Existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters
aa r X i v : . [ m a t h . O C ] M a y EXISTENCE AND ALMOST EVERYWHERE REGULARITYOF ISOPERIMETRIC CLUSTERS FOR FRACTIONAL PERIMETERS
M. COLOMBO AND F. MAGGI
Abstract.
The existence of minimizers in the fractional isoperimetric problem with multiplevolume constraints is proved, together with a partial regularity result. Introduction
The goal of this paper is establishing basic existence and partial regularity results for thefractional isoperimetric problem with multiple volume constraints. If E ⊂ R n , n ≥
2, and s ∈ (0 , fractional perimeter of order s of E is defined as P s ( E ) = Z R n w E ( x ) dx = Z E dx Z E c dy | x − y | n + s . (1.1)The kernel z
7→ | z | − n − s is not integrable near the origin, and the potential w E ( x ) := 1 E ( x ) Z E c dy | x − y | n + s x ∈ R n explodes like dist( x, ∂E ) − s as x ∈ E approaches ∂E . Since t − s is integrable near 0, by decomposingthe integral of w E on a small layer around ∂E as the integral along the normal rays t p − t ν E ( p ), p ∈ ∂E , then we see that P s ( E ), at leading order, is measuring the perimeter P ( E ) = H n − ( ∂E )of E . This idea is made precise by the fact that, as s → − , (1 − s ) P s ( E ) → c ( n ) P ( E ) for everyset of finite perimeter E , see [BBM01, D´av02], and (1 − s ) P s → P in the sense of Γ-convergence[ADPM11].The last few years has seen a great effort by many authors towards the understanding ofgeometric variational problems in the fractional setting. This line of research has been initiatedin [CRS10] with the regularity theory for the fractional Plateau’s problem (see [SV13, FV13,BFV14, CG10, CV11] for further developments in this direction), while fractional isoperimetricproblems have been the subject of [FLS08, KM13, KM14, DCNRV15, FFM + N ∈ N ,a N -cluster (or simply a cluster) E is a family E = {E ( h ) } Nh =1 of disjoint Borel subsets of R n . Thesets E ( h ), h = 1 , ..., N , are called the chambers of E , while E (0) = R n \ S Nh =1 E ( h ) is called theexterior chamber of E . When |E ( h ) | < ∞ for h = 1 , ..., N , then the volume vector m ( E ) of E isdefined as m ( E ) = ( |E (1) | , ..., |E ( N ) | ) ∈ R N while the fractional s -perimeter of E is given by P s ( E ) = 12 N X h =0 P s ( E ( h )) . (1.2)Given m ∈ R N + (that is, m h > h = 1 , ..., N ), we consider the following isoperimetric probleminf n P s ( E ) : m ( E ) = m o . (1.3)Every minimizer in (1.3) is called an isoperimetric cluster. The following theorem is our mainresult. Theorem 1.1.
For every m ∈ R N + there exists an isoperimetric cluster E with m ( E ) = m . If weset ∂ E = n x ∈ R n : ∃ h = 1 , ..., N such that < |E ( h ) ∩ B r ( x ) | < | B r ( x ) | ∀ r > o (1.4) then ∂ E is bounded and there exists a closed set Σ( E ) ⊂ ∂ E such that H n − (Σ( E )) = 0 if n ≥ , Σ( E ) is discrete if n = 2 , and ∂ E \ Σ( E ) is a C ,α -hypersurface in R n for some α ∈ (0 , . Let us review the theory of isoperimetric clusters when the classical perimeter, not the frac-tional one, is minimized. This theory has been initiated by Almgren [Alm76] with the proof of theanalogous result to Theorem 1.1, namely an existence and C ,α -regularity theorem out of a closedsingular set of Hausdorff dimension n −
1. When n = 2 the only singular minimal cone consists ofthree half-lines meeting at 120 degrees at a common end-point, so that, by a standard dimensionreduction argument, the singular set has Hausdorff dimension at most n −
2, and is discrete when n = 2. (This estimate is of course sharp.) Taylor [Tay76] has proved that, if n = 3, then the onlysingular cones are obtained either by the union of three half-planes meeting at 120 degrees alonga common line, or as cones spanned by the edges of regular tetrahedra over their barycenters;and that, moreover, ∂ E is locally C ,α -diffeomorphic to its tangent cone at every point, includingsingular ones. The regular part ∂ E \ Σ( E ) has constant mean curvature and is real analytic, indimension n = 3 up to the singular set [Nit77, KS78]. Regularity of and near the singular set indimension n ≥ N = 2) the only isoperimetric clusters are double-bubbles, whose boundaries consistof three ( n − n − +
93] in dimension n = 2, [HMRR02] ( n = 3) and [Rei08, RHLS03] ( n ≥ N = 3) one can define a candidate isoperimetric cluster, the so-called triple bubble, enclosing three given volumes. When n = 2, the minimality of this triplebubble was proved in [Wic04]. Another important isoperimetric problem is partitioning a flattorus into chambers of equal volumes. In the case n = 2 this problem has been solved in [Hal01],where the minimality of hexagonal honeycomb partitions is proved. Global stability inequalities forplanar double bubbles and for hexagonal honeycombs have been obtained in [CLM12] and [CM16],together with quantitative descriptions of minimizers in the presence of a small potential term.The present paper naturally opens two kind of questions, which are actually closely related:first, understanding singularities of fractional isoperimetric clusters and, second, characterizingfractional isoperimetric clusters in some basic cases. Thinking about the arguments used to achievethese goals in the local theory, the extension to the fractional case is necessarily going to requirethe introduction of new arguments and ideas.One may speculate that the fractional theory may be helpful in advancing the local theory: onthe one hand, depending on the question under study, the rigidity of nonlocal perimeters may endup bringing in some simplifications with respect to the local case; on the other hand, information in the classical setting can be recovered from the fractional case in the limit s → − . In any case,at present, the viability of this idea has not been really tested on specific examples.The paper is divided into two sections. In section 2 we prove the existence part of Theorem 1.1by adapting to the fractional setting Almgren’s original proof (as presented in [Mag12, Part IV]).In section 3 we prove the partial regularity assertion in Theorem 1.1. Similarly to what done in[CRS10] for fractional perimeter minimizing boundaries, we exploit an extension problem to obtaina monotonicity formula, showing that nearby most points of the boundary only two chambers ofthe isoperimetric cluster are present. When this happens we can show that the two neighboringchambers locally almost-minimize fractional perimeter, and we can thus apply the main result in[CG10] to prove their C ,α -regularity. Acknowledgment:
Work partially supported by NSF DMS Grant 1265910 and DMS-FRG Grant1361122. 2.
Existence theorem
The goal of this section is proving the existence of isoperimetric clusters of any given volume.Precisely, given m ∈ R N + we discuss the existence of isoperimetric sets of volume m , that is,minimizers in γ = inf (cid:8) P s ( E ) : E is an N -cluster in R n with m ( E ) = m (cid:9) . (2.1)Every minimizing sequence {E k } k ∈ N is compact in L ( R n ) (see section 2.1 for the terminology usedhere and in the sequel) and fractional perimeters are trivially lower semicontinuous with respect tothis convergence, so that the only difficulty in showing the existence of minimizers is the possibilitythat minimizing sequences do not converge in L ( R n ) (loss of volume at infinity). Almgren’sstrategy to fix this problem [Alm76] (which predates by a decade the formalization of this kind ofargument in the theory of concentration-compactness!) consists in nucleating , truncating , volume-fixing and translating a given minimizing sequence. The nucleation step consists in decomposingthe cluster E k into finitely many “chunks” which contain most of the volume. These chunks aredefined by intersecting the chambers of E k with a finite collection of balls of equal radii, eachchunk having bounded diameter and possibly diverging from the others. In the truncation step thechambers of E k are “chopped” by a slight enlargement of the nucleating balls in such a way thatthe perimeter is decreased by an amount which is proportional to the volume left out. The volumeis then restored by slight deformations of the clusters. By these operations one has obtained a newminimizing sequence, localized into finitely many regions of bounded diameter. In the classicalcase, where local perimeter is minimized, one can finally translate these nuclei so to obtain a newminimizing sequence entirely contained in a bounded region. In the nonlocal case one cannotfreely translate disconnected parts of the cluster without changing in a complex way its fractionalperimeter. However, in section 2.2 we show that once a sequence of clusters have bounded fractionalperimeter and is localized into finitely many (possibly diverging) regions of bounded diameter,then the sequence is actually bounded (see Lemma 2.1). In sections 2.3, 2.4, 2.5 we take care,respectively, of the volume-fixing, truncation and nucleation steps of the argument, highlightingthe differences brought in Almgren’s argument by the nonlocality of fractional perimeters. Finally,in section 2.6 we combine these tools to prove the existence of isoperimetric sets.2.1. Notation and terminology.
Given disjoint Borel sets
E, F ⊂ R n and s ∈ (0 , s between E and F by setting I s ( E, F ) = Z E Z E c dx dy | x − y | n + s . M. COLOMBO AND F. MAGGI
The fractional s -perimeter is then given by P s ( E ) = I s ( E, E c ), see (1.1). The s -perimeter of E ⊂ R n relative to an open set Ω ⊂ R n is defined by the formula P s ( E ; Ω) = I s ( E ∩ Ω , E c ∩ Ω) + I s ( E ∩ Ω , E c ∩ Ω c ) + I s ( E ∩ Ω c , E c ∩ Ω) . The motivation for this definition lies in the fact that if P s ( E ) and P s ( F ) are both finite and E ∩ Ω c = F ∩ Ω c , then P s ( E ) − P s ( F ) = P s ( E ; Ω) − P s ( F ; Ω).A N -cluster, or simply a cluster, is a family E = {E ( h ) } Nh =1 of disjoint sets, called the chambersof E . The set E (0) = R n \ S Nh =1 E ( h ) is called the exterior chamber of E . The volume of E is thevector m ( E ) = ( |E (1) | , ..., |E ( N ) | ). The relative distance between the N -clusters E and E ′ in Ω ⊆ R n is defined by d Ω ( E , E ′ ) = N X h =0 | Ω ∩ ( E ( h )∆ E ′ ( h )) | . The relative s -perimeter P s ( E ; Ω) of the cluster E in Ω is defined as P s ( E ; Ω) = 12 M X i =1 P s ( E ( i ); Ω) , so that P s ( E ) = P s ( E ; R n ), see (1.2). We say that a sequence {E k } k ∈ N of N -clusters converges in L (Ω) to a N -cluster E if 1 E k ( h ) → E ( h ) in L (Ω) for every h = 1 , ..., N . If sup k ∈ N P s ( E k ; Ω) < ∞ ,then one can find a subsequence of {E k } k ∈ N which admits an L (Ω) limit. Finally, the boundaryof a Borel set E ⊂ R n is defined as ∂E = n x ∈ R n : 0 < | E ∩ B r ( x ) | < | B r ( x ) | ∀ r > o . (2.2)In this way (1.4) is equivalent to ∂ E = N [ h =1 ∂ E ( h ) . A boundedness criterion.
The following lemma exploits the rigidity of fractional perime-ters to show that a cluster consisting of finitely many pieces localized in different bounded regionshas actually bounded diameter.
Lemma 2.1.
Let {E k } k ∈ N be a minimizing sequence for (2.1) . Let us assume that there existpositive constants R and c and, for every k ∈ N , finitely many points { x k ( i ) } i =1 ,...,L ( k ) , with theproperty that E k ( h ) ⊆ L ( k ) [ i =1 B R ( x k ( i )) , ∀ k ∈ N , h = 1 , ..., N , (2.3)sup k ∈ N L ( k ) < ∞ , (2.4) N X h =1 |E k ( h ) ∩ B R ( x k ( i )) | ≥ c , ∀ i = 1 , ..., L ( k ) , k ∈ N . (2.5) Then there exists R > and a subsequence (not relabelled) such that E k ( h ) ⊆ B R ( x k (1)) forevery h = 1 , ..., N and for every k ∈ N . Before proving the lemma, we recall that the s -perimeter is subadditive, and more preciselyfor every couple of disjoint measurable sets E, F ⊆ R n we have P s ( E ) + P s ( F ) − | E | | F | dist( E, F ) n + s ≤ P s ( E ∪ F ) ≤ P s ( E ) + P s ( F ) − | E | | F | diam( E ∪ F ) n + s . (2.6) Indeed, we have that P s ( E ∪ F ) = P s ( E ) + P s ( F ) − I s ( E, F ) and | E | | F | diam( E ∪ F ) n + s ≤ I s ( E, F ) ≤ | E | | F | dist( E, F ) n + s . This observation will be applied to estimate the perimeter of a sequence of clusters with a finitenumber of “components” which are moving away from each other.
Lemma 2.2.
Let E k ⊆ R n be a sequence of measurable sets such that E k ⊆ L [ i =1 B R ( x k ( i )) ∀ k ∈ N , where R > , L ∈ N and, for each i = 1 , ..., L , { x k ( i ) } k ∈ N are sequences of points such that lim k →∞ inf ≤ i The inequality P s ( E k ) ≤ L X i =1 P s (cid:0) E k ∩ B R ( x k ( i )) (cid:1) follows from the subadditivity of the s -perimeter. Moreover, by induction over (2.6), given L sets F ,..., F L whose mutual distances are bigger than D > 0, one has P s (cid:16) L [ i =1 F i (cid:17) ≥ L X i =1 P s ( F i ) − L max i =1 ,...,L | F i | D n + s . (2.9)Given k ∈ N , we apply this inequality to the sets F i = E k ∩ B R ( x k ( i ))). Since in this case we have | F i | ≤ | B R | and D ≥ min i = j | x k ( i ) − x k ( j ) | − R , we obtain P s ( E k ) ≥ L X i =1 P s (cid:0) E k ∩ B R ( x k ( i )) (cid:1) − L | B R | min i = j ( | x k ( i ) − x k ( j ) | − R ) n + s . By (2.7) we obtain (2.8). (cid:3) Proof of Lemma 2.1. We argue by contradiction, assuming that there exists a minimizing sequence {E k } k ∈ N in (2.1) such that (2.3), (2.4), and (2.5) hold, but withlim k →∞ max ≤ h ≤ N diam ( E k ( h )) = + ∞ . Up to extracting a subsequence, we may assume that L ( k ) = L independent on k . Step one : We claim that there exist L ∈ { , .., L } and S ≥ R such that, up to extracting asubsequence in k and up to reordering the set { , ..., L } , we havelim k →∞ | x k ( i ) − x k ( j ) | = ∞ , ∀ i, j ∈ { , ..., L } , i = j, (2.10) E k ( h ) ⊆ L [ i =1 B S ( x k ( i )) , ∀ k ∈ N , h = 1 , ..., N , (2.11) N X h =1 |E k ( h ) ∩ B S ( x k ( i )) | ≥ c , ∀ i = 1 , ..., L, (2.12)where the constant c is the one appearing in (2.5). Indeed, up to extracting subsequences, we mayassume that for every i, j ∈ { , ..., L } there exists S ( i, j ) = lim k →∞ | x k ( i ) − x k ( j ) | ∈ [0 , ∞ ]. We M. COLOMBO AND F. MAGGI then say that { x k ( i ) } k ∈ N and { x k ( j ) } k ∈ N are asymptotically close if S ( i, j ) < ∞ , and introduce anequivalence relation ∼ on { , ..., L } so that i ∼ j if and only if { x k ( i ) } k ∈ N and { x k ( j ) } k ∈ N areasymptotically close. Up to reordering { , ..., L } , we may assume that L is such that { , ..., L } contains exactly one representative of each equivalence class. Hence, (2.10) follows by the factthat representatives of different classes cannot be asymptotically close. Finally, by taking S :=sup i,j ∈{ ,...,L } ,i ∼ j S ( i, j ) + R , we clearly have B R ( x k ( i )) ⊆ B S ( x k ( j )) for every i, j ∈ { , ..., L } with i ∼ j , so that (2.3) implies (2.11). Finally, (2.12) follows from (2.5) since B R ( x k ( i )) ⊆ B S ( x k ( i )). Step two : Up to further extracting subsequences and reordering indices, we may assume that | x k (1) − x k (2) | ≤ | x k ( i ) − x k ( j ) | , ∀ k ∈ N , i, j ∈ { , ..., L } , i = j . and that lim k →∞ P s ( E k ( h )) exists ∀ h = 1 , ..., L . Moreover, up to a translation and a rotation, we may assume that x k (1) = 0 , x k (2) | x k (2) | = e . We now define a new sequence {E ′ k } k ∈ N so that E ′ k coincides with E k in the balls B S ( x k ( i )) with i = 2, whereas the part of E k inside B S ( x k (2)) is translated at distance 3 S from x k (1) = 0: moreprecisely, for every h = 1 , ..., N we set E ′ k ( h ) = (cid:16) ( E k ( h ) ∩ B S ( x k (2))) + (3 S − | x k (2) | ) e (cid:17) ∪ [ i =2 (cid:0) E k ( h ) ∩ B S ( x k ( i )) (cid:1) . By Lemma 2.2 applied to each chamber of E k and to E k (0) c we have2 γ = 2 lim k →∞ P s ( E k ) = lim k →∞ (cid:16) P s ( E k (0) c ) + N X h =1 P s ( E k ( h )) (cid:17) = lim k →∞ h L X i =1 P s (cid:0) E k (0) c ∩ B S ( x k ( i )) (cid:1) + N X h =1 L X i =1 P s (cid:0) E k ( h ) ∩ B S ( x k ( i )) (cid:1)i . We can use the same argument on the chambers of E ′ k which are contained in the balls { B S (0) , B S ( x k ( i )) :3 ≤ i ≤ L } , in order to obtain2 lim sup k →∞ P s ( E ′ k ) = lim sup k →∞ (cid:16) P s ( E ′ k (0) c ) + N X h =1 P s ( E ′ k ( h )) (cid:17) = lim sup k →∞ h P s (cid:0) E ′ k (0) c ∩ B S (cid:1) + L X i =3 P s (cid:0) E k (0) c ∩ B S ( x k ( i )) (cid:1) + N X h =1 P s (cid:0) E ′ k ( h ) ∩ B S (cid:1) + N X h =1 L X i =3 P s (cid:0) E k ( h ) ∩ B S ( x k ( i )) (cid:1)i . By combining these identities we get2 γ − k →∞ P s ( E ′ k ) = lim sup k →∞ h − P s (cid:0) E ′ k (0) c ∩ B S (cid:1) + X i =1 , P s (cid:0) E k (0) c ∩ B S ( x k ( i )) (cid:1) − N X h =1 P s (cid:0) E ′ k ( h ) ∩ B S (cid:1) + X i =1 , N X h =1 P s (cid:0) E k ( h ) ∩ B S ( x k ( i )) (cid:1)i . (2.13) By the subadditivity of the s -perimeter, for every k ∈ N and h = 1 , ..., N one has P s (cid:0) E ′ k ( h ) ∩ B S (cid:1) ≤ P s (cid:0) E k ( h ) ∩ B S ) + P s ( E k ( h ) ∩ B S ( x k (2)) (cid:1) . (2.14)At the same time, for every k ∈ N , P s (cid:0) E ′ k (0) c ∩ B S (cid:1) ≤ P s (cid:0) E k (0) c ∩ B S (cid:1) + P s (cid:0) E k (0) c ∩ B S ( x k (2)) (cid:1) − c (8 S ) n + s . (2.15)To prove (2.15), we exploit the upper bound in (2.6) with E = E ′ k (0) c ∩ B S and F = E ′ k (0) c ∩ B S (3 Se ). Since E ∪ F ⊆ B S and | E | , | F | ≥ c by (2.5), we find that P s (cid:0) E ′ k (0) c ∩ B S (cid:1) = P s (cid:0) ( E ′ k (0) c ∩ B S ) ∪ ( E ′ k (0) c ∩ B S (3 Se )) (cid:1) ≤ P s (cid:0) E ′ k (0) c ∩ B S (cid:1) + P s (cid:0) E ′ k (0) c ∩ B S (3 Se ) (cid:1) − c (8 S ) n + s . Since E ′ k (0) c ∩ B S (3 Se ) is a translation of E k (0) c ∩ B S ( x k (2)), we have prove (2.15). By combining(2.14) and (2.15) with (2.13), and taking into account that each E ′ k is a competitor in (2.1), wefinally find a contradiction, namely γ ≤ lim sup k →∞ P s ( E ′ k ) ≤ γ − c (8 S ) n + s . (cid:3) Volume-fixing variations. In studying isoperimetric problems with multiple volume con-straints one needs to use local diffeomorphic deformations to adjust volumes of competitors. (Scal-ing is not useful here, as it can just be used to fix the volume of a chamber per time.) This basictechnique is found in Almgren’s work [Alm76, VI-10,11,12]. Here we follow the presentation of[Mag12, Sections 29.5-29.6], and discuss the adaptations needed to work in the fractional setting.Given a reference N -cluster E , our goal is proving that for every cluster E ′ which is sufficiently L -close to E and for every volume m ′ sufficiently close to m ( E ′ ) there exists a deformation of E ′ with volume m ′ and perimeter which has increased, at most, proportionally to the small quantity | m ′ − m ( E ′ ) | ; see Proposition 2.6 below.The first step to achieve this is proving that, in any ball where the two chambers E ( i ) and E ( j ) are present, one can build a compactly supported vector field whose flow increases the volumeof E ( i ) with speed 1, decreases the volume of E ( j ) with speed − 1, and leaves the volumes of theother chambers infinitesimally unchanged. In the local case this is done in a geometrically explicitway by exploiting the notion of reduced boundary to push E ( i ) along its (measure-theoretic) outerunit normal, compare with [Mag12, Section 29.5]. In the fractional case we are not dealing withsets of finite perimeter, and we thus resort to a more abstract approach, which in fact simplifiesthe construction. In the following we set V = (cid:8) a ∈ R N +1 : a (0) + ... + a ( N ) = 0 (cid:9) . Lemma 2.3. If E is an N -cluster in R n , ≤ i < j ≤ N , and z ∈ ∂ E ( i ) ∩ ∂ E ( j ) , then for every R > there exists a vector field T ij ∈ C ∞ c ( B R ( z ); R n ) such that Z E ( i ) div( T ij ) dx = 1 = − Z E ( j ) div( T ij ) dx, Z E ( h ) div( T ij ) dx = 0 ∀ h = i, j. Proof. Step one : Given R > z ∈ R n , let H ⊂ { , ..., N } be such that h ∈ H if and onlyif 0 < |E ( h ) ∩ B R ( z ) | < B R ( z ). Let us consider the linear operator L : C ∞ c ( B R ( z ); R n ) → R N +1 defined by L ( T ) = (cid:16) Z E (0) div( T ) dx, ..., Z E ( N ) div( T ) dx (cid:17) , M. COLOMBO AND F. MAGGI and consider the linear spaces I = n L ( T ) : T ∈ C ∞ c ( B R ( z ); R n ) o V ′ = (cid:8) a ∈ V : a ( h ) = 0 ∀ h H (cid:9) . We claim that I = V ′ . Trivially, I ⊂ V ′ . Since I is the intersection of all the hyperplanes thatcontain it, it is enough to show that if J is an hyperplane in R N +1 which contains I , then V ′ ⊂ J .Indeed, let { λ h } Nh =0 be such that a ∈ J if and only if P Nh =0 λ h a ( h ) = 0. The condition I ⊂ J implies that0 = X h ∈ H λ h Z E ( h ) div( T ) dx = Z R n (cid:16) X h ∈ H λ h E ( h ) (cid:17) div( T ) dx , ∀ T ∈ C ∞ c ( B R ( z ); R n ) , so that P h ∈ H λ h E ( h ) is constant in B R ( z ). As the chambers E ( h ) are disjoint, this means thatthere exists λ ∈ R such that λ h = λ for every h ∈ H , and thus V ′ ⊂ J holds. Step two : Now let z ∈ ∂ E ( i ) ∩ ∂ E ( j ) for some 0 ≤ i < j ≤ N , and given R > H ⊂ { , ..., N } be defined as in step one, so that { i, j } ⊂ H . Since I = V ′ and the equations a ( i ) = 1, a ( j ) = − a ( h ) = 0 for h = i, j define an element a ∈ V ′ , we conclude the existence of T ij ∈ C ∞ c ( B R ( z ); R n )with the required properties. (cid:3) The subsequent step is checking that the flows generated by the vector-fields T ij found in theprevious lemma have the required properties. We notice that the constant C below depends alsoon k T k C (and therefore on our particular cluster), so the dependence on s is not explicit here. Lemma 2.4 (Infinitesimal volume exchange between two chambers) . Let s ∈ (0 , and E be an N -cluster in R n . If ≤ h < k ≤ N , z ∈ ∂ E ( h ) ∩ ∂ E ( k ) , and r, δ > , then there exist positiveconstants ε , ε , C depending only on E , z, r, δ , and a family of diffeomorphisms { f t } | t |≤ ε suchthat (cid:8) x ∈ R n : x = f t ( x ) (cid:9) ⊂⊂ B r ( z ) , ∀| t | ≤ ε , (2.16) which satisfies the following properties: (i) if E ′ is a cluster, d ( E , E ′ ) < ε (in particular, if E ′ = E ), and | t | < ε , then (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:12)(cid:12)(cid:12) f t (cid:0) E ′ ( h ) (cid:1) ∩ B r ( z ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) < δ, (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:12)(cid:12)(cid:12) f t (cid:0) E ′ ( k ) (cid:1) ∩ B r ( z ) (cid:12)(cid:12)(cid:12) + 1 (cid:12)(cid:12)(cid:12)(cid:12) < δ, (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:12)(cid:12)(cid:12) f t (cid:0) E ′ ( i ) (cid:1) ∩ B r ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ ∀ i = h, k, (cid:12)(cid:12)(cid:12)(cid:12) d dt (cid:12)(cid:12)(cid:12) f t (cid:0) E ′ ( i ) (cid:1) ∩ B r ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C ∀ i = 0 , ..., N . (notice that f t ( E ) ∩ B r ( z ) = f t ( E ∩ B r ( z )) for every E ⊂ R n ). (ii) if E is a set of finite s -perimeter and | t | < ε , then | P s ( f t ( E )) − P s ( E ) | ≤ C | t | P s ( E ) . Proof. Given z ∈ ∂ E ( h ) ∩ ∂ E ( k ) and r > 0, let T ∈ C ∞ c ( B r ( z )) be the vector field given byLemma 2.3, which satisfies Z E ( h ) div( T ) dx = 1 = − Z E ( k ) div( T ) dx, Z E ( i ) div( T ) dx = 0 ∀ i = h, k. (2.17)For every t ∈ (0 , 1) we define f t ( x ) = x + tT ( x ), x ∈ R n . Since f ( x ) = x and spt T ⊂ B r ( z ),there exists ε > { f t } | t |≤ ε is a family of diffeomorphisms satisfying (2.16). By thearea formula, for every Borel set E ⊂ R n (cid:12)(cid:12) f t ( E ) ∩ B r ( z ) (cid:12)(cid:12) = Z E ∩ B r ( z ) Jf t ( x ) dx. Noticing that Jf t ( x ) = 1 + t div T ( x ) + O ( t ), we deduce that ddt (cid:12)(cid:12)(cid:12) t =0 (cid:12)(cid:12) f t ( E ) ∩ B r ( z ) (cid:12)(cid:12) = Z E ∩ B r ( z ) div T ( x ) dx and statement (i) follows, possibly further reducing the value of ε , by (2.17) and by the fact that t → (cid:12)(cid:12) f t ( E ) ∩ B r ( z ) (cid:12)(cid:12) is a smooth function when t is small. By the change of variable formula wehave also that P s ( f t ( E )) = Z E Z E c Jf t ( x ) Jf t ( y ) | f t ( x ) − f t ( y ) | n + s dx dy. Since Jf t ( x ) J t ( y ) = 1 + t (div T ( x ) + div T ( y )) + o ( t ) there exist C > n and T onlysuch that | Jf t ( x ) J t ( y ) − | ≤ C | t | ;moreover, up to considering larger values of C , we have1 | f t ( x ) − f t ( y ) | n + s ≤ | x − y | − | t || T ( x ) − T ( y ) | ) n + s ≤ | x − y | n + s (1 − | t |k∇ T k L ∞ ) n + s ≤ C | t || x − y | n + s | f t ( x ) − f t ( y ) | n + s ≥ | x − y | + | t || T ( x ) − T ( y ) | ) n + s ≥ | x − y | n + s (1 + | t |k∇ T k L ∞ ) n + s ≥ − C | t || x − y | n + s , so that (cid:12)(cid:12)(cid:12) | f t ( x ) − f t ( y ) | n + s − | x − y | n + s (cid:12)(cid:12)(cid:12) ≤ C | t || x − y | n + s for t small enough. Hence, up to reducing ε we deduce that | P s ( f t ( E )) − P s ( E ) | ≤ Z E Z E c (cid:12)(cid:12)(cid:12) Jf t ( x ) Jf t ( y ) | f t ( x ) − f t ( y ) | n + s − | x − y | n + s (cid:12)(cid:12)(cid:12) dx dy ≤ Z E Z E c (cid:12)(cid:12)(cid:12) Jf t ( x ) Jf t ( y ) | f t ( x ) − f t ( y ) | n + s − Jf t ( x ) Jf t ( y ) | x − y | n + s (cid:12)(cid:12)(cid:12) dx dy + Z E Z E c (cid:12)(cid:12)(cid:12) Jf t ( x ) Jf t ( y ) | x − y | n + s − | x − y | n + s (cid:12)(cid:12)(cid:12) dx dy ≤ C | t | Z E Z E c | x − y | n + s dx dy, which proves statement (ii). (cid:3) Lemma 2.4 gives us a way to exchange volume between the chambers E ( h ) and E ( k ) at apoint z ∈ ∂ E ( h ) ∩ ∂ E ( k ), without significantly change the volume of other chambers. The nextstep is choosing where to pick the points z so to have enough freedom to achieve any small volumeadjustment. To this end we introduce the following terminology: E ( h ) and E ( k ) are neighboringchambers if H n − ( ∂ E ( h ) ∩ ∂ E ( k )) > 0. Let S be the set of the indexes corresponding to neighboringchambers of E , S = n ( h, k ) ∈ { , ..., N } : h < k, H n − ( ∂ E ( h ) ∩ ∂ E ( k )) > o , let M ∈ { N, ..., N } be the cardinality of S , and let φ = ( φ , φ ) : { , ..., M } → S be a bijection(so that φ is an enumeration of S ). A finite family of distinct points { z α } α =1 ,...,M is a system ofinterface points of E if for every α ∈ { , ..., M } we have that z α ∈ ∂ E ( φ ( α )) ∩ ∂ E ( φ ( α )). Thefollowing lemma states the existence of a system of interface points of E and shows that a certainmatrix, which keeps into account the links between different chambers, has rank N . Lemma 2.5. (i) If E is an N -cluster in R n and M and φ are as above, then the matrix L =( L jα ) j =0 ,...N, α =1 ,...,M ∈ R ( N +1) × M defined as L jα = if j = φ ( α ) , − if j = φ ( α ) , if j = φ ( α ) , φ ( α ) , ≤ α ≤ M has rank N .(ii) If δ > and A is an open set in R n such that for every h = 0 , ..., N there exists a connectedcomponent A ′ of A with |E (0) ∩ A ′ | > and |E ( h ) ∩ A ′ | > , then there exists systems of interfacepoints { z α } α =1 ,...,M ⊂ A and { y α } α =1 ,...,M ⊂ A with | z α − y β | > δ for every α, β = 1 , ..., M .Proof. See [Mag12, Proof of Theorem 29.14, Step 1]. (cid:3) By combining the previous lemma we obtain the following proposition on volume-fixing vari-ations. Proposition 2.6 (Volume-fixing variations) . Let s ∈ (0 , , E be an N -cluster with < |E ( h ) | < ∞ for every h = 1 , ..., N , { z α } α =1 ,...,M be a system of interface points of E , and let < r < min {| z α − z β | / ≤ α < β ≤ M } .Then there exist positive constants η, ε , ε , C ( s , E , { z α } α =1 ,...,M and r ) with the followingproperty: for every N -cluster E ′ with d ( E , E ′ ) < ε there exists a C -function Φ : (( − η, η ) N +1 ∩ V ) × R n → R n such that (i) if a ∈ ( − η, η ) N +1 ∩ V then Φ( a , · ) : R n → R n is a diffeomorphism with { x ∈ R n : Φ( a , x ) = x } ⊂ M [ α =1 B r ( z α ) ⊂⊂ R n (ii) if a ∈ ( − η, η ) N +1 ∩ V then for ≤ h ≤ N (cid:12)(cid:12)(cid:12) Φ( a , E ′ ( h )) ∩ { x ∈ R n : Φ( a , x ) = x } (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E ′ ( h ) ∩ { x ∈ R n : Φ( a , x ) = x } (cid:12)(cid:12)(cid:12) + a ( h );(iii) if a ∈ ( − η, η ) N +1 ∩ V and F is a set of finite s -perimeter, then | P s (Φ( a , F )) − P s ( F ) | ≤ CP s ( F ) N X h =0 | a ( h ) | . Proof. Given Lemma 2.4 and Lemma 2.5 the proof is basically the same as in [Mag12, Proof ofTheorem 29.14], so we just give a sketch for the sake of clarity. By Lemma 2.4 given positiveconstants δ and r , there exist positive constants ε , ε , C (depending on E , r , δ and { z α } Mα =1 ) anddiffeomorphisms { f αt } α =1 ,...,M , | t | <ε such that (cid:8) x ∈ R n : x = f αt ( x ) (cid:9) ⊂⊂ B r ( z α ) , ∀| t | ≤ ε , α = 1 , ..., M , (2.18)and, if E ′ is a cluster with d ( E , E ′ ) < ε , | t | < ε , α = 1 , ..., M , and ( h, k ) = ( φ ( α ) , φ ( α )), then (cid:12)(cid:12)(cid:12) ddt (cid:12)(cid:12)(cid:12) f αt (cid:0) E ′ ( h ) (cid:1) ∩ B r ( z α ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) < δ, (cid:12)(cid:12)(cid:12) ddt (cid:12)(cid:12)(cid:12) f αt (cid:0) E ′ ( k ) (cid:1) ∩ B r ( z α ) (cid:12)(cid:12)(cid:12) + 1 (cid:12)(cid:12)(cid:12) < δ, (2.19) (cid:12)(cid:12)(cid:12) ddt (cid:12)(cid:12)(cid:12) f αt (cid:0) E ′ ( i ) (cid:1) ∩ B r ( z α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ for i = h, k, (2.20) (cid:12)(cid:12)(cid:12) d dt (cid:12)(cid:12)(cid:12) f αt (cid:0) E ′ ( i ) (cid:1) ∩ B r ( z α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C for 0 ≤ i ≤ N (2.21)and such that, whenever E is a set of finite s -perimeter, | P s ( f αt ( E )) − P s ( E ) | ≤ C | t | P s ( E ) . (2.22)Since r < min {| z α − z β | / ≤ α < β ≤ M } , if we define Ψ : ( − ε , ε ) M × R n → R n by settingΨ( t , x ) = ( f t ◦ f t ◦ ... ◦ f Mt M )( x ) , ( t , x ) ∈ ( − ε , ε ) M × R n , then Ψ( t , · ) is a diffeomorphisms with { Ψ( t , · ) = Id } compactly contained in the union of thedisjoint balls { B r ( z α ) } Mα =1 . We claim the existence of η > ζ : ( − η, η ) N +1 ∩ V → R M suchthat Φ( a , x ) = Ψ( ζ ( a ) , x ) ( a , x ) ∈ (( η, η ) N +1 ∩ V ) × R n , (2.23)satisfies all the required properties. To this end, we consider first the function ψ : ( − ε , ε ) M → V ⊆ R N +1 defined by setting, for every h = 0 , ..., N and t ∈ ( − ε , ε ) M , ψ h ( t ) = (cid:12)(cid:12) Ψ( t , E ′ ( h )) ∩ (cid:8) x ∈ R n : x = Ψ( t , x ) (cid:9)(cid:12)(cid:12) − (cid:12)(cid:12) E ′ ( h ) ∩ (cid:8) x ∈ R n : x = Ψ( t , x ) (cid:9)(cid:12)(cid:12) = M X α =1 (cid:12)(cid:12) f αt α (cid:0) E ′ ( h ) (cid:1) ∩ B r ( z α ) (cid:12)(cid:12) − (cid:12)(cid:12) E ′ ( h ) ∩ B r ( z α ) (cid:12)(cid:12) . (2.24)By (2.19), (2.20), (2.21), we see that ψ (0) = 0, |∇ ψ ( t ) | ≤ C for every t ∈ ( − ε , ε ) M , with | ∂ α ψ h (0) − L hα | ≤ C ( N, M ) δ for every h = 0 , ..., N and α = 1 , ..., M . Since the rank of ( L hα ) h,α is N (Lemma 2.5), by arguing as in [Mag12, Proof of Theorem 29.14, Step 3] we find that provided δ is small enough then there exists κ > ∇ ψ (0) e ≥ κ | e | for every e ∈ ker ∇ ψ (0) ⊥ . By theimplicit function theorem (with the same statement as in [Mag12, Proof of Theorem 29.14, Step 2]for having a quantitative dependence of η on E and ε but not on E ′ ) we deduce that there existsa class C function ζ : ( − η, η ) N +1 ∩ V → R M such that ψ ( ζ ( a )) = a , | ζ ( a ) | ≤ κ | a | . With this definition at hand, it is clear that Φ defined in (2.23) satisfies (i). Thanks to the definitionof ζ and ψ , it satisfies also (ii). We are left to check (iii), which requires a computation specific tothe fractional setting. If a ∈ ( − η, η ) N +1 ∩ V and F is a set of finite s -perimeter, then we have | P s (Φ( a , F )) − P s ( F ) | = | P s (( f ζ ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) − P s ( F ) | = M − X α =1 | P s (( f αζ α ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) − P s (( f α +1 ζ α +1 ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) | + | P s ( f Mζ M ( a ) ( F )) − P s ( F ) | . (2.25)By (2.22), we deduce that for every α = 1 , ..., M − | P s (( f αζ α ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) − P s (( f α +1 ζ α +1 ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) | ≤ C | ζ α ( a ) | P s (( f α +1 ζ α +1 ( a ) ◦ ... ◦ f Mζ M ( a ) )( F ))(2.26)and similarly | P s ( f Mζ M ( a ) ( F )) − P s ( F ) | ≤ C | ζ M ( a ) | P s ( F ) . In particular, for every α = 1 , ..., M − 1, since | ζ α ( a ) | ≤ ε ≤ 1, we obtain P s (( f αζ α ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) ≤ P s (( f α +1 ζ α +1 ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) + | P s (( f αζ α ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) − P s (( f α +1 ζ α +1 ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) |≤ (1 + C ) P s (( f α +1 ζ α +1 ( a ) ◦ ... ◦ f Mζ M ( a ) )( F ))and P s ( f Mζ M ( a ) ( F )) ≤ (1 + C ) P s ( F );an easy induction shows then that P s (( f αζ α ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) ≤ (1 + C ) M P s ( F ) . (2.27) By (2.25), (2.26), and (2.27), we deduce that | P s (Φ( a , F )) − P s ( F ) | = C M − X α =1 | ζ α ( a ) | P s (( f α +1 ζ α +1 ( a ) ◦ ... ◦ f Mζ M ( a ) )( F )) + C | ζ M ( a ) | P s ( F ) ≤ (1 + C ) M +1 P s ( F ) M X α =1 | ζ α ( a ) | ≤ M / (1 + C ) M +1 κ P s ( F ) N X h =0 | a ( h ) | , (2.28)so that also (iii) is satisfied. (cid:3) In the local case Proposition 2.6 would be sufficient for showing that isoperimetric clustersare locally almost-minimizing perimeter (a key step in the regularity theory) and for modifyingminimizing sequences in the existence argument. In the fractional case, the latter application willneed the following version of Proposition 2.6. Proposition 2.7 (Volume-fixing variations of a minimizing sequence) . Let s ∈ (0 , , m ∈ R N + , {E k } k ∈ N be a sequence of N -clusters with m ( E k ) = m for every k ∈ N , and define S > by setting ω n S n = 2( m (1) + ... + m ( N )) . Finally, let us assume that there exist c > and sequences { x k (1) } k ∈ N , ..., { x k ( N ) } k ∈ N such that |E k ( h ) ∩ B S ( x k ( h )) | ≥ c for every k ∈ N and h = 1 , ..., N. (2.29) Then there exist positive constants η, C such that for every k ∈ N (up to a not relabeled subsequence)there exists a C -function Φ k : (( − η, η ) N ∩ V ) × R n → R n such that (i) if a ∈ ( − η, η ) N +1 ∩ V then Φ k ( a , · ) : R n → R n is a diffeomorphism with { x ∈ R n : Φ k ( a , x ) = x } ⊂ N [ h =0 B S ( x k ( h )) ⊂⊂ R n (ii) if a ∈ ( − η, η ) N +1 ∩ V then for ≤ h ≤ N (cid:12)(cid:12)(cid:12) Φ k ( a , E k ( h )) ∩ { x ∈ R n : Φ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E k ( h ) ∩ { x ∈ R n : Φ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) + a ( h );(iii) if a ∈ ( − η, η ) N +1 ∩ V and F is a set of finite s -perimeter, then | P s (Φ k ( a , F )) − P s ( F ) | ≤ CP s ( F ) N X h =0 | a ( h ) | . In the course of the proof we shall need the following basic property of fractional perimeters:for every measurable set E and for every ball B it holds P s ( E ∩ B ) ≤ I s ( E ∩ B, E c ) + I s ( E ∩ B, B c ) ≤ I s ( E, E c ) + I s ( B, B c ) = P s ( E ) + P s ( B ) . (2.30) Proof. Up to extracting a not relabelled subsequence, we may assume that there exist lim k →∞ x k ( h ) − x k ( h ′ ) for every h, h ′ ∈ { , ..., N } . Moreover, we can partition { , ..., N } into ℓ disjoint sets Λ , ..., Λ ℓ such that for every j = 1 , ..., ℓ there exists lim k →∞ x k ( h ) − x k ( h ′ ) ∈ B NS if h, h ′ ∈ Λ j , lim inf k →∞ x k ( h ) − x k ( h ′ ) > S for every h, h ′ ∈ Λ j . The construction of the sets Λ j is performed in [Mag12, Section 29.7, Step 1]. Then we haveisolated ℓ disjoint nuclei in E k , each of them of the form E k ( h ) ∩ [ h ′ ∈ Λ j B S ( x k ( h ′ )) for every h = 1 , ..., N, j = 1 , ..., ℓ. By setting v j = 8( N + 1) Sje n and by selecting an element h j in each set Λ j , we define a newsequence of clusters E ∗ k by setting for every h = 1 , ..., N E ∗ k ( h ) = ℓ [ i =1 v j − x k ( h j ) + (cid:16) E k ( h ) ∩ [ h ′ ∈ Λ j B S ( x k ( h ′ )) (cid:17) . For every h = 1 , ..., N , by (2.30) we obtain P s ( E ∗ k ( h )) ≤ ℓ X j =1 X h ′ ∈ Λ j P s (cid:0) E ∗ k ( h ) ∩ B S ( v j − x k ( h j ) + x k ( h ′ )) (cid:1) = ℓ X j =1 X h ′ ∈ Λ j P s (cid:0) E k ( h ) ∩ B S ( x k ( h ′ )) (cid:1) = N X h ′ =1 P s (cid:0) E k ( h ) ∩ B S ( x k ( h ′ )) (cid:1) ≤ N P s (cid:0) B S (0) (cid:1) + N X h =1 P s (cid:0) E k ( h ) (cid:1) . By the bound on the perimeters of E ∗ k above, which are all contained in B N +1) S (0), we deducethat there exists a cluster E ⊂ B N +1) S (0) such that, up to a subsequence, each chamber of E ∗ k converges to the corresponding chamber of E ∗ in L ( B N +1) S (0)). Moreover, by (2.29), if h ∈ Λ j for some j , we have that |E ∗ ( h ) ∩ B S ( v j − x k ( h j ) + x k ( h )) | ≥ c . We apply Lemma 2.5 to obtain a system of interface points for E ∗ in ∪ Nh =1 B S ( v j − x k ( h j ) + x k ( h ))(we use the open set A given by a union of balls). Following the proof of Proposition 2.6 appliedto the reference cluster E , we find η, ε and C (independent on k ), one-parameter families ofdiffeomorphisms { f αt } α =1 ,...,M and ζ : ( − η, η ) N +1 ∩ V → R M (the latter two depend on k , asin the previous proof they depended on E ′ , but for simplicity we omit this dependence) with thefollowing properties. For every α = 1 , ..., M there exists a j ∈ { , ..., ℓ } and h ′ ∈ Λ j such that (cid:8) x ∈ R n : x = f αt ( x ) (cid:9) ⊂⊂ B S ( v j − x k ( h j ) + x k ( h ′ )) , for every | t | ≤ ε , (2.31)the sets (cid:8) x ∈ R n : x = f αt ( x ) (cid:9) are all disjoint as α ranges in 1 , ..., M , | P s ( f αt ( E )) − P s ( E ) | ≤ C | t | P s ( E ) , (2.32)and settingΦ ∗ k ( a , x ) = ( f ζ ( a ) ◦ f ζ ( a ) ◦ ... ◦ f Mζ M ( a ) )( x ) ( a , x ) ∈ (( η, η ) N +1 ∩ V ) × R n , (2.33)we have (cid:12)(cid:12)(cid:12) Φ ∗ k ( a , E ∗ k ( h )) ∩ { x ∈ R n : Φ ∗ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E ∗ k ( h ) ∩ { x ∈ R n : Φ ∗ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) + a ( h ) . (2.34)Now we suitably translate the functions f ζ ( a ) , ..., f Mζ M ( a ) in such a way that they act on the cluster E k rather than on its translation E ∗ k ; more precisely, we define for every α = 1 , ..., Mg αζ α ( a ) ( x ) = f αζ α ( a ) ( x + v j − x k ( h j )) − v j + x k ( h j ) (once more we omit the dependence on k for ease of notation; here j ∈ { , ..., ℓ } and h ′ ∈ Λ j arechosen to satisfy (2.31)) and for every k ∈ N Φ k ( a , x ) = ( g ζ ( a ) ◦ g ζ ( a ) ◦ ... ◦ g Mζ M ( a ) )( x ) ( a , x ) ∈ ( η, η ) N +1 × R n . It is clear that, since f αζ α ( a ) is the identity outside B S ( v j − x k ( h j ) + x k ( h )), the diffeomorphism g αζ α ( a ) is the identity outside B S ( x k ( h )); moreover { x ∈ R n : x = g αt ( x ) (cid:9) = x k ( h j ) − v j + { x ∈ R n : x = f αt ( x ) (cid:9) . It is easily checked by the definition of E ∗ k that for every h = 1 , ..., N the set g αζ α ( a ) ( E k ( h )) ∩ (cid:8) x ∈ R n : x = g αt ( x ) (cid:9) is a translation of f αζ α ( a ) ( E ∗ k ( h )) ∩ (cid:8) x ∈ R n : x = f αt ( x ) (cid:9) , so that the volumechange induced on E ∗ k by f αζ α ( a ) is the same volume change induced on E k by g αζ α ( a ) : in other words, (cid:12)(cid:12) g αζ α ( a ) ( E k ( h )) ∩ (cid:8) x ∈ R n : x = g αt ( x ) (cid:9)(cid:12)(cid:12) = (cid:12)(cid:12) f αζ α ( a ) ( E ∗ k ( h )) ∩ (cid:8) x ∈ R n : x = f αt ( x ) (cid:9)(cid:12)(cid:12) . Since the diffeomorphisms f αζ α ( a ) act (as α varies) on nonintersecting sets, and the same happensto g αζ α ( a ) , by composing the diffeomorphisms when α varies by (2.34) we deduce that (cid:12)(cid:12)(cid:12) Φ k ( a , E k ( h )) ∩ { x ∈ R n : Φ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Φ ∗ k ( a , E ∗ k ( h )) ∩ { x ∈ R n : Φ ∗ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E ∗ k ( h ) ∩ { x ∈ R n : Φ ∗ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) + a ( h )= (cid:12)(cid:12)(cid:12) E k ( h ) ∩ { x ∈ R n : Φ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) + a ( h );hence (ii) holds true. To prove (iii), we repeat word by word the argument between (2.25) and(2.28) with g αζ α ( a ) replacing f αζ α ( a ) at every occurrence (by the nonlocality of the s -perimeter, thefact that (iii) holds with Φ ∗ k replacing Φ k does not allow directly to conclude the statement; weneed to repeat the argument on each g αζ α ( a ) ). (cid:3) Truncation lemma. We now state and prove the truncation lemma for fractional perimetersneeded in the existence proof. In the case of sets ( N = 1) this lemma has already appeared as[FFM + 15, Lemma 4.5]. Lemma 2.8. Let n ≥ , s ∈ (0 , , τ ∈ (0 , , let E be an N -cluster in R n , and F ⊆ R n be aclosed set with u ( x ) = dist( x, F ) for x ∈ R n . If N X h =1 |E ( h ) \ F | ≤ τ , then there exists r ∈ [0 , C τ /n ] such that the N -cluster E ′ in R n defined by E ′ ( h ) = E ( h ) ∩ { u ≤ r } ≤ h ≤ N satisfies (1 − s ) P s ( E ′ ) ≤ (1 − s ) P s ( E ) − dist( E , E ′ ) C ( n, s ) τ s/n , (2.35) where C ( n, s ) := 2 n − s ) /s (cid:16) | B | ( n − s ) /n P ( B ) s (1 − s ) P s ( B ) (cid:17) /s , C ( n, s ) := 2 | B | ( n − s ) /n (1 − s ) P s ( B ) . (2.36) In particular, sup { C ( n, s ) + C ( n, s ) : s ≤ s < } < ∞ for every fixed s ∈ (0 , . Remark 2.9. Here we pay some attention to the dependency of constants from s , as the constantscan be shown to be uniform in the limit s → − . Proof. For every r ≥ 0, let us call F r = { u ≤ r } the r -enlargement of F and let us define thecluster E r whose chambers are E r ( h ) = E ( h ) ∩ F r for every 1 ≤ h ≤ N . Without loss of generalitywe may assume that N X h =1 |E ( h ) \ F C τ /n | > r = C τ /n and (2.35) holds. If we set m ( r ) = P Nh =1 |E ( h ) \ F r | , r > 0, then m is a nonincreasing function with[0 , C τ /n ] ⊂ spt m m (0) ≤ τ , m ′ ( r ) = − N X h =1 H n − ( E ( h ) ∩ ∂F r ) for a.e. r > . (2.37)Arguing by contradiction, we now assume that(1 − s ) P s ( E ) ≤ (1 − s ) P s ( E r ) + m ( r ) C τ s/n , ∀ r ∈ (0 , C τ /n ) . (2.38)First, for every r > h = 1 , ..., N we have the identity P s ( E ( h ) ∩ F r ) − P s ( E ( h )) = 2 P s ( F r ; E ( h )) − P s ( E ( h ) \ F r )= 2 Z E ( h ) ∩ F r Z E ( h ) ∩ F cr dx dy | x − y | n + s − P s ( E ( h ) \ F r ) . Since E ( h ) ∩ F r ⊆ B u ( y ) − r ( y ) for every y ∈ E ( h ) ∩ F cr and by the coarea formula, for every r > Z E ( h ) ∩ F r Z E ( h ) ∩ F cr dx dy | x − y | n + s ≤ Z E ( h ) ∩ F cr dy Z B u ( y ) − r ( y ) dx | x − y | n + s = P ( B ) Z E ( h ) ∩ F cr dy Z ∞ u ( y ) − r dρρ s = P ( B ) s Z E ( h ) ∩ F cr dy ( u ( y ) − r ) s = P ( B ) s Z ∞ r − m ′ ( t )( t − r ) s dt . Finally, by the nonlocal isoperimetric inequality, N X h =1 P s ( E ( h ) \ F r ) ≥ P s (cid:16) N [ h =1 E ( h ) \ F r (cid:17) ≥ P s ( B ) | B | ( s − n ) /n m ( r ) ( n − s ) /n . We may thus combine these three remarks with (2.38) to conclude that, if r ∈ (0 , C τ /n ), then0 ≤ P ( B ) s Z ∞ r − m ′ ( t )( t − r ) s dt − P s ( B ) | B | ( n − s ) /n m ( r ) ( n − s ) /n + m ( r )(1 − s ) C τ s/n ≤ P ( B ) s Z ∞ r − m ′ ( t )( t − r ) s dt − P s ( B )2 | B | ( n − s ) /n m ( r ) ( n − s ) /n , (2.39)where in the last inequality we have used our choice of C and the fact that m ( r ) ≤ τ for every r > 1. We rewrite (2.39) in the more convenient form m ( r ) ( n − s ) /n ≤ C Z ∞ r − m ′ ( t )( t − r ) s dt , ∀ r ∈ (1 , C τ /n ) , (2.40)where we have set C ( n, s ) := 4 | B | ( n − s ) /n P ( B ) s P s ( B ) . Proceeding as in [FFM + 15, Lemma 4.5] one can show that any function m satisfying the previousdifferential inequality satisfies m ( r ) → r → C τ /n . This gives a contradiction. (cid:3) Nucleation lemma. The following nucleation lemma is obtained by combining part of theargument leading to its local analogous (see [Mag12, Lemma 29.10]) with a lemma for fractionalperimeters already appeared in [FFM + 15, Lemma 4.3]. Lemma 2.10. Let n ≥ and s ∈ (0 , . If P s ( E ) < ∞ , < | E | < ∞ , and ε ≤ min n | E | , − sχ χ P s ( E ) o (2.41) then there exists a finite family of points I ⊂ R n such that (cid:12)(cid:12)(cid:12) E \ [ x ∈ I B ( x ) (cid:12)(cid:12)(cid:12) < ε, (cid:12)(cid:12)(cid:12) E ∩ B ( x ) (cid:12)(cid:12)(cid:12) ≥ (cid:16) χ ε (1 − s ) P s ( E ) (cid:17) n/s ∀ x ∈ I, (2.42) where χ ( n, s ) := (1 − s ) P s ( B )4 | B | ( n − s ) /n ξ ( n ) , χ ( n, s ) := 2 n/s ) | B | ( n − s ) /n P ( B ) s (1 − s ) P s ( B ) , (2.43) and where ξ ( n ) is Besicovitch’s covering constant (see for instance [Mag12, Theorem 5.1] ). Inparticular, < inf { χ ( n, s ) , χ ( n, s ) − : s ∈ [ s , } < ∞ for every s ∈ (0 , . Moreover, | x − y | > for every x, y ∈ I , x = y , and I ≤ | E | (cid:16) (1 − s ) P s ( E ) χ ε (cid:17) n/s . Proof. In [FFM + 15, Proof of Lemma 4.3, Step 1] it is proved that if x ∈ E (1) with | E ∩ B ( x ) | ≤ (cid:16) (1 − s ) P s ( B )2 | B | ( n − s ) /n α (cid:17) n/s (2.44)for some α satisfying α ≥ n/s ) P ( B ) s , (2.45)then there exists r x ∈ (0 , 1] such that | E ∩ B r x ( x ) | ≤ (1 − s ) α Z E ∩ B rx ( x ) Z E c dz dy | z − y | n + s . (2.46)This statement is in turn the basic step for proving the following claim: if F ⊆ R n is closed, ε satisfies (2.41), and (cid:12)(cid:12)(cid:12)(cid:8) x ∈ E : dist( x, F ) > (cid:9)(cid:12)(cid:12)(cid:12) ≥ ε, (2.47)then there exists x ∈ E (1) with dist( x, F ) > (cid:12)(cid:12)(cid:12) E ∩ B ( x ) (cid:12)(cid:12)(cid:12) ≥ (cid:16) χ ε (1 − s ) P s ( E ) (cid:17) n/s . Indeed, by contradiction, assume that if x ∈ E (1) with dist( x, F ) > (cid:12)(cid:12)(cid:12) E ∩ B ( x ) (cid:12)(cid:12)(cid:12) < (cid:16) χ ε (1 − s ) P s ( E ) (cid:17) n/s = (cid:16) (1 − s ) P s ( B )2 | B | ( n − s ) /n α (cid:17) n/s . In the last equality we chose α = 2(1 − s ) P s ( E ) ξ ( n ) /ε . Thanks to our assumption (2.41) on ε ,we see that (2.45) holds. Hence, by (2.46) for every x ∈ E (1) with dist( x, F ) > r x such that (2.46) holds. Applying the Besicovitch covering theorem to F = { B r x ( x ) : x ∈ E (1) , dist( x, F ) > } we find a countable disjoint subfamily F ′ of F such that (cid:12)(cid:12)(cid:12)(cid:8) x ∈ E : dist( x, F ) > (cid:9)(cid:12)(cid:12)(cid:12) ≤ ξ ( n ) X B rx ( x ) ∈F ′ | E ∩ B r x ( x ) |≤ (1 − s ) ξ ( n ) α X B rx ( x ) ∈F ′ Z E ∩ B rx ( x ) Z E c dz dy | z − y | n + s ≤ (1 − s ) ξ ( n ) α P s ( E ) . Thanks to our choice of α and to (2.41), the right-hand side equals ε/ { x i } i ∈ I inductively. First, we define x applying the claim with F = ∅ .Then, inductively, we assume that we have chosen I = { x i } i =1 ,...,s and we consider whether (cid:12)(cid:12)(cid:12) E \ [ x ∈ I B ( x ) (cid:12)(cid:12)(cid:12) < ε holds or not. If this holds, the set I satisfies the properties required by our lemma; otherwise,we apply the claim with F = ∪ ij =1 B ( x j ), to find x s +1 such that (2.42) holds and such that itsdistance from { x , ..., x s } is at least 2. Since | E | < ∞ , this process ends in finitely many steps. (cid:3) Existence of isoperimetric clusters. In this section we prove the existence statement inTheorem 1.1: Theorem 2.11. If n, N ≥ , s ∈ (0 , , and m ∈ R N + , then there exist minimizers in the variationalproblem γ = inf (cid:8) P s ( E ) : E is an N -cluster in R n with m ( E ) = m (cid:9) . (2.48) Moreover, if E is a minimizer, then diam( ∂ E ) < ∞ .Proof. By explicit comparison with a cluster whose chambers are N disjoint balls with suitablevolumes we find that γ < ∞ . Let us consider a minimizing sequence sequence {E k } k ∈ N such thatlim k →∞ P s ( E k ) = γ m ( E k ) = m ∀ k ∈ N . Let us set m min = min { m ( h ) : 1 ≤ h ≤ N } m max = max { m ( h ) : 1 ≤ h ≤ N } p min = inf { P s ( E k ( h )) : 1 ≤ h ≤ N, k ∈ N } m max = sup { P s ( E k ( h )) : 1 ≤ h ≤ N, k ∈ N } so that p min ≥ P s ( B ) m ( n − s ) /n min / | B | ( n − s ) /n > p max < ∞ since γ < ∞ . Step one: first nucleation and construction of volume-fixing diffeomorphisms . We apply the nu-cleation Lemma 2.10 with E = E ( h ) and ε = min { m min , − sχ χ p min } n/s (where χ and χ dependonly on n and s and are defined in (2.43)). We obtain that there exist sequences { x k ( h ) } k ∈ N (1 ≤ h ≤ N ), such that for every k ∈ N and 1 ≤ h ≤ N (cid:12)(cid:12)(cid:12) E k ( h ) ∩ B ( x k ( h )) (cid:12)(cid:12)(cid:12) ≥ c, (2.49)where c depends only on n, s, m min , p max . If we define S by ω n S n = 2( m (1) + ... + m ( N )), then atleast half of the volume in B S ( x k ( h )) is occupied by the exterior chamber E k (0), that is (cid:12)(cid:12) E k (0) ∩ B S ( x k ( h )) (cid:12)(cid:12) ≥ ω n S n . We apply Proposition 2.7 to obtain the existence of positive constants η < c / C such that,up to extracting a not-relabeled subsequence in k , there exist C functionsΨ k : (( − η, η ) N ∩ V ) × R n → R n such that for every a ∈ ( − η, η ) N +1 ∩ V the map Ψ k ( a , · ) : R n → R n is a diffeomorphism with { x ∈ R n : Ψ k ( a , x ) = x } ⊂ ∪ Nh =1 B S ( x k ( h )) ⊂⊂ R n (2.50) (cid:12)(cid:12)(cid:12) Ψ k ( a , E k ( h )) ∩ { x ∈ R n : Ψ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E k ( h ) ∩ { x ∈ R n : Ψ k ( a , x ) = x } (cid:12)(cid:12)(cid:12) + a ( h ); (2.51) | P s (Ψ k ( a , F )) − P s ( F ) | ≤ CP s ( F ) N X h =0 | a ( h ) | , (2.52)whenever 1 ≤ h ≤ N , k ∈ N , and F is a set of finite s -perimeter. Step two: Fine nucleation of the cluster . Let χ and χ be the constants in (2.43). We prove thatthere exists a sequence of clusters {E ′′ k } k ∈ N such that for k large enough P s ( E ′′ k ) ≤ P s ( E ) (2.53)and there are r , ε > { x k ( h, i ) } i =1 ,...,L ( k,h ) with the property that E ′′ k ( h ) ⊆ B r ( x k ( h )) ∪ L ( k,h ) [ i =1 B r ( x k ( h, i )) (2.54) (cid:12)(cid:12)(cid:12) E ′′ k ( h ) ∩ B S ( x k ( h )) (cid:12)(cid:12)(cid:12) ≥ c h = 1 , ..., N (2.55) N X j =1 (cid:12)(cid:12)(cid:12) E ′′ k ( j ) ∩ B r ( x k ( h, i )) (cid:12)(cid:12)(cid:12) ≥ min n c , (cid:16) χ ε (1 − s ) p max (cid:17) n/s o for i = 1 , ..., L ( k, h ) , h = 1 , ..., N, (2.56) L ( k, h ) ≤ m max (cid:16) (1 − s ) p max χ ε (cid:17) n/s . To this end, let ε > ε ≤ min n η, m min , − sχ χ p min o (2.57)and, for every k ∈ N and h = 1 , ..., N , let us apply Lemma 2.10 to each chamber E k ( h ) for findingfinitely many points { x k ( h, i ) } i =1 ,...,L ( h,i ) with the property that (cid:12)(cid:12)(cid:12) E k ( h ) \ L ( k,h ) [ i =1 B ( x k ( h, i )) (cid:12)(cid:12)(cid:12) < ε , (2.58) (cid:12)(cid:12)(cid:12) E k ( h ) ∩ B ( x k ( h, i )) (cid:12)(cid:12)(cid:12) ≥ (cid:16) χ ε (1 − s ) p max (cid:17) n/s ∀ i = 1 , ..., L ( h, i ) , (2.59) L ( k, h ) ≤ |E k ( h ) | (cid:16) (1 − s ) P s ( E k ( h )) χ ε (cid:17) n/s ≤ m max (cid:16) (1 − s ) p max χ ε (cid:17) n/s . Next, for every k ∈ N we consider the closed set F k ⊂ R n given by F k := N [ h =1 (cid:16) B S ( x k ( h )) ∪ L ( k,h ) [ i =1 B S ( x k ( h, i )) (cid:17) and then we apply Lemma 2.8 with τ = ε to each E k and F k . We set C and C as in (2.36)depending only on n and s , and we introduce the function u k = dist( x, F k ) to find a sequence { r k } k ∈ N ⊂ [0 , C ε /n ] such that the clusters E ′ k defined by E ′ k ( h ) = E k ( h ) ∩ { u k ≤ r k } , ≤ h ≤ N satisfy (1 − s ) P s ( E ′ k ) ≤ (1 − s ) P s ( E k ) − dist( E k , E ′ k ) C ε s/n (2.60)(in particular lim k →∞ P s ( E ′ k ) = γ ). Finally, we set a k ( h ) := |E k ( h ) | − |E ′ k ( h ) | = |E k ( h ) ∩ { u k > r k }| ≤ h ≤ N, a k (0) := N X h =1 a k ( h ) . By (2.58) we have that a k ( h ) ≤ ε ≤ η , hence we can define E ′′ k ( h ) := Ψ k ( a k , E ′ k ( h )) 1 ≤ h ≤ N. By (2.50) it follows that { x ∈ R n : Ψ k ( a , x ) = x } ⊂ F k ⊂ { u k ≤ r k } , and thus for every k ∈ N and h = 1 , .., N we haveΨ k ( a k , E k ( h )) ∩ { u k ≤ r k } = Ψ k ( a k , E ′ k ( h )) ∩ { u k ≤ r k } = Ψ k ( a k , E ′ k ( h )) , and | Ψ k ( a k , E ′ k ( h )) | = | Ψ k ( a k , E k ( h )) ∩ { u k ≤ r k }| = | Ψ k ( a k , E k ( h )) | − |E k ( h ) ∩ { u k > r k }| = | Ψ k ( a k , E k ( h )) | − a k = |E k ( h ) | , that is, m ( E ′′ k ) = m ( E k ) = m . We notice that (2.54) holds with r = 2 S + 1 + C ε /n . To prove(2.55), we observe that (cid:12)(cid:12)(cid:12) E ′′ k ( h ) ∩ B S ( x k ( h )) (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) E k ( h ) ∩ B S ( x k ( h )) (cid:12)(cid:12)(cid:12) − a k ( h ) ≥ c − η ≥ c h = 1 , ..., N and i = 1 , ..., L ( k, h ), we consider two separate cases:if B ( x k ( h, i )) intersects a ball B S ( x k ( l )) for some l = 1 , ..., N , then B S ( x k ( l )) ⊆ B r ( x k ( h, i )) andtherefore N X j =1 (cid:12)(cid:12)(cid:12) E ′′ k ( j ) ∩ B r ( x k ( h, i )) (cid:12)(cid:12)(cid:12) ≥ |E ′′ k ( l ) ∩ B r ( x k ( l )) (cid:12)(cid:12)(cid:12) ≥ c B ( x k ( h, i )) does not intersect any of the balls B S ( x k ( l )), l = 1 , ..., N , then(2.59) gives (cid:12)(cid:12)(cid:12) E ′′ k ( h ) ∩ B r ( x k ( h, i )) (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) E ′′ k ( h ) ∩ B ( x k ( h, i )) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E k ( h ) ∩ B ( x k ( h, i )) (cid:12)(cid:12)(cid:12) ≥ (cid:16) χ ε (1 − s ) p max (cid:17) n/s and thus (2.56) holds. Finally, we apply (2.52) to E ′ k ( h ) and, using also (2.60) and the equality P Nh =0 | a ( h ) | = dist( E k , E ′ k ), we find that P s ( E ′′ k ) = P s (Ψ k ( a , E ′ k )) ≤ P s ( E ′ k ) + | P s (Ψ k ( a , E ′ k )) − P s ( E ′ k ( h )) |≤ P s ( E ′ k ) + CP s ( E ′ k ) N X h =0 | a ( h ) |≤ P s ( E k ) − dist( E k , E ′ k ) C (1 − s ) ε s/n + CP s ( E ′ k ) N X h =0 | a ( h ) |≤ P s ( E k ) − dist( E k , E ′ k ) C (1 − s ) ε s/n + 2 Cγ dist( E k , E ′ k )which proves (2.53) provided that we choose ε small enough. Step 3: boundedness of the new minimizing sequence, compactness and lower semicontinuity argu-ment . We conclude the proof. Lemma 2.1 applied to the sequence of clusters E ′′ k with R = r 00 M. COLOMBO AND F. MAGGI and c = min { c / , [ χ ε (1 − s ) − p − ] n/s } implies that there exists R > E k ⊆ B R for every k ∈ N . Therefore, each chamber E k ( h ), h = 1 , ..., N , converges in L to a set E ( h ) which has volume m ( E ( h )) = m ( h ) and perimeter P s ( E ( h )) ≤ lim inf k →∞ P s ( E k ( h )), by the lower semicontinuity of P s with respect to L conver-gence of sets. Hence P s ( E ) = N X h =0 P s ( E ( h )) ≤ N X h =0 lim inf k →∞ P s ( E k ( h )) ≤ lim inf k →∞ N X h =0 P s ( E k ( h )) = γ, which proves that E is a minimizer for problem (2.48). (cid:3) Almost everywhere regularity We now address the regularity statements in Theorem 1.1, with the goal of proving the fol-lowing statement: Theorem 3.1. If n ≥ and E is an isoperimetric N -cluster in R n (that is, P s ( E ) ≤ P s ( F ) whenever m ( F ) = m ( E ) ), then there exists α ∈ (0 , and a closed set Σ( E ) ⊂ ∂ E such that H n − (Σ( E )) = 0 if n ≥ , Σ( E ) is discrete if n = 2 , and for every x ∈ ∂ E \ Σ( E ) there exists r x > such that ∂ E ∩ B r x ( x ) is a C ,α -hypersurface in R n . In particular, ∂ E is a locally H n − -rectifiableset in R n \ Σ( E ) and it has Hausdorff dimension n − . The proof is divided in two parts. In section 3.1 we prove the C ,α -regularity of ∂ E nearbypoints where E blows-up two complementary half-spaces. In section 3.2, following the approach of[CRS10], we estimate the dimensionality of the subset of ∂ E where this blow-up property does nothold.3.1. Regular part of the boundary. Given a N -cluster E , x ∈ R n and r > blow-up of E at x at scale r is the N -cluster E x,r defined by E x,r ( h ) = E ( h ) − xr , h = 1 , ..., N . The regular set Reg( E ) of E is the set of those x ∈ ∂ E such that there exist an open half-space H ⊂ R n and h , k ∈ { , ..., N } such that, as r → + and for every j = h, k , E x,r ( h ) → H E x,r ( k ) → R n \ H E x,r ( j ) → ∅ in L ( R n ). (3.1)Our goal is proving that if E is an isoperimetric cluster, then Reg( E ) is a C ,α -hypersurface in R n which is relatively open in ∂ E .We shall actually prove this fact for a larger class of clusters. Given an open set A ⊂ R n , Λ ≥ r ∈ (0 , ∞ ], we say that an N -cluster E is (Λ , r ) -minimizing in A (it is tacitly understoodthat the word minimizing refers to s -perimeter) P s ( E ; A ) ≤ P s ( F ; A ) + Λ1 − s d ( E , F ) , (3.2)whenever E ∆ F ⊂⊂ B r ( x ) ⊂⊂ A , r < r . The use of perturbed minimality conditions such as (3.2)has been introduced in [Alm76] as a natural point of view for unifying regularity theorems. Forexample, as shown below, every isoperimetric cluster is (Λ , r )-minimizing in R n , but also everyminimizer in the nonlocal partitioning probleminf n P s ( E ; A ) + N X h =1 Z E ( h ) g h ( x ) dx : E ( h ) \ A = E ( h ) \ A h = 1 , ..., N o (where E is a given N -cluster with P s ( E ; A ) < ∞ and where { g h } Nh =1 ⊂ L ∞ ( A )) is (Λ , r )-minimizing in A ′ for every A ′ ⊂⊂ A (with Λ and r depending on the functions g h and on the distance between A ′ and A ). So minimizers in different variational problems satisfy analogous localalmost-minimality conditions, which in turn imply several basic regularity properties. Proposition 3.2. If E is an isoperimetric cluster in R n , then there exist constants Λ ≥ and r > (depending on E ) such that E is (Λ , r ) -minimizing in R n .Proof. Immediate from Proposition 2.6 and Lemma 2.5. (cid:3) As explained at the beginning of the section, we aim to prove the following result. Theorem 3.3. If E is a (Λ , r ) -minimizing cluster in R n , then there exists α ∈ (0 , such that Reg( E ) is a C ,α -hypersurface in R n which is relatively open in ∂ E . The next infiltration lemma (compare with [Mag12, Lemma 30.2]) is a key step in provingTheorem 3.3. Lemma 3.4. If E is a (Λ , r ) -minimizing N -cluster in R n , then there exist positive constants σ = σ ( n, s, N ) > , and r ≤ r (depending on n, s, Λ , r ) such that, if x ∈ R n , r < r , h = 0 , ..., N and |E ( h ) ∩ B r ( x ) | ≤ σ r n , then |E ( h ) ∩ B r/ ( x ) | = 0 . Proof. We directly assume that x = 0 and define an increasing function u : (0 , ∞ ) → (0 , ∞ ) by u ( r ) = | B r ∩ E ( h ) | r > , sot that u ′ ( r ) = H n − ( ∂B r ∩ E ( h )) for a.e. r > 0. For every r > i = 0 , ..., N , i = h , we considerthe cluster obtained by giving part of the h -th chamber, namely B r ∩ E ( h ), to the i -th chamber F r,i ( j ) = E ( h ) \ B r if j = h E ( i ) ∪ (cid:0) E ( h ) ∩ B r (cid:1) if j = i E ( j ) if j ∈ { , ..., N } \ { i, h } . Since E is (Λ , r )-minimizing in R n and since each F r,i is an admissible competitor in (3.2), wefind that for every r ≤ r , i = 0 , ..., N , i = h ,Λ1 − s u ( r ) ≥ P s ( E ) − P s ( F r,i ) = P s ( E ( i )) + P s ( E ( h )) − P s ( F r,i ( i )) − P s ( F r,i ( h )) . (3.3)To estimate the right-hand side in (3.3) we compute P s ( F r,i ( i )) − P s ( E ( i )) = I s (cid:0) E ( i ) ∪ (cid:0) E ( h ) ∩ B r (cid:1) , E ( i ) c ∩ (cid:0) E ( h ) ∩ B r (cid:1) c (cid:1) − I s ( E ( i ) , E ( i ) c )= I s (cid:0) E ( h ) ∩ B r , E ( i ) c ∩ (cid:0) E ( h ) ∩ B r (cid:1) c (cid:1) + I s (cid:0) E ( i ) , E ( i ) c ∩ (cid:0) E ( h ) ∩ B r (cid:1) c (cid:1) − I s ( E ( i ) , E ( i ) c )= I s (cid:0) E ( h ) ∩ B r , E ( i ) c ∩ (cid:0) E ( h ) ∩ B r (cid:1) c (cid:1) − I s (cid:0) E ( i ) , E ( h ) ∩ B r (cid:1) (3.4)and P s ( F r,i ( h )) − P s ( E ( h )) = I s (cid:0) E ( h ) ∩ B cr , E ( h ) c ∪ B r (cid:1) − I s ( E ( h ) , E ( h ) c )= I s (cid:0) E ( h ) ∩ B cr , E ( h ) c ∪ B r (cid:1) − I s ( E ( h ) ∩ B cr , E ( h ) c ) − I s ( E ( h ) ∩ B r , E ( h ) c )= I s (cid:0) E ( h ) ∩ B cr , E ( h ) ∩ B r (cid:1) − I s ( E ( h ) ∩ B r , E ( h ) c ) (3.5)We notice that I s ( A, B ) − I s ( A, C ) = I s ( A, B \ C ) − I s ( A, C \ B ) (3.6) for every triple of measurable sets A, B, C ⊆ R d . Hence the difference between the first term inthe right-hand side of (3.4) and the second term in the right-hand side of (3.5) equals I s (cid:0) E ( h ) ∩ B r , E ( i ) c ∩ (cid:0) E ( h ) ∩ B r (cid:1) c (cid:1) − I s ( E ( h ) ∩ B r , E ( h ) c ) = I s (cid:0) E ( h ) ∩ B r , E ( h ) ∩ B cr (cid:1) − I s (cid:0) E ( i ) , E ( h ) ∩ B r (cid:1) . We add the previous equations (3.4) and (3.5), plugging them into (3.3), and then we apply thelast equality to find that, for every r > i = 0 , ..., N , i = h ,Λ1 − s u ( r ) ≥ I s (cid:0) E ( h ) ∩ B r , E ( i ) c ∩ (cid:0) E ( h ) ∩ B r (cid:1) c (cid:1) − I s (cid:0) E ( i ) , E ( h ) ∩ B r (cid:1) + I s (cid:0) E ( h ) ∩ B cr , E ( h ) ∩ B r (cid:1) − I s ( E ( h ) ∩ B r , E ( h ) c )= 2 (cid:16) I s ( E ( h ) ∩ B r , E ( i )) − I s ( E ( h ) ∩ B r , E ( h ) ∩ B cr ) (cid:17) . Averaging over i = h we obtain thatΛ1 − s u ( r ) ≥ N X i = h I s ( E ( h ) ∩ B r , E ( i )) − I s ( E ( h ) ∩ B r , E ( h ) ∩ B cr )= 2 N I s ( E ( h ) ∩ B r , E ( h ) c ∪ B cr ) − (cid:16) N (cid:17) I s ( E ( h ) ∩ B r , E ( h ) ∩ B cr ) (3.7)where the last equality follows from the fact that E ( h ) c ∪ B cr = (cid:0) E ( h ) ∩ B cr (cid:1) ∪ [ i = h E ( i ) . By the isoperimetric inequality, we have that I s ( E ( h ) ∩ B r , E ( h ) c ∪ B cr ) = P s ( E ( h ) ∩ B r ) ≥ P s ( B ) | B | ( n − s ) /n u ( r ) ( n − s ) /n By the coarea formula and the fact that u ′ ( t ) = H n − ( E ∩ ∂B t ), we find I s ( E ( h ) ∩ B r , E ( h ) ∩ B cr ) ≤ Z E ( h ) ∩ B r dx Z B ( x,r −| x | ) c dy | x − y | n + s = P ( B ) s Z E ( h ) ∩ B r dx ( r − | x | ) s = P ( B ) s Z r u ′ ( t )( r − t ) s dt. Hence, from (3.7) we deduce that2 P s ( B ) N | B | ( n − s ) /n u ( r ) ( n − s ) /n ≤ (cid:16) N (cid:17) P ( B ) s Z r u ′ ( t )( r − t ) s dt + Λ1 − s u ( r ) . (3.8)Setting r = min n r , (cid:16) (1 − s ) P s ( B ) N Λ | B | (cid:17) /s o , c = (cid:16) s N + 1) | B | n/s (1 − s ) P s ( B ) P ( B ) (cid:17) n/s , we find that for every r ≤ r Λ1 − s u ( r ) ≤ Λ1 − s u ( r ) ( n − s ) /n u ( r ) s/n ≤ Λ1 − s u ( r ) ( n − s ) /n | B | s/n r s ≤ P s ( B ) N | B | ( n − s ) /n u ( r ) ( n − s ) /n . Therefore, (3.8) implies that u ( r ) ( n − s ) /n ≤ N + 1) P ( B ) | B | ( n − s ) /n sP s ( B ) Z r u ′ ( t )( r − t ) s dt. By a De Giorgi-type iteration lemma (see [FFM + 15, Lemma 3.2 and Proof of Lemma 3.1]) thisimplies that u ( r ) > c | B | r n for every r ≤ r and concludes the proof of the lemma. (cid:3) Corollary 3.5. If E is a (Λ , r ) -minimizing cluster in R n , then there exist positive constants σ = σ ( n, s, N ) and r ≤ r (depending on n, s, Λ , r ) such that, if x ∈ R n , r < r , S ⊆ { , ..., N } and |E ( h ) ∩ B r ( x ) | ≤ σ r n for every h ∈ S, then |E ( h ) ∩ B − N r ( x ) | = 0 for every h ∈ S. Proof. Take the new σ to be the one given by the previous lemma divided by 2 nN . Then we canapply the Lemma 3.4 iteratively to deduce that k chambers in S are not present in B − k r ( x ). (cid:3) Corollary 3.6. If E is a (Λ , r ) -minimizing cluster in R n , then there exist positive constants r and C (depending on n , s , Λ and r ) and c , c ∈ (0 , (depending on n only), such that for every r < r , x ∈ R n and h = 0 , ..., N one has N X h =0 P s ( E ( h ) ∩ B r ( x )) ≤ C r n − s , (3.9) c ω n r n ≤ |E ( h ) ∩ B r ( x ) | ≤ c ω n r n . (3.10) Proof. Clearly (3.10) follows from Lemma 3.4, so we focus on (3.9). Comparing E to the clusterwhich is obtained from E by giving B r ( x ) to the exterior chamber in the (Λ , r )-minimality, wehave that P s ( E (0)) − P s ( E (0) ∪ B r ( x )) + N X h =1 (cid:0) P s ( E ( h )) − P s ( E ( h ) \ B r ( x )) (cid:1) ≤ Λ1 − s d ( E , F ) , Since for every measurable sets E, F we have that P s ( E ∩ F ) + P s ( E \ F ) ≤ I s ( E ∩ F, E c ) + I s ( E ∩ F, F c ) + I s ( E ∩ F c , E c ) + I s ( E ∩ F c , F ) ≤ P s ( E ) + 2 P s ( F ) (3.11)and similarly P s ( E ∩ F ) + P s ( E ∪ F ) ≤ P s ( E ) + P s ( F ) , (3.12)applying (3.11) to each chamber E = E ( h ) with F = B r ( x ) and applying (3.12) to E = E (0) with F = B r ( x ) we deduce that N X h =0 P s ( E ( h ) ∩ B r ( x )) ≤ (2 N + 1) P s ( B r ( x )) + Λ1 − s d ( E , F ) , Since by scaling P s ( B r ) = P s ( B ) r n − s and d ( E , F ) ≤ ω n r n ≤ ω n r s r n − s , we have proved (3.9). (cid:3) Proof of Theorem 3.3. Step one : We show that if x ∈ Reg( E ), then there exist h = 0 , ..., N and s x > x ∈ ∂ E ( h ) and E ( h ) is (Λ ′ , s x )-minimizing in B s x ( x ). Indeed, by definition ofReg( E ), if x ∈ Reg( E ), then lim r → + |E ( h ) ∩ B r ( x ) || B r ( x ) | + |E ( k ) ∩ B r ( x ) || B r ( x ) | = 1for some h, k ∈ { , ..., N } , h = k . Thus, by Corollary 3.5, there exists r x > E ( j ) ∩ B r x ( x ) = ∅ if j = h, k . We now claim that if s x = min { r x , r } / 2, then there exists Λ ′ ≥ Λ suchthat P s ( E ( h ); B s x ( x )) ≤ P s ( F ; B s x ( x )) + Λ ′ − s | F ∆ E ( h ) | (3.13) whenever F ∆ E ( h ) ⊂⊂ B s x ( x ). Indeed, given such a set F , let F be the cluster defined by F ( h ) = F , F ( k ) = ( E ( k ) ∩ B s x ( x ) c ) ∪ ( F c ∩ B s x ( x )) , F ( j ) = E ( j ) ∀ j = h, k . (3.14)Since E ∆ F ⊂⊂ B s x ( x ) and s x < r we have P s ( E ) ≤ P s ( F ) + (1 − s ) − Λ d ( E , F ), which in turngives P s ( E ( h )) + P s ( E ( k )) ≤ P s ( F ( h )) + P s ( F ( k )) + 2Λ1 − s (cid:16) |E ( h )∆ F ( h ) | + |E ( k )∆ F ( k ) | (cid:17) . We want to rewrite this condition in terms of E ( h ) and F ( h ) only: to this end, we set R = E (0) ∪ E (3) ∪ ... ∪ E ( N ), and since E ( h ) = ( F ( h ) ∪ R ) c we thus find0 ≤ P s ( F ( h )) + P s ( F ( h ) ∪ R ) − P s ( E ( h )) − P s ( E ( h ) ∪ R ) + 4Λ1 − s |E ( h )∆ F ( h ) | . (3.15)We have that P s ( F ( h ) ∪ R ) = I s ( F ( h ) , F ( h ) c ∩ R c ) + I s ( R, F ( h ) c ∩ R c ) = P s ( F ( h )) − I s ( F ( h ) , R ) + P s ( R )and similarly P s ( E ( h ) ∪ R ) = I s ( E ( h ) , E ( h ) c ∩ R c ) + I s ( R, E ( h ) c ∩ R c ) = P s ( E ( h )) − I s ( E ( h ) , R ) + P s ( R ) . Plugging the last two equations in (3.15) and dividing by 2 we obtain0 ≤ P s ( F ( h )) − P s ( E ( h )) + I s ( E ( h ) , R ) − I s ( F ( h ) , R ) + 2Λ1 − s |E ( h )∆ F ( h ) | . Moreover, since R and F ( h ) \ E ( h ) are at distance r / 2, by (3.6) I s ( E ( h ) , R ) − I s ( F ( h ) , R ) = I s ( E ( h ) \ F ( h ) , R ) − I s ( F ( h ) \ E ( h ) , R ) ≤ I s ( E ( h ) \ F ( h ) , R ) ≤ Z E ( h ) \F ( h ) Z | x − y | >r / dx | x − y | n + s dy = P ( B ) |E ( h ) \ F ( h ) | Z ∞ r / drr s ≤ s P ( B ) sr s |E ( h ) \ F ( h ) | . (3.16)Hence we are left with0 ≤ P s ( F ( h )) − P s ( E ( h )) + 2Λ1 − s |E ( h )∆ F ( h ) | + 2 s P ( B ) sr s |E ( h ) \ F ( h ) | , which in turn proves the (Λ ′ , s x )-minimality of E ( h ) in B s x ( x ) with Λ ′ = 2Λ + (1 − s ) P ( B ) /sr s . Step two : Let x ∈ Reg( E ) and let s x and h as in step one. By (3.1) there exists an half-space H such that E ( h ) x,r → H as r → + . By (3.10), given δ > s x depending on δ , we may entail that B s x ( x ) ∩ ∂ E ( h ) ⊂ (cid:8) y ∈ R n : dist( y, ∂H ) < δ (cid:9) . By the main result in [CG10] (see [CRS10] for the case Λ ′ = 0), if we take a suitable value of δ (depending on n , s and Λ ′ ), then (3.13) implies that B s x / ( x ) ∩ E ( h ) is contained in the epigraphof a C ,α function defined of ( n − B s x / ( x ) ∩ ∂ E ( h ) ⊂ Reg( E ), that B s x / ( x ) ∩ ∂ E ( h ) = B s x / ( x ) ∩ ∂ E , and that B s x / ( x ) ∩ ∂ E ( h ) is a C ,α -hypersurface. The theoremis proved. (cid:3) Blow-ups and monotonicity formula. We now come to the problem of addressing the sizeof the singular set Σ( E ) = ∂ E \ Reg( E ), consisting of those x ∈ ∂ E such that E x,r do not convergeto a pair complementary half-spaces as r → + . The first step in this direction is showing thatsequential blow-ups of minimizing clusters are conical (and still minimizing). In order to state theresult, we introduce the following terminology.A conical M -cluster K in R n is a M -cluster with the property that r K ( i ) = K ( i ) for every r > i = 1 , ..., M . We notice that, for conical clusters, being (Λ , r )-minimizing for someΛ , r > , ∞ )-minimizing. We thus simply speak of minimizing conicalclusters . Finally, for any open set A and for any pair of sets E, F ⊆ R n , we define the Hausdorffdistance between E and F relative to A ashd A ( E, F ) = inf { ε > E ∩ A ⊆ F ε and F ∩ A ⊆ E ε } , where E ε denotes the ε -enlargement of a set E ⊆ R n . We aim to prove the following theorem. Theorem 3.7. If E is a (Λ , r ) -minimizing N -cluster in R n , x ∈ ∂ E , and r j → + , then thereexist a conical minimizing M -cluster K (with M ≤ N ) and an injective function σ : { , ..., M } →{ , ..., N } such that, up to extracting subsequences E x,r j ( σ ( i )) → K ( i ) in L ( R n ) ∀ i = 0 , ..., M (3.17)hd B R ( ∂ E x,r j ( σ ( h )) , ∂ K ( h )) → ∀ R > , i = 0 , ..., M , (3.18) as j → ∞ . As usual, the key ingredient in proving Theorem 3.7 is obtaining a monotonicity formula.Following [CRS10], this is obtained at the level of a degenerate Dirichlet energy associated to anextension problem, see Theorem 3.10 below. The argument follow very closely [CRS10], so we limitourselves to a quick review.We start by introducing the extension problem and the Dirichlet form. Let a = 1 − s andembed R n into { z ≥ } = { X = ( x, z ) ∈ R n +1 : z ≥ } . We set U R = { X ∈ R n +1 : | X | < R } , A + = A ∩ { z > } , A = A ∩ { z = 0 } , ∀ A ⊂ R n +1 . Given a measurable set E we define u E : { z ≥ } → R by solving ( div( z a ∇ u E ) = 0 in { z ≥ } u E = 1 E − E c on { z = 0 } . Notice that u E is obtained by convolution with the Poisson kernel, u E ( · , z ) = P ( · , z ) ∗ (1 E − E c ) , P ( x, z ) = c ( n, a ) z − a ( | x | + z ) n +1 − a . If E ⊂ R n is such that P s ( E ) < ∞ , then there exists a unique minimizer u E ininf n I s ( u ) = Z { z> } z a |∇ u | : tr ( u ) = 1 E − E c o , where tr denotes the trace operator from T R> W , ( B R × (0 , R )) to L loc ( R n ), and one has I s ( u E ) = c P s ( E ) , for some c = c ( n, s ). The following lemma relates minimality for the nonlocal perimeter to mini-mality for the degenerate Dirichlet energy. Lemma 3.8 (Lemma 7.2 in [CRS10]) . There exists a constant c depending only on n and s ,and having the following property. If E, F ⊂ R n are such that P s ( E ; B ) , P s ( F, B ) < ∞ and E ∆ F ⊂⊂ B , then inf Ω ,v Z Ω + z a ( |∇ v | − |∇ u E | ) dz = c (cid:0) P s ( F ; B ) − P s ( E ; B ) (cid:1) (3.19) where Ω ⊂ R n +1 is any bounded Lipschitz domain with Ω ⊆ B an v ∈ W , (Ω) is such that spt( v − u E ) ⊂ Ω and tr v = 1 F − F c on Ω . Corollary 3.9. A cluster E is (0 , ∞ ) -minimizing in B if and only if for every N -cluster F with E ∆ F ⊂⊂ B the extensions u E ( h ) of E ( h ) satisfy N X h =0 Z Ω + z a |∇ u E ( h ) | dz ≤ N X h =0 Z Ω + z a |∇ v h | dz (3.20) for all bounded Lipschitz domains Ω ⊂ R n +1 with Ω ⊆ B and all functions v h such that spt( v h − u E ( h ) ) ⊂⊂ Ω and tr v h = 1 F ( h ) − F ( h ) c on Ω .Proof. Immediate from Lemma 3.8. (cid:3) We can now prove the following monotonicity formula. Theorem 3.10 (Monotonicity formula) . If E is a (Λ , r ) -minimizing cluster with ∈ ∂ E , thenthere exists Λ ′ ≥ (of the form Λ ′ = C ( n, s )Λ ) such that Φ E ( r ) + Λ ′ r s is increasing on (0 , r ) where we have set Φ E ( r ) = 1 r n − s N X h =0 Z U + r z a |∇ u E ( h ) | . (3.21) Moreover, if r = ∞ and Λ = 0 then Φ E is constant if and only if E ( h ) is a cone with vertex at for every h = 0 , ..., N .Proof. The proof is again a simple adaptation of the argument in [CRS10]. Given λ ∈ (0 , 1) and r > v h = u E ( h ) , v λh ( X ) = v h ( X/λ ) , if | X | < λ r ,v h ( rX/ | X | ) , if λ r < | X | < r ,v h ( X ) , if | X | > r , In this way Z z> z a |∇ v λh | − Z z> z a |∇ v h | = ( λ n − s − Z U + r |∇ v h | + Z rλr ds Z ( ∂U s ) + z a |∇ τ v λh | , where ∇ τ v ( X ) = ∇ v ( X ) − | X | − ( ∇ v ( X ) · X ) X . We now notice thattr v λh = 1 F λ ( h ) − F λ ( h ) c , where F λ ( h ) \ B r = E ( h ) \ B r , F λ ( h ) ∩ B λr = λ E ( h ) ∩ B λr , F λ ( h ) ∩ ( B r \ B λr ) = ( R + ( E ( h ) ∩ ∂B r )) ∩ ( B r \ B λr ) . Since F λ ∆ E ⊂ B r if r < r then we find P s ( E ) ≤ P s ( F λ ) + Λ1 − s d ( E , F λ ) , which by (3.19) takes the form N X h =0 Z U + r z |∇ v h | ≤ Z U r + z |∇ v λh | + 2Λ(1 − s ) c | m ( E ) − m ( F λ ) | , that is(1 − λ n − s ) N X h =0 Z U + r |∇ v h | ≤ Z rλr ds N X h =0 Z ( ∂U s ) + z a |∇ λτ v h | + 2Λ(1 − s ) c | m ( E ) − m ( F λ ) | . Now | m ( E ) − m ( F λ ) | = N X h =1 ||E ( h ) | − |F λ ( h ) || ≤ (1 − λ n ) N X h =1 |E ( h ) ∩ B r | + | B r \ B rλ | ≤ C r n (1 − λ n ) . In this way since λ ∈ (0 , − λ n − s − λ N X h =0 Z U + r |∇ v h | ≤ − λ Z rλr ds N X h =0 Z ( ∂U s ) + z a |∇ τ v h | + C ( n, s ) Λ r n − λ n − λ . We notice that |∇ v λh ( X ) | = |∇ v h ( rX/ | X | ) | for every X with λ r < | X | < r and we let λ → − tofind, ( n − s ) N X h =0 Z U + r |∇ v h | ≤ r N X h =0 Z ( ∂U r ) + z a |∇ τ v h | + C ( n, s ) Λ r n . Therefore r n − s ) Φ ′E ( r ) = r n − s − (cid:26) r N X h =0 Z ( ∂U r ) + z a |∇ v h | − ( n − s ) N X h =0 Z U + r |∇ v h | (cid:27) ≥ r n − s − (cid:26) r N X h =0 Z ( ∂U r ) + z a |∇ v h · ˆ X | − C ( n, s )Λ r n (cid:27) , where ˆ X = X/ | X | . Rearranging terms we find (cid:16) Φ E ( r ) + C ( n, s )Λ s r s (cid:17) ′ ≥ r n − s N X h =0 Z ( ∂U r ) + z a |∇ v h · ˆ X | , for every r < r . This proves that Φ E ( r ) + Λ r s is increasing on (0 , r ). Assume now that r = ∞ and Λ = 0. In this case Φ E is increasing on (0 , ∞ ), and Φ E is constant on (0 , ∞ ) if and only if ∇ v h is homogeneous of degree 0 for every h = 0 , ..., N , that is if and only if E ( h ) is a cone with vertexat the origin for every h = 0 , ..., N . (cid:3) Proof of Theorem 3.7. Without loss of generality let us assume that x = 0, so that E ,r ( h ) = r − E ( h ). By the upper perimeter estimate of Lemma 3.6, for every R, r > h = 0 , ..., N wehave P s (cid:0) ( r − E ( h )) ∩ B R (cid:1) = r s − n P s ( E ( h ) ∩ B R r ( x )) ≤ C R n − s . In particular for every h = 0 , ..., N there exists F ( h ) ⊂ R n such that, up to extracting subsequences, r − j E ( h ) → F ( h ) in L ( R n ) as j → ∞ . Define M ≤ N so that there are exactly M + 1 indexes h = 0 , ..., N such that |F ( h ) | > 0. Then we can find a injective function σ : { , ..., M } → { , ..., N } such that, setting K ( i ) = F ( σ ( i )), we have r − j E ( σ ( i )) → K ( i ) in L ( R n ) as j → ∞ . This proves(3.17), which in turn implies (3.18) thanks to the volume density estimates in Lemma 3.6. Since r j E is (Λ r j , r j /r )-minimizing in R n , by a simple variant of [CRS10, Theorem 3.3] we see that K is (0 , ∞ )-minimizing in R n . Moreover, by scalingΦ E ( r j r ) = Φ r − j E ( r ) ∀ r > j →∞ Φ r − j E ( r ) = Φ E (0 + ) , ∀ r > . At the same time, by arguing as in [CRS10, Proposition 9.1], we getlim j →∞ Φ r − j E ( r ) = Φ K ( r ) , ∀ r > . In conclusion, Φ K ( r ) is constant over r > 0, and since K is (0 , ∞ )-minimizing in R n we can exploitTheorem 3.10 to deduce that K is conical. (cid:3) We conclude this section with a last result that can be proved with the aid of the extensionproblem and that it is useful in the dimension reduction argument (see next section). Proposition 3.11. A cluster E is (0 , ∞ ) -minimizing in R n if and only if E× R is (0 , ∞ ) -minimizingin R n +1 . Here, by definition, ( E × R )( h ) = E ( h ) × R for every h = 1 , ..., N .Proof. This is an immediate adaptation of [CRS10, Theorem 10.1]. (cid:3) Dimension reduction argument. Given Theorem 3.7 and Proposition 3.11 we can exploitthe standard dimension reduction argument of Federer to give estimates on the Hausdorff dimensionof Σ( E ). Theorem 3.12 (Dimension reduction) . If K is a minimizing conical M -cluster in R n , x = e n ∈ ∂ K and λ k → ∞ as k → ∞ , then there exists a minimizing conical cluster K ′ in R n − such that,up to extracting subsequences, λ k ( K − x ) → K ′ × R in L ( R n ) as k → ∞ .Proof. By Theorem 3.7 there exists a conical minimizing M -cluster K such that, up to extractingsubsequences, λ k ( K − e n ) → K in L ( R n ). We want to prove that K = K ′ × R for some conicalcluster K ′ in R n − , and the fact that K ′ is minimizing will then follow by Proposition 3.11. Since ∂ K is a closed set of measure 0 thanks to the density estimates, it is enough to prove that theinterior of each chamber is constant in the x n -direction, namely that for every chamber K ( h ) andfor every ball B ε ( x ) ⊆ K ( h ) we have B ε ( x ) + R e n ⊆ K ( h ) . (3.22)To prove this claim, we notice that the cone with vertex in − λ k e n generated by B ε ( x ) convergeslocally to B ε ( x ) + R e n . Moreover, setting K k = λ k ( K − e n ), we have that B ε ( x ) ∩ K k ( h ) convergesto B ε ( x ) ∩ K ( h ) = B ε ( x ). As a consequence, the difference between the indicator of the cones withvertex in − λ k e n generated by B ε ( x ), and by B ε ( x ) ∩ K k ( h ) respectively, converges in L ( R n )to 0. Putting together these facts, we deduce that the cone with vertex in − λ k e n generated by B ε ( x ) ∩ K k ( h ) (which is contained in K k ( h ) because by assumption K k ( h ) is a cone with vertex − λ k e n ) converges in L ( R n ) to B ε ( x ) + R e n . By the convergence of K k ( h ) to K ( h ), we find that(3.22) holds. (cid:3) Theorem 3.13 (Dimension of the singular set) . If E is a (Λ , r ) -minimizing N -cluster in R n , thenthe singular set Σ( E ) is a closed set of Hausdorff dimension at most n − , that is, H ℓ (Σ( E )) = 0 ∀ ℓ > n − . As a consequence, ∂ E has Hausdorff dimension n − , namely H ℓ ( ∂ E ) = 0 ∀ ℓ > n − . Proof. From Theorem 3.12 and Proposition 3.11 it follows that the singular set of any minimizingcluster E has Hausdorff dimension n − 2. This is a classical argument, which can be repeated verbatim from [CRS10, Proof of Theorem 10.4]: first, one proves that H ℓ (Σ( E )) = 0 for any ℓ suchthat H ℓ (Σ( K )) = 0 for every conical minimizing cluster K ; next, one shows that H ℓ (Σ( K )) = 0for every conical minimizing cluster K ⊆ R n , then H ℓ +1 (Σ ˜ K ) = 0 for every conical minimizingcluster ˜ K ⊆ R n +1 . In proving both claims one uses a compactness argument to say that for every x ∈ Σ( E ) there exists δ ( x ) > δ ≤ δ ( x ) and any set D ⊆ Σ( E ) ∩ B δ ( x ) thereexists a covering of D with balls B r i ( x i ) such that x i ∈ D and P r ℓi ≤ δ ℓ / 2. Finally, since ∂ E is a C -hypersurface in a neighborhood of each x ∈ Reg( E ), we conclude that ∂ E has Hausdorffdimension n − (cid:3) In the planar case n = 2 we can say more by exploiting the fact, proved in [SV13], that everyconical minimizing 2-cluster in R is given by two complementary half-spaces. By definition ofReg( E ), this fact implies that if x ∈ Σ( E ) for a (Λ , r )-minimizing cluster in R and K is a conicalminimizing M -cluster arising as a blow-up limit of E at x , then M ≥ K has at leastthree non-trivial conical sectors). With this remark in mind we can prove the following fact. Proposition 3.14. The singular set Σ( E ) of a (Λ , r ) -minimizing cluster E in R is locally discrete.Proof. Assume by contradiction that there exists a sequence { x k } k ∈ N ⊆ Σ( E ) such that x k con-verges to x ∈ Σ( E ) as k → ∞ . Set λ k = | x k − x | − E k = E x ,λ − k , and assume up to rotations that x k − x | x k − x | = v ∈ S ∀ k ∈ N . In this way, E k is (Λ /λ k , r λ k )-minimizing in R with 0 , v ∈ Σ( E k ) for every k ∈ N . By Theo-rem 3.7, up to extracting subsequences, λ k ( E − x ) → K in L ( R ) with 0 , v ∈ Σ( K ). The factthat v ∈ Σ( K ) is based on the fact that, as notice above, by [SV13] x k ∈ Σ( E k ) implies (up toextracting a subsequence in k and up to reordering the chambers of E ) that |E k ∩ B r ( v ) | > h = 1 , , r > k ∈ N . Moreover, by the density estimates of Lemma 3.6 (notethat they are uniform with respect to k , since they are applied to a blow-up of a single cluster andso they hold at every scale less than r ( E ) as k increases) |E k ∩ B r ( v ) | ≥ cr n for every h = 1 , , 3, for every r and for every k large enough (depending on r ). Thus thereare at least three chambers of K which have positive volume nearby v , so that v Reg( K ). ByTheorem 3.12 any blow-up of K at v has the form K ′ × R for some conical cluster K ′ in R . 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Institute for Theoretical Studies, ETH Z¨urich, Clausiusstrasse 47, CH-8092 Z¨urich, Switzerland,Institut f¨ur Mathematik, Universitaet Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland E-mail address : [email protected] Department of Mathematics, The University of Texas at Austin, 2515 Speedway Stop C1200, Austin,Texas 78712-1202, USA E-mail address ::