Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion

We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small $C^{1,\gamma}$ perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the $1-$capacity where both terms in the energy are of the same order.


Introduction
In this paper, we consider a geometric variational problem motivated by models for charged liquid drops recently studied in a series of papers [7,19,20,[30][31][32].One of the main features of these problems is the strong competition between surface tension and charge repulsion.In particular, as opposed to the much studied Gamow liquid drop model (see [5,22]), the non-local effects often dominate the cohesive forces leading to singular behaviors.The aim of the paper is to consider the case of very strong short-range repulsion between the charges, thus completing the program started in [19].
We now introduce the model.Given α ∈ (0, N ) and a measurable set E ⊂ R N , we define the Riesz interaction energy (1.1) This energy coincides with the inverse of the α−capacity, see (2.9).Letting P (E) denotes the perimeter of E (see [27]), we consider for every charge Q > 0 the functional While postponing the discussion about the precise class in which we are minimizing, the aim of the paper is to study for m > 0 the problem By a scaling argument, up to renaming the constant Q, it is enough to consider the case m = ω N , where ω N is the volume of the unit ball B 1 .
This question is motivated by the model for an electrically charged liquid drop in absence of gravity, introduced by Lord Rayleigh [34] in the physically relevant case N = 3 and α = 2, and later investigated by many authors (see for instance [12,19,21,30,31,38,40]).We proved in [19] that, quite surprisingly, for every N ≥ 2 and α ∈ (1, N ) (in particular in the Coulombic case α = 2), the problem is ill-posed.Indeed, in that case, we can show that inf |E|=ωN ,E smooth F α,Q (E) = P (B 1 ).
In words, starting from smooth sets the lower semicontinuous envelope of the energy F α,Q in L 1 (R N ) reduces to the perimeter.To restore well-posedness of the problem one needs to impose some extra regularity conditions such as bounds on the curvature [19], entropic terms [30] or the convexity of competitors [20].
Later on, it was shown in [31] that at least if N = 2 and α = 1, the problem admits the ball as unique minimizer as long as Q lies below an explicit threshold, and that no-existence occurs otherwise.
The aim of this paper is to complement the picture in the case α ∈ (0, 1] for N ≥ 2. 1.1.Main results.As already mentioned above, the first difficulty with this model is to properly define the class of competitors.Indeed, while both the perimeter and I α are well-defined in the class of smooth compact sets, this class does not enjoy good compactness properties.For variational problems involving the perimeter, the usual setup is the one of sets of finite perimeter (see [27]) where we identify two sets E and F if they are equal up to a Lebesgue-negligible set.However, it is not hard to see that I α is not well-behaved under such identification (we have I α (E) = I α (F ) if E = F outside a set of zero α−capacity), see [19].As advocated in [31,32] for N = 2 and α = 1, we will consider here the class S = E ⊂ R N : E is compact and P (E) = H N −1 (∂E) < +∞ .
We will always identify sets in S which differ only on a set of Lebesgue measure zero (and thus actually agree H N −1 a.e.), see Remark 2.1.The variational problem we consider is thus min |E|=ωN ,E∈S F α,Q (E). (1. 2) The main result of this paper is the following.
We recall that by [19], if α ∈ (1, N ), problem (1.2) does not admit minimizers for any value of Q > 0. The proof of Theorem 1.1 follows the same general scheme as in [22] (see also [1,3,4,11,29] where similar strategies have been used).Inspired by the proof [6] of the quantitative isoperimetric inequality, the idea is to prove first existence of (generalized) minimizers for the problem.Then, the challenge is to prove regularity estimates for minimizers which are uniform in Q.This allows by compactness to reduce the problem to a second order Taylor expansion of the energy close to the ball (this is the so-called Fuglede type argument).As we will now see, in our case all three steps present serious difficulties.Let us point out that when N = 2 and α = 1, the proof in [31] is of totally different nature.Indeed, it uses a combination of convexification and Brunn-Minkowski inequalities.While it could be interesting to see if this argument could be extended to the case α ∈ (0, 1), it is intrinsically limited to N = 2.We start with existence of minimizers: Theorem 1.2.For every N ≥ 2 and α ∈ (0, 1], there exists Q 1 = Q 1 (N, α) > 0 such that for every Q ≤ Q 1 minimizers of (1.2) exist.
This result is proven in Theorem 3.11 where we prove actually a bit more.Indeed, we show that if α < 1, minimizers exist for every Q > 0, at least in a generalized sense (see Definition 2.2).As in [3,10,17,20], a classical first step is to transform the volume constraint into a penalization, see Lemma 3.1.Following for example [3,14,17,23,33] we would then like to prove Theorem 1.2 through a concentration-compactness argument.However, since the class S is not closed under L1 convergence and because of the issues related to the lower semi-continuity of I α raised above, it is not clear how to argue directly for F α,Q .The idea is thus to first regularize the functional by penalizing concentration of the charge.While we believe that the precise choice of regularization is not essential, in line with [30], for ε > 0, we replace I α by In particular, this functional is well defined in L 1 i.e.I α,ε (E) = I α,ε (F ) if E = F a.e. and we can prove in Proposition 3.7 the existence of (generalized) minimizers.
In order to conclude the proof of Theorem 1.2 and send ε to zero, we show in Proposition 3.10 that minimizers enjoy density estimates which are uniform in ε.This is a consequence of a first almost-minimality property of minimizers proven in Proposition 3.8.Indeed, using a relatively simple lower bound from [31] on the Riesz interaction energy of the union of two disjoint sets, we show that there exists a constant C > 0 such that if E is a minimizer for the regularized functional and F is such that E∆F ⊂ B r , then The desired density bounds then follow from [18], see also [27,37] when α < 1.
We then turn to regularity: Theorem 1.3.For every N ≥ 2 and α ∈ (0, 1], there exist This result is contained in Proposition 3.12 for α < 1 (see also Remark 3.15) and Proposition 3.21 for α = 1.On the one hand, we see from (1.3) that when α < 1, we may directly appeal to the classical regularity theory for almost-minimizers of the perimeter, see [27,37], and there is nothing to prove.On the other hand, when α = 1 the situation is much more delicate.In fact, Proposition 3.21 may be seen as one of the main achievements of this paper.When α = 1, while (1.3) is in general too weak to obtain C 1,γ regularity, it is still strong enough to yield Reifenberg flatness of E (see Definition 3.13) when Q ≪ 1 as recently shown in 1 [18].In order to improve it to the full C 1,γ regularity we rely on a second almost-minimality property.We show in Proposition 3.16 that if µ E is the optimal charge distribution for E i.e. µ E is the minimizer in (1.1), and if E∆F ⊂ B r , then The proof is inspired by [7,Proposition 4.5].Notice however that we have to deal with difficulties which are quite different from the ones in [7].Indeed, on the one hand our operator is smooth (here it is just the half Laplacian, see (2.5)) as opposed to [7] where the heart of the problem is the presence of irregular coefficients.On the other hand in our case the charge distribution µ E is, a priori, just a measure while in [7] it is known to be a function in L ∞ .In fact, in light of (1.4), the main point here is to prove good integrability properties of the charge distribution µ E .This is done in Lemma 3.17 and Lemma 3.18 where we prove that for any γ ∈ (0, 1/2), if E is a sufficiently flat Reifenberg sets E, By comparing this result with the case of the ball, we can see that this estimate is optimal.It may be seen as an extension to irregular domains of the boundary regularity for the fractional Laplacian developed in [35].As in the case of the Laplacian considered in [25], the main ingredient for the proof is the monotonicity formula of Alt-Caffarelli-Friedman. Combining (1.4) and (1.5) we find that also for α = 1, minimizers of (1.2) are actually classical almost-minimizers of the perimeter and thus Theorem 1.3 follows.
Let us point out that our proof actually applies to a more general class of functionals.Indeed, for Λ ≥ 0 we say that E is a Λ−minimizer of F α,Q if for every set F , Arguing as in Theorem 1.3 we can prove the following result (we state it only for α = 1 since α ∈ (0, 1) is simpler): In particular, this answers a question left open in [32].
Thanks to Theorem 1.3 and the quantitative isoperimetric inequality, if Q is small enough then up to translation any minimizer of (1.2) is nearly spherical.By this we mean that fixing γ ∈ (0, 1) (which remains implicit), |E| = ω N , the barycenter of E is in zero and there is φ : The proof of Theorem 1.1 is thus concluded once we show the minimality of the ball among nearly spherical sets: Theorem 1.5.Let α ∈ (0, 2).There exist Q 3 = Q 3 (N, α, γ) > 0 and ε = ε(N, α, γ) > 0 such that for every nearly spherical set E with φ W 1,∞ (∂B1) ≤ ε, and every

Moreover, equality is attained only if
This shows in particular the stability of the ball under small C 1,γ perturbations if Q is small enough.The counterpart of Theorem 1.5 for the Coulomb case α = 2 has been obtained in [19,Theorem 1.7].The main point of the proof is to show in Proposition 4.5 that .
The proof of this quantitative estimate for I α follows the same general strategy as in [19].As in [19], the difficulty comes from the fact that the optimal measure µ E is not explicitly given in terms of E. There are however some important differences between the Coulomb case α = 2 and the non-local case α ∈ (0, 2).In the Coulomb case, the charges are concentrated on the boundary.This makes it easier to express the Riesz interaction energy in terms of φ with respect to the case α ∈ (0, 2) where the optimal measure µ E has support equal to E. Another difficulty here comes from the blow-up of µ E near ∂E.
The last result of this paper is a nonexistence result in dimension 2.
Theorem 1.6.Let N = 2 and α ∈ (0, 1].Then, there exists The paper is divided into four parts.In Section 2 we collect the precise notation and definitions used in the paper.In Section 3, we prove Theorem 1.2 and Theorem 1.3 about existence and regularity of minimizers for (1.2).We then prove Theorem 1.5 in Section 4. A last short section is dedicated to the proof of the nonexistence Theorem 1.6.

Notation
We will use the notation A B to indicate that there exists a constant C > 0, typically depending on the dimension N and on α such that A ≤ CB (we will specify when C depends on other quantities).We write A ∼ B if A B A. We use in hypotheses the notation A ≪ B to indicate that there exists a (typically small) universal constant ε > 0 depending only on N and α such that if A ≤ εB then the conclusion of the statement holds.
For a measurable set E ⊂ R N and an open set Ω ⊂ R N , we denote by |E| the Lebesgue measure of E and P (E, Ω) its relative perimeter in Ω.When Ω = R N we simply write P (E, Ω) = P (E) (see [27]).We use ∂E to indicate the topological boundary of a set E, ∂ M E the measure-theoretic boundary and ∂ * E for the reduced boundary.

2.1.
Fractional Sobolev spaces, Laplacians and capacities.We collect here some standard notation and basic properties about fractional Sobolev spaces, Laplacians and capacities.We refer for instance to [9,24,26] for more information on this topic.For the Fourier transform we use the convention For s ∈ R we then define the (homogeneous) H s semi-norm as When there is no risk of confusion we simply write [u] H s for [u] H s (R N ) .We define the s−fractional Laplacian by its Fourier transform: For s ∈ (0, 1), there exists C(N, s) > 0 such that where the integral is intended in the principal value sense.We then have the alternative formula for the H s semi-norm We will also use fractional Sobolev spaces defined on the unit sphere ∂B.For these we take (2.2) as starting point and define for φ : ∂B → R and s ∈ (0, 1), Let us point out that we use here a slight abuse of notation and do not distinguish between the volume measure on R N and the one on the sphere.We recall that for 0 < s < s ′ < 1, if we denote φ = 1 P (B) ∂B φ we have where we write ∇φ for the tangential gradient and where the implicit constants depend on N , s and s ′ .Indeed, the first inequality follows from Cauchy-Schwarz as The second inequality in (2.4) is immediate while the third can be deduced by [8, Proposition 2.4 and Remark 2.8].
For α ∈ (0, N ) and µ a Radon measure, we define the Riesz interaction energy as With this notation, definition (1.1) becomes Remark 2.1.Recalling the definition (1.1) of S, we see that for α ∈ (0, 1], I α is well defined in S in the sense that if E, F ∈ S with |E∆F | = 0 then actually H N −1 (E∆F ) = 0 and thus [28,Theorem 8.7]).
For every α ∈ (0, N ), there exists a constant C ′ (N, α) > 0 such that for any radon measure µ, the associated potential (2.5) In particular, using that we have Similarly, see [19,24] for a precise justification.We finally point out that if we define the fractional capacity as then it is not hard to check that at least for smooth enough sets E, u E /I α (E) is the minimizer of so that by (2.7) and (2.2) (2.9) We refer to [24,28] for further information on fractional capacities.
2.2.Generalized sets and minimizers.For a (possibly finite) sequence of sets E = (E i ) i≥1 and measures µ = (µ i ) i≥1 , we define (2.11) Notice that since I α ( µ + µ ′ ) ≥ I α ( µ), when minimizing over µ, we may assume without loss of generality that µ i is concentrated on E i .
Definition 2.2.We call generalized set a collection of sets E = (E i ) i≥1 as above, and we set (2.12) For Q > 0, we define the energy of a generalized set as We say that E ∈ S N is a (volume constrained) generalized minimizer for F α,Q if, for any collection of sets 3. Existence and regularity of minimizers 3.1.Relaxation of the volume constraint.For Λ > 0, we relax the volume constraint by considering Our first result is that for Λ large enough the relaxed problem coincides with the constrained one.
Proof.Since the fact that the left-hand side of (3.2) is larger than the right-hand side, it is enough to prove the remaining inequality.Let Λ ≫ 1 + Q 2 and assume that there exists Using B 1 as competitor we find We now use t E = (tE i ) i≥1 , which satisfies |t E| = ω N , as competitor and obtain using Taylor expansion, In the case δ ≤ 0 we reach the same contradiction since (3.4) yields this time 3.2.The regularized functional.As mentioned in the introduction, since the capacitary term I α is not well defined in L 1 , which is the natural setting to minimize the perimeter, we will first show existence of (generalized) minimizers for a regularized energy.For ε > 0 and a positive measure µ we define We then define for µ = (µ i ) i≥1 and E = (E i ) i≥1 , in analogy with (2.10) and (2.11), Indeed, every measure µ with I α,ε (µ) < ∞ is in L 2 and thus satisfies µ(E) = µ(F ).When considering I α,ε instead of I α we can therefore identify sets which agree Lebesgue a.e. ) Proof.We start with the upper bound.Let m = | E|. and for every i ≥ 1 let B i be a ball such that m in the definition of I α,ε ( E) and recalling the Riesz rearrangement inequality, we get To obtain the lower bound we simply observe that for every µ = (µ i ) i≥1 with i µ i (E i ) = 1, we have by Cauchy-Schwarz The desired bound follows by minimizing in µ.
As a consequence, of Lemma 3.3, we can prove the existence of an optimal measure for I α,ε ( E).
Proof.Uniqueness follows from strict convexity of the energy so we only focus on the existence part of the statement.We first notice that from the definition (3.5) of I α,ε we have Hence, the existence of an optimal µ follows if we can prove that on the one hand, for every fixed set E of finite volume, there exists an optimal measure for I α,ε (E) and on the other hand that there exists an optimal distribution of charges (q i ) i≥1 for (3.8).We thus start by considering a fixed set E with |E| + I α,ε (E) < ∞ and prove the existence of an optimal charge µ.If µ n is a minimizing sequence, arguing as in (3.7) we find that for every R > 0, α,ε (µ n ).Therefore µ n is tight and we can extract a sequence converging weakly in L 2 (R N ) to a measure µ with µ(E) = 1.By lower semi-continuity of I α,ε (see [24, (1.4.5)]), µ is a minimizer for I α,ε (E).We now turn to the existence of an optimal charge distribution (q i ) i≥1 .For this we first observe that from the first inequality in (3.6) and thus in particular lim I→∞ i≥I I −1 α,ε (E i ) = 0. Now for every (q i ) i≥1 and every I ∈ N, by Cauchy-Schwarz so that tightness of minimizing sequences follows leading to the existence of an optimal distribution (q i ) i≥1 .

3.3.
Existence of generalized minimizers for the regularized energy.In line with (3.1), we introduce the regularized energy The aim of this section is to prove the existence of minimizers for this functional.We start with the simple observation that for F α,Q,Λ,ε , minimizing among classical or generalized sets gives the same infimum energy.
Lemma 3.6.For every α ∈ (0, N ), Q, Λ, ε > 0, we have Proof.Since the left-hand side of (3.11) is larger than the right-hand side, it is enough to prove that for every δ > 0 and every generalized set We first observe that for each fixed i, lim Combining this with the fact that i |E i | < ∞ we see that we can choose I and R large enough so that Moreover, thanks to the co-area formula we may further assume that We now turn to the last term in the energy.Let µ = (µ i ) i≥1 be the optimal measure for I α,ε ( E) given by Corollary 3.4.We then set for i ≤ I, ν i = ) and ν i = 0 otherwise so that ν = (ν i ) i≥1 is admissible for I α,ε ( F ). Using that I i=1 µ i (F i ) converges to 1 as I → ∞ and R → ∞, we can also assume that I and R are chosen such that in addition to (3.12) and (3.13) we have We finally choose for every i ≤ I a point Since F i ⊂ B R by construction, the sets F i + x i are pairwise disjoint and from (3.12) and (3.13) we have Finally we observe that ν is admissible for I α,ε (E) with We can now prove the existence of generalized minimizers for F α,Q,Λ,ε .This will be proven by a concentration-compactness argument which relies on isoperimetric effects to avoid the loss of mass together with the semi-continuity of I α,ε with respect to L 1 loc convergence.This type of arguments is relatively standard by now (see for instance [3] which we closely follow or [14,17,23,33]).However, we face here the additional difficulty that we need to avoid not only loss of volume but also loss of charge in the limit.
Proof.Let (E n ) n≥1 be a (classical) minimizing sequence for F α,Q,Λ,ε .By Lemma 3.6 it is also a minimizing sequence in the class of generalized sets.Using for instance the ball B 1 as competitor we have and assume without loss of generality that for every n, m i,n is decreasing in i. Moreover we tacitly consider from now on only the indices i such that m i,n > 0. We let µ n be the optimal measure for I α,ε (E n ) and set We start by proving tightness of (m i,n ) i≥1 and (q i,n ) i≥1 .For m i,n , we argue as usual that thanks to the relative isoperimetric inequality (recall that with our choice of L, Using that m i,n ≤ m i we conclude that for every I ∈ N, For q i,n we argue as in (3.7) and obtain invoking twice Cauchy-Schwarz, Therefore, up to extraction we have lim n→∞ m i,n = m i with i m i = m and lim n→∞ q i,n = q i with i q i = 1.
We now construct a generalized set E which will be our generalized minimizer.By the perimeter bound, up to extraction we have for every i, ) converges weakly in L 2 to some µ i .We can further assume that for every i, j, We now say that i ∼ j if a ij < ∞ and denote by [i] the equivalence class of i.Notice that if i ∼ j then E i and E j are translated of each other.For each class of equivalence we denote For every i, using the convergence of E n − z i,n to E i and of µ i n to µ i , and the definition of the equivalence relation, we have Up to relabeling, we may now assume that each class of equivalence [i] is made of a single element.
If we set E = (E i ) i≥1 and µ = (µ i ) i≥1 we have just shown that µ is admissible for I α,ε ( E).Let us finally prove that We consider separately each term of the energy.Since | E| = m = lim n→∞ |E n |, the volume term is not a problem.For the first term, we fix I ∈ N and R > 0. If n is large enough, we can assume that for i, j ≤ I with i = j, |z i,n − z j,n | ≫ R. By the co-area formula we can find for every i ≤ I From this bound we see that For the last term, we use similarly that for every fixed I ∈ N and R > 0, 3.4.First almost minimality property and existence of minimizers for the original problem.In this section we use Lemma 3.5 to prove a first almost minimality property for generalized minimizers of F α,Q,Λ,ε .In order to pass to the limit ε → 0 it is crucial that the estimates are uniform in ε.
Proposition 3.8.There exists C > 0 depending only on N and α ∈ (0, N ) with the following property.For every Q > 0, ε ∈ (0, 1) and Λ ∼ 1 + Q 2 for which Lemma 3.1 applies, every generalized minimizer E = (E i ) i≥1 of F α,Q,Λ,ε is an almost minimizer of the perimeter in the sense that for every i ≥ 1, x ∈ R N and r ≪ 1, Proof.Without loss of generality we may assume that i = 1 and x = 0. To simplify a bit notation we denote E = E 1 .Using F = F × (E i ) i≥2 as competitor and the minimality of E we have after simplifications We are left with the case F ⊂ E. Writing E = F ∪ (E\F ) and appealing to (3.9) from Lemma 3.5, we have Now on the one hand, since by Lemma 3.1, (recall that r ≪ 1) and thus by (3.6) of Lemma 3.3, On the other hand, since E\F ⊂ B r we have Putting these two things together, (3.18) yields Plugging this in combination with |E∆F | r N in (3.17) concludes the proof of (3.16).
Remark 3.9.From (3. 19) we see that we can improve (3.16) to This means that for every α ∈ (0, N ), if r ≤ ε 1/α then the classical regularity theory for perimeter almost-minimizers applies (see [27]).In particular, for α = 2, this gives a very elementary proof of the regularity of minimizers for the functional considered in [7,30] if the permittivity of the droplet is assumed to coincide with the permittivity of the vacuum (see however [7,Remark 4.6] where it is observed that this assumption would also simplify their proof ).
At this point we see the difference between the case α > 1 and α ≤ 1.Indeed, in the latter case, thanks to (3.16), we may appeal to the regularity theory for almost-minimizers of the perimeter (since N − α ≥ N − 1).We start with the simpler part which consists of the density estimates.Since the cases α < 1 and α = 1 are treated differently, we introduce the notation 1 α=1 = 1 if α = 1 and 1 α=1 = ∞ if α ∈ (0, 1).Proposition 3.10.For every α ∈ (0, 1] and Q > 0 let Λ ∼ 1 + Q 2 be such that Proposition 3.8 applies.Then, for every ε ∈ (0, 1], and every generalized minimizer and (3.21)As a consequence, there exists Q 1 > 0 such that for Q ≤ Q ≤ Q 1 1 α=1 , up to the choice of a representative, every generalized minimizer is made of finitely many E i , each of which is connected with E i ∈ S and for which ∂E i = ∂ M E i .Moreover, the number of such components as well as their diameter depends only on Q.
Proof.Estimates (3.20) and (3.21) follow directly from [18, Proposition 3.1].For α < 1 they can also be obtained (under slightly stronger hypothesis on r) from the more classical theory, see for instance [27].The regularity of the minimizers as well as the bound on the number and diameter of the connected components is classical (see e.g.[22]) once we observe that for every Q ≤ Q ≤ Q 1 1 α=1 there is r depending only on Q such that max(Q 2 r1−α , r) ≪ 1 for every Q ≤ Q.The fact that we may assume that each component E i of E is made of a single connected component follows from Before stating the full conclusions of the regularity theory for perimeter almost minimizers, let us conclude the proof of the existence of generalized volume-constrained minimizers of F α,Q .Theorem 3.11.Let Q 1 be given by Proposition 3.10.Then for every 0 Moreover, for each i ≤ I, E i is a perimeter almost minimizer in the sense of (3.16) and both I and diam(E i ) are bounded by a constant depending only on Q.
Proof.Let Λ ∼ 1 + Q2 be such that both Lemma 3.1 and Proposition 3.7 apply.By the latter, for every ε ∈ (0, 1] and Q ≤ Q, there exists a generalized minimizer E ε of F α,Q,Λ,ε .Moreover, by Proposition (3.10), E ε = (E i ε ) I i=1 for some connected sets E i ∈ S, where both I and their diameters depend only on Q. Thanks to the uniform density bounds (3.20) and (3.21) we can extract a subsequence for which E i ε converges both in L 1 and in the Kuratowski sense to some E i ∈ S. By compactness of perimeter almost minimizers, see [18], E i satisfies (3.16).We set We now prove that inf which combined with the previous inequality would show that E is a generalized minimizer of F α,Q,Λ as Arguing exactly as in Lemma 3.6, we see that it is enough to prove that for every F ∈ S, there exists a sequence F ε such that By [36] applied to F c , we can find smooth compact sets F δ with F ⊂ F δ , P (F δ ) ≤ P (F ) + δ and and we can thus further assume that F is smooth in the proof of (3.22).For smooth sets, by [19, Proposition 2.16] 2 , we can find for every δ > 0 a function Since for every δ > 0, lim ε→0 I α,ε (f δ ) = I α (f δ ), a diagonal argument shows that I α (F ) = lim ε→0 I α,ε (F ).Using F ε = F we conclude the proof of (3.22).
As E is a generalized minimizer of F α,Q,Λ , Lemma 3.1 implies that | E| = ω N and thus E is also a volume-constrained generalized minimizer of F α,Q .
We end this section by recalling the regularity properties of generalized minimizers and show in particular that for small charge Q they are actually classical minimizers.We start with the case α < 1. Proposition 3.12.For α ∈ (0, 1) and The conclusion follows from the classical regularity theory for perimeter almost minimizers, see [27,37] and the fact that by the quantitative isoperimetric inequality, up to translation and relabeling,  which implies in conjunction with (3.20) that for Q small enough, E i = ∅ for i ≥ 2 (so that E = E 1 is a classical minimizer) together with the convergence to B 1 .
2 The statement of [19,Proposition 2.16] requires F to be connected but the proof works for disconnected sets as well.
For α = 1 it is well-known that in general (3.16) does not even imply C 1 regularity.In order to state the counterpart of Proposition 3.12 in this case, let us first recall the definition of Reifenberg flat sets.Definition 3.13.Let δ, r 0 > 0 and x ∈ R N .We say that E is (δ, r 0 )−Reifenberg flat in B r0 (x) if for every B r (y) ⊂ B r0 (x), there exists an hyperplane H y,r containing y and such that • we have where d denotes the Hausdorff distance; • one of the connected components of is included in E and the other in E c .We say that E is uniformly (δ, r 0 )−Reifenberg flat if the above conditions hold for every x ∈ ∂E.Proposition 3.14.Let α = 1.There exists Q 2 > 0 such that for every Q ≤ Q 2 , every volumeconstrained generalized minimizer of F α,Q is a classical minimizer.Moreover, for every δ > 0, there exist Q δ , r δ > 0 such that for every Q ≤ Q δ , every volume-constrained minimizer E Q of F α,Q is uniformly (δ, r δ )−Reifenberg flat and up to translation, Proof.The proof is exactly as for Proposition 3.12, replacing the classical regularity theory by [18, Corollary 1.4].

3.5.
Second almost minimality property and regularity of minimizers.The aim of this section is to prove that in the case α = 1, we can pass from the Reifenberg flatness of volumeconstrained minimizers of F α,Q stated in Proposition 3.14 to almost C 1, 1 2 regularity.This will be obtained by proving a second almost minimality property for minimizers together with a higher integrability result for the optimal measure µ.Remark 3.15.Let us point out that using a similar proof for α ∈ (0, 1), it would be possible to improve the C 1, 1 2 (1−α) regularity from Proposition 3.12 to almost C 1, 1 2 .However, in this case, the proof of the integrability of µ can be greatly simplified by appealing directly to [35] (see also (4.5) below).Moreover, we expect that any C 1,β regularity may be improved to higher regularity through the Euler-Lagrange equation, see [32].
We start with the quasi-minimality property.Proposition 3.16.There exists C > 0 depending only on N with the following property.If Q ≤ 1 and E is a volume-constrained minimizer of F 1,Q with µ E the corresponding 1/2-harmonic measure i.e.I 1 (E) = I 1 (µ E ), then for every x ∈ R N and 0 < r ≪ 1, Proof.Without loss of generality we may assume that x = 0 and µ E ∈ L 2N N +1 (B r ) since otherwise there is nothing to prove.By Lemma 3.1, there exists a universal constant Λ > 0 (recall that Q ≤ 1) such that E is a minimizer of Arguing as in the proof of Proposition 3.8, we see that it is enough to prove that for every F ⊂ E with E\F ⊂ B r , In order to prove (3.23) we follow the general strategy of [7, Proposition 4.5] and use as a competitor for I 1 (F ).We define the potentials associated to µ E and µ.We recall from (2.5) that u solves on R N the equation and that by (2.7), 1 Let us notice that since u E = I 1 (E) on E, recall (2.8), and since µ Since I 1 (F ) ≤ I 1 (µ) and I 1 (E) = I 1 (µ E ) we have Using Fubini we have and we get .
We now estimate [u . For this, using Hölder inequality and Sobolev embedding we write Finally, by Hölder inequality, This concludes the proof of (3.23).
From Proposition 3.16, we see that in order to prove that E is a perimeter almost-minimizer in the classical sense, it is enough to show decay estimates for µ E L 2N/(N +1) (Br (x)) for x ∈ ∂E.We start by proving the following Hölder estimate for the potentials.Lemma 3.17.For every δ > 0, there exists γ ∈ (0, 1  2 ) with γ → 1 2 as δ → 0 such that if E is a bounded (δ, r 0 )−Reifenberg flat domain then where Proof.By scaling we may assume that r 0 = 1.We follow ideas from the proofs of [25,39] and use the Alt-Caffarelli-Friedman monotonicity formula to show (3.25).
for some exponent γ > 0 with γ → 1 2 as δ → 0 and sup Provided (3.26) and (3.27) hold, we can conclude the proof of (3.25) using Poincaré inequality, Campanato's criterion and v = 0 in E × {0}.Eventually we show that γ → 1/2 as δ → 0. We devote a step for each of these three claims Step 1: Proof of (3.26).To show that (3.26) holds, we first observe that it is enough to consider x ∈ ∂E × {0}.Indeed, assume the statement is proven in that case.Then for x / ∈ ∂E × {0}, using either an odd reflection or an even reflection with respect to x N +1 = 0 we may assume that v is harmonic in B r (x) for every r ≤ r = min (1, d is increasing (this follows for instance from sub-harmonicity of |∇v| 2 which is itself a consequence of Bochner formula).Therefore for any 0 ≤ γ ≤ 1, which proves (3.26) also in this case.
Let now x ∈ ∂E × {0}.Without loss of generality we may assume that x = 0.For every r > 0, let 3 be the first eigenvalue of the Laplacian on the half-sphere with Dirichlet boundary conditions on E. Define then the function and then We claim that for r ∈ (0, 1], the function is increasing.For this we follow almost verbatim the proof of [39, Theorem 2.6] (see also [39,Lemma 2.10 & Lemma 2.11]).In particular, a regularization argument is required to make rigorous all the computations below but we refer the reader to [39] for the details.Computing the logarithmic derivative of Φ, we have and it is therefore enough to prove that We first claim that For this we notice that since Γ = |x| 1−N is the Green function of the Laplacian on R N +1 , we have ∆Γ ≤ 0 and moreover, since it is radially symmetric, we have Using integration by parts we have (using that on This proves (3.29).We thus have A direct computation shows that the above minimum is attained for t min = 1 r γ(r 2 λ(r)) and that min t>0 r γ(r 2 λ(r)) so that eventually This concludes the proof of (3.28).By monotonicity of Φ we have and the proof of (3.26) with γ = γ is completed.
Step 3: Asymptotic on γ and conclusion.We finally show that γ → 1/2 as δ → 0. Since γ(λ) is an increasing function of λ and since for every r > 0, λ(r) is monotone under inclusion (i.e. if we make the dependence in E explicit, then If δ = 0, then γ H0 (r) = 1 2 , see [39, Proposition 2.12].Since γ H δ (r) does not depend on r, it is enough to consider r = 1 and drop the dependence in r.The proof is then concluded observing that δ → λ H δ is continuous as δ → 0. Indeed, this can be proven by an easy Γ−convergence argument.If u δ is a minimizer for λ H δ , then up to normalization we may assume that ∂ + B1 u 2 δ = 1 so that u δ is bounded in H 1 (∂ + B 1 ) and its trace on H δ is bounded in H 1/2 .Therefore, up to extraction it converges weakly in H 1 (∂ + B 1 ) to a function u 0 which vanishes on H 0 (by compact embedding of H in L 2 for instance).Therefore u 0 is admissible for λ H0 and we have We now convert estimate (3.25) on the potential into the desired statement on µ E .
Lemma 3.18.For every γ ∈ (0, 1 2 ), there exists δ 0 > 0 such that for every r 0 > 0 and every loc (R N ) and for every x ∈ R N and r < r 0 /2 there holds where the implicit constant depends on N , γ, r 0 and |E|.
Proof.Let γ = γ(δ) be given by Lemma 3.17.We first derive from (3.25) the following estimate on µ E : If u E denotes the associated potential, = (−∆) = C(N, 1/2) where in the last line we used that if B is a ball of measure |E| then I 1 (E) ≤ I 1 (B).This follows for instance from the fractional Polya-Szëgo inequality [15] and the capacitary definition (2.9) of I 1 (see also [2]).
We now prove (3.30).We may assume without loss of generality that x = 0 and |γ − 1 2 | ≪ 1.For P > 0, we set µ P = min{µ E , P }.Clearly µ P is an integrable function and µ P → µ a.e. in E.Moreover, since 0 ≤ µ P ≤ µ, it satisfies inequality (3.31).We first claim that there exist C 0 , C 1 > 0 such that for every x ∈ ∂E and every r ≤ r 0 /2, there exists a set A(x) ⊂ ∂E such that and Again, there is no loss of generality by restricting ourselves to x = 0. Recall that by Definition 3.13, since E is (δ, r 0 )−Reifenberg flat, for every r ≤ r 0 /2, there exists an hyperplane H r such that In particular, if N r = {y ∈ B r : d(y, H r ) > 2δr} we have for y ∈ N r , d(y, ∂E) ∼ d(y, H r ) so that We now estimate the last term on the right-hand side.By triangle inequality, for every x ∈ N c r ∩B r , d(x, ∂E ∩B r ) ≤ 3δr and thus setting r 1 = 5δr we have that {B r1 (y)} y∈∂E∩Br is a covering of N c r ∩ B r .By Vitali covering Lemma we can extract a finite subset of points A ⊂ ∂E ∩ B r such that • B r1/5 (y) y∈A is made up of pairwise disjoint balls, and

32). The second condition gives
concluding the proof of (3.33).
For k ≥ 0, we set r k = (5δ) k r and define recursively A 0 = {0} and, From (3.32) we have and thus applying recursively (3.33), we find for K ≥ 0, By definition of µ P we have where and thus provided δ is small enough, the sum converges and we have (notice that all the constants involved are independent of P ) Sending P → ∞ concludes the proof of (3.30).Remark 3.19.We point out that this estimate is essentially optimal as can be seen from the case E = B 1 , see [24,Chapter II.13] and Section 4 below.Remark 3.20.A quick inspection of the proof shows that for every q < 2, µ E ∈ L q loc (R N ) if E is δ−Reifenberg flat with δ small enough.This is again optimal in light of the boundary behavior of the 1/2−harmonic measure of the ball see [24,Chapter II.13] and Section 4 below.This higher integrability (with respect to L 2N N +1 ) would however by itself not be sufficient to obtain the regularity of volume-constrained minimizers of F α,Q so that we need the more precise estimate (3.30).
Combining Proposition 3.14 together with Proposition 3.16 and Lemma 3.18 we obtain that for small charge Q every volume constrained minimizer of F 1,Q is also a perimeter almost-minimizer for which the classical regularity theory applies, see [27] so that we have the counterpart of Proposition 3.12.

Rigidity of the ball for small charges
In this section we prove Theorem 1.5 i.e. we show that for every α ∈ (0, 2) and small enough charge Q, the ball is the unique minimizer of F α,Q under volume constraints in the class of nearly spherical sets.
Before embarking in the proofs let us set some notation and make a few preliminary remarks.First, recall that fixing an arbitrary γ ∈ (0, 1), we say that E is nearly spherical if |E| = |B| (where B = B 1 is the unit ball), E has barycenter in 0 and there exists φ : With a slight abuse of notation we still denote by φ its 0−homogeneous extension outside ∂B that is, the function R N ∋ x → φ(x/|x|).We recall from [16] that if E is nearly spherical, we have In particular, if φ W 1,∞ (∂B) ≪ 1, recalling the notation φ = 1 P (B) ∂B φ, we have for s ∈ (0, 1), For µ and ν two Radon measures on R N , we define the positive bilinear operator (see [24]) In particular we have I α (µ) = I α (µ, µ).We let µ E be the optimal measure of E and then u E (x) = E dµE |x−y| N −α be its associated potential.When there is no risk of confusion we drop the index E from both.In the specific case of the unit ball B we have by [24,Chapter II.13] We sometimes use the notation φ x = φ(x).In particular, if E is nearly spherical and φ is the corresponding parametrization, we set for x ∈ B, T (x) = (1 + φ x )x so that E = T (B).We then define g = T −1 #µ E (which is a probability measure on B) so that We can now begin the proofs.We first prove that g has the same behavior as µ B close to ∂B.
Lemma 4.1.Let α ∈ (0, 2) and E be a nearly spherical set.Then its optimal measure satisfies for x ∈ ∂B g(x) (4.4) Proof.The proof resemble the proof of Lemma 3.18 taking advantage of the regularity of E to obtain a sharp estimate.We first show that µ = µ E satisfies µ(x) Recall that by (2.5) and (2.8), Thus, by the boundary regularity theory for the fractional Laplacian developed in [35], Hence, arguing as in the proof of Lemma 3.18 we compute for x ∈ E, To ge the lower bound we may assume that |1 − |x|| ≪ 1. Squaring we get Lemma 4.2.For x, y ∈ B, we have where Proof.Expanding the squares we get Comparing with (4.7) we are left with the proof of For this we write that (φ x + φ y ) 2 = (φ x − φ y ) 2 + 4φ x φ y to get As a consequence we get the following Taylor expansion of and where ψ is the function defined in (4.8).
Proof.Let us first point out that under our hypothesis we have ψ L ∞ ≪ 1.Indeed, this follows from φ W 1,∞ (∂B) ≪ 1 and This inequality may be easily seen using for instance polar coordinates.That is if x = rσ and y = sv then We then obtain the result by (4.7) and Taylor expansion.
The next result contains one of the key linearization estimates we need to obtain our rigidity result.
Proof.In view of (4.9) and (4.3), it is enough to prove that .
By symmetry in x and y we have y so that by (4.13) and the definition (4.8) of ψ, we just need to prove . Using (4.11) and (4.4) we have where we used Young inequality and formula (4.13) to estimate the second term in the last inequality.Thanks to (4.2), we may further reduce the proof of (4.12) to Recalling that µ B (x) (1 − |x|) − α 2 and writing x and y in polar coordinates x = rσ and y = sv, with r, s ∈ R and σ, v ∈ ∂B, we get where We claim that for 0 < θ ≤ 2, It is enough to prove this estimate for θ ≪ 1.To this aim we first estimate where in the last line we did the change of variables s = r + t.We now write that by the change of variables t = rθs, R dt where we used that since θ ≤ 1, s 2 + 1 + θs (s + 1) 2 + |s|.We thus conclude that We now focus on the integral between 1/2 and 1 which we split as , where in the last line we made the change of variables r = 1 − t and s = 1 − t + w.Now for every w ∈ (−1, 1), and thus This proves which together with (4.16) concludes the proof of (4.15).We thus find , which is (4.14).
We may now conclude the proof of the stability inequality for nearly spherical sets. .
We further decompose the term I α (g, φg) as follows: We first estimate I α (g − µ B , φg).We notice that (4.20) We now turn to I α (µ B , φ(g − µ B )).For this we use that u B is constant on B to write Let ρ be a smooth, positive cut-off function with ρ = 1 on B and ρ = 0 on B c 2 .We then set Φ = φρ so that .
Using Young inequality we conclude the proof of (4.17).
We can now conclude the proof of Theorem 1.5.
This implies that either P (E) = P (B) and thus E = B by the isoperimetric inequality or 1 Q 2 which proves the claim.

Non existence in dimension 2
We show here a nonexistence result in dimension 2. Namely, that in N = 2 minimizers in S (and hence classical minimizers) cannot exist for large Q. (5.1) Proof.Let us point out that although the case α = 1 is already covered by [31] (with an explicit threshold between existence and non-existence) we will still include it in the proof.We follow the ideas of [22,Theorem 3.3] in the streamlined version of [13].For ν ∈ ∂B Since |E| = ω N , this yields the estimate 1 Q 2 d 1−α where d = diam(E).If α = 1 this already gives the conclusion so that we are left with the case α < 1.Since for N = 2, P (E) d, we get the lower bound For a generalized set E r made of n copies of the ball of radius r = n −1/2 , we have Optimizing in r by choosing r = Q − 2 1+α , we find by minimality of E, Remark 5.2.While we believe that the same result holds for N ≥ 3, it is well-known that this kind of arguments gives useful information only when α > N − 2, which is compatible with α ≤ 1 only if N = 2.

] 2 H
(4.10) we are left with the proof of B×B ψ(x, y) |x − y| N −α dg x dg y [φ