid:48) τ given by Step 2 and the bound for (cid:98) K (cid:48) τ given in Step 3. Step 6:
As a consequence of the convergence of A (cid:48) τ in Step 5, one has thatlim τ → sup {| h − ¯ h ∗ | : e (cid:48)∞ ,τ ( h ) = 0 } = 0 Step 7:
The function ¯ e ∞ , defined in (4.3) is strictly convex in a neighborhood of the uniqueminimizer. Indeed, ¯ e (cid:48)(cid:48)∞ , ( h ) = 1 h (cid:104) ¯ C q ( q − q h q − − (cid:105) which on the minimizer ¯ h ∗ = (( q −
1) ¯ C q ) / ( q − gives h ∗ ( q − > Conclusion:
By Step 6, critical points of e ∞ ,τ for τ small enough have to be contained in aconnected neighborhood of ¯ h ∗ where ¯ e (cid:48)(cid:48)∞ , ≥ α >
0. Since by Step 5 A (cid:48)(cid:48) τ converges uniformly to A (cid:48)(cid:48) = ¯ e (cid:48)(cid:48)∞ , , then also e (cid:48)(cid:48)∞ ,τ > e (cid:48)∞ ,τ = 0 for τ small. Therefore the critical points of e (cid:48)∞ ,τ arelocal minima, and since the neighborhood of ¯ h ∗ where they are contained is connected, then theyare also unique. 22he main ingredient to prove Theorem 4.1 is the following estimate, which recalls [30, Lemma 6.3]. Lemma 4.2.
For every L − periodic set E of locally finite perimeter, it holds F τ,L ( E ) ≥ L (cid:88) x ∈ ∂E ∩ [0 ,L ) ( x + − x ) e ∞ ,τ ( x + − x ) + ( x − x − ) e ∞ ,τ ( x − x − ) , (4.5) where x + , x − have been defined in (2.8) . Theorem 4.1 follows from (4.5) in a few lines as in [30].Lemma 4.2 is a direct consequence of the following two Lemmas.The first, analogous to [30, Lemma 6.6], rewrites the second term of the functional F τ,L as a Laplacetransform. Lemma 4.3.
For every E ∈ [0 , L ) , τ > , F τ,L ( E ) = 1 L Per( E, [0 , L )) (cid:16) − ˆ R (cid:98) K τ ( z ) | z | d z (cid:17) − ˆ + ∞ f ( α ) (cid:16) ˆ [0 ,L ) ˆ R | χ E ( x ) − χ E ( y ) | e − α | x − y | d y d x (cid:17) d α, (4.6) where f is a nonnegative integrable function, the inverse Laplace transform of (cid:98) K τ .Proof. Notice that, in comparison with the functional F ,L studied in [30], the part of F τ,L whichmultiplies the perimeter of E has finite energy (due to the presence of τ β in the kernel) and thenit does not need to be transformed through Laplace transform together with the last term.For the last term, it is sufficient as in Section 2 to notice that, due to assumption (2.6), f isnonnegative. Lemma 4.4 ([30, Lemma 6.10]) . For every α > and every L − periodic set E − ˆ [0 ,L ] ˆ R | χ E ( x ) − χ E ( y ) | e − α | x − y | d y d x ≥ (cid:88) x ∈ ∂E ∩ [0 ,L ) ( x + − x ) e α, ∞ ( x + − x )+( x − x − ) e α, ∞ ( x − x − ) . (4.7) where, for α, h > , e α, ∞ ( h ) := − h ˆ h ˆ R | χ E h ( x ) − χ E h ( y ) | e − α | x − y | d y d x = lim L → + ∞ − L ˆ [0 ,L ] ˆ R | χ E h ( x ) − χ E h ( y ) | e − α | x − y | d y d x. For the proof of this result, based on the so-called reflection positivity technique, see [30, Lemma 6.10].
Proof of Theorem 4.1.
Theorem 4.1 follows in the same way as in [30] once Lemma 4.2 is shown.We refer to Section 6 in [30] for the details. 23
Discrete Γ -convergence Result In this section, we will show how to adapt the proof of Theorem 3.1 in order to obtain the analogousresult for the discrete setting. In order to state the precise result we need two preliminary steps.
Continuous representation of a discrete set E ⊂ ε Z d . Given E ⊂ ε Z d ε >
0, we will associate to it ˜ E ε ⊂ R d via˜ E ε := (cid:91) i ∈ E (cid:16) i + [ − ε/ , ε/ d (cid:17) . Per ,ε ( E, [0 , L ) d ) := (cid:88) x ∈ [0 ,L ) d ∩ ε Z d (cid:88) y ∼ x | χ E ( x ) − χ E ( y ) | ε d − . (5.1)We call Per ,ε the (1 , ε )-perimeter. Notice thatPer ,ε ( E, [0 , L ) d ) = Per ( ˜ E ε , [0 , L ) d ) . Let also denote by B ε := (cid:110) F ⊂ R d : ∃ E ⊂ ε Z d such that χ ˜ E ε ( x ) = χ F ( x ) for a.e. x ∈ R d (cid:111) . Since for every F ∈ B ε there exists only one E ⊂ ε Z d such that χ ˜ E ε ( x ) = χ F ( x ) for almost everypoint x , we will use the above relation to identify the discrete E ⊂ ε Z d set with the correspondingcontinuous set ˜ E ε .Letting E ⊂ Z d and ˜ E ⊂ R d as in (5.1) (for ε = 1), then the discrete functional defined in (1.1)can be rewritten as˜ F dsc J,L ( E ) := 1 L d (cid:16) J Per , ( E, [0 , L ) d ) − ˆ [0 ,L ) d (cid:88) ζ ∈ Z d \{ } K dsc ( ζ ) | χ ˜ E ( x + ζ ) − χ ˜ E ( x ) | d x (cid:17) , where K dsc ( ζ ) := | ζ | p for some p ≥ d + 2. Scaling of the functional.
Let us denote by K dsc ε ( ζ ) := ε d | ζ | p and let τ := J dsc c − J for J < J dsc c , where J dsc c has been defined in(1.4).The factor ε d in the definition of K dsc ε , comes from the rescaling in the spatial variables. Indeed ε d corresponds to the volume of [ − ε/ , ε/ d . It is not difficult to see that the measure µ ε = (cid:80) ζ ∈ ε Z d K dsc ε ( ζ ) δ ζ (by δ ζ we denote the Dirac measure concentrated in ζ ) converge weakly* to K dsc ( ζ ) d ζ in R d \ { } . Moreover the piecewise constant function associated to µ ε x → (cid:88) ζ ∈ ε Z d \{ } | ζ | p χ Q ε ( ζ ) ( x )converges in L ( R d \ { } ) to | x | p .As in the continuous case, if one optimizes in the discrete setting on the family of periodicstripes, one finds that the optimal stripes have a width of order τ − / ( p − d − and energy of or-der − τ ( p − d ) / ( p − d − (see [26]). Letting β := p − d −
1, this motivates the rescaling x := τ − /β (cid:98) x, L := τ − /β (cid:98) L and ˜ F dsc J,L ( E ) := τ ( p − d ) /β F dsc τ, (cid:98) L ( (cid:98) E ) , (5.2)24here now ˆ E ⊂ τ /β Z d .In these variables, the optimal stripes have width of order one. For simplicity of notation we willdenote by κ = τ /β . After rescaling the functional (similarly to the continuum) we obtain a newfunctional defined for [0 , L ) d -periodic sets E ⊂ κ Z d (where L is a multiple of κ ) via the formula F dsc τ,L ( E ) = 1 L d (cid:16) − Per ,κ ( E, [0 , L ) d ) + (cid:88) ζ ∈ κ Z d \{ } K dsc κ ( ζ ) (cid:104) ˆ ∂ ˜ E κ ∩ [0 ,L ) d d (cid:88) i =1 | ν ˜ E κ i ( x ) || ζ i | d H d − ( x ) − ˆ [0 ,L ) d | χ ˜ E κ ( x ) − χ ˜ E κ ( x + ζ ) | d x (cid:105)(cid:17) , (5.3)where ν ˜ E κ is the generalized normal of ∂ ˜ E κ .Before stating the main result of this section, define the functional F d → c τ,L ( F ) := (cid:40) F dsc τ,L ( E ) if F = ˜ E κ , for some E ⊂ κ Z d + ∞ otherwiseOur main result is the following: Theorem 5.1.
Let p ≥ d + 2 and L > . Then one has that F d → c τ,L Γ -converge in the L -topologyas τ → to a functional (cid:98) F ,L which is invariant under permutation of coordinates and finite onsets of the form E = F × R d − where F ⊂ R is L -periodic with { ∂F ∩ [0 , L ) } < ∞ . On such setsthe functional is defined by (cid:98) F ,L ( E ) = 1 L (cid:16) − { ∂F ∩ [0 , L ) } + ˆ R d K dsc ( ζ ) (cid:104) (cid:88) x ∈ ∂F ∩ [0 ,L ) | ζ | − ˆ L | χ F ( x ) − χ F ( x + ζ ) | d x (cid:105) d ζ (cid:17) . Moreover, if there exists M and a family of sets E τ ⊂ τ /β Z d such that for every τ one has that F d → c τ,L ( ˜ E κτ ) < M , then up to a relabeling of the coordinate axes, one has that there is a subsequencewhich converges in L to some set E = F × R d − with { ∂F ∩ [0 , L ) } < + ∞ .Proof. As the method used in the continuum applies almost identically to the discrete setting,instead of repeating the same arguments we will give the main steps.
Step 1:
Lower bound of the functional.As in the continuum (see Section 2), we find a lower bound for the discrete functionals very similarto (2.16). In order to obtain it let us define quantities analog to those in Section 2.Let G i, d → c τ,L ( ˜ E κ ) := G i, dsc τ,L ( E ) := ˆ [0 ,L ) d − G d, d → c τ,L ( ˜ E κx ⊥ i ) d x ⊥ i , (5.4)where for every E ⊂ Z and thus ˜ E κ ⊂ R , one has that G d, d → c τ,L ( ˜ E κ ) := (cid:88) z ∈ κ Z \{ } (cid:98) K dsc κ ( z ) (cid:16) Per( ˜ E κ , [0 , L )) | z | − ˆ L | χ ˜ E κ ( x ) − χ ˜ E κ ( x + z ) | d x (cid:17) , (5.5)and (cid:98) K dsc κ ( z ) = (cid:80) ζ (cid:48) ∈ κ Z d − K dsc κ ( z, ζ (cid:48) ). 25otice that when z = 0, (cid:98) K dsc κ ( z ) = + ∞ . However in (5.5) (cid:98) K dsc κ (0) is multiplied by 0. Thus, byusing the standard convention in analysis + ∞ · I i, d → c τ,L ( ˜ E κ ) := 2 d (cid:88) ζ ∈ κ Z d \{ } ˆ [0 ,L ) d K dsc κ ( ζ ) | χ ˜ E κ ( x ) − χ ˜ E κ ( x + ζ i ) || χ ˜ E κ ( x ) − χ ˜ E κ ( x + ζ ⊥ i ) | d x, ˜ I d → c τ,L ( ˜ E κ ) := d (cid:88) i =1 I i, d → c τ,L ( ˜ E κ ) . (5.6)We also define for E ⊂ R d [0 , L ) d -periodic G i, d → c0 ,L ( E ) := ˆ [0 ,L ) d − G d, d → c0 ,L ( E x ⊥ i ) d x ⊥ i , G d, d → c0 ,L ( E ) := ˆ R (cid:98) K dsc ( z ) (cid:16) Per( E, [0 , L )) | z | − ˆ L | χ E ( x ) − χ E ( x + z ) | d x (cid:17) d z,I i, d → c0 ,L ( E ) := ˆ [0 ,L ) d ˆ R d K dsc ( ζ ) | χ E ( x + ζ i ) − χ E ( x ) || χ E ( x + ζ ⊥ i ) − χ E ( x ) | d ζ d x In a similar fashion to Section 2, one obtains that F d → c τ,L ( ˜ E κ ) = F dsc τ,L ( E ) ≥ L d (cid:16) − Per ( ˜ E κ , [0 , L ) d ) + d (cid:88) i =1 G i, d → c τ,L ( ˜ E κ ) + d (cid:88) i =1 I i, d → c τ,L ( ˜ E κ ) (cid:17) . (5.7)and for E ⊂ κ Z G d, d → c τ,L ( ˜ E κ ) ≥ C d,L (cid:88) x ∈ ∂ ˜ E κ ∩ [0 ,L ) min(( x + − x ) − β , τ − ) + min(( x − x − ) − β , τ − ) . (5.8)where x + , x − are defined in (2.8). Inequality (5.8) follows also with the discrete kernel in the sameway as in the continuum. Step 2:
Estimate of the perimeter.From Step 1 and by using a similar reasoning to the one leading to (2.24), one can obtain that for E ⊂ κ Z d it holds Per ( ˜ E κ , [0 , L ) d ) ≤ C d,L L d max(1 , F d → c τ,L ( ˜ E κ )) . (5.9)The above in particular implies a compactness result, namely given E τ ⊂ κ Z d such that F d → c τ,L ( ˜ E κτ ) < + ∞ , then there exists a set of locally finite perimeter E ⊂ R d and a subsequence { ˜ E κ n τ n } such that˜ E κ n τ n → E in L . Step 3:
Lower semicontinuity of I i, d → c τ,L .For E ⊂ κ Z d , let us define f i ˜ E κ ( x, ζ ) = | χ ˜ E κ ( x + ζ i ) − χ ˜ E κ ( x ) || χ ˜ E κ ( x + ζ ⊥ i ) − χ ˜ E κ ( x ) | (cid:29) δ (cid:29) κ and let x, ζ ∈ κ Z d with | ζ | > δ . Then, for every ζ (cid:48) ∈ Q κ ( ζ ), by using ζ > δ (cid:29) κ and Taylor’s remainder theorem, one has that | ζ | p ≤ | ζ (cid:48) | p + C d κ | ζ | p +1 , and thus ˆ Q κ ( x ) κ d f i ˜ E κ ( x (cid:48) , ζ ) | ζ | p d x (cid:48) = ˆ Q κ ( x ) ˆ Q κ ( ζ ) f i ˜ E κ ( x (cid:48) , ζ ) | ζ | p d ζ (cid:48) d x (cid:48) = ˆ Q κ ( x ) ˆ Q κ ( ζ ) f i ˜ E κ ( x (cid:48) , ζ (cid:48) ) K dsc ( ζ (cid:48) ) d ζ (cid:48) d x (cid:48) + C d κ d +1 / | ζ | p +1 . (5.10)By summing over x ∈ [0 , L ) d and ζ ∈ κ Z d \ { } and using the definition of I i, d → c τ,L (see (5.6)) andusing the fact that κ d , L ) d ∩ κ Z d ) (cid:39) L d , one has that I i, d → c τ,L ( ˜ E κ ) ≥ ˆ [0 ,L ) d ˆ R d χ B cδ (0) ( ζ ) f i ˜ E κ ( x, ζ ) K dsc ( ζ ) d ζ d x + L d C d κ/δ p − d +1 . Finally, let us assume that ˜ E κτ ⊂ R d is such that there exists E ⊂ R d such that ˜ E κτ converges to E in L . Then lim inf τ ↓ I i, d → c τ,L ( ˜ E κτ ) ≥ lim inf τ ↓ ˆ [0 ,L ) d ˆ R d χ B cδ (0) ( ζ ) f i ˜ E κτ ( x, ζ ) K dsc ( ζ ) d ζ d x ≥ ˆ [0 ,L ) d ˆ R d χ B cδ (0) ( ζ ) f iE ( x, ζ ) K dsc ( ζ ) d ζ d x, thus from the arbitrariness of δ one has thatlim inf τ ↓ I i, d → c τ,L ( ˜ E κτ ) ≥ ˆ [0 ,L ) d ˆ R d f iE ( x, ζ ) K dsc ( ζ ) d ζ d x, Step 4:
Semicontinuity of G i, d → c τ,L .Let E τ ⊂ κ Z d . Moreover, assume that ˜ E κτ converges to E in L . Thenlim τ ↓ (cid:88) z ∈ κ Z d | z | >δ ˆ [0 ,L ) d κ d | χ ˜ E κτ ( x + z ) − χ ˜ E κτ ( x ) || z | p d x = ˆ R d ˆ [0 ,L ) d χ R d \ B δ (0) ( ζ ) | χ E ( x + ζ ) − χ E ( x ) | K dsc ( ζ ) d ζ d x. (5.11)In order to prove the above notice that given x, ζ ∈ κ Z d , then whenever | ζ | > δ (cid:29) κ one has that(similarly to (5.10)) ˆ Q κ ( x ) κ d | χ ˜ E κτ ( x (cid:48) + ζ ) − χ ˜ E κτ ( x (cid:48) ) || ζ | p d x (cid:48) = ˆ Q κ ( x ) ˆ Q κ ( ζ ) | χ ˜ E κτ ( x (cid:48) + ζ (cid:48) ) − χ ˜ E κτ ( x (cid:48) ) | K dsc ( ζ (cid:48) ) d ζ (cid:48) d x (cid:48) + C d κ d +1 / | ζ | p +1 Thus summing over x and ζ one obtains (5.11).We would like to show the semicontinuity G i, d → c τ,L . Let E τ ⊂ κ Z d , such that ˜ E κτ converges to E ⊂ R d in L . Then lim inf τ ↓ G i, d → c τ,L ( ˜ E κτ ) ≥ G i, d → c0 ,L ( E ) .
27o prove it, it is sufficient to show a similar statement for the G d, d → c τ,L . Namely let E τ ⊂ κ Z suchthat ˜ E κτ converges to E ⊂ R in L . Thenlim inf τ ↓ G d, d → c τ,L ( ˜ E κτ ) ≥ G d, d → c0 ,L ( E ) . (5.12)Let us now show (5.12). Given that (see the comment after (2.20))Per( ˜ E κτ , [0 , L )) | z | − ˆ L | χ ˜ E κτ ( x ) − χ ˜ E κτ ( x + z ) | d x ≥ , one has that for every δ >
0, it holds G d, d → c τ,L ( ˜ E κτ ) ≥ (cid:88) z ∈ κ Z | z |≥ δ (cid:98) K dsc κ ( z ) (cid:16) Per( ˜ E κτ , [0 , L )) | z | − ˆ L | χ ˜ E κτ ( x ) − χ ˜ E κτ ( x + z ) | d x (cid:17) . Thus by passing to the liminf in the above, and using the lower semicontinuity of the perimeterand (5.11) one has thatlim inf τ ↓ G d, d → c τ,L ( ˜ E κτ ) ≥ lim inf τ ↓ (cid:88) z ∈ κ Z | z |≥ δ (cid:98) K dsc κ ( z ) (cid:16) Per( ˜ E κτ , [0 , L )) | z | − ˆ L | χ ˜ E κτ ( x ) − χ ˜ E κτ ( x + z ) | d x (cid:17) ≥ lim inf τ ↓ ˆ R χ B cδ (0) ( z ) (cid:98) K dsc ( z ) (cid:16) Per( ˜ E κτ , [0 , L )) | z | − ˆ L | χ ˜ E κτ ( x ) − χ ˜ E κτ ( x + z ) | d x (cid:17) d z ≥ ˆ R χ B cδ (0) ( z ) (cid:98) K dsc ( z ) (cid:16) Per( E, [0 , L )) | z | − ˆ L | χ E ( x ) − χ E ( x + z ) | d x (cid:17) d z, where (cid:98) K dsc ( z ) = ´ R d − K dsc ( ζ (cid:48) , z ) d ζ (cid:48) . By the arbitrarity of δ one has the desired claim. Step 5:
Gamma limit.Let E τ ⊂ Z d such that sup τ F d → c τ,L ( ˜ E κτ ) < + ∞ . From Step 2, one has that ˜ E κτ is a sequence of sets of locally finite perimeter, thus there exists E ⊂ R d such that E is a set of locally finite perimeter and ˜ E κτ converges to E in L . Fromthe lower semicontinuity Steps 3 and 4, one has that for every i ∈ { , . . . , d } it holds G i, d → c0 ,L ( E ), I i, d → c0 ,L ( E ) < + ∞ .Thus from the rigidity estimate (Proposition 3.2), one has that E is a union of stripes.The Γ-limit result consists in two parts:(i) Γ-liminf(ii) Γ-limsupThe Γ-liminf is obtained by Step 4 above. In order to obtain the Γ-limsup, we need to find arecovery sequence E τ such that F d → c τ,L ( ˜ E κτ ) → (cid:98) F ,L ( E ). To do so it is sufficient to consider for every τ , a union of stripes ˜ E κτ such that ˜ E κτ is [0 , L ) d -periodic and such that it is close in L to E .28 Structure of minimizers
In this section we prove Theorem 1.2, asserting that minimizers of F τ,L are periodic stripes ofperiod h τ , provided τ is small enough.The idea of the proof of Theorem 1.2 is to start from (2.16), namely F τ,L ( E ) ≥ L d (cid:16) − d (cid:88) i =1 Per i ( E, [0 , L ) d ) + d (cid:88) i =1 G iτ,L ( E ) + d (cid:88) i =1 I iτ,L ( E ) (cid:17) . (6.1)Since I iτ,L ( E ) = 0 if and only if E is a union of stripes, one has that the l.h.s. and r.h.s. of theabove are equal whenever E is a union of stripes. Thus, it is sufficient to show that optimal stripesare minimizers of the r.h.s. of the above.For the r.h.s., in order to show optimality of stripes we will initially use Theorem 3.1 in order toreduce ourselves to a situation in which the minimizer of F τ,L are close to optimal stripes S . Thisholds for τ < ¯ τ , where ¯ τ depends on L . In this situation we will show that oscillations of thecharacteristic function of the set E in directions which are orthogonal to the direction of S increasenecessarily the r.h.s. of (6.1) (this is done in Lemma 6.1). Thus the r.h.s. of (6.1) can be furtherbounded from below by 1 L d (cid:16) − Per i ( E, [0 , L ) d ) + G iτ,L ( E ) (cid:17) , (6.2)where e i is the orientation of the stripes. Finally, optimal stripes minimize (6.2), since it correspondsto the one-dimensional problem of Section 4.Informally, the next lemma says the following: suppose that E is L -close to a set S which is a unionof periodic stripes in the direction e (see Figure 2 (a)), then the contribution of the functionalfrom directions other than e are nonnegative. This claim is formally expressed in (6.4). Moreover,these contribution are equal to zero if and only if the set E is also union of stripes. (a) t i + εt i − εt i (b) Figure 2: In (a) a set which is close to optimal periodic stripes is depicted. In (b) a zoomed-inportion where the set deviates from being a union of stripes is depicted.29 emma 6.1 (Stability) . Let E ⊂ R d be a [0 , L ) d -periodic set of locally finite perimeter, and S be aset which is a union of periodic stripes, i.e. (up to exchange of coordinates and translations) thereexists ˆ E ⊂ R such that S = ˆ E × R d − and ˆ E = (cid:91) k ∈ Z [2 kh, (2 k + 1) h ) , (6.3) for a suitable h . Then, there exist ¯ ε, ¯ τ > such that if (cid:107) χ E − χ S (cid:107) L < ¯ ε and τ < ¯ τ , one has that,for i ∈ { , . . . , d } , − Per i ( E, [0 , L ) d ) + G iτ,L ( E ) + I iτ,L ( E ) ≥ . (6.4) Moreover, in (6.4) equality holds if and only if E is a union of stripes in direction e .Proof. Let i ∈ { , . . . , d } . From (2.1) and (2.19), − Per i ( E, [0 , L ) d ) + G iτ,L ( E ) = ˆ [0 ,L ) d − (cid:104) − ∂E x ⊥ i + G dτ,L ( E x ⊥ i ) (cid:105) d x ⊥ i . From (2.21), − Per i ( E, [0 , L ) d ) + G iτ,L ( E ) ≥ ˆ [0 ,L ) d − (cid:88) s ∈ ∂E x ⊥ i [ − C d,L min (cid:0) ( s + − s − ) − β , τ − (cid:1) (6.5)+ C d,L min (cid:0) ( s − s − ) − β , τ − (cid:1) ] d x ⊥ i , (6.6)where s + and s − are defined after (2.8).Let τ > η > τ < τ and ρ ≤ η − C d,L min( ρ − β , τ − ) > . Thus, given that there are at most
L/η points s ∈ ∂E x ⊥ i in the slice with s + − s, s − s − > η , onehas that the r.h.s. of (6.5), can be bounded from below by − L/η .Consider the following decomposition[0 , L ) d − = A i ( η ) ∪ A i ∪ A i ( η ) , where A i ( η ) = { x ⊥ i ∈ [0 , L ) d − : min s ∈ ∂E x ⊥ i ( s + − s, s − s − ) ≥ η } ; (6.7) A i = { x ⊥ i ∈ [0 , L ) d − : ∂E x ⊥ i = ∅} ; (6.8) A i ( η ) = { x ⊥ i ∈ [0 , L ) d − : ∃ s ∈ ∂E x ⊥ i s.t. s + − s < η or s − s − < η } (6.9)where η ≤ η is such that, for τ ≤ ¯ τ with 0 < ¯ τ < τ , and for ρ ≤ η − C d,L min (cid:0) ρ − β , τ − (cid:1) > Lη . (6.10)The integrand in the r.h.s. of (6.5) can be estimated as follows:30. if x ⊥ i ∈ A i ( η ) it can be estimated from below by − L/η ,2. if x ⊥ i ∈ A i it is zero,3. if x ⊥ i ∈ A i ( η ) it is positive. Indeed, if x ⊥ i ∈ A i ( η ), then there exists a point s ∈ ∂E x ⊥ i suchthat for ρ equal to either s + − s or s − s − , (6.10) holds. Since for the choice of η , the sum ofthe negative terms in r.h.s. of (6.5) is bigger than or equal to − L/η , one has that for every x ⊥ i ∈ A i ( η ) the integrand in the r.h.s. of (6.5) is positive.Hence, − Per i ( E, [0 , L ) d ) + G iτ,L ( E ) ≥ − Lη H d − ( A i ( η )) + c H d − ( A i ( η )) , for some c > ε < η s.t. I iτ,L ( E ) > Lε H d − ( A i ( ε )) for τ smallenough. In order to do so, we will estimate via slicing the contribution in I iτ,L ( E ) for fixed x ⊥ i ∈ A i ( ε ) and show that it is larger than 2 Lε for a certain choice of parameters ε and τ .Let us now estimate I iτ,L ( E ). Recall that d I iτ,L ( E ) = ˆ R d − ˆ R ˆ [0 ,L ) d − ˆ [0 ,L ) K τ ( ζ ) f E ( x ⊥ i , x i , ζ ⊥ i , ζ i ) d x i d x ⊥ i d ζ i d ζ ⊥ i , where f E was defined in (3.3), namely f E ( x ⊥ i , x i , ζ ⊥ i , ζ i ) = | χ E ( x ⊥ i + x i + ζ i ) − χ E ( x i + x ⊥ i ) || χ E ( x ⊥ i + x i + ζ ⊥ i ) − χ E ( x i + x ⊥ i ) | . Choose (16¯ ε ) /d < η , (16¯ ε ) /d < ε < η and fix t ⊥ i ∈ A i ( ε ). The choice of ε, ¯ ε is made in order tohave (6.11).Because of the definition of A i ( ε ), there exists t i ∈ ∂E t ⊥ i such that one of the following holds(i) ( t i − ε, t i ) ⊂ E t ⊥ i and ( t i , t i + ε ) ⊂ E ct ⊥ i (ii) ( t i − ε, t i ) ⊂ E ct ⊥ i and ( t i , t i + ε ) ⊂ E t ⊥ i ,W.l.o.g., we may assume that (i) above holds (see Figure 2 (b)) and that i = d . We recall fromSection 2 that for ε > t ⊥ d ∈ [0 , L ) d − we let Q ⊥ ε ( t ⊥ d ) = { z ⊥ d ∈ [0 , L ) d − : | t ⊥ d − z ⊥ d | ≤ ε } .Since (cid:107) χ E − χ S (cid:107) L ([0 ,L ) d ) < ¯ ε , choosing ¯ ε as above, one hasmax (cid:16) | Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) ∩ E c || Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) | , | Q ⊥ ε ( t ⊥ d ) × ( t d , t d + ε ) ∩ E || Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) | (cid:17) ≥ . (6.11)For a set which is a union of stripes in the direction e d the above is trivial with 1 / / E , we use the hypothesis that it is close to stripes in the direction e less than ε d /
16 with respect to the distance induced by L ([0 , L ) d ).Indeed, we have that L d (cid:16)(cid:0) Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) (cid:1) ∩ E c (cid:17) ≥ L d (cid:16)(cid:0) Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) (cid:1) ∩ S c (cid:17) − ˆ Q ⊥ ε ( t ⊥ d ) × ( t d − ε,t d ) | χ E ( x ) − χ S ( x ) | d x ≥ L d (cid:16)(cid:0) Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) (cid:1) ∩ S c (cid:17) − ˆ [0 ,L ) d | χ E ( x ) − χ S ( x ) | d x ≥ L d (cid:16)(cid:0) Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) (cid:1) ∩ S c (cid:17) − ¯ ε L d (cid:16)(cid:0) Q ⊥ ε ( t ⊥ d ) × ( t d , t d + ε ) (cid:1) ∩ E (cid:17) ≥ L d (cid:16)(cid:0) Q ⊥ ε ( t ⊥ d ) × ( t d , t d + ε ) (cid:1) ∩ S (cid:17) − ˆ Q ⊥ ε ( t ⊥ d ) × ( t d ,t d + ε ) | χ E ( x ) − χ S ( x ) | d x. ≥ L d (cid:16)(cid:0) Q ⊥ ε ( t ⊥ d ) × ( t d , t d + ε ) (cid:1) ∩ S (cid:17) − ¯ ε Given that for every t ⊥ d ∈ [0 , L ) d − and S periodic union of stripes of period h (actually it wouldsuffice that S are stripes) it holdsmax (cid:16) L d (cid:16)(cid:0) Q ⊥ ε ( t ⊥ d ) × ( t d , t d + ε ) (cid:1) ∩ S (cid:17) , L d (cid:16)(cid:0) Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) (cid:1) ∩ S c (cid:17)(cid:17) ≥ ε d / , one has the desired claim.Hence, from the above, we can further assume that( t d − ε, t d ) ⊂ E t ⊥ d and | Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) ∩ E c || Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) | ≥ . (6.12)For every s ∈ ( t d − ε, t d ), ( ζ ⊥ d , s ) (cid:54)∈ E and ζ d + s ∈ ( t d , t d + ε ) we have that f E ( t ⊥ d , s, ζ ⊥ d , ζ d ) = 1.Thus by integrating initially in ζ d and estimating by (2.3) K τ ( ζ ) with Cε p + τ p/β , we have that ˆ t d + εt d − ε ˆ t d + ε − st d − s ˆ Q ⊥ ε ( t ⊥ d ) f E ( t ⊥ d , s, ζ ⊥ d , ζ d ) K τ ( ζ ) d ζ ⊥ d d ζ d d s ≥≥ Cε p + τ p/β ε ˆ Q ⊥ ε ( t ⊥ d ) ˆ t d t d − ε | χ E t ⊥ d ( s ) − χ E t ⊥ d + ζ ⊥ d ( s ) | d s d ζ ⊥ d ≥ Cε p + τ p/β ε ˆ Q ⊥ ε ( t ⊥ d ) ˆ t d t d − ε | − χ E t ⊥ d + ζ ⊥ d ( s ) | d s d ζ ⊥ d ≥ Cε p + τ p/β ε | Q ⊥ ε ( t ⊥ d ) × ( t d − ε, t d ) ∩ E c | ≥ C/ ε d +1 ε p + τ p/β . In order to conclude we notice that d I dτ,L ( E ) ≥ ˆ A d ( ε ) (cid:16) (cid:88) t d ∈ ∂E t ⊥ d ˆ t d + εt d − ε ˆ t d + ε − st d − s ˆ Q ⊥ ε ( t ⊥ d ) f E ( t ⊥ d , s, ζ ⊥ d , ζ d ) d ζ ⊥ d d ζ d d s (cid:17) d t ⊥ d ≥ ˆ A d ( ε ) (cid:16) (cid:88) t d ∈ ∂E t ⊥ d Cε d +1 ε p + τ p/β (cid:17) d t ⊥ d . (6.13)Finally by choosing ε and τ small we have the desired result, namely that there exists ε < η s.t. I dτ,L ( E ) > Lε H d − ( A d ( ε )) for τ small enough. Up to a permutation of coordinates, this naturallyholds also for i = 2 , . . . , d − Proof of Theorem 1.2.
Before proceeding to the proof, let us explain its strategy. In the first stepwe show that the minimizers are L ([0 , L ) d )-close to some set S that consists of optimal periodicstripes. W.l.o.g. let us assume that S = ˆ E × R d − , whereˆ E = (cid:91) k ∈ Z [2 kh, (2 k + 1) h ) . − ˆ [0 ,L ) d − Per( E x ⊥ , [0 , L )) d x ⊥ + ˆ [0 ,L ) d − G dτ,L ( E x ⊥ ) d x ⊥ + I τ,L ( E ) (6.14)and one which is larger than − d (cid:88) i =2 Per i ( E, [0 , L ) d ) + d (cid:88) i =2 G iτ,L ( E ) + d (cid:88) i =2 I iτ,L ( E ) . (6.15)Afterwards, we notice that optimal stripes minimize the first term (6.14) and are such that (6.15)is equal to zero. On the other side, for anything that does not consists of stripes and is L ([0 , L ) d )-close to S , one has that the contribution given from (6.15) is strictly positive. Thus optimal stripesare minimizers for F τ,L . To prove the positivity of (6.15) in case of nonoptimal stripes, we useLemma 6.1. Step 1:
From the Γ-convergence result (see Theorem 3.1), we have that for every ε >
0, thereexists a τ = τ ( ε ) > < τ < τ and for every minimizer E τ of F τ,L , onehas that E τ is ε -close to S in L ([0 , L ) d ), where S is a periodic stripe of size 2 h . W.l.o.g., we mayassume that S = ˆ E × [0 , L ) d − , whereˆ E = j (cid:91) k =0 [2 kh, (2 k + 1) h ) , for some j ∈ N .We fix ε = ¯ ε and τ < ¯ τ as in Lemma 6.1. Step 2:
Let us consider the original functional F τ,L , for τ ≤ ¯ τ as in Step 1 and set E = E τ .Recall that one has F τ,L ( E ) = 1 L d (cid:16) − Per ( E, [0 , L ) d ) + ˆ R d K τ ( ζ ) (cid:104) ˆ ∂E ∩ [0 ,L ) d d (cid:88) i =1 | ν Ei ( x ) || ζ i | d H d − ( x ) − ˆ [0 ,L ) d | χ E ( x ) − χ E ( x + ζ ) | d x (cid:105) d ζ (cid:17) ≥ L d (cid:16) − Per ( E, [0 , L ) d ) + ˆ R d K τ ( ζ ) (cid:104) ˆ ∂E ∩ [0 ,L ) d | ν E ( x ) || ζ | d H d − ( x ) − ˆ [0 ,L ) d | χ E ( x ) − χ E ( x + ζ ) | d x (cid:105) dζ (cid:17) (6.16) − d (cid:88) i =2 Per i ( E, [0 , L ) d ) + d (cid:88) i =2 G iτ,L ( E ) + d (cid:88) i =2 I iτ,L ( E ) . (6.17)One notices immediately that, thanks to Theorem 4.1, if E = ˆ E τ × [0 , L ) d − with ˆ E τ one-dimensionalperiodic set of period h τ , then the first term (6.16) of F τ,L is minimized, while the terms in (6.17)are equal to zero.On the other hand, from Lemma 6.1, if a minimizer E does not have such a structure, or more ingeneral E is not a stripe in direction e , then the last term (6.17) is strictly positive.33 roof of Theorems 1.3. This claim is implied by the one-dimensional result of Theorem 4.1 oncefrom Theorem 1.2 one knows that the minimizers are stripes. τ from L The main purpose of this section will be to prove Theorem 1.4.At the end of this section, we will briefly say how to deal with the discrete setting, namely how toprove Theorem 1.5.
Let us first give an idea of the proof.We say that a union of stripes S is oriented along the direction e i , if S is invariant with respect toevery translation orthogonal to e i .As in Section 6, instead of the functional F τ,L it is convenient to consider the r.h.s. of (6.1) andshow that its minimizers are stripes.The main ingredient in the proof of Theorem 1.2 was to use the rigidity argument in order to showthat the minimizers of F τ,L are close to being stripes (say oriented along e i ). Once in this situationwe showed that on slices E t ⊥ j in directions e j (cid:54) = e i having points in ∂E t ⊥ j increases necessarily theenergy. Thus we are left to optimize on the slices in direction e i . On the slices in direction e i theenergy contribution is bounded from below by the energy contribution of the periodic stripes ofperiod h ∗ τ .The main difficulty in proving Theorem 1.4 when compared to Theorem 1.2 lies in the fact thatthe rigidity result (Proposition 3.2 and Theorem 3.1) can not be applied directly in order to implythe closeness of the minimizers in L ([0 , L ) d ) to optimal stripes or even stripes. This is due to thefact that the rigidity argument works for fixed L and τ ↓ L is rewrittenas an average of local contributions on smaller cubes of size l (the functionals ¯ F τ ( E, Q l ( z )) definedin (7.15)), where l is fixed independently of L and l < L ( l depends only on a constant dependingonly on the dimension which comes out of the estimates, as explained at the end of this outline).Namely, we will show that (see (7.18)) F τ,L ( E ) ≥ L d ˆ [0 ,L ) d ¯ F τ ( E, Q l ( z )) d z. (7.1)The aim of this section will be to show that the minimizers of the r.h.s. of the above are optimalperiodic stripes and on optimal periodic stripes the above inequality is an equality. In this outline,we will speak of contributions to the energy of subsets of [0 , L ) d . For a generic subset B ⊂ [0 , L ) d ,such contribution is ˆ B ¯ F τ ( E, Q l ( z )) d z. When a set J is contained in a one-dimensional slice then its contribution is given by ˆ J ¯ F τ ( E, Q l ( z )) d H ( z ) . (7.2)34here H is the usual one-dimensional Hausdorff measure. Indeed, since we use slicing argumentsin the proof, often the contribution on a set B will be recovered by integrating the contributionsgiven by its slices.Let E τ be a minimizer. The functional ¯ F τ ( E τ , Q l ( z )) will contain by construction a term of theform 1 l d d (cid:88) i =1 ˆ Q l ( z ) ˆ R d | χ E τ ( x + ζ i ) − χ E τ ( x ) || χ E τ ( x + ζ ⊥ i ) − χ E τ ( x ) | K τ ( ζ ) d ζ d x, (7.3)which is a kind of local version (on scale l ) of the cross-interaction term I τ,L ( E τ ) defined in (2.15).As we prove in the local rigidity lemma, namely Lemma 7.6, such term will be large, for E notclose in L to stripes in Q l ( z ) for τ < τ ( l ) (our measure of closeness will be quantified in Definition7.3). This is the local counterpart of the rigidity Proposition 3.2. A first consequence of this fact isthat, from average arguments, one has that for “most” of the z contained in [0 , L ) d , it holds that E τ ∩ Q l ( z ) has to be close to a set which is a union of stripes. The aim is then to prove that oneis L -close to stripes in some fixed direction on the whole cube [0 , L ) d .A clearer picture of what happens in [0 , L ) d is given by the following decomposition:[0 , L ) d = A − ∪ A ∪ . . . ∪ A d where • A i with i > z such that there is only one direction e i , such that E τ ∩ Q l ( z ) is close to stripes oriented in direction e i . • A − is the set of points z such that there exist directions e i and e j ( i (cid:54) = j ) and stripes S i (oriented in direction e i ) and S j (stripes oriented in direction e j ) such that E τ ∩ Q l ( z ) is closeto both S i ∩ Q l ( z ) and S j ∩ Q l ( z ). In particular, this implies that either | E τ ∩ Q l ( z ) | (cid:28) l d or | E cτ ∩ Q l ( z ) | (cid:28) l d (see Remark 7.4 (ii)). • A is the set of points z where none of the above points is true.The aim is then to show that A ∪ A − = ∅ and that there exists only one A i with i >
0. Thus,by the local version of the Stability Lemma 6.1, namely Lemma 7.8, minimizers must be stripes,which by Theorem 1.3 are periodic of period h ∗ τ (with τ depending on l and not on L ).As we will show in the proof of Theorem 1.4, A ∪ A − separates the different A i , namely everycontinuous curve γ : [0 , T ] → [0 , L ) d intersecting A i and A j has necessarily to intersect A ∪ A − .Let us initially explain what is intuitively expected:(i) for any z ∈ A i with i >
0, when slicing in directions orthogonal to e i , alternation betweenregions in E and regions in E c should increase the energy (similarly to Lemma 6.1). Thusone expects the contribution of A i to be C ∗ τ | A i | /L d − C l | ∂A i | /L d where C l is a constant depending on l and C ∗ τ is the energy of periodic stripes of width h ∗ τ .(ii) for any z ∈ A or z ∈ A − we expect sub-optimal contributions, namely larger than C ∗ τ . Thushaving A ∪ A − is not energetically convenient. Since A ∪ A − separates the different A i ,one has that | A ∪ A − | acts like a boundary term and compensates the boundary term | ∂A i | in (i). 35e will show that by choosing τ small but independent of L , the contribution of A ∪ A − balancesthe contribution due to the presence of ∂A i .Let us now give more specific technical details as a guideline in the reading of the proof.In a similar way to the proof of Theorem 1.1 (see Lemma 6.1), we first show that, once we are ina region A i with i >
0, alternations between regions in E and regions in E c on slices in directionsperpendicular to e i increase necessarily the value of the functional (see Lemma 7.8).Therefore, we can ignore for regions A i contributions along e j for j (cid:54) = i .Thus we are left to bound from below the contributions of the slices of A i s in direction e i , for all i > , L ) d in the direction e i . There are two cases:(i) all the slice is contained in A i ;(ii) there are points in the slice belonging to ∂A i .In the first case, we show that the contribution of the slice to the energy is bigger or equal to C ∗ τ L ,which would be the contribution of periodic stripes of period h ∗ τ .In the second case, points in the slice which belong to the boundary of A i necessarily belong to A ∪ A − and we prove the following estimates. Let I ⊂ A i be a maximal interval on the slice indirection e i . The optimal contribution of I whenever ∂I ∩ A (cid:54) = ∅ is bigger than ( C ∗ τ | I | − M l )where | I | is the length of the interval, C ∗ τ is the optimal energy density for stripes of width h ∗ τ and M is a constant not depending on τ but depending only on the dimension (see Lemma 7.9). Theconstant M comes from the nonlocal interactions close to the boundary. Analogously, if I ⊂ A i satisfies ∂I ⊂ A − one has that the contribution has the better lower bound ( C ∗ τ | I | − M /l ) (dueto the fact that, close to the boundary and then to A − , one is close also to stripes in a directiondifferent from i and then Lemma 7.8 can be applied).In order to balance the negative term ( − M l or − M /l ) coming from the presence of the boundaryof I , we will use the fact that the adjacent regions in A or A − are “thick” enough. For A , thisis a consequence of the fact that the map E (cid:55)→ “ L -distance of E from the stripes” . is Lipschitz in L (see Remark 7.4 (i)). For A − this follows from the continuity of the L measurew.r.t. translations.If ∂I ∩ A (cid:54) = ∅ , choosing τ = τ ( l ) sufficiently small but finite, the term (7.3) in such a region willgive a large positive contribution M (see the Local Rigidity Lemma 7.6), that will compensate theterm − M l .If ∂I ∩ A − (cid:54) = ∅ , the total negative contribution of an interval I ⊂ A i and the neighbouring J ⊂ A − will be at most of the order of C ∗ τ | I | − max( M , | J | /l , due to the fact that A − is “thick”, i.e. | J | ≥ L d , the energy will be boundedfrom below by C ∗ τ d (cid:88) i =1 | A i | /L d − M | A ∪ A − | lL d ≥ C ∗ τ − C ∗ τ | A ∪ A − | L d − M | A ∪ A − | lL d , which, provided l is chosen bigger than some constant depending only on the dimension chosen atthe beginning, is greater than C ∗ τ and strictly greater than C ∗ τ if | A ∪ A − | > A i , i >
0, with | A i | >
0. Then, as a consequence of Lemma 7.8,when one slices [0 , L ) d in directions orthogonal to e i , any set that deviates from being exactly astripe in direction e i gives a positive contribution (see (7.36)) while slicing in direction e i one hasthat periodic unions of stripes of period h ∗ τ in direction e i are optimal. Therefore, as in the proofof Theorem 1.2 one gets that the minimizers must be periodic unions of stripes of period h ∗ τ indirection e i . Let t ∈ R d . We recall that we denote by t i = (cid:104) t, e i (cid:105) e i and t ⊥ i = t − t i . We will also denote by Q ⊥ l ( z ⊥ i ) the cube of size l centered at z in the subspace which is orthogonal to e i . As explained inSection 2, with a slight abuse of notation with might identify points x i with their e i -coordinates in R and points x ⊥ i with points of R d − . In order to simplify notation, we will use A (cid:46) B , whenever there exists a constant ¯ C d dependingonly on the dimension d such that A ≤ ¯ C d B . In the definitions (7.5), (7.12) and (7.13) below, we introduce the different terms which, summedtogether, give rise to the “local contribution” ¯ F τ ( E, Q l ( z )) to the energy on a square Q l ( z ) (definedin (7.15)). Let us see how these terms come out naturally from the lower bound (6.1) on thefunctional F τ,L .Let us recall that (see (6.1) and (2.16)) F τ,L ( E ) ≥ L d (cid:16) − d (cid:88) i =1 Per i ( E, [0 , L ) d ) + d (cid:88) i =1 G iτ,L ( E ) + d (cid:88) i =1 I iτ,L ( E ) (cid:17) = − L d d (cid:88) i =1 Per i ( E, [0 , L ) d ) + 1 L d d (cid:88) i =1 (cid:104) ˆ [0 ,L ) d ∩ ∂E ˆ R d | ν Ei ( x ) || ζ i | K τ ( ζ ) d ζ d H d − ( x ) − ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ i ) − χ E ( x ) | K τ ( ζ ) d ζ d x (cid:105) + 2 d L d d (cid:88) i =1 ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ i ) − χ E ( x ) || χ E ( x + ζ ⊥ i ) − χ E ( x ) | K τ ( ζ ) d ζ d x (7.4)and recall that in the above equality holds whenever the set E is a union of stripes. Thus, provingthat optimal stripes are the minimizers of the r.h.s. of (7.4) implies that they are the minimizersfor F τ,L .Now we want to further express the r.h.s. of (7.4) as a sum of contributions obtained first by slicingand then considering interactions with neighbouring points on the slice, namely − L d Per i ( E, [0 , L ) d ) + 1 L d (cid:104) ˆ [0 ,L ) d ∩ ∂E ˆ R d | ν Ei ( x ) || ζ i | K τ ( ζ ) d ζ d H d − ( x ) − ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ i ) − χ E ( x ) | K τ ( ζ ) d ζ d x (cid:105) = 1 L d ˆ [0 ,L ) d − (cid:88) s ∈ ∂E t ⊥ i ∩ [0 ,L ] r i,τ ( E, t ⊥ i , s ) d t ⊥ i , s ∈ ∂E t ⊥ i , r i,τ ( E, t ⊥ i , s ) := − ˆ R | ζ i | (cid:98) K τ ( ζ i ) d ζ i − ˆ ss − ˆ + ∞ | χ E t ⊥ i ( u + ρ ) − χ E t ⊥ i ( u ) | (cid:98) K τ ( ρ ) d ρ d u − ˆ s + s ˆ −∞ | χ E t ⊥ i ( u + ρ ) − χ E t ⊥ i ( u ) | (cid:98) K τ ( ρ ) d ρ d u (7.5)and s + = inf { t (cid:48) ∈ ∂E t ⊥ i , with t (cid:48) > s } s − = sup { t (cid:48) ∈ ∂E t ⊥ i , with t (cid:48) < s } (7.6)are as in (2.8).Indeed, given that E is a set of locally finite perimeter, we can use slicing arguments (see Section 2).Let us slice in the directions e i , i = 1 , . . . , d . One has that − L d Per i ( E, [0 , L ) d ) + 1 L d (cid:104) ˆ [0 ,L ) d ∩ ∂E ˆ R d | ν Ei ( x ) || ζ i | K τ ( ζ ) d ζ d H d − ( x ) − ˆ [0 ,L ) d ˆ R d | χ E ( x + ζ i ) − χ E ( x ) | K τ ( ζ ) d ζ d x (cid:105) = 1 L d ˆ [0 ,L ) d − (cid:16) − Per( E t ⊥ i , [0 , L )) + (cid:88) s ∈ ∂E t ⊥ i ˆ R | ρ | (cid:98) K τ ( ρ ) d ρ − ˆ L ˆ −∞ | χ E t ⊥ i ( ρ + u ) − χ E t ⊥ i ( u ) | (cid:98) K τ ( ρ ) d ρ d u − ˆ L ˆ + ∞ | χ E t ⊥ i ( ρ + u ) − χ E t ⊥ i ( u ) | (cid:98) K τ ( ρ ) d ρ d u (cid:17) d t ⊥ i . Now, given a measurable and L -periodic function f in R , it is immediate to notice that ˆ L f ( u ) d u = (cid:88) s ∈ ∂E t ⊥ i ˆ ss − f ( u ) d u = (cid:88) s ∈ ∂E t ⊥ i ˆ s + s f ( u ) d u. (7.7)Therefore we have that ˆ L ˆ −∞ | χ E t ⊥ i ( ρ + u ) − χ E t ⊥ i ( u ) | (cid:98) K τ ( ρ ) d ρ d u = (cid:88) s ∈ ∂E t ⊥ ˆ s + s ˆ −∞ | χ E t ⊥ i ( ρ + u ) − χ E t ⊥ i ( u ) | (cid:98) K τ ( ρ ) d ρ d u and analogously ˆ L ˆ + ∞ | χ E t ⊥ i ( ρ + u ) − χ E t ⊥ i ( u ) | (cid:98) K τ ( ρ ) d ρ d u = (cid:88) s ∈ ∂E t ⊥ ˆ ss − ˆ + ∞ | χ E t ⊥ i ( ρ + u ) − χ E t ⊥ i ( u ) | (cid:98) K τ ( ρ ) d ρ d u. For notational reasons it is convenient to introduce the one-dimensional analogue of (7.5). Namely,let E ⊂ R be a set of locally finite perimeter and let s − , s, s + ∈ ∂E . We define r τ ( E, s ) := − ˆ R | ρ | (cid:98) K τ ( ρ ) d ρ − ˆ ss − ˆ + ∞ | χ E ( ρ + u ) − χ E ( u ) | (cid:98) K τ ( ρ ) d ρ d u − ˆ s + s ˆ −∞ | χ E ( ρ + u ) − χ E ( u ) | (cid:98) K τ ( ρ ) d ρ d u. (7.8)38he quantities defined in (7.5) and (7.8) are related via r i,τ ( E, t ⊥ i , s ) = r τ ( E t ⊥ i , s ).The next Remark is the analogue of (2.21), where now instead of considering all the contributionsfrom the points in ∂E t ⊥ i , we restrict to the points neighbouring s . Remark 7.1.
There exists η > and τ > such that, for E ⊂ R d , s − , s, s + ∈ ∂E t ⊥ i threeconsecutive points, whenever τ < τ and min( | s − s − | , | s + − s | ) < η , then r i,τ ( E, t ⊥ i , s ) > .Indeed, since ∀ ρ ∈ (0 , + ∞ ) , it holds: ˆ ss − | χ E t ⊥ i ( u + ρ ) − χ E t ⊥ i ( u ) | d u ≤ min( ρ, | s − s − | ) ∀ ρ ∈ ( −∞ , , it holds: ˆ s + s | χ E t ⊥ i ( u + ρ ) − χ E t ⊥ i ( u ) | d u ≤ min( − ρ, | s − s + | ) , (see (2.17) ), thus r i,τ ( E, t ⊥ i , s ) ≥ − ˆ + ∞ (cid:0) ρ − min( ρ, | s − s − | ) (cid:1) (cid:98) K τ ( ρ ) d ρ + ˆ −∞ (cid:0) − ρ − min( − ρ, | s − s + | ) (cid:1) (cid:98) K τ ( ρ ) d ρ. ≥ − ˆ + ∞ | s − s − | ρ (cid:98) K τ ( ρ ) d ρ + ˆ − | s − s + |−∞ − ρ (cid:98) K τ ( ρ ) d ρ. From the above formula the claim follows directly. Moreover, by using the elementary inequality min (cid:16) ˆ − α −∞ − ρ (cid:98) K τ ( ρ ) d ρ, ˆ + ∞ α ρ (cid:98) K τ ( ρ ) d ρ (cid:17) (cid:38) min( α − β , τ − ) , one can further estimate the above as r i,τ ( E, t ⊥ i , s ) ≥ − C min( | s − s + | − β , τ − ) + C min( | s − s − | − β , τ − ) (7.9) where C is a constant depending on the kernel and β = p − d − . The above is the “local” analogueof (2.21) . Recalling the notation in (3.3) f E ( t ⊥ i , t i , ζ ⊥ i , ζ i ) = | χ E ( t i + t ⊥ i + ζ i ) − χ E ( t i + t ⊥ i ) || χ E ( t i + t ⊥ i + ζ ⊥ i ) − χ E ( t i + t ⊥ i ) | , (7.10)we can also rewrite the last term on the r.h.s. of (7.4) as2 d L d ˆ [0 ,L ) d ˆ R d f E ( t ⊥ i , t i , ζ ⊥ i , ζ i ) K τ ( ζ ) d ζ d t = 1 L d ˆ [0 ,L ) d − (cid:88) s ∈ ∂E t ⊥ i ∩ [0 ,L ] v i,τ ( E, t ⊥ i , s ) d t ⊥ i + 1 L d ˆ [0 ,L ) d w i,τ ( E, t ⊥ i , t i ) d t (7.11)where w i,τ ( E, t ⊥ i , t i ) = 1 d ˆ R d f E ( t ⊥ i , t i , ζ ⊥ i , ζ i ) K τ ( ζ ) d ζ. (7.12)39nd v i,τ ( E, t ⊥ i , s ) = 12 d ˆ s + s − ˆ R d f E ( t ⊥ i , u, ζ ⊥ i , ζ i ) K τ ( ζ ) d ζ d u (7.13)and s + , s − as in (7.6).Notice that w i,τ is closely related to I iτ,L in (2.15). Indeed, w i,τ can be seen as a localization ordensity of I iτ,L . More precisely, I iτ,L ( E ) = 2 ˆ [0 ,L ) d w i,τ ( E, t ⊥ i , t i ) d t ⊥ i d t i . Let us now show the decomposition claimed in (7.11). By using (7.7), one has that (cid:88) s ∈ ∂E t ⊥ i v i,τ ( E, t ⊥ i , s ) = 12 d (cid:88) s ∈ ∂E t ⊥ i ˆ s + s − ˆ R d f E ( t ⊥ i , u, ζ ⊥ i , ζ i ) K τ ( ζ ) d ζ d u = 12 d (cid:88) s ∈ ∂E t ⊥ i ˆ s + s ˆ R d f E ( t ⊥ i , u, ζ ⊥ i , ζ i ) K τ ( ζ ) d ζ d u + 12 d (cid:88) s ∈ ∂E t ⊥ i ˆ ss − ˆ R d f E ( t ⊥ i , u, ζ ⊥ i , ζ i ) K τ ( ζ ) d ζ d u = 1 d ˆ L ˆ R d f E ( t ⊥ i , u, ζ ⊥ i , ζ i ) K τ ( ζ ) d ζ d u Finally, integrating over t ⊥ i one has that1 L d ˆ [0 ,L ) d − (cid:88) s ∈ ∂E t ⊥ i v i,τ ( E, t ⊥ i , s ) d t ⊥ i = 1 d L d ˆ [0 ,L ) d ˆ R d f E ( t ⊥ i , t i , ζ ⊥ i , ζ i ) d ζ d t. (7.14)To conclude the proof of the decomposition claimed in (7.11), it is sufficient to combine (7.14) andthe definition of w i,τ (see (7.12)).The intuition of the role of the terms r i,τ and v i,τ is the following. The term r i,τ first penalizesoscillations with high frequency in direction e i , namely sets whose slices in direction i have boundarypoints at small minimal distance. Indeed, fix t ⊥ i and consider s (cid:55)→ χ E t ⊥ i ( s ). If this function oscillateswith high frequency, there exist s, s + ∈ ∂E t ⊥ i such that | s − s + | is small. Hence by (7.9) thecontribution of r i,τ will be positive and large.The term v i,τ penalizes oscillations in direction e i whenever the neighbourhood of the point ( t ⊥ i + se i )is close in L to a stripe oriented along e j . This last statement will be made precise in Lemma 7.8.Finally, for every Q l ( z ), define¯ F i,τ ( E, Q l ( z )) := 1 l d (cid:104) ˆ Q ⊥ l ( z ⊥ i ) (cid:88) s ∈ ∂E t ⊥ i t ⊥ i + se i ∈ Q l ( z ) ( v i,τ ( E, t ⊥ i , s ) + r i,τ ( E, t ⊥ i , s )) d t ⊥ i + ˆ Q l ( z ) w i,τ ( E, t ⊥ i , t i ) d t (cid:105) , ¯ F τ ( E, Q l ( z )) := d (cid:88) i =1 ¯ F i,τ ( E, Q l ( z )) . (7.15)40he above consists in the “local contribution” to the energy in a cube Q l ( z ) mentioned in theoutline. More precisely, we will write the r.h.s. of (7.4) in terms of ¯ F τ ( E, Q l ( · )) via an averagingprocess. In order to do this we will need the following lemma. Lemma 7.2.
Let µ be a [0 , L ) d -periodic locally finite measure, namely a measure invariant undertranslations in L Z d . Then one has that ˆ [0 ,L ) d d µ ( x ) = 1 l d ˆ [0 ,L ) d ˆ Q l ( z ) d µ ( x ) d z. (7.16) Proof.
The proof of (7.16) is done by changing order of integration (namely Fubini): first integratingin z .Indeed, ˆ [0 ,L ) d ˆ Q l ( z ) d µ ( x ) d z = ˆ [0 ,L ) d ˆ R d χ Q l ( z ) ( x ) d µ ( x ) d z = ˆ [0 ,L ) d ˆ R d χ Q l ( x ) ( z ) d z d µ ( x )= l d ˆ [0 ,L ) d d µ ( x ) . By Lemma 7.2, we have that the r.h.s. of (7.4) is equal to1 L d ˆ [0 ,L ) d ¯ F τ ( E, Q l ( z )) d z. (7.17)Indeed, since E is [0 , L ) d -periodic we can see that the r.h.s. of (7.4) as an integration with respectto a [0 , L ) d -periodic measure. This implies that F τ,L ( E ) ≥ L d ˆ [0 ,L ) d ¯ F τ ( E, Q l ( z )) d z. (7.18)Given that, in the above inequality, equality holds for stripes, if we show that the minimizers of(7.17) are periodic optimal stripes, then the same claim holds for F τ,L .We will say that a set which is a union of stripes, S , is oriented along the direction e i , if S isinvariant with respect to every translation orthogonal to e i .In the next definition we define a quantity which measures the distance of a set from being a unionof stripes. Such a quantity being small means for us to be ( L -)“close” to stripes in a given cube. Definition 7.3.
For every η we denote by A iη the family of all sets F such that(i) they are union of stripes oriented along the direction e i (ii) their connected components of the boundary are distant at least η .We denote by D iη ( E, Q ) := inf (cid:110) Q ) ˆ Q | χ E − χ F | : F ∈ A iη (cid:111) and D η ( E, Q ) = inf i D iη ( E, Q ) . (7.19) Finally, we let A η := ∪ i A iη . S be union of stripes, namely S = ˆ S × R d − , with ˆ S = (cid:83) i ∈ Z ( α i , β i ). Then condition (ii) inDefinition 7.3, says that inf i,j | α i − β j | > η . This corresponds to the minimal distance between theconnected components of the boundaries of the stripes in S .In the following remark, we first notice that the local distance (7.19) from a family of stripes in acertain direction is a Lipschitz function w.r.t. the centre of the cube we consider. In particular,what we need in the proof of Theorem 1.4 is that, if at a point we are far from being stripes, thenin a neighbourhood of it we also have approximately the same distance (“thickness” of the set A mentioned in the outline and that will be defined in (7.54)).Moreover, in point (ii) of the next remark we notice that, if we are sufficiently close to stripes indifferent directions, then either | E ∩ Q l ( z ) | (cid:28) l d or | E c ∩ Q l ( z ) | (cid:28) l d (property of the set A − mentioned in the Outline and that will be defined in (7.55)). Remark 7.4. (i) Let
E, F ⊂ R d . Then the map z (cid:55)→ D η ( E, Q l ( z )) is Lipschitz, with Lipschitz constant C d /l ,where C d is a constant depending only on the dimension d . In order to see this for fixed F consider the map T F : z (cid:55)→ l d ˆ Q l ( z ) | χ E ( x ) − χ F ( x ) | d x. Then T F ( z (cid:48) ) = 1 l d ˆ Q l ( z (cid:48) ) | χ E ( x ) − χ F ( x ) | d x ≤ l d ˆ Q l ( z ) | χ E ( x ) − χ F ( x ) | d x + 1 l d | Q l ( z )∆ Q l ( z (cid:48) ) |≤ T F ( z ) + C d l | z − z (cid:48) | , where C d is a constant depending only on the dimension d and Q l ( z ) (cid:52) Q l ( z (cid:48) ) = ( Q l ( z ) \ Q l ( z (cid:48) )) ∪ ( Q l ( z (cid:48) ) \ Q l ( z )) . Finally given that D iη ( E, Q l ( · )) and D η ( E, Q l ( · )) are the infimaof T F ( · ) for F ∈ A iη and A η respectively, we have that D iη ( E, Q l ( · )) and D η ( E, Q l ( · )) areLipschitz with Lipschitz constant C d /l .In particular, whenever D η ( E, Q l ( z )) > α and D η ( E, Q l ( z (cid:48) )) < β , then | z − z (cid:48) | > l ( α − β ) /C d .(ii) For every ε there exists δ = δ ( ε ) such that for every δ ≤ δ whenever D jη ( E, Q l ( z )) ≤ δ and D iη ( E, Q l ( z )) ≤ δ with i (cid:54) = j for some η > , it holds min (cid:0) | Q l ( z ) \ E | , | E ∩ Q l ( z ) | (cid:1) ≤ ε. (7.20) The above claim follows easily by contradiction. Indeed, suppose that there exist ε > , asequence of sets in { E n } , and sequences δ n ↓ and η n > such that D jη n ( E n , Q l ( z )) ≤ δ n and D iη n ( E n , Q l ( z )) ≤ δ n (with i (cid:54) = j ) (7.21) and such that min (cid:0) | Q l ( z ) \ E | , | E ∩ Q l ( z ) | (cid:1) > ε . W.l.o.g. we assume that z = 0 . From (7.21) ,we have that there exist two sets S in and S jn such that the distance of E n is δ n -close in L to S in and S jn . Thus ˆ Q l (0) | χ S in ( x ) − χ S in ( x ) | d x ≤ δ n . (7.22)42 t is not difficult to see that (7.22) holds if and only if both S in ∩ Q l (0) and S jn ∩ Q l (0) are L -close (depending on δ n ) either to Q l (0) or to ∅ . Indeed, without lost of generality we canassume that d = 2 , thus S n = F n × R and S n = R × F n where F n , F n ⊂ R .Rewriting (7.22) and using Fubini, we have that ˆ Q l (0) | χ S in ( x ) − χ S in ( x ) | d x = ˆ l/ − l/ ˆ l/ − l/ | χ F n ( x ) − χ F n ( x ) | d x d x ≤ δ n . Noticing that χ F n ( x ) does not depend on x and that χ F n ( x ) ∈ { , } , we immediately deducethat χ F ∩ ( − l/ , l/ is close (depending on δ n ) in L (( − l/ , l/ to either ( − l/ , l/ or to ∅ , which in turn implies that E n is close in L ( Q l (0)) to Q l (0) or to ∅ . Notice that the abovereasoning does not depend on η , since the only thing used is that χ S iε and χ S iε are invariantwith respect to two different directions and take values in { , } . The following lemma is a technical lemma that is used in the Local Rigidity Lemma (Lemma 7.6).In particular, it says that if a family of sets E τ of locally finite perimeter in R converges in L toa set E of locally finite perimeter and the local contributions given by r τ ( E τ , s ) defined in (7.8)(which for slices E τ,t ⊥ i of E τ ⊂ R d coincides with r i,τ ( E, t ⊥ i , s ) in (7.5)) are uniformly bounded, then(7.23) holds. This is one of the preliminary steps used in Lemma 7.6 to show that E is a union ofstripes. Lemma 7.5.
Let E , { E τ } ⊂ R be a family of sets of locally finite perimeter and I ⊂ R be an openinterval. Moreover, assume that E τ → E in L ( I ) . If we denote by { k , . . . , k m } = ∂E ∩ I , then lim inf τ ↓ (cid:88) s ∈ ∂E τ s ∈ I r τ ( E τ , s ) ≥ m (cid:88) i =1 ( − C | k i − k i +1 | − β ) , (7.23) where r τ is defined in (7.8) .Proof. Let us denote by { k τ , . . . , k τm τ } = ∂E τ ∩ I . We will also denote by k τ = sup { s ∈ ∂E τ : s < k τ } and k τm τ +1 = inf { s ∈ ∂E τ : s > k τm τ } . Denote by A the r.h.s. of (7.23). From (7.9), one has that r τ ( E τ , k τi ) ≥ − C max( | k τi − k τi +1 | − β , τ − ). Thus, there exist η and ¯ τ > τ < ¯ τ , whenevermin i ∈{ ,...,m τ } | k τi +1 − k τi | < η then (cid:88) s ∈ ∂E τ s ∈ I r τ ( E τ , s ) (cid:38) A. Hence, assume there exists a subsequence τ k such that | k τ k i +1 − k τ k i | > η for all i ≤ m τ k . Up torelabeling, let us assume that it holds true for the whole sequence of E τ .Since min i | k τi +1 − k τi | > η the convergence E τ → E in L ( I ) can be upgraded to the convergenceof the boundaries, namely one has that there exists a ¯ τ such that for τ < ¯ τ , it holds ∂E τ ∩ I ) = ∂E ∩ I ) and k τi → k i . 43hen because of the convergence of the boundaries, we have thatlim inf τ ↓ (cid:88) s ∈ ∂E τ s ∈ I r τ ( E τ , s ) ≥ lim inf τ ↓ m (cid:88) j =1 (cid:0) − C max( | k τi − k τi +1 | − β , τ − ) (cid:1) ≥ m (cid:88) j =1 (cid:0) − C | k i − k i +1 | − β (cid:1) . (7.24)The following lemma contains the local version of Theorem 3.1, namely the rigidity estimate men-tioned in the outline.Its content can be summarized as follows. Given a sequence of sets E τ ⊂ R d of bounded local energy,by Remark 7.1 their boundary points on the slices are not too close and then they converge to a setof locally finite perimeter E . Then, using the lower semicontinuity result of Lemma 7.5 and themonotonicity in τ of the kernel, one gets as τ → (cid:80) di =1 G i ,L ( E )+ I ,L ( E ) < + ∞ ,but with G i ,L and I ,L substituted by their local counterparts ( r i,τ and w i,τ ). Then, applyingProposition 3.2, which has been already proved without using any periodicity assumption on E (see Remark 3.4), one has that the set E has to be a union of stripes. Therefore, for τ > l , the sets E τ will be close to E in the sense of Definition7.3. Lemma 7.6 (Local Rigidity) . For every
M > , l, δ > , there exist ¯ τ , ¯ η > such that whenever τ < ¯ τ and ¯ F τ ( E, Q l ( z )) < M for some z ∈ [0 , L ) d and E ⊂ R d [0 , L ) d -periodic, with L > l , then itholds D η ( E, Q l ( z )) ≤ δ for every η < ¯ η . Moreover ¯ η can be chosen independent on δ . Notice that ¯ τ and ¯ η are independent of L .Proof. The proof will follow by contradiction. Assume that the claim is false. This implies thatthere exists
M > , l , δ > { τ k } , { η k } , { L k } , { z k } , { E τ k } such that:(i) one has that τ k ↓ L k > l , η k ↓ z k ∈ [0 , L k ) d ;(ii) the family of sets E τ k is [0 , L k ) d -periodic(iii) one has that D η k ( E τ k , Q l ( z k )) > δ and ¯ F τ k ( E τ k , Q l ( z k )) < M .Given that η (cid:55)→ D η ( E, Q l ( z )) is monotone increasing, we can fix ¯ η sufficiently small instead of η k with D ¯ η ( E τ k , Q l ( z k )) > δ . In particular, ¯ η will be chosen at the end of the proof depending onlyon M, l .W.l.o.g. (taking e.g. E τ k − z k instead of E τ k ) we can assume there exists z ∈ R d such that z k = z for all k ∈ N .Because of Remark 7.1, one has that sup k Per ( E τ k , Q l ( z )) < + ∞ . Thus up to subsequences thereexists E such that E τ k → E in L ( Q l ( z )) with D ¯ η ( E , Q l ( z )) > δ .In order to keep the notation simpler, we will write τ → τ k → E τ → E insteadof E τ k → E .In this proof we will denote by J i the interval ( z i − l/ , z i + l/ τ such that for almost every t ⊥ i ∈ Q ⊥ l ( z ⊥ i )one has that E τ,t ⊥ i ∩ J i converges to E ,t ⊥ i ∩ J i in L ( Q l ( z )).44y using (7.15) and the fact that v i,τ ≥
0, we have that M ≥ ¯ F τ ( E τ , Q l ( z )) ≥ l d d (cid:88) i =1 ˆ Q ⊥ l ( z ⊥ i ) (cid:88) s ∈ ∂E τ,t ⊥ i s ∈ J i r i,τ ( E τ , t ⊥ i , s ) d t ⊥ i + ˆ Q l ( z ) w i,τ ( E τ , t ⊥ i , t i ) d t ⊥ i d t i . (7.25)By the Fatou lemma we have that l d M ≥ lim inf τ ↓ d (cid:88) i =1 ˆ Q ⊥ l ( z ⊥ i ) (cid:88) s ∈ ∂E τ,t ⊥ i s ∈ J i r i,τ ( E τ , t ⊥ i , s ) d t ⊥ i ≥ d (cid:88) i =1 ˆ Q ⊥ l ( z ⊥ i ) lim inf τ ↓ (cid:88) s ∈ ∂E τ,t ⊥ i s ∈ J i r i,τ ( E τ , t ⊥ i , s ) d t ⊥ i ≥ d (cid:88) i =1 ˆ Q ⊥ l ( z ⊥ i ) (cid:88) s ∈ ∂E ,t ⊥ i s ∈ J i (cid:0) − C ( s + − s ) − β + C ( s − s − ) − β (cid:1) d t ⊥ i , where in the last inequality we have used Lemma 7.5 applied to r i,τ ( E τ , t ⊥ i , s ) = r ( E τ,t ⊥ i , s ).The same type of inequality holds also for the last term in (7.25), namelylim inf τ ↓ ˆ Q l ( z ) w i,τ ( E τ , t ⊥ i , t i ) d t ⊥ i d t i ≥ lim inf τ ↓ d ˆ Q l ( z ) ˆ Q l ( z ) f E τ ( t ⊥ i , t i , t (cid:48)⊥ i − t ⊥ i , t (cid:48) i − t i ) K τ ( t − t (cid:48) ) d t d t (cid:48) ≥ d ˆ Q l ( z ) ˆ Q l ( z ) f E ( t ⊥ i , t i , t (cid:48)⊥ i − t ⊥ i , t (cid:48) i − t i ) K ( t − t (cid:48) ) d t d t (cid:48) (7.26)Indeed, in order to prove (7.26) we fix τ (cid:48) > E τ → E in L ( Q l ( z )) andafterwards the monotonicity of τ (cid:55)→ (cid:98) K τ ( s ) we have thatlim inf τ ↓ ˆ Q l ( z ) w i,τ ( E τ , t ⊥ i , t i ) d t ⊥ i d t i ≥ sup τ (cid:48) lim inf τ ↓ ˆ Q l ( z ) w i,τ (cid:48) ( E τ , t ⊥ i , t i ) d t ⊥ i d t i ≥ sup τ (cid:48) d ˆ Q l ( z ) ˆ Q l ( z ) f E ( t ⊥ i , t i , t (cid:48)⊥ i − t (cid:48)⊥ i , t i − t (cid:48) i ) K τ (cid:48) ( t − t (cid:48) ) d t d t (cid:48) ≥ d ˆ Q l ( z ) ˆ Q l ( z ) f E ( t ⊥ i , t i , t (cid:48)⊥ i − t (cid:48)⊥ i , t i − t (cid:48) i ) K ( t − t (cid:48) ) d t d t (cid:48) . To summarize, we have shown that d (cid:88) i =1 d ˆ Q l ( z ) ˆ Q l ( z ) f E ( t ⊥ i , t i , t (cid:48)⊥ i − t (cid:48)⊥ i , t i − t (cid:48) i ) K ( t − t (cid:48) ) d t d t (cid:48) + d (cid:88) i =1 ˆ Q ⊥ l ( z ⊥ i ) (cid:88) s ∈ ∂E ,t ⊥ i s ∈ J i (( s + − s ) − β + ( s − s − ) − β ) d t ⊥ i (cid:46) l d M + Per ( E , Q l ( z )) (7.27)45imilarly to (2.24), one can show thatPer ( E τ , Q l ( z )) (cid:46) l d max(1 , ¯ F τ ( E τ , Q l ( z ))) (cid:46) l d max(1 , M ) (cid:46) l d M, thus from the lower semicontinuity of the perimeter we have thatPer ( E , Q l ( z ))) ≤ lim inf τ ↓ Per ( E τ , Q l ( z ))) (cid:46) l d M. In particular, d (cid:88) i =1 ˆ Q ⊥ l ( z ⊥ i ) (cid:88) s ∈ ∂E ,t ⊥ i s ∈ J i (( s + − s ) − β + ( s − s − ) − β ) d t ⊥ i (cid:46) l d M (7.28) d (cid:88) i =1 ˆ Q l ( z ) ˆ Q l ( z ) f E ( t ⊥ i , t i , t (cid:48)⊥ i − t (cid:48)⊥ i , t i − t (cid:48) i ) K ( t − t (cid:48) ) d t d t (cid:48) (cid:46) l d M. (7.29)At this point, the same reasoning as in Proposition 3.2 can be applied in order to obtain that (7.28)and (7.29) hold if and only if E ∩ Q l ( z ) is a union of stripes. It is indeed sufficient to substitute[0 , L ) d with Q l ( z ) in the proof.Moreover, since the l.h.s. of (7.28) explodes for stripes with minimal width tending to zero, one hasthat there exists ¯ η = ¯ η ( M, l ) such that D ¯ η ( E , Q l ( z )) = 0. This contradicts that D ¯ η ( E , Q l ( z )) > δ ,which was assumed at the beginning of the proof.The following is a technical lemma needed in the proof of Lemma 7.9. It says that given a set E ⊂ R , and I ⊂ R an interval, then the one-dimensional contribution to the energy, namely (cid:80) s ∈ ∂E ∩ I r τ ( E, s ), is comparable to the periodic case up to a constant C depending only on thedimension.More precisely, C depends on η , which is the minimal distance between boundary points of E so that r τ ( E, s ) <
0, and it is what one can lose by extending periodically the set E outside theinterval I . For periodic sets then, by the reflection positivity Theorem 4.1, the energy contributionin (7.30) is bigger than or equal to the contribution of periodic stripes of width h ∗ τ , namely C ∗ τ times the length of the interval I . Lemma 7.7.
There exists C > such that the following holds. Let E ⊂ R be a set of locallyfinite perimeter and I ⊂ R be an open interval. Let s − , s and s + be three consecutive points onthe boundary of E and r τ ( E, s ) defined as in (7.8) . Then for all τ < τ , where τ is given inRemark 7.1, it holds (cid:88) s ∈ ∂Es ∈ I r τ ( E, s ) ≥ C ∗ τ | I | − C . (7.30) Proof.
Let us denote by k < . . . < k m the points of ∂E ∩ I , and k = sup { s ∈ ∂E : s < k } and k m +1 = inf { s ∈ ∂E : s > k m } W.l.o.g. we may assume that r τ ( E, k ) < r τ ( E, k m ) < I (cid:48) ⊂ I such that r τ ( E, k (cid:48) ) < r τ ( E, k (cid:48) m (cid:48) ) <
0, where k (cid:48) , · · · , k (cid:48) m (cid:48) are the points of ∂E ∩ I (cid:48) . Thus if estimate (7.30) holds for I (cid:48) then it holdsalso for I by the following chain of inequalities (cid:88) s ∈ ∂Es ∈ I r τ ( E, s ) ≥ (cid:88) s ∈ ∂Es ∈ I (cid:48) r τ ( E, s ) ≥ C ∗ τ | I (cid:48) | − C ≥ C ∗ τ | I | − C , where in the last inequality we have used that C ∗ τ < r τ ( E, k ) < r τ ( E, k m ) < η > τ ≤ τ ) such thatmin( | k − k | , | k − k | , | k m − − k m | , | k m +1 − k m | ) > η . We claim that m (cid:88) i =1 r τ ( E, k i ) ≥ m (cid:88) i =1 r τ ( E (cid:48) , k i ) − ¯ C (7.31)where E (cid:48) is obtained by extending periodically E with the pattern contained in E ∩ ( k , k m ) and¯ C = ¯ C ( η ) >
0. The construction of E (cid:48) can be done as follows: if m is odd we repeat periodically E ∩ ( k , k m ), and if m is even we repeat periodically ( k − η , k m ).Thus we have constructed a set E (cid:48) which is periodic of period k m − k or k m − k + η . Therefore m (cid:88) i =1 r τ ( E (cid:48) , k i ) ≥ C ∗ τ ( k m − k + η ) ≥ C ∗ τ | I | − ˜ C , (7.32)where ˜ C = ˜ C ( η ). Inequality (7.32) follows by reflection positivity (Theorem 4.1). Indeed,reflection positivity implies that optimal [0 , L )-periodic set E L must be a union of periodic stripesand C ∗ τ was the minimal energy density by further optimizing in L .Inequality (7.32) combined with (7.31) yields (7.30).To show (7.31), notice that the symmetric difference between E and E (cid:48) satisfies E ∆ E (cid:48) ⊂ ( −∞ , k − η ) ∪ ( k m + η , + ∞ ) , where η is the constant defined in Remark 7.1. To obtain (7.31), we need to estimate | (cid:80) mi =1 r τ ( E, k i ) − (cid:80) mi =1 r τ ( E (cid:48) , k i ) | . Let m (cid:88) i =1 r τ ( E, k i ) − m (cid:88) i =1 r τ ( E (cid:48) , k i ) = A + B, where A = m − (cid:88) i =0 ˆ k i +1 k i ˆ + ∞ ( s − | χ E ( s + u ) − χ E ( u ) | ) (cid:98) K τ ( s ) d s d u − m − (cid:88) i =0 ˆ k i +1 k i ˆ + ∞ ( s − | χ E (cid:48) ( s + u ) − χ E (cid:48) ( u ) | ) (cid:98) K τ ( s ) d s d uB = m (cid:88) i =1 ˆ k i +1 k i ˆ −∞ ( s − | χ E ( s + u ) − χ E ( u ) | ) (cid:98) K τ ( s ) d s d u − m (cid:88) i =1 ˆ k i +1 k i ˆ −∞ ( s − | χ E (cid:48) ( s + u ) − χ E (cid:48) ( u ) | ) (cid:98) K τ ( s ) d s d u. (cid:98) K (see (4.1)), we have that | A | ≤ ˆ k m k ˆ + ∞ χ E ∆ E (cid:48) ( u + s ) (cid:98) K τ ( s ) d s d u ≤ ˆ k m k ˆ ∞ k m + η (cid:98) K τ ( u − v ) d v d u ≤ C , where C is a constant depending only on η . Similarly, | B | ≤ C / (cid:12)(cid:12)(cid:12) m (cid:88) i =1 r τ ( E, k i ) − m (cid:88) i =1 r τ ( E (cid:48) , k i ) (cid:12)(cid:12)(cid:12) ≤ ˆ k m k ˆ ∞ k m + η (cid:98) K τ ( u − v ) d u d v + ˆ k m +1 k ˆ k − η −∞ (cid:98) K τ ( u − v ) d v d u. Since for every periodic set we have that C ∗ τ is the infimum of all the energy densities for periodicsets (of any period), we have the desired result.The next lemma is the local analog of the Stability Lemma 6.1. Informally, it shows that if weare in a cube where the set E ⊂ R d is close to a set E (cid:48) which is a union of stripes in direction e i (according to Definition 7.3), then it is not convenient to oscillate in direction e j with j (cid:54) = i (namely, on the slices in direction e i to have points in ∂E t ⊥ i ). We show that in such a case eitherthe local contribution given by r i,τ or the one given by v i,τ are large. We recall that the first termpenalizes alternation between regions in E t ⊥ i and regions in E ct ⊥ i with high frequency and the secondpenalizes oscillations in direction i whenever the neighbourhood of the point ( t ⊥ i , t i ) is close in L to a stripe in a perpendicular direction j . This second fact (about the role of v i,τ ) is justified inan analogous way to (6.11)-(6.13), with the only difference that there the interaction term I τ,L wasglobal (on [0 , L ) d ) and here we consider the localized version. Lemma 7.8 (Local Stability) . Let ( t ⊥ i + se i ) ∈ ( ∂E ) ∩ [0 , l ) d and η , τ as in Remark 7.1. Thenthere exist ˜ τ , ˜ ε > (independent of l ) such that for every τ < ˜ τ and ε < ˜ ε the following holds:assume that(a) min( | s − l | , | s | ) > η (b) D jη ( E, [0 , l ) d ) ≤ ε d l d for some η > and with j (cid:54) = i (this condition expresses that E ∩ [0 , l ) d isclose to stripes oriented along a direction orthogonal to e i )Then r i,τ ( E, t ⊥ i , s ) + v i,τ ( E, t ⊥ i , s ) ≥ .Proof. Let s − , s, s + be three consecutive points for ∂E t ⊥ i . By Remark 7.1, there exists η , τ > τ < τ min( | s − s − | , | s + − s | ) < η then r i,τ ( E, t ⊥ i , s ) > . Thus without loss of generality we may assume that min( | s − s − | , | s + − s | ) ≥ η .We choose ˜ ε, ˜ τ < τ with the following properties: ˜ ε < η , where η is defined in Remark 7.1 and˜ ε, ˜ τ are such that 7 C/ ε d +1 ˜ ε p + τ p/β ≥ , where K τ ( ζ ) ≥ C | ζ | p + τ p/β , (7.33)for every τ < ˜ τ (see (2.3) for the estimates on the kernel).48ecause of condition (a) in the statement of the lemma, there exists a cube Q ⊥ ˜ ε ( t (cid:48)⊥ i ) ⊂ R d − of size˜ ε , such that t ⊥ i ∈ Q ⊥ ˜ ε ( t (cid:48)⊥ i ) and ( s − ˜ ε, s + ˜ ε ) × Q ⊥ ˜ ε ( t (cid:48)⊥ i ) ⊂ [0 , l ) d .By definition, one has that r i,τ ( E, t ⊥ i , s ) + v i,τ ( E, t ⊥ i , s ) ≥ − ˆ s +˜ εs − ˜ ε ˆ ε − ε ˆ Q ⊥ ˜ ε ( t (cid:48)⊥ i ) f E ( t ⊥ i , t i , ζ ⊥ i , ζ i ) K τ ( ζ ) d ζ ⊥ i d ζ i d t i . (7.34)In order to estimate the r.h.s. of (7.34) from below, one proceeds exactly as in the proof of Lemma6.1 (using now that assumption (b) telling that E is L -close to stripes on the cube [0 , l ) d insteadof [0 , L ) d , see Figure 2) and obtains ˆ ss − ˜ ε ˆ ε − ε ˆ Q ⊥ ˜ ε ( t (cid:48)⊥ i ) f E ( t ⊥ i , u, ζ ⊥ i , ζ i ) K τ ( ζ ) d ζ d u ≥ C/ ε d +1 ˜ ε p + τ p/β . Then, by (7.34) and (7.33), r i,τ ( E, t ⊥ i , s ) + v i,τ ( E, t ⊥ i , s ) ≥ J on aslice in direction e i , we want to estimate from below the contribution of the energy on J , namely ´ J ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s .Point (i) below aims at estimating the contribution on J assuming that on l -cubes around thepoints in J the set E is close to stripes in another direction e j , j (cid:54) = i . Considering only points s ∈ J which are far from the boundary, like in ( a ) of Lemma 7.8, then by Lemma 7.8 we would have thatthe contribution is positive. The possibly negative contribution in (7.35) is due to the presence ofthe boundary terms, and therefore is absent in (7.36) when J = [0 , L ).Point (ii) wants to estimate the contribution on J without making assumptions on the behaviouraround points in J . One uses here, far from ∂J , the one-dimensional estimate of Lemma 7.7. Closeto the boundary, one has eventually some additional negative contribution, which is smaller inabsolute value when around the points of ∂J one is close to stripes in another direction (see (7.37)and (7.38) and Remark 7.10). Lemma 7.9.
Let ˜ ε, ˜ τ > as in Lemma 7.8. Let δ = ε d / (16 l d ) with < ε ≤ ˜ ε , τ ≤ ˜ τ and l > C / ( − C ∗ τ ) , where C is the constant appearing in Lemma 7.7. Let t ⊥ i ∈ [0 , L ) d − and η > .The following statements hold: there exists M constant independent of l (but depending on thedimension) such that(i) Let J ⊂ R an interval such that for every s ∈ J one has that D jη ( E, Q l ( t ⊥ i + se i )) ≤ δ with j (cid:54) = i . Then ˆ J ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ − M l . (7.35) Moreover, if J = [0 , L ) , then ˆ J ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ . (7.36)49 ii) Let J = ( a, b ) ⊂ R . If for s = a and s = b it holds D jη ( E, Q l ( t ⊥ i + se i )) ≤ δ with j (cid:54) = i , then ˆ J ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ | J | C ∗ τ − M l , (7.37) otherwise ˆ J ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ | J | C ∗ τ − M l. (7.38) Moreover, if J = [0 , L ) , then ˆ J ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ | J | C ∗ τ . (7.39) Remark 7.10.
Note that since C ∗ τ < , one has that (7.35) is a stronger inequality when comparedto (7.37) . This is due to the additional hypothesis D jη ( E, Q l ( t ⊥ i + se i )) ≤ δ , j (cid:54) = i , for all s ∈ J .Proof. Let us now prove (i). Since the result is valid for a general set E , we may assume withoutloss of generality that a = 0 and b = l (cid:48) , namely J = [0 , l (cid:48) ). We consider the decomposition Q l ( t ⊥ i + t i e i ) = Q ⊥ l ( t ⊥ i ) × Q il ( t i ), where Q ⊥ l ( t ⊥ i ) ⊂ R d − is the cube of size l centered at t ⊥ i and Q il ( t i ) ⊂ R is the interval of size l centered in t i . From the definition of ¯ F i,τ (see 7.15) and since w i,τ ≥
0, we have that ˆ J ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ l d ˆ J ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ Q il ( s ) (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i d s = 1 l d ˆ J ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i χ Q il ( s ) ( s (cid:48) ) (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i d s = 1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( − l ,l (cid:48) + l ) | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i . (7.40)where in order to obtain the last line, we have used Fubini to integrate first in s .Let us now estimate the last term in (7.40).For every point s (cid:48) ∈ ∂E t (cid:48)⊥ i such that min( | s (cid:48) + l/ | , | s (cid:48) − l (cid:48) − l/ | ) > η , using the fact that τ ≤ ˜ τ and δ = ε d / (16 l d ), 0 < ε < ˜ ε , where ˜ ε, ˜ τ are given in the statement of Lemma 7.8, and η >
0, wecan apply such Lemma and obtain that r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) ≥ s (cid:48) when s (cid:48) is close to the boundary, namely min( | s (cid:48) + l/ | , | s (cid:48) − l (cid:48) − l/ | ) < η . Since r i,τ is positive when the neighbouring points are closer than η , by using r i,τ ≥ − v i,τ ≥
0, for every t (cid:48)⊥ i we have that (cid:88) { s (cid:48) : min( | s (cid:48) + l | , | s (cid:48) − l (cid:48) − l | ) <η } (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) ≥ − . s (cid:48) is such that min( | s (cid:48) + l/ | , | s (cid:48) − l (cid:48) − l/ | ) < η we have that | Q il ( s (cid:48) ) ∩ J | l ≤ η l , by pluggingthe above on the r.h.s. of (7.40), we have the first claim.Let us now prove (7.36). ˆ L ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ l d ˆ L ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ Q il ( s ) (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i d s = 1 l d ˆ L ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i χ Q il ( s ) ( s (cid:48) ) (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i d s = 1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ [0 ,L ) (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i , where the last step follows from using initially Fubini as in (7.40) and afterwards the L -periodicity.Hence (7.36) holds since when J = [0 , L ) by periodicity we do not have to care for points s (cid:48) ∈ ∂E t (cid:48)⊥ i close to ∂J and we can directly apply Lemma 7.8 giving r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) ≥ J = (0 , l (cid:48) ).As in (7.40) one has that ˆ J ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( − l ,l (cid:48) + l ) | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i = 1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( l ,l (cid:48) − l ) | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i + 1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( − l , l ] ∪ [ l (cid:48) − l ,l (cid:48) + l ) | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i (7.41)where if l (cid:48) ≤ l , we have that ( l/ , l (cid:48) − l/
2) is empty and then1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( l/ ,l (cid:48) − l/ (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i = 0 . Fix t (cid:48)⊥ i . We will now estimate the contributions for ( t (cid:48)⊥ i , s (cid:48) ) ∈ Q l ( t (cid:48)⊥ i ) and ( t (cid:48)⊥ i , s (cid:48) ) ∈ Q l ( t (cid:48)⊥ i + l (cid:48) e i ).If the condition D jη ( E, Q l ( t (cid:48)⊥ i )) ≤ δ or D jη ( E, Q l ( t (cid:48)⊥ i + l (cid:48) e i )) ≤ δ is missing, then we will estimate r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) from below with − η , otherwise r i,τ ≥
0. Hence, the last term in (7.41) can be estimated by1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( − l , l ] | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i (cid:38) − M l l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ [ l (cid:48) − l ,l (cid:48) + l ) | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i (cid:38) − M l. If l (cid:48) > l , we have that1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( l/ ,l (cid:48) − l/ r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) d t (cid:48)⊥ i ≥ C ∗ τ | J | − lC ∗ τ − C , where in the last inequality we have used Lemma 7.7 for E = E t (cid:48)⊥ i and I = ( l/ , l (cid:48) − l/ l (cid:48) < l , we have that1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( l/ ,l (cid:48) − l/ r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) d t (cid:48)⊥ i ≥ (cid:0) C ∗ τ | J | − lC ∗ τ − C (cid:1) χ (0 , + ∞ ) ( | J | − l ) , Thanks to the fact that l > C / ( − C ∗ τ ), one has (7.38).Let us now turn to the proof of (7.37). Given that D jη ( E, Q l ( t ⊥ i )) ≤ δ and D jη ( E, Q l ( t ⊥ i + l (cid:48) e i )) ≤ δ for some j (cid:54) = i , by Lemma 7.8 with δ = ε d / (16 l d ) we have that r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) ≥ | s (cid:48) − l (cid:48) + l/ | , | s (cid:48) − l (cid:48) − l/ | ) ≥ η and ( t (cid:48)⊥ i + s (cid:48) e i ) ∈ Q l ( t ⊥ i + l (cid:48) e i ) or min( | s (cid:48) + l/ | , | s (cid:48) − l/ | ) ≥ η and ( t (cid:48)⊥ i + s (cid:48) e i ) ∈ Q l ( t ⊥ i ).Fix t (cid:48)⊥ i . Then (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( − l , l ) | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) ≥ (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( − l , l )min( | s (cid:48) + l/ | , | s (cid:48) − l/ | ) ≥ η | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) + (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( − l , l )min( | s (cid:48) + l/ | , | s (cid:48) − l/ | ) <η | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) r i,τ ≥ η and otherwise r i,τ ≥ −
1. Moreover, given that | Q il ( s (cid:48) ) ∩ J | l < η l for s (cid:48) ∈ ( − l/ , l/ ∪ ( l (cid:48) − l/ , l (cid:48) + l/ − M /l . Finallyintegrating over t (cid:48)⊥ i we obtain that1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( − l/ ,l/ ∪ ( l (cid:48) − l/ ,l (cid:48) + l/ | Q il ( s (cid:48) ) ∩ J | l (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i (cid:38) − M l . By using the above inequality in (7.41) and the fact that for every s (cid:48) ∈ ( l/ , l (cid:48) − l/
2) it holds | Q l ( s (cid:48) ) ∩ J | l = 1, we have that ˆ J ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( l/ ,l (cid:48) − l/ (cid:16) r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) + v i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) (cid:17) d t (cid:48)⊥ i − M l To conclude the proof of (7.37), as for (7.38), we notice that1 l d − ˆ Q ⊥ l ( t ⊥ i ) (cid:88) s (cid:48) ∈ ∂E t (cid:48)⊥ i s (cid:48) ∈ ( l/ ,l (cid:48) − l/ r i,τ ( E, t (cid:48)⊥ i , s (cid:48) ) d t (cid:48)⊥ i ≥ (cid:0) C ∗ τ | J | − lC ∗ τ − C (cid:1) χ (0 , + ∞ ) ( | J | − l ) , where in the last inequality we have used Lemma 7.7 for E = E t (cid:48)⊥ i , I = ( l/ , l (cid:48) − l/ l > C / ( − C ∗ τ ), one gets (7.37).If | J | ≤ l , then the first sum on the r.h.s. of (7.41) is performed on an empty set. Therefore, inboth (7.37) and (7.38) one has only the boundary terms and can conclude in a similar way.The proof of (7.39) proceeds using the L -periodicity of the contributions as done for (7.36).The next lemma gives a simple lower bound on the energy in the case almost all the volume of Q l ( z ) is filled by E or E c (this will be the case on the set A − defined in (7.55)). Lemma 7.11.
Let E be a set of locally finite perimeter such that min( | Q l ( z ) \ E | , | E ∩ Q l ( z ) | ) ≤ δl d ,for some δ > . Then ¯ F τ ( E, Q l ( z )) ≥ − δdη , where η is defined in Remark 7.1.Proof. By assumption, w.l.o.g. | Q l ( z ) \ E | ≤ δl d .Fix t ⊥ i and consider the slice E t ⊥ i . Then one has that E t ⊥ i = (cid:83) nj =1 ( u j , s j ). Since whenever u j + i − s j ≤ η one has that r i,τ ( E, t ⊥ i , s j ) ≥
0, and otherwise r i,τ ( E, t ⊥ i , s j ) ≥ −
1, one has that n (cid:88) j =1 r i,τ ( E, t ⊥ i , s j ) ≥ (cid:88) { u j +1 − s j >η } − ≥ − | Q il ( z ) \ E t ⊥ i | η . t ⊥ i and using the definition of ¯ F i,τ ( E, Q l ( z )) (7.15) one has that¯ F i,τ ( E, Q l ( z )) ≥ − l d ˆ Q ⊥ l ( z ⊥ i ) | Q il ( z ) \ E t ⊥ i | η d t ⊥ i ≥ − δη . The sets defined in the proof and the main estimates will depend on a set of parameters l, δ, ρ, M, η and τ . If suitably chosen, they lead to the proof of the theorem for any L > l of the form 2 kh ∗ τ with k ∈ N . Recall that h ∗ τ is the width of the periodic stripes which minimize the energy density F τ,L among all periodic stripes as L varies.Let us first specify how the parameters are chosen, and their dependence on each other. The reasonfor such choices will be clarified during the proof.Let 0 < σ < − C ∗ /
2, where C ∗ is the energy density of optimal periodic stripes for F ,L (defined in(3.1)) over all L . Notice that C ∗ < τ ↓ C ∗ τ = C ∗ . • We first fix l > l > max (cid:110) dC d − C ∗ − σ , C − C ∗ − σ (cid:111) , (7.43)where C d is a constant (depending only on the dimension d ) that appears in (7.56), and C is the constant which appears in the statement of Lemma 7.7. • Let η and τ as in Remark 7.1. Then from Lemma 7.8, have the parameters ˜ ε = ˜ ε ( η , τ )and ˜ τ = ˜ τ ( η , τ ). • We then fix ε < ˜ ε , τ < ˜ τ as in Lemma 7.9. Thus we obtain δ defined by δ = ε d . Moreover,by choosing ε sufficiently small we can additionally assume that if for some η > D iη ( E, Q l ( z )) ≤ δ and D jη ( E, Q l ( z )) ≤ δ, i (cid:54) = j ⇒ min {| E ∩ Q l ( z ) | , | E c ∩ Q l ( z ) |} ≤ l d − . (7.44)This follows from Remark 7.4 (ii). • Thanks to Remark 7.4 (i), we then fix ρ ∼ δl. (7.45)in such a way that we have that for any η the following holds ∀ z, z (cid:48) s.t. D η ( E, Q l ( z )) ≥ δ, | z − z (cid:48) | ∞ ≤ ρ ⇒ D η ( E, Q l ( z (cid:48) )) ≥ δ/ , (7.46)where for x ∈ R d , | x | ∞ = max i | x i | . • Afterwards, we fix M such that M ρ d > M l. (7.47) • By applying Lemma 7.6, we obtain ¯ η = ¯ η ( M, l ) and ¯ τ = ¯ τ ( M, l, δ/ < η < ¯ η, ¯ η = ¯ η ( M, l ) (7.48)54
Finally, we choose further τ > τ < τ as in Remark 7.1, (7.49) τ < ˜ τ , ˜ τ as in Lemma 7.8 and Lemma 7.9 , (7.50) τ < ¯ τ , ¯ τ as in Lemma 7.6 depending on M, l, δ/ η (7.51)and τ s.t. C ∗ τ < C ∗ + σ. (7.52)Notice that, by the Γ-convergence result of Theorem 2.3, ∃ ˆ τ s.t. if τ < ˆ τ , then (7.52) holds. Inparticular, (7.43) is satisfied with − C ∗ τ instead of − C ∗ − σ .Given such parameters, let us prove the theorem for any L > l of the form L = 2 kh ∗ τ , with k ∈ N .Let E be a minimizer of F τ,L . Since E is L -periodic, we can consider E ⊂ T dL , where T dL is the d -dimensional torus of size L . Thus the problem is naturally defined on the torus. Hence with aslight abuse of notation, we will denote by [0 , L ) d the cube of size L with the usual identificationof the boundary. Decomposition of [0 , L ) d : We define ˜ A := (cid:110) z ∈ [0 , L ) d : D η ( E, Q l ( z )) ≥ δ (cid:111) . Hence, by Lemma 7.6, for every z ∈ ˜ A one has that ¯ F τ ( E, Q l ( z )) > M .Let us denote by ˜ A − the set of points˜ A − := (cid:110) z ∈ [0 , L ) d : ∃ i, j with i (cid:54) = j s.t. D iη ( E, Q l ( z )) ≤ δ, D jη ( E, Q l ( z )) ≤ δ (cid:111) . One can easily see that ˜ A and ˜ A − are closed.By the choice of ρ made in (7.45), (7.46) holds, namely for every z ∈ ˜ A and | z − z (cid:48) | ∞ ≤ ρ one hasthat D η ( E, Q l ( z (cid:48) )) > δ/ δ satisfies (7.44), when z ∈ ˜ A − , then one has that min( | E ∩ Q l ( z ) | , | Q l ( z ) \ E | ) ≤ l d − .Thus, using Lemma 7.11 with δ = 1 /l , one has that¯ F τ ( E, Q l ( z )) (cid:38) − l . Moreover, let now z (cid:48) such that | z − z (cid:48) | ∞ ≤ z ∈ ˜ A − . It is not difficult to see that if | Q l ( z ) \ E | ≤ l d − then | Q l ( z (cid:48) ) \ E | (cid:46) l d − . Thus from Lemma 7.11, one has that¯ F τ ( E, Q l ( z (cid:48) )) ≥ − ˜ C d l . (7.53)The above observations motivate the following definitions A := (cid:110) z (cid:48) ∈ [0 , L ) d : ∃ z ∈ ˜ A with | z − z (cid:48) | ∞ ≤ ρ (cid:111) (7.54) A − := (cid:110) z (cid:48) ∈ [0 , L ) d : ∃ z ∈ ˜ A − with | z − z (cid:48) | ∞ ≤ (cid:111) , (7.55)55 l A − A A A e e Figure 3: Example of decomposition of [0 , L ) d according to the distance of E to stripes on squares Q l ( z ). The darker regions within the same color represent the set E .By the choice of the parameters and the observations above, for every z ∈ A one has that¯ F τ ( E, Q l ( z )) > M and for every z ∈ A − , ¯ F τ ( E, Q l ( z )) ≥ − ˜ C d /l .For simplicity of notation let us denote by A := A ∪ A − .The set [0 , L ) d \ A has the following property: for every z ∈ [0 , L ) d \ A , there exists i ∈ { , . . . , d } such that D iη ( E, Q l ( z )) ≤ δ and for every k (cid:54) = i one has that D kη ( E, Q l ( z )) > δ .Given that A is closed, we consider the connected components C , . . . , C n of [0 , L ) d \ A . The sets C i are path-wise connected.Let us now show the following claim: given a connected component C j one has that there exists i suchthat D iη ( E, Q l ( z )) ≤ δ for every z ∈ C j and for every k (cid:54) = i one has that D kη ( E, Q l ( z )) > δ . Indeed,suppose that there exists z, z (cid:48) ∈ C j such that D iη ( E, Q l ( z )) ≤ δ and D kη ( E, Q l ( z (cid:48) )) ≤ δ with i (cid:54) = k and take a continuous path γ : [0 , → C j such that γ (0) = z and γ (1) = z (cid:48) . From our hypothesis,we have that { s : D iη ( E, Q l ( γ ( s ))) ≤ δ } (cid:54) = ∅ and there exists ˜ s ∈ ∂ { s : D iη ( E, Q l ( γ ( s ))) ≤ δ } ∩ ∂ { s : D jη ( E, Q l ( γ ( s ))) ≤ δ } for some j (cid:54) = i . Let ˜ z = γ (˜ s ). Thus there are points arbitraryclose to ˜ z in C j such that D jη ( E, Q l ( · )) ≤ δ and D iη ( E, Q l ( · )) ≤ δ . From the continuity of the maps z (cid:55)→ D iη ( E, Q l ( z )), z (cid:55)→ D jη ( E, Q l ( z )), we have that ˜ z ∈ A − , which contradicts our assumption.We will say that C j is oriented in direction e i if there is a point in z ∈ C j such that D iη ( E, Q l ( z )) ≤ δ .Because of the above being oriented along direction e i is well-defined.We will denote by A i the union of the connected components C j such that C j is oriented along thedirection e i .An example of such a partition of [0 , L ) d for a periodic set E is given in Figure 3.Let us now summarize the important properties that will be used in the following(i) The sets A = A − ∪ A , A , A , . . . , A d form a partition of [0 , L ) d .(ii) The sets A − , A are closed and A i , i >
0, are open.(iii) For every z ∈ A i , we have that D iη ( E, Q l ( z )) ≤ δ .(iv) There exists ρ (independent of L, τ ) such that if z ∈ A , then ∃ z (cid:48) s.t. Q ρ ( z (cid:48) ) ⊂ A and z ∈ Q ρ ( z (cid:48) ). If z ∈ A − then ∃ z (cid:48) s.t. Q ( z (cid:48) ) ⊂ A − and z ∈ Q ( z (cid:48) ).(v) For every z ∈ A i and z (cid:48) ∈ A j one has that there exists a point ˜ z in the segment connecting z to z (cid:48) lying in A ∪ A − . 56or simplicity we will denote by B = (cid:83) i> A i . Main Claim:
For every i , and τ as in (7.50) and (7.51), we will show that the following holds1 L d ˆ B ¯ F i,τ ( E, Q l ( z )) d z + 1 dL d ˆ A ¯ F τ ( E, Q l ( z )) d z ≥ C ∗ τ | A i | L d − C d | A | lL d (7.56)for some constant C d depending on the dimension d . Assuming (7.56), we can sum over i andobtain that, since l satisfies (7.43) and τ satisfies (7.52), F τ,L ( E ) ≥ d (cid:88) i =1 L d ˆ [0 ,L ) d ¯ F i,τ ( E, Q l ( z )) d z ≥ C ∗ τ L d d (cid:88) i =1 | A i | − dC d | A | lL d ≥ C ∗ τ − C ∗ τ | A | L d − dC d lL d | A | ≥ C ∗ τ , where in the above C ∗ τ is the energy density of optimal stripes of stripes h ∗ τ and we have used that C ∗ τ < | A | + (cid:80) di =1 | A i | = | [0 , L ) d | = L d .Notice that, in the inequality above, equality holds only if | A | = 0 and therefore by ( v ) only if thereis just one A i , i > | A i | > t ⊥ i ∈ [0 , L ) d − , it holds1 L d ˆ B t ⊥ i ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s + 1 dL d ˆ A t ⊥ i ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s ≥ C ∗ τ | A i,t ⊥ i | L d − C d | A t ⊥ i | lL d (7.57)Indeed by integrating (7.57) over t ⊥ i we obtain (7.56).Notice also that B t ⊥ i is a finite union of intervals. Indeed, being a union of intervals follows from(ii) and finiteness follows from condition (v) on the decomposition. Indeed, for every point thatdoes not belong to B t ⊥ i because of (iv) there is a neighbourhood of fixed positive size that is notincluded in B t ⊥ i . Let { I , . . . , I n } such that (cid:83) nj =1 I j = B t ⊥ i with I j ∩ I k = ∅ whenever j (cid:54) = k . Wecan further assume that I i ≤ I i +1 , namely that for every s ∈ I i and s (cid:48) ∈ I i +1 it holds s ≤ s (cid:48) . Byconstruction there exists J j ⊂ A t ⊥ i such that I j ≤ J j ≤ I j +1 .Whenever J j ∩ A ,t ⊥ i (cid:54) = ∅ , we have that | J j | > ρ and whenever J j ∩ A − ,t ⊥ i (cid:54) = ∅ then | J i | > L d ˆ B t ⊥ i ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s + 1 dL d ˆ A t ⊥ i ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s ≥ n (cid:88) j =1 L d ˆ I j ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s + 1 dL d n (cid:88) j =1 ˆ J j ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s ≥ L d n (cid:88) j =1 (cid:16) ˆ I j ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s + 12 d ˆ J j − ∪ J j ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s (cid:17) , where in order to obtain the third line from the second line, we have used periodicity and J := J n .Let first I j ⊂ A i,t ⊥ i . By construction, we have that ∂I j ⊂ A t ⊥ i .57f ∂I j ⊂ A − ,t ⊥ i , by using our choice of parameters, namely (7.43) and (7.52), we can apply (7.37)in Lemma 7.9 and obtain1 L d ˆ I j ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ L d (cid:16) | I j | C ∗ τ − M l (cid:17) . If ∂I j ∩ A ,t ⊥ i (cid:54) = ∅ , by using our choice of parameters, namely (7.43) and (7.52), we can apply (7.38)in Lemma 7.9, and obtain1 L d ˆ I j ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ L d (cid:16) | I j | C ∗ τ − M l (cid:17) . On the other hand, if ∂I j ∩ A ,t ⊥ i (cid:54) = ∅ , we have that either J j ∩ A ,t ⊥ i (cid:54) = ∅ or J j − ∩ A ,t ⊥ i (cid:54) = ∅ . Thus12 dL d ˆ J j − ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s + 12 dL d ˆ J j ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s ≥ M ρ dL d − | J j − ∩ A − ,t ⊥ i | ˜ C d dlL d − | J j ∩ A − ,t ⊥ i | ˜ C d dlL d , where ˜ C d is the constant in (7.53).Since M satisfies (7.47), in both cases ∂I j ⊂ A − ,t ⊥ i or ∂I j ∩ A ,t ⊥ i (cid:54) = ∅ , we have that1 L d ˆ I j ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s + 12 dL d ˆ J j − ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s + 12 dL d ˆ J j ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s ≥ C ∗ τ | I j | L d − | J j − ∩ A − ,t ⊥ i | ˜ C d dlL d − | J j ∩ A − ,t ⊥ i | ˜ C d dlL d . If I j ⊂ A k,t ⊥ i with k (cid:54) = i from Lemma 7.9 Point (i) it holds1 L d ˆ I j ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s ≥ − M lL d . In general for every J j we have that1 dL d ˆ J j ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s ≥ | J j ∩ A ,t ⊥ i | MdL d − ˜ C d dlL d | J j ∩ A − ,t ⊥ i | . For I j ⊂ A k,t ⊥ i such that ( J j ∪ J j − ) ∩ A ,t ⊥ i (cid:54) = ∅ with k (cid:54) = i , we have that1 L d ˆ I j ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s + 12 dL d ˆ J j − ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s + 12 dL d ˆ J j ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s ≥ − M lL d + M ρ dL d − | J j − ∩ A − ,t ⊥ i | ˜ C d dlL d − | J j ∩ A − ,t ⊥ i | ˜ C d dlL d . ≥ − | J j − ∩ A − ,t ⊥ i | ˜ C d dlL d − | J j ∩ A − ,t ⊥ i | ˜ C d dlL d . where the last inequality is true due to (7.47). 58or I j ⊂ A k,t ⊥ i such that ( J j ∪ J j − ) ⊂ A − ,t ⊥ i with k (cid:54) = i , we have that1 L d ˆ I j ¯ F i,τ ( E, Q l ( t ⊥ i + se i )) d s + 12 dL d ˆ J j − ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s + 12 dL d ˆ J j ¯ F τ ( E, Q l ( t ⊥ i + se i )) d s ≥ − M lL d − | J j − ∩ A − ,t ⊥ i | ˜ C d dlL d − | J j ∩ A − ,t ⊥ i | ˜ C d dlL d . ≥ − max (cid:16) M , ˜ C d d (cid:17)(cid:18) | J j − ∩ A − ,t ⊥ i | lL d + | J j ∩ A − ,t ⊥ i | lL d (cid:19) . where in the last inequality we have used that | J j ∩ A − ,t ⊥ i | ≥ , | J j − ∩ A − ,t ⊥ i | ≥ j , and taking C d = max (cid:16) M , ˜ C d d (cid:17) , one obtains (7.57) as desired.Therefore, it has been proved that there exists i > A i = [0 , L ) d . Finally, let us consider1 L d ˆ [0 ,L ) d ¯ F τ ( E, Q l ( z )) d z = 1 L d ˆ [0 ,L ) d ¯ F i,τ ( E, Q l ( z )) d z (7.58)+ 1 L d (cid:88) j (cid:54) = i ˆ [0 ,L ) d ¯ F j,τ ( E, Q l ( z )) d z (7.59)We will now apply Lemma 7.9 with j = i and slice the cube [0 , L ) d in direction e i . From (7.36),one has that (7.59) is nonnegative and strictly positive unless the set E is a union of stripes indirection e i . On the other hand, from (7.39), one has the r.h.s. of (7.58) is minimized by a periodicunion of stripes in direction e i and with width h ∗ τ . Proof of Theorem 1.5.
Let us recall the notation κ = τ /β , which was introduced in Section 5. Byusing the continuous representation ˜ E κ of a discrete set E ⊂ κ Z d , which is described in Section 5,one can see the rescaled discrete functional F dsc τ,L ( E ) as the functional F d → c τ,L ( ˜ E κ ). Therefore thediscrete problem is transformed into the continuous problem into a continuous one.All the statements that have been used to prove Theorem 1.4, are obtained by using(i) qualitative properties of the kernel, namely (2.3)(ii) one-dimensional optimization via the reflection positivity (see assumption 2.6).In the discrete, the kernel is K dsc κ ( ζ ) = κ d | ζ | p . The measure defined by µ κ = (cid:80) ζ ∈ κ Z d \{ } K dsc κ ( ζ )converges to the measure | ζ | p d ζ and the piecewise constant function associated to µ κ , namely x → (cid:88) ζ ∈ κ Z d \{ } | ζ | p χ Q κ ( ζ ) ( x )converges in L ( R d \ { } ) to | x | p .Thus in a similar way to the continuous setting, one can define similar quantities to r i,τ , v i,τ , w i,τ and ¯ F i,τ and obtain the same type of estimates. Indeed, the only two lemmas that would dependon the specific form of the kernel are Lemma 7.7 and Lemma 7.9 in which Lemma 7.7 is used.They depend on the specific form of the kernel as the reflection positivity technique is used. Giventhat reflection positivity holds for the discrete kernel, one can obtain the same type of results forLemma 7.7 and consequently for Lemma 7.9.As the proof of Theorem 1.4 is a consequence of the previous lemmas, one has that the proof followsby the same reasoning. 59 cknowledgements We would like to thank M. Cicalese, A. Giuliani and E. Spadaro for useful comments.
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