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An isoperimetric problem with a competing nonlocal singular term

In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular we prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either split into smaller sets that drift to infinity or develop fingers of a prescribed width. Finally, we connect these two alternatives to a related minimization problem for the optimal constant in a classical interpolation inequality (a Gagliardo-Nirenberg type inequality for fractional perimeter).


Setting of the problem
Given a monotone decreasing radially symmetric kernel K(x) satisfying 0 ≤ K(x) ≤ 1 |x| n+s ∀x ∈ R n , for some s ∈ (0, 1) (1.1) we consider the volume constrained minimization of the following energy functional: where χ E is the characteristic function of the set E ⊂ R n and α is a positive parameter. This functional involves the classical perimeter P (E) =ˆR n |Dχ E | := sup ˆR n χ E div g dx ; g ∈ [C 1 c (R n )] n , |g| ≤ 1 and the nonlocal perimeter is the usual s-perimeter (up to a constant), which has received a lot of attention [3,4,11,1,20] and appears in many applications. The functional (1.2) models phenomena in which the classical perimeter competes with a nonlocal interaction term. While the perimeter tries to aggregate the set E into a ball, the nonlocal term has the opposite effect. This competition leads to a non trivial problem in which even the existence of minimizers is not obvious. This problem is reminiscent of several recent works, including many results on the classical Gamow's liquid drop model of an atomic nucleus [17,5,18,16,15]. More recently, some results were obtained in [8] in a framework similar to ours, when P K = P s . We will see that for small volume (or small α), the perimeter dominates the nonlocal effect and that the minimizers exist and are balls (see Theorems 2.1 and 2.2), while for larger volume, the ball cannot be the global minimizer (see Theorem 2.3). Our results will be uniform in K under assumption (1.1) and consistent with the natural scaling of the problem when P K = P s . This is important for the application we have in mind, which is discussed in Section 1.2 below, in which the kernel K itself depends on a small parameter ε.
For large volume, the situation is less clear, and we show that two things might happen: Either the minimizers still exists but are no longer balls. Instead the set E develops finger of width of the order of α − 1 1−s . Or the minimizer no longer exists because the minimizing sequence splits into smaller sets. Indeed, we recall the following important properties of nonlocal perimeters: For disjoint sets A, B, we have P K (A ∪ B) = P K (A) + P K (B) − 4ˆR While we do not determine whether minimizers exist (and form fingers) or not (and split) for large volume, we show (see Proposition 2.7) that when P K = P s this question is directly related to the optimal constant in the following classical interpolation inequality for sets of finite perimeter, which plays an important role in our paper: We denote by µ n,s ≤ 2 1−s nω n the optimal constant for a given s, that is . (1.5) The proof of this proposition is classical and presented in Appendix A for the sake of completeness. To the authors' knowledge, it is an open question whether the optimal constant in (1.4) is reached (i.e. there is equality for a set of finite perimeter) or not (minimizing sequences for (1.5) have unbounded perimeters).
Finally, note that we include the constant s(1 − s) in front of P K in (1.2) so that we can easily make sense of our result when s → 0 + and s → 1 − . Indeed, it is known (see for instance [9,1]) that In Section 1.2 below, we will briefly motivate the problem by relating it to a more classical minimization problem. Our main results are presented and discussed in Section 2 and the rest of the paper is devoted to their proofs.

Motivation and related functional
To motivate this work, we consider the functional where V K is a nonlocal repulsive interaction energy defined by This energy functional has been intensively studied with the kernel K(x) = 1 |x| n−a for some a ∈ (0, n). The case a = n − 1 and n = 3 in particular corresponds to the classical Gamow's liquid drop model of an atomic nucleus [17,5,18,16,15]. We note that the two terms in J have opposite effects: The perimeter tries to keep the mass together in a ball, while the repulsion potential V K has the tendency to spread the mass around. In [10], a detailed analysis of the related functional P s + αV K is done.
The motivation for our study is very different and stems from some model for cell motility recently introduced in [6]. In that framework, the kernel K depends on a small parameter ε ≪ 1 and is the solution of K ǫ + ε s (−∆) s/2 K ε = δ(x).
The functional V K is related to nonlocal perimeters in the following way: (1.8) Since we are interested in the minimization of J under the volume constraint |E| = m, it is equivalent to the minimization ofJ (E) = P (E) − β σ P Kε (E) (1.9) with the same constraint. Finally, when s ∈ (0, 1), the interesting regime corresponds to ε ≪ 1 and β σ ∼ ε −s (this is the regime in which the two terms inJ have the same order in ε). If we set α = βk0 σs(1−s) ε s , we are led to investigate the functional Importantly, the kernelK ε satisfy the upper bound (1.1) uniformly with respect to ε (up to a constant), with equality in the limit ε → 0. The results derived in this paper therefore apply to the functionalJ α,ε uniformly in ε. Remark 1.3. We note that we only work with s ∈ (0, 1) in this paper, even though Lemma 1.2 holds for s ∈ (0, 2). Indeed, when s > 1, the interesting regime corresponds to ε ≪ 1 and β σ ∼ ε −1 : If we set α = β σ ε, we get the functional and a classical result (see [2,7]) implies that for some constant σ 0 . So the analysis in that case is somewhat different and in the limit, the behavior depends on whether ασ 0 < 1 (in which case the ball is the unique minimizer) or ασ 0 > 1 (in which case minimizers do not exist).
The Γ-convergence of PK ε to P s when s ≤ 1 is the object of a forthcoming paper.

Main results
The paper is thus devoted to the following minimization problem in R n (expect for Theorem 2.2, all of our results hold for any n ≥ 2): This problem has two parameters, α and m. When we have equality in (1.1), that is when P K = P s , these parameters are related by a natural scaling. Indeed, if we define then E is a minimizer of F s,α with |E| = m if and only ifẼ = 1 Even though this is no longer valid when we only have the inequality (1.1), our results below will be consistent with this scaling.
Next, we note that the existence of minimizing sequence, with bounded perimeter, is easy to establish. Indeed, using (1.4) and Young's inequality, we find for all s ∈ (0, 1)). In particular, we have which guarantees the existence of a minimizing sequence {F k } k∈N with volume constraint |F k | = m and satisfying P (F k ) ≤ C. The difficulty in proving the existence of a minimizer for (2.1) is that the sets F k might not be bounded in R n . In particular, F k might split into two (or more) connected components moving away from each others (since translating the components away from each others does not change the perimeter but increases the nonlocal perimeter, and thus decreases the functional J α ). Our first result shows that this does not happen when α (or equivalently m) is small enough. More precisely, we show the existence of a bounded minimizer for J α when the α m 1−s n is small enough (which is consistent with the scaling (2.2)): We note that a similar result is proved in [8] for the functional F s,α (that is when P K = P s ).
When α = 0, the only minimizer of J 0 = P is the ball, and we next show that when α m 1−s n is small enough, the unique bounded minimizer of (2.1) is also the ball of volume m (again, we note that a similar result is proved in [8] for the functional F s,α ): Theorem 2.2. Assume 2 ≤ n ≤ 7 and let K(x) be a radially symmetric, non-increasing function satisfying (1.1) for some s ∈ (0, 1). There exists γ 1 > 0 depending on n and s such that if is large? and if so, is this minimizer still a ball? We start with the second part of this question, which is much simpler to answer. First we prove that when α m 1−s n is large, the global minimizer (if it exists) cannot be the ball, and in fact that this global minimizer must (in some sense) develop fingers. We note that such a result cannot hold in the general framework considered so far, since we can take K = 0 (in which case the ball is the unique minimizer for all α). We thus assume that K satisfy the lower bound: for some c ≥ 0.
Theorem 2.3. Let K(x) be a radially symmetric, non-increasing function satisfying (1.1) and (2.5) for some s ∈ (0, 1), and let E be a minimizer of (2.1). There exists a constant c > 0 depending only on n such that if In particular there exists γ 2 > 0 depending only on n, s such that if α m 1−s n ≥ max{γ 2 , cαρ 1−s 0 } then the ball of volume m cannot be a global minimizer of (2.1).
The first part of the theorem says that the largest ball that can fit inside the set E has radius c max{ρ 0 , α −1 1−s } for some c depending only on n and s (note that when P K = P s , we can take ρ 0 = 0). This shows that when m increases with α fixed, minimizers, if they exists, have a maximum width determined by ρ 0 and α. Note that the non-degeneracy estimate (4.1) suggests that this is optimal and that the fingers have width comparable to α −1 From the point of view of many applications (in particular the cell motility applications discussed in this paper), global minimizers of J α are not necessarily the most relevant objects one would like to study. In particular, the stationary solutions obtained by gradually increasing the value of α (or obtained as the long time limit for some related evolution problem) might be a stable critical point (or a local minimizer) but not a global energy minimizer. In the theorem below, we characterize the range of values of α m  The result implies that a bifurcation phenomena takes place when γ = γ * 2 . While we do not pursue this in this paper, it is indeed possible to show, by a Crandall-Rabinovitch type argument, that a branch of non-radially symmetric stationary solutions appear for γ = γ * 2 . This branch of solution has a lower energy than the ball and is thus important in application, even if the fact that γ * 2 > γ * 1 implies that this branch is not a global minimizer for γ close to γ * 2 (two balls of half the size far away from each other have a lower energy).
Finally, we come back to our first question: the existence (or non existence) of global minimizer when α m 1−s n is large. First, we recall that (2.4) implies that any minimizing sequence {F k } k∈N satisfies P (F k ) ≤ C for some constant C(m, α). In particular χ F k is bounded in BV (R n ) and the obstacle to the existence of a minimizer is the issue of decomposability: A set of finite perimeter F is decomposable if there exists a partition F = F 1 ∪ F 2 with |F i | = 0 and P (F ) = P (F 1 ) + P (F 2 ). Otherwise the set is indecomposable (which is the measure theoretic equivalent of the notion of a connected set). In Proposition 4.3 we will prove that any global minimizer of J α must be bounded and indecomposable since otherwise we can decrease the energy by moving two components away from each others (see (1.3)).
In many applications, splitting of the minimizing sequence into smaller sets can be an interesting feature and once the components are far enough from each others we can typically neglect the interactions and treat the resulting set as a minimizer of the problem. Mathematically, this can be done by assuming that K(x) = 0 for |x| large enough, or by replacing the original minimization problem with the following: Given a collection E = {E i } i∈I of sets of finite perimeter, let and |E| = i∈I |E i |. We then consider the minimization problem: For that problem we can prove that a minimizer exists for all m and all α. More precisely, we have the following result: Theorem 2.5. Let K(x) be a radially symmetric, non-increasing function satisfying (1.1). Then the minimization problem (2.6) has a global minimizer E = {E i } i∈I for all m > 0 and α ≥ 0. Furthermore, there exists C depending only on s and n such that any global minimizer E has at most Cα When P K = P s , we can prove the following proposition which relates our minimization problem to the interpolation inequality (1.4): Proposition 2.6. Let µ n,s ≥ 0 be the optimal constant in the interpolation equality (1.4), defined by .
Then for all α > 0, we have: The proof of this proposition makes use of sets that are almost minimizers for (2.7) and it allows us to prove the following result: Proposition 2.7.
(i) If there exists a set E 0 with |E 0 | = 1 and P (E 0 ) < ∞ for which there is equality in (2.7), then for all k ∈ N, the set with k components is a minimizer of G s,α with mass constraint This proposition suggests that the existence of bounded minimizer for F s,α is directly related to whether minimizing sequences (2.7) have bounded perimeter or not.
In particular, if equality is achieved in (2.7), then the first part of Proposition 2.7 suggests that for large enough m minimizing sequences for F s,α have several connected components. In that case global minimizers of F s,α do not exist, although from the point of view of application, since we can often neglect the interaction between different connected components if those are far enough from each others, it means that global minimizers are obtained by splitting the mass into smaller sets.
The question of whether equality can be achieved in (2.7) is, as far as the authors know, an open and interesting question.
Notation: Throughout the paper, we denote ω n = |B 1 | the volume of the unit ball and recall that nω n = P (B 1 ) and we denote ν n = nω 1/n n the isoperimetric constant which is such that Outline of the rest of the paper: The existence of bounded minimizers for small γ := αm 1−s n (Theorem 2.1) is proved in Section 3. In Section 4, we establish two key properties of these minimizers: a classical non-degeneracy estimate (for all γ) and the regularity of the minimizer (for small γ). These results are used to prove Theorem 2.2 (minimizers are balls for small γ) in Section 5. Section 6 is devoted to further investigation of the role of the ball when γ is large (Theorems 2.3 and 2.4). In Section 7, we prove the existence of a minimizer for all γ for the generalized minimization problem (2.6) (Theorem 2.5). Finally, Section 8 is devoted to Propositions 2.6 and 2.7 which relate our minimization problem to the interpolation inequality (2.7).

Proof of Theorem 2.1
We already explained in the introduction that inequality (2.4) implies the existence of a minimizing sequence F k such that |F k | = m and lim k→∞ J α (F k ) = inf{J α (F ) ; |F | = m}. Inequality (2.4) also implies that P (F k ) is bounded with respect to k. The key step in the proof of Theorem 2.1 is thus to show that we can always modify the sequence F k to construct a minimizing sequence which is bounded. This will follow from the following proposition: It is not difficult to prove that F k convergence strongly in L 1 (R n ) to E and Theorem 2.1 follows from the lower semi-continuity of the perimeter and the following classical lemma (whose proof is presented in Appendix B): Assume that χ FK converges strongly in L 1 (R n ) to χ E and that P (F K ) ≤ C. Then Before proving Proposition 3.1, we prove the following preliminary lemma: Then there exists a constant C depending only on n such that:

3)
and up to a translation In particular, if γ ≤ (nω n ) −1 , then for some constant C depending only on n.
Proof of Lemma 3.3. Using (3.1) and (2.4), we get: Next, using (1.4), (3.1) and (3.2) we can write: 3) follows. We now conclude by using the following quantitative isoperimetric inequality [13]: Proof of Proposition 3.1. Throughout the proof, we fix r 0 > 0 such that |B r0 | = m and we recall that γ := α m 1−s n . Since we want to prove the result for small enough γ, we will always assume that γ ≤ (nω n ) −1 . Given ρ ∈ (r 0 , 1), we define (we drop the dependence of F 1 and F 2 on ρ below) and define by A(ρ) the area of F ∩ ∂B ρ : We note that for a.e. ρ ∈ (r 0 , 2r 0 ), we have We now adapt an argument from [16]: The proof of Proposition 3.1 is divided in two cases, depending on whether Σ(ρ) and so (3.6) together with the isoperimetric inequality (2.8) imply for all ρ ∈ (r 0 , 1).
In particular, this implies that there exists when γ ≤ 1, and so U (r 0 ) ≤ c 1 r n 0 provided γ ≤ γ 0 small enough (depending only on n). In that case, we find U (2r 0 ) = 0, and so F ⊂ B 2r0 (up to a set of measure 0) and the Proposition follows with F = F .
We now denote Furthermore, the fact that χ F = χ F1 + χ F2 and the triangle inequality imply and so (using (2.3)) Finally, denoting m 2 = |F 2 | and using the isoperimetric inequality (2.8), we get provided nω n γ ≤ νn 8 1−s , which is satisfied for all s if γ ≤ γ 0 small enough (depending only on n).
So, the set F 1 has a lower energy than F 1 . However, we cannot take Note that we have in particular t = m m1 − 1 = m2 m1 . Inequality (3.5) implies m 2 ≤ m/2 ifγ 0 is small enough and so m 1 = m − m 2 ≥ m/2. In turn, we get (using (3.5) again) and we have the following lemma (proved below): For all set F 1 with finite perimeter and for all t ∈ [0, 1], there holds: for some constant C depending only on n.
Combining all of those inequalities yield we can write Using the bounds Using the fact that ρ = (1 + t) 1/n with 0 ≤ t ≤ 1, we deduce Optimizing with respect to R by taking R = 2|F1| P (F1) yields which proves the right hand side inequality of Lemma 3.4.

Non-degeneracy and regularity for the minimizer
In this section, we prove some classical properties of the minimizers of J α , namely non degeneracy and regularity. These properties play a key role in proof of Theorem 2.2. We start with the following non-degeneracy result: Let E be a minimizer of (2.1). There exists a constant c depending only on n such that for every x 0 ∈ ∂E and r > 0. In particular, we have: Note that the result also holds, for small enough r, if E is a local minimizer.
Proof. Given r > 0 we define the functions: where t is chosen such that |F | = |E| = m, that is: .
Givenr > 0, we assume that V (r) ≤ ε 0 min 1, 1 (nω n γ) n 1−s m for some ε 0 < 1/2 to be chosen later. Since r → V (r) is non-decreasing, we then have and so in particular We thus need to bound the last two term in the right hand side. First, using Lemma 3.4, Young's inequality and (4.3) we can write Next, we write (using the definition of P K and (2.3)): Going back to (4.4), we deduce: Finally, we have (using (3.2)): In order to conclude, we combine (4.6) with the isoperimetric inequality (2.8), which gives and the fact that V ′ (r) = A(r) for a.e. r > 0, to conclude: Choosing ε 0 small enough, we thus have Proceeding similarly, we can also prove the following non-degeneracy lemma for E c . Note that the result also holds, for small enough r, if E is a local minimizer.
Proof. Given r > 0, we define and define the set With Lemma 3.4 (for t ≤ 1) and λ(r) ≤ V (r) m we have Therefore, using the definitions of S(r), A(r) and V (r), Thus, Finally, we have (using (3.2)): The non-degeneracy estimate allows us to prove: Proposition 4.3. Let E be a minimizer of (2.1). Then E is bounded, indecomposable and its diameter satisfies Proof. Letr 0 = min{1, (nω n γ) − 1 1−s }m 1/n , then the non-degeneracy estimate (4.1) implies In particular we can cover E with at most N := C m r n 0 = C max{1, (nω n γ) 1 1−s } n ball Br 0 and E is thus bounded.
To prove that the minimizer E is indecomposable, suppose the opposite is true and that there exist two sets of finite perimeter E 1 and E 2 such that E 1 ∩E 2 = ∅ and E = E 1 ∪E 2 with P (E) = P (E 1 )+P (E 2 ). Since E is bounded when γ < γ 0 , we have E 1 and E 2 are bounded. Define E R = E 1 ∩ (E 2 + e 1 R). We have |E R | = m and P (E R ) = P (E) for R > 0 sufficiently large, and the nonlocal energy decreases which contradicts the minimizing property of E.
Finally, proceeding as in [16] Lemma 7.2 we can show that the diameter of E is bounded by CNr 0 which leads to (4.8).
Classically, the non-degeneracy results also imply the following bound, which will be useful in the proof of Theorem 4.5 below: Proof. Using (4.7), we obtain (using the fact that V (r) ≤ m and V (r) ≤ |B r |): P (E, B r (x)) = S(r) ≤ 3A(r) + (nω n α) Finally, we end this section with the following regularity result: Theorem 4.5. Let E be a minimizer of (2.1) with |E| = m and let r 0 be such that |B r0 | = m. Then the reduced boundary ∂ * E is a C 1, 1 2 (1−s) hypersurface in R n , and for all k > n − 8 (where H k denotes the k−Hausdorff measure).
Furthermore, when n ≤ 7 and α m 1−s n ≤ γ ′ 1 (small enough, but depending only on n and s), we can write for some function u ∈ C 1, 1 2 (1−s) (∂B 1 ) with the C 1, 1 2 (1−s) regularity constant depending on n and s only.
The first part of Theorem 4.5 follows from Theorem 1 in [19] if we prove that In order to prove (4.9), we first note that we have the following scaling property with x = r 0 x * . We then fix a set F ⊂ R n such that E∆F ⊂ B ρ (x). Without loss of generality, we may assume since otherwise (4.9) is trivial. We now set F ′ = tF with t chosen so that |F ′ | = |tF | = |E| = m. In particular, we have and since E is a minimizer of J α in E m , we can write: P (E) − P (tF ) ≤ αs(1 − s)(P K (E) − P K (tF )). and so For the first term, we use (4.11) and Corollary 4.4 to get for some constant C depending on n and s. The second term is bounded by Lemma 3.4 and (4.12):  We deduce: and since this holds for all F ⊂ R n such that E∆F ⊂ B ρ (x), we get The scaling property (4. which completes the proof of (4.9) and the first part of the theorem follows from Theorem 1 in [19].

Proof of Theorem 2.2
Throughout this section, we assume that E is a minimizer of J α with |E| = m and that γ = α m for some function u ∈ C 1, 1 2 (1−s) (∂B 1 ) with the C 1, 1 2 (1−s) regularity constant depending on n and s only (see Theorem 4.5).
If u(x 0 ) ≥ 0, then there exists a unit vector e and universal constant C > 0 such that in the set Then there exists universal constant ρ 0 > 0 such that when |x − x 0 | < ρ 0 and e · (x − x 0 ) ≥ 1 2 |x − x 0 |, If u(x 0 ) < 0, then there exists unit vector e, universal constants C, ρ 0 > 0 such that in the set Λ

In both cases, we can thus write
Cm αm which implies: and leads to a contradiction if γ < γ ′′ 1 small constant (depending only on n and s). This completes the proof of (5.3).
With Lemma 5.1, we can apply Theorem 2.1 in [12] and obtain Proposition 5.2. Let E be a minimizer of (2.1) such that for some function u ∈ C 1,α (∂B 1 ). There exists a universal constantγ 1 > 0 so that when γ <γ 1 , In order to get an upper bound on D(E), we will show the following proposition: for some constant C depending only on n.
Before turning to the proof of this proposition, we will use Propositions 5.2 and 5.3 to prove Theorem 2.2: Proof of Theorem 2.2. When γ ≤ γ ′ 1 , Theorem 4.5 implies ∂E ∈ C 1, 1 2 (1−s) and and thus by Proposition 5.3, On the other hand, Proposition 5.2 gives the lower bound: ) as long as γ <γ 1 . Thus, we obtain When γ = αm 1−s n ≤ γ 1 with γ 1 small enough (but depending only on n and s), this inequality, together with Sobolev embedding implies u = 0 and so E = B r0 .
Proof of Proposition 5.3. The proof follows similar computations as in [10], although the fact that P K = P s complicates things a bit. First, we write P K (F ) as By using the identityˆb , and the symmetric property of f , we have For the first integral, we use the fact that for every x, y ∈ ∂B 1 we have and therefore, .

(5.8)
For the second integral, fix x ∈ ∂B 1 , and by the symmetric properties of K, the integral ∂B1 dH n−1 (y)ˆr is a function of u(x) only, so for every η > 0, we define and write Note that (5.7) implies Together with (5.8), (5.7) now implies and using (5.9), we deduce To complete the proof, we use the following technical lemma, which is proved in Appendix C: With β as in Lemma 5.4, we get We thus havê The result now follows from (5.11).
6 The ball is not a minimizers for large αm 1−s n

Proof of Theorem 2.3
Theorem 2.3 states in particular that the ball is not a global minimizer for large αm 1−s n , but also gives an upper bound on the width the minimizer. To prove the result, we assume, by contradiction that there exists x 0 ∈ R n such that B ρ/2 (x 0 ) ⊂ B ρ (x 0 ) ⊂ E (and so |B ρ (x 0 ) \ E| = 0) for some ρ ≥ 4ρ 0 . We then construct a competitor F by removing the ball B ρ/2 (x 0 ) from E and placing it "at infinity". More precisely, given y 0 such that |x 0 − y 0 | = R ≫ 1, and B ρ/2 (y 0 ) ∩ E = ∅, we consider the set and therefore (using the fact that K(x) → 0 as |x| → ∞): as R → ∞.
Since |F R | = |E| = m, we must have J α (F R ) ≥ J α (E) and so where we used the lower bound (2.5) and the fact that ρ ≥ 4ρ 0 . Passing to the limit R → ∞, this implies that ρ ≤ (Cαs(1 − s)) −1 1−s and the result follows.
In particular, if the ball of volume m is a minimizer, we must have which is equivalent to αm 1−s n ≤ max{γ 2 , cαρ 1−s 0 } for some γ 2 and c depending on s and n.

Proof of Theorem 2.4
We note that E is a minimizer F s,α with the constraint |E| = m, if and only if E m = 1 m 1/n E is a minimizer of energy F s,β under the constraint |F | = |B 1 | with The first part of the theorem relies on the following lemma: Lemma 6.1. If B 1 is the global minimizer of F s,β (F ) with constraint |F | = |B 1 | for β = β 1 , then it is the global minimizer of F s,β (F ) for all β < β 1 .
Proof. Let E be such that |E| = |B 1 | and for some β < β 1 . This implies Since B 1 is the global minimizer of F β1 , we have so (6.2) (and the fact that β 1 − β > 0) yields The isoperimetric inequality for the fractional perimeter thus implies that E = B 1 and the lemma follows. Lemma 6.1, together with Theorems 2.2 and 2.3 imply that there exists β * 1 ∈ (0, ∞) such that {β ; B 1 is a global minimizer of F s,β } = [0, β * 1 ] and thus prove the first part of Theorem 2.4.
The proof of the second part of the Theorem 2.4 (local minimizers) relies on an analysis of the first and second variations of the energy F s,β . Following [10], given a vector field X ∈ C ∞ c (R n ; R n ) we denote by {Φ t } t∈R the flow induced by X (solution of ∂ t Φ t (x) = X(Φ t (x)), Φ 0 (x) = x) and we then define E t := Φ t (E). We say that X induces a volume-preserving flow on E if |E t | = |E| for small t. This implies in particular where we denoted ζ := X · ν with ν normal unit vector to ∂E. With these notations, we have the following formula for the first and second variation of the perimeters along a volume-preserving flow (see for example Section 6 in [10]): We used here the following classical notations: H ∂E denotes the scalar mean-curvature of ∂E, c 2 ∂E denotes the sum of the squares of the principal curvatures of ∂E, X τ = X − ζν is the tangential projection of X along ∂E and ∇ τ , div τ denotes the tangential gradient and divergence operators.
When E = B 1 , all curvatures are constant and using (6.3), we deduce: Similar formula are derived in [10] for nonlocal perimeters. In particular we can write: where H s,∂E denotes the s−curvature defined by When E = B 1 , using (6.3), we get: Using (6.4) and (6.6), we see that the unit ball B 1 is always a critical point of the energy F s,β for volume preserving variation, since Furthermore, the second variation formula yield: We will say that B 1 is a volume-constrained stable set for F s,β if δ 2 F s,β (B 1 )[X] ≥ 0 for every X inducing a volume-preserving flow on B 1 . We then have the following result: .
Note that the definition of β, (6.1), then yields Proof of Proposition 6.2. We introduce the set and we denote Using (6.8), we see that the condition δ 2 F s,β (B 1 )[X] ≥ 0 for all X is equivalent to . (6.10) In order to find the value of β given by (6.9), we follow [10] again by introducing an orthogonal basis for is an orthogonal basis of where the fine dimensional subspace of spherical harmonics of degree k). Denoting a i k (ζ) =´∂ B1 ζY i k dH n−1 , we then have: Lemma 6.4 below states that this minimum is reached when k = 2 and so Finally, using formula (2.11), (2.12) and Proposition 7.2 in [10], we conclude: .
Lemma 6.4. For all n and s ∈ (0, 1), we have: Proof. To prove for every k ≥ 2, it is equivalent to prove for all k ≥ 2. We prove by deduction. When k = 2, A 2 = 1. Assume A k ≥ 1 which implies and we aim to prove A k+1 ≥ 1. With (6.11) and After simplification, it is equivalent to In order to complete the proof of Theorem 2.4, it only remains to show that β 0 < β 2 . This follows from the following Lemma: Lemma 6.5. For all n and s, we have Proof of Lemma 6.5. To prove the lemma, we will show that for β >β = 2 It is thus a simple exercise to show that F s,β (E R ) < F s,β (B 1 ) if β >β and R is large enough.

Proof of Theorem 2.5
In order to prove Theorem 2.5, we first add a confinement term to the energy functional J α : We then have the following result: Proof. For every m, α, β > 0, using (2.3), we can show that J α,β (E) has a lower bound In particular, there exists a minimizing sequence {E j } j∈N with |E j | = |B r0 | = m and we can always assume that This implies for all j ∈ N. So χ Ej is uniformly bounded in BV (R n ) and thus relatively compact in L 1 loc (R n ). By a diagonal extraction argument, we deduce the existence of a subsequence χ Ej k and a set E such that χ Ej k converges to χ E weakly in L 1 (R n ) and strongly in L 1 (B R ) for all R > 0. The lower semicontinuity of the perimeter implies and Lemma 3.2 gives P K (E) = lim j→∞ P K (E kj ).
Finally, the strong L 1 convergence implieŝ and using the fact that and therefore |E| = m. Altogether, we deduce that E is a minimizer of J α,β with |E| = m.
Next, for N > 0, we introduce Proceeding as in Proposition 7.1, we can show that for all α > 0, β > 0, m > 0, and N the minimization problem has a minimizer. Note that if |E i0 | = 0 for some i 0 , then that component of E does not contribute to the energy and we can set E i0 = ∅. In the sequel, we will always assume that either E i = ∅ or |E i | > 0.
We then prove: Proof. We fix m 0 such that for all m ≤ m 0 the minimizer of J α,β with constraint |E| = m is the ball and such that the energy of two balls of mass 2m in total is bigger than that of one ball of mass 2m. It is then clear that E can have at most 1 component with Lebesgue measure smaller than m 0 and therefore we can take N 0 = m m0 + 1. From now on, we fix N ≥ N 0 + 1 so that at least one component of E has zero Lebesgue measure (and since we can remove such components without changing the energy, we can assume that one of the component is the empty set). We then have ..,N be a global minimizer of (7.2). Then each component E i is indecomposable.
Proof. If one component E i0 is decomposable, that it can be written as E i0 = F 1 ∪ F 2 with P (E i0 ) = P (F 1 ) + P (F 2 ) and |F i | > 0, then the set obtained by replacing E i0 by F 1 and replacing an empty component by F 2 will have (strictly) lower energy.
Next, the crucial tool to pass to the limit β → 0 and prove our result is the following non-degeneracy estimate: ..,N be a global minimizer of (7.2). There exists a constant c depending only on n and s such that for every x 0 ∈ ∂E i and r > 0.
Before proving this proposition, we state and prove the following corollary which allows us to pass to the limit β → 0 and prove Theorem 2.5: ..,N be a global minimizer of (7.2). If β ≤ 1 m 2/n , then In particular, we can assume that E i ⊂ B R0 for some R 0 depending on m and α but not on β.
Proof of Corollary 7.5. Without loss of generality, we can assume that there exists x, y in E i such that Proof of Proposition 7.4. We denote E = E i0 and m = m i0 and we note that E must minimize J α with the constraints E ⊂ B R and |E| = m. Given r > 0 we define the functions: Proceeding as in the proof of Lemma 4.1, we then have Next, we note that for λ > 0, and E minimizer of J α,β with |E| = m, we have Using (7.1) together with (4.5) we get: Choosing λ such that (1 + λ)|E ′ | = m, that is we deduce from (7.4) that for some ε 0 < 1/2 to be chosen later. Since r → V (r) is non-decreasing, we then have 1 m 2 β n m for all r ≤r, (7.5) and so in particular Putting everything together, we deduce: Choosing ε 0 small enough, we thus have Proof of Proposition 2.6. Using the scaling properties of F s,α , it is enough to prove the result when α = 1. We denote F = P − s(1 − s)P s and G the corresponding generalized functional. Next we note that we clearly have inf |F |=m F (F ) ≥ inf |F|=m G (F) (since we can take F = {F }, and the other inequality is obtained by recalling (Theorem 2.5) that the minimizer of G has finitely many bounded component. We can thus construct a sequence F k by taking those components and sending them to infinity is different direction to get F (F k ) → inf |F|=m G (F).
We recall that µ n,s > 0, defined in (1.5), denotes the optimal constant in the interpolation inequality (1.4). We thus have Since the minimum of the function g( We also note that we have equality if E achieves the infimum in (1.5) and if P (E) = (sµ n,s ) 1 1−s m. Since we do not know whether there exists a set E which achieves the optimal constant in (1.4), we consider, given ε > 0, a set E 0 such that |E 0 | = 1 and In order to find a set for which there is almost equality in (8.1), we take k ∈ N and consider the set E k made of k copies of the rescaled set ( m k ) 1/n E 0 (with mass m k ) positioned in R n so that the distance between any two copies is at least R ≫ 1. The set E k satisfy |E k | = m and Since we can take R → ∞ to find inf F (E), we will discard the O(1/R n+s ) below (this amount to consider the set E k in which the k copies of ( m k ) 1/n E 0 have been sent to infinity in different directions). Furthermore, we note that the function g(x) is minimum when x =x = sPs(E0) P (E0) Together with (8.1), it shows that E k is a minimizer for G .
This shows that whenever m ∈ k P (E 0 ) sP s (E 0 ) n 1−s ; k ∈ N then a minimizing sequence for F with volume constraint |E| = m can be constructed by taking k copies of the set ( m k ) 1/n E 0 and sending them to infinity in different directions. (ii) Assume there exists a sequence {m k } ∞ k=1 with m k → ∞ such that there exist bounded minimizers E k of F s,α (F ) with |F | = m k . Using the first part of Proposition 2.6, we have  First, we write in the second integral, we get: Optimizing with respect to R by taking R = 2|E| P (E) yields the result.

B Proof of Lemma 3.2
We write The strong convergence in L 1 (R n ) implies that the first term converges to 0 as k → ∞. Using (A.1), we deduce lim sup Since this holds for all η > 0, the result follows.

C Proof of Lemma 5.4
It is equivalent to prove We can see The first integral