Michael Goldman
email: goldman@math.univ-paris-diderot.fr

AND Matteo Novaga
email: novaga@dm.unipi.it

Berardo Ruffini
email: berardo.ruffini@umontpellier.fr

On minimizers of an isoperimetric problem with long-range interactions under a convexity constraint

We study a variational problem modeling the behavior at equilibrium of charged liquid drops under convexity constraint. After proving well-posedness of the model, we show C 1,1 -regularity of minimizers for the Coulombic interaction in dimension two. As a by-product we obtain that balls are the unique minimizers for small charge. Eventually, we study the asymptotic behavior of minimizers, as the charge goes to infinity.

Introduction

In this paper we are interested in the existence and regularity of minimizers of the following problem:

min F Q,α (E) : E ⊂ R N convex body, |E| = V . (1.1)
where, for E ⊂ R N , V, Q > 0 and α ∈ [0, N ), we have set

F Q,α (E) := P (E) + Q 2 I α (E).
(1.2)

Here P (E) := H N -1 (∂E) stands for the perimeter of E and, letting P(E) be the set of probability measures supported on the closure of E, we set for α ∈ (0, N ),

I α (E) := inf µ∈P(E) E×E dµ(x) dµ(y) |x -y| α , (1.3)
and for α = 0,

I 0 (E) := inf µ∈P(E) E×E log 1 |x -y| dµ(x) dµ(y). (1.4)
Notice that, up to rescaling, we can assume, as we shall do for the rest of the paper, that V = 1.

Starting from the seminal work of Lord Rayleigh [START_REF] Rayleigh | On the equilibrium of liquid conducting masses charged with electricity[END_REF] (in the Coulombic case N = 3, α = 1), the functional (1.2) has been extensively studied in the physical literature to model the shape of charged liquid drops (see [START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF] and the references therein). In particular, it is known that the ball is a linearly stable critical point for (1.1) if the charge Q is not too large (see for instance [START_REF] Fontelos | Symmetry-breaking bifurcations of charged drops[END_REF]). However, quite surprisingly, the authors showed in [START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF] that, without the convexity constraint, (1.2) never admits minimizers under volume constraint for any Q > 0 and α < N -1. In particular, this implies that in this model a charged drop is always nonlinearly unstable. This result is in sharp contrast with experiments (see for instance [START_REF] Zeleny | Instability of electricfied liquid surfaces[END_REF][START_REF] Taylor | Disintegration of Water Drops in an Electric Field[END_REF]), where there is evidence of stability of the ball for small charges. This suggests that the energy F Q,α (E) does not include all the physically relevant contributions.

As shown in [START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF], a possible way to gain well-posedness of the problem is requiring some extra regularity of the admissible sets. In this paper, we consider an alternative type 1 of constraint, namely the convexity of admissible sets. This assumption seems reasonable as long as the minimizers remain strictly convex, that is for small enough charges. Let us point out that in [START_REF] Muratov | On well-posedness of variational models of charged drops[END_REF], still another regularizing mechanism is proposed. There, wellposedness is obtained by adding an entropic term which prevents charges to concentrate too much on the boundary of E.

Using the good compactness properties of convex sets, our first result is the existence of minimizers for every charge Q > 0.

Theorem 1.1. For every α ∈ [0, N ) and every Q, (1.1) admits a minimizer.

We then study the regularity of minimizers. As often in variational problems with convexity constraints, regularity (or singularity) of minimizers is hard to deal with in dimension larger than two (see [START_REF] Lamboley | Regularity and singularities of optimal convex shapes in the plane[END_REF][START_REF] Lamboley | Estimates of First and Second Order Shape Derivatives in Nonsmooth Multidimensional Domains and Applications[END_REF]). We thus restrict ourselves to N = 2. Since our analysis strongly uses the regularity of equilibrium measures (i.e. the minimizer of (1.3)), we are further reduced to study the case α = N -2 (that is α = 0 in this case). The second main result of the paper is then Theorem 1.2. Let N = 2 and α = 0, then for every Q > 0, the minimizers of (1.1) are of class C 1,1 .

Since we are able to prove uniform C 1,1 estimates as Q goes to zero, building upon our previous stability results established in [START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF], we get Corollary 1.3. If N = 2 and α = 0, for Q small enough, the only minimizers of (1.1) are balls.

The proof of Theorem 1.2 is based on the natural idea of comparing the minimizers with a competitor made by "cutting out the angles". However, the non-local nature of the problem makes the estimates non-trivial. As already mentioned, a crucial point is an estimate on the integrability of the equilibrium measures. This is obtained by drawing a connection with harmonic measures (see Section 3). Let us point out 1 that, up to proving the regularity of the shape functional I 0 and computing its shape derivative, one could have obtained a proof of Theorem 1.2 by applying the abstract regularity result of [START_REF] Lamboley | Regularity and singularities of optimal convex shapes in the plane[END_REF]. Nevertheless, since our proof has a nice geometrical flavor and since regularity of I 0 is not known in dimension two (in higher dimension, one can exploit the connection with the capacity to prove it [START_REF] Jerison | A Minkowski problem for electrostatic capacity[END_REF][START_REF] Crasta | On a long-standing conjecture by Pólya-Szegö and related topics[END_REF]), we decided to keep it as it is. We remark that, differently from the two-dimensional case, when N = 3 we expect the onset of singularities at a critical value Q c > 0, with the shape of a spherical cone with a prescribed angle. Such singularities are also observed in experiments and are usually called Taylor cones (see [START_REF] Taylor | Disintegration of Water Drops in an Electric Field[END_REF][START_REF] Zeleny | Instability of electricfied liquid surfaces[END_REF]). At the moment we are not able to show the presence of such singularities in our model, and this will be the subject of future research.

Eventually, in Section 6, we study the behavior of the optimal sets when the charge goes to infinity. Even though this regime is less significant from the point of view of the applications, we believe that it is still mathematically interesting. Building on Γ-convergence results, we prove 1 this was suggested to us by J. Lamboley Theorem 1.4. Let α ∈ [0, 1) and N ≥ 2. Then, every minimizers E Q of (1.1) satisfies (up to a rigid motion)

Q - 2N (N -1) 1+(N -1)α E Q → [0, L N,α ] × {0} N -1 ,
where the convergence is in the Hausdorff topology and where

L N,α := α(N -1)I α ([0, 1]) N (N -2)/(N -1) ω 1/(N -1) N -1 (N -1) 1+α(N -1)
for α ∈ (0, 1)

and

L N,0 := (N -1) N -1 ω N -1 N N -2 ,
ω N being the volume of the unit ball in R N . For α = 1 and N = 2, 3, we have

Q -2(N -1) N (log Q) -1+1/N E Q → [0, L N,1 ] × {0} N -1 ,
where

L N,1 := 4(N -1)
N (N -2)/(N -1) ω 1/(N -1) N -1 (N -1)/N .
Acknowledgements. The authors wish to thank Guido De Philippis, Jimmy Lamboley and Antoine Lemenant for useful discussions on the subject of this paper. They also thank Cyrill Muratov for suggesting the proof of (6.4). M. Novaga and B. Ruffini were partially supported by the Italian CNR-GNAMPA and by the University of Pisa via grant PRA-2015-0017.

Existence of minimizers

In this section we show that the minimum in (1.1) is achieved. We begin with a simple lemma linking estimates on the energy with estimates on the size of the convex body. Lemma 2.1. Let N ≥ 2, and λ 1 , .., λ N > 0.

Letting E := N i=1 [0, λ i ], V := |E| and Φ := V -N -2 N -1 P (E), it holds 2 max i λ i Φ N -1 and min i λ i ∼ V 1 N -1 Φ -1 , (2.1)
where the involved constants depend only on the dimension. Moreover, letting i max be such that λ imax = max i λ i , it holds for α > 0,

λ imax I α (E) -1/α and λ i I α (E) 1/α Φ N -2 V 1 N -1 for i = i max , (2.2)
and for α = 0,

λ imax exp (-I 0 (E)) and λ i exp (I 0 (E)) Φ N -2 V 1 N -1 for i = i max , (2.3)
where the constants implicitly appearing in (2.2) and (2.3) depend only on N and α.

2 here and in the rest of the paper, we write f g if there exists C > 0 such that f ≤ Cg. If f g and g f , we will simply write f ∼ g Proof. Without loss of generality, we can assume that

λ 1 ≥ λ 2 ≥ • • • ≥ λ N . Then, since V = N i=1 λ i and P (E) N -1
i=1 λ i , taking the ratio of these two quantities, we obtain

λ N V P (E) -1 = V 1 N -1 Φ -1 . Now, since the λ i are decreasing (in particular λ i ≥ λ N for all i), this implies Φ V -N -2 N -1 N -1 i=1 λ i = V -N -2 N -1 λ 1 N -1 i=2 λ i V -N -2 N -1 λ 1 V N -2 N -1 Φ -(N -2) , yielding (2.1). Assume now that α > 0. Then, from diam(E) ∼ λ 1 , we get I α (E) λ -α 1 . If N = 2, together with λ 1 λ 2 = V , this implies (2.2). If N ≥ 3, we infer as above that Φ V -N -2 N -1 λ 1 λ 2 N -1 i=3 λ i V -N -2 N -1 I α (E) -1 α λ 2 V N -3 N -1 Φ -(N -3) V -1 N -1 Φ -(N -3) I α (E) -1 α λ 2 .
This gives (2.2). The case α = 0 follows analogously, using the fact that

I 0 (E) ≥ C - log λ 1 .
The next result follows directly from John's lemma [START_REF] John | Extremum problems with inequalities as subsidiary conditions[END_REF].

Lemma 2.2.

There exists a dimensional constant C N > 0 such that for every convex body E ⊂ R N , up to a rotation and a translation, there exists

R := N i=1 [0, λ i ], such that R ⊆ E ⊆ C N R.
As a consequence diam(E) ∼ diam(R), |E| ∼ |R|, P (E) ∼ P (R) and I α (E) ∼ I α (R) for α > 0 (and exp(-I 0 (E)) ∼ exp(-I 0 (R))).

With these two preliminary results at hand, we can prove existence of minimizers for (1.1).

Theorem 2.3. For every Q > 0 and α ∈ [0, N ), (1.1) has a minimizer.

Proof. Let E n be a minimizing sequence and let us prove that diam(E n ) is uniformly bounded. Let R n be the parallelepipeds given by Lemma 2.2. Since diam(E n ) ∼ diam(R n ), it is enough estimating diam(R n ) from above. Let us begin with the case α > 0. In this case, since I α (R n ) ≥ 0, by (2.1), applied with V = 1, we get

diam(R n ) P (R n ) N -1 F Q,α (E n ) N -1 .
In the case α = 0, from (2.1) and (2.3) applied to V = 1, we get

P (R n ) exp - I 0 (R n ) N -1 so that F Q,0 (R n ) exp - I 0 (R n ) N -1 + I 0 (R n ),
from which we obtain that |I 0 (R n )| is bounded and thus also P (R n ) is bounded, whence, arguing as above, we obtain a uniform bound on diam(R n ).

Since the E n 's are convex sets, up to a translation, we can extract a subsequence which converges in the Hausdorff (and L 1 ) topology to some convex body E of volume one. Since the perimeter functional is lower semicontinuous with respect to the L 1 convergence, and the Riesz potential I α is lower semicontinuous with respect to the Hausdorff convergence (see [START_REF] Landkof | Foundations of modern potential theory[END_REF][START_REF] Saff | Logarithmic potentials with external fields[END_REF] and [11, Prop. 2.2]), we get that E is a minimizer of (1.1).

Regularity of the planar charge distribution for the logarithmic potential

In this section we focus on the case N = 2 and α = 0. Relying on classical results on harmonic measures, we show that for every convex set E, the corresponding optimal measure µ for I 0 (E) is absolutely continuous with respect to H 1 ∂E with L p estimates. Upon making that connection between µ and harmonic measures, this fact is fairly classical. However, since we could not find a proper reference, we recall (and slightly adapt) few useful results. Let us point out that most definitions and results of this section extend to the case N ≥ 3 and α = N -2, and to more general classes of sets. In particular, for bounded Lipschitz sets, the fact that harmonic measures are absolutely continuous with respect to the surface measure with L p densities for p > 2 was established in [START_REF] Dahlberg | Estimates of harmonic measure[END_REF], and extended later to more general domains (see for instance [START_REF] Kenig | Free boundary regularity for harmonic measures and Poisson kernels[END_REF][START_REF] Kenig | Harmonic measure on locally flat domains[END_REF][START_REF] Jerison | Boundary behavior of harmonic functions in nontangentially accessible domains[END_REF]). The interest for harmonic measures stems from the fact that they bear a lot of geometric information (see in particular [START_REF] Alt | Existence and regularity for a minimum problem with free boundary[END_REF][START_REF] Kenig | Free boundary regularity for harmonic measures and Poisson kernels[END_REF]). The main result of this section is the following. Theorem 3.1. Let E n be a sequence of compact convex bodies converging to a convex body E and let µ n be the associated equilibrium measures. Then, µ n = f n H 1 ∂E n and there exists p > 2 and M > 0 (depending only on E) such that

f n ∈ L p (∂E n ) with f n L p (∂En) ≤ M.
Moreover, if E is smooth, then p can be taken arbitrarily large. Remark 3.2. By applying the previous result with E n = E, we get that that the equilibrium measure of a convex set is always in some L p (∂E) with p > 2. We stress also that the exponent p and the bound on the L p norm of its equilibrium measure depend indeed on the set: for instance, a sequence of convex sets with smooth boundaries converging to a square cannot have equilibrium measures with densities uniformly bounded in L p for p > 4.

We will denote here Ω := E c . Let us recall the definition of harmonic measures (see [START_REF] Garnett | of New Mathematical Monographs[END_REF][START_REF] Kenig | Free boundary regularity for harmonic measures and Poisson kernels[END_REF]). Definition 3.3. Let Ω be a Lipschitz open set (bounded or unbounded) such that R 2 \∂Ω has two connected components, and let X ∈ Ω, we denote by G X Ω the Green function of Ω with pole at X i.e. the unique distributional solution of -∆G X Ω = δ X in Ω and G X Ω = 0 on ∂Ω, and by ω X Ω the harmonic measure of Ω with pole at X, that is the unique (positive) measure such that for every f ∈ C 0 (∂Ω), the solution u of

-∆u = 0 in Ω and u = f on ∂Ω, satisfies u(X) = ∂Ω f (y)dω X Ω (y).
If Ω is unbounded with ∂Ω bounded and 0 ∈ Ω c , we call ω ∞ Ω the harmonic measure of Ω with pole at infinity, that is the unique probability measure on ∂Ω satisfying

∂Ω φdω ∞ = Ω u∆φ ∀φ ∈ C ∞ c (R 2 )
where u is the solution of

         -∆u = 0 in Ω u > 0 in Ω u = 0 on ∂Ω lim |z|→+∞ u(z) -1
2π log |z| exists and is finite.

(3.1)
When it is clear from the context, we omit the dependence of G X , ω X or ω ∞ on the domain Ω.

Remark 3.4. For smooth domains, it is not hard to check that ω X = ∂ ν G X H 1 ∂Ω, and that ω ∞ = ∂ ν uH 1 ∂Ω where ν is the inward unit normal to Ω. Moreover, for Ω unbounded, if h ∞ is the harmonic function in Ω with h ∞ (z) = -1 2π log |z| on ∂Ω, then the function u from (3.1) can also be defined by u(z) = 1 2π log |z| + h ∞ (z). We can now make the connection between harmonic measures and equilibrium measures. For E a Lipschitz bounded open set containing 0, let µ be the optimal measure for I 0 (E) and let

v(x) := ∂E -log(|x -y|)dµ(y). Since -∆v = 2πµ in R 2 , v < I 0 (E) in E c and v = I 0 (E) on ∂E, if we let u := (2π) -1 (I 0 (E) -v), we see that it satisfies (3.1) for Ω = E c . Therefore, µ = ω ∞ E c (recall that µ(∂E) = 1)
. For Lipschitz sets Ω, it is well-known that ω ∞ is absolutely continuous with respect to H 1 ∂Ω with density in L p (∂Ω) for some p > 1 (see [START_REF] Garnett | of New Mathematical Monographs[END_REF]Th. 4.2]). However, we will need a stronger result, namely that it is in L p (∂Ω) for some p > 2, with estimates on the L p norm depending only on the geometry of Ω.

Given a convex body E and a point x ∈ ∂E, we call angle of ∂E at x the angle spanned by the tangent cone ∪ λ>0 λ(E -x).

We now state a crucial lemma which relates in a quantitative way the regularity of E with the integrability properties of the corresponding harmonic measure. This result is a slight adaptation of [28, Thm. 2]. Lemma 3.5. Let E be a convex body containing the origin in its interior, let ζ ∈ (0, π] be the minimal angle of ∂E, and let

p c := π π-ζ + 1 if ζ < π and p c := +∞ if ζ = π.
Let also E n be a sequence of convex bodies converging to E in the Hausdorff topology. Then, for every 1 ≤ p < p c , there exists C(p, ∂E) such that for n large enough (depending on p), every conformal map

ψ n : E c n → B 1 with ψ n (∞) = 0 satisfies ∂En |ψ n | p ≤ C(p, ∂E), (3.2)
where we indicate by |ψ n | the absolute value of the derivative of ψ n (seen as a complex function). In particular, for n large enough, ψ n ∈ L p (∂E n ) for some p > 2.

Proof. The scheme of the proof follows that of [28, Thm. 2, Eq. ( 9)], thus we limit ourselves to point out the main differences. We begin by noticing that although [START_REF] Warschawski | On conformal mapping of certain classes of Jordan domains[END_REF]Thm. 2] is written for bounded sets, up to composing with the map z → z -1 this does not create any difficulty. We first introduce some notation from [START_REF] Warschawski | On conformal mapping of certain classes of Jordan domains[END_REF]. Given a convex body E we let ∂E = {γ(s) : s ∈ [0, L]} be an arclength parametrization of ∂E. Notice that, for every s, the left and right derivatives γ ± (s) exist and the angle v(s) between γ (s) and a fixed direction, say e 1 , is a function of bounded variation. Up to changing the orientation of ∂E, we can assume that v is increasing. We then let

η := max s [v(s + ) -v(s -)] ≥ 0. Notice that ζ = π -η is the minimal angle of ∂E.
Letting ϕ n := ψ -1 n , we want to prove that there exists C(p, ∂E) such that

∂B 1 |ϕ n | -p ≤ C(p, ∂E),
for n large enough and for p < π/η By a change of variables, this yields (3.2). Let p < p < π/η, and let as in [START_REF] Warschawski | On conformal mapping of certain classes of Jordan domains[END_REF],

h := 1 2π (pη + π) and h := 1 2π (p η + π), so that πh p > πh p > η.
Let now v n (resp. v)be the angle functions corresponding to the sets E n (resp. E). As in [START_REF] Warschawski | On conformal mapping of certain classes of Jordan domains[END_REF], there exists δ > 0 such that for s -

s ≤ δ, v(s) -v(s ) ≤ πh p .
By the convexity of E n and by the convergence of E n to E, for n large enough and for s -s ≤ δ we get that

v n (s) -v n (s ) ≤ πh p . Let L n := H 1 (∂E n ) and let us extend v n to R by letting for s ≥ 0, v n (s) := v n (L n s/L n )+ v n (s -L n s/L n )
, and similarly for s ≤ 0, so that v n is an increasing function with (v n ) periodic of period L n . Let now k n := L n /δ ∈ N and δ n := L/k n . By the convergence of E n to E, k n and δ n are uniformly bounded from above and below. For t ∈ [0, δ n ], and

0 ≤ j ≤ k n , let s t j := t + jδ n . Since δn 0 kn-1 j=0 s t j+1 s t j v n (s) -v n (s t j ) s -s t j dsdt = kn-1 j=0 δn 0 δn 0 v n (s + t + jδ n ) -v n (t + jδ n ) s dtds = δn 0 1 s kn-1 j=0 δn 0 v n (s + t + jδ n ) -v n (t + jδ n )dtds = δn 0 1 s Ln+s Ln v n (t)dt - s 0 v n (t)dt ds ≤ 2δ n sup [0,2Ln] |v n | δ n v ∞ , we can find t ∈ (0, δ n ) such that kn-1 j=0 s t j+1 s t j v n (s) -v n (s t j ) s -s t j ds v ∞ .
For notational simplicity, let us simply denote s j := s t j . Arguing as above, we can further assume that

kn-1 j=0 s j+1 s j v n (s j+1 ) -v n (s) s j+1 -s ds v ∞ .
The proof then follows almost exactly as in [28, Thm. 2], by replacing the pointwise quantity

G n j := sup s j <s<s j+1 v n (s) -v n (s j ) s -s j ,
by the integral ones. There is just one additional change in the proof: letting 0 ≤ λ n j := v n (s j+1 ) -v n (s j ) ≤ πh p , we see that in the estimates of [28, Thm. 2], the quantity max λ n j =0 1/λ n j appears and could be unbounded in n. Let γ n (s) be the arclength parametrization of ∂E n and let θ n (s) be such that γ n (s) = ϕ n (e iθn(s) ). For 0 < r < 1 and

j ∈ [0, k n -1], if λ n j = 0, we have 1 λ n j s j+1 s j dv n (s) θn(s j+1 ) θn(s j ) dt |e iθn(s) -re it | h 1 1 -h .
Using this estimate, the proof can be concluded exactly as in [28, Thm. 2].

We can now prove Theorem 3.1.

Proof of Theorem 3.1. Without loss of generality we can assume that the sets E n and E contain the origin in their interior. As observed above, we then have

µ n = ω ∞ E c n . Let ψ n be a conformal mapping from E c n to B 1 with ψ n (∞) = 0. We have µ n = ω ∞ E c n = (ψ -1 n ) ω 0 B 1 = (ψ -1 n ) H 1 ∂B 1 2π = |ψ n | 2π H 1 ∂E n .
Then, Lemma 3.5 gives the desired estimate.

We will also need a similar estimate for C 1,β sets.

Lemma 3.6. Let E be a convex set with boundary of class C 1,β . Then, the optimal charge distribution µ is of class C 0,β and in particular it is in L ∞ (∂E). Moreover, µ C 0,β depends only on the C 1,β norm of ∂E.

Proof. Up to translation we can assume that 0 ∈ E with dist(0, ∂E) ≥ c (with c depending only on the C 1,β character of ∂E). By [24, Thm. 3.6], there exists a conformal mapping

ψ of class C 1,β which maps E c into B 1 with ψ(∞) = 0 and ψ C 1,β (E c ) controlled by the C 1,β character of ∂E.
Since, as before, µ = (ψ -1 ) ω 0 B 1 , and the claim follows by Lemma 3.5. In this section we show that any minimizer of (1.1) has boundary of class C 1,1 . We begin by showing that we can drop the volume constraint, by adding a volume penalization to the functional. This penalization is commonly used in isoperimetric type problems (see for instance [START_REF] Esposito | A remark on a free interface problem with volume constraint[END_REF][START_REF] Goldman | Volume-constrained minimizers for the prescribed curvature problem in periodic media[END_REF] and references therein). Let Λ be a positive number and define the functional

G Λ (E) := P (E) + Q 2 I 0 (E) + Λ ||E| -1| . Lemma 4.1. For every Q 0 > 0, there exists Λ > 0 such that, if Λ > Λ and Q ≤ Q 0 , the minimizers of min E⊆R 2 , E convex G Λ (E) (4.1)
are also minimizers of (1.1) and vice-versa.

Proof. By repeating the proof of Theorem 2.3 it is easy to show that for any Λ > 0, G Λ admits a minimizer. Notice that the minimum in (4.1) is always less or equal than the minimum in (1.1). We are thus left to prove the opposite inequality. Let Λ > 0 be given and let E be a minimizer of G Λ . Assume that E is not a minimizer for F Q,0 . In this case we get that σ := ||E| -1| > 0.

Let B be a ball with

|B| = 1. Then diam(E) -Q 2 log(diam(E)) ≤ G Λ (E) ≤ G Λ (B) = F Q,0 (B) 1,
where the constant involved depends only on Q 0 . Therefore diam(E) is bounded by a constant depending only on Q 0 . From this, we deduce that Λσ is itself also bounded by a constant (again depending only on Q 0 ). From now on we assume that |E| < 1, or equivalently, |E| = 1 -σ, since the other case is analogous. Let us define

F := 1 (1 -σ) 1 2 E, so that |F | = 1.
Then, by the minimality of E, the homogeneity of the perimeter and recalling that

I 0 (λE) = I 0 (E) -log(λ), a Taylor expansion gives Λσ = G Λ (E) -F Q,0 (E) ≤ G Λ (F ) -F Q,0 (E) = P (E) (1 -σ) -1 2 + Q 2 I 0 (E) + 1 2 log(1 -σ) -F Q,0 (E) ≤ P (E)((1 -σ) -1 2 -1) ≤ P (E) 2 σ, so that Λ ≤ P (E) 2 1.
Therefore, if Λ is large enough, we must have σ = 0 or equivalently that E is also a minimizer of F Q,0 .

Let now E be a minimizer of (4.1). In order to prove the regularity of E, we shall construct a competitor in the following way: since E is a convex body, there exists ε 0 such that for ε ≤ ε 0 , and every

x 0 ∈ ∂E, we have ∂E ∩ ∂B ε (x 0 ) = {x ε 1 , x ε 2 } (in particular |x 0 -x ε i | = ε). Let us fix x 0 . For ε ≤ ε 0 , let x ε 1 ,
x ε 2 be given as above and let L ε be the line joining x ε 1 to x ε 2 . Denote by H + ε the half space with boundary L ε containing x 0 (and H - ε be its complementary). We then define our competitor as

E ε := E ∩ H - ε .
Let us fix some further notation (see Figure 1):

-We denote by Π : ∂E ∩ H + ε → L ε the projection of the cap of ∂E inside H + ε , on L ε . We shall extend Π to the whole ∂E as the identity, outside ∂E ∩ H + ε . -If f H 1 ∂E is the optimal measure for I 0 (E), we let f ε := Π f (which is defined on ∂E ε ) so that µ ε := f ε H 1 ∂E ε is a competitor for I 0 (E ε ). -For x, y ∈ ∂E, we denote by γ ε (x, y) the acute angle between the line L x,y joining

x to y and

L ε (if L x,y is parallel to L ε , we set γ ε (x, y) = 0). -If y = x 0 , then we denote γ ε (x) := γ ε (x, x 0 ). -We let γ ε := γ ε (x ε 1 ) = γ ε (x ε 2 ). -Let ∂B 3ε (x 0 )∩∂E = {x 3ε 1 , x 3ε 2 }.
As before, we define H + 3ε as the half space bounded by L x 3ε 1 ,x 3ε 2 containing x 0 and H - 3ε its complementary. We then let Σ

ε := ∂E ∩ H + ε , Σ 3ε := ∂E ∩ H + 3ε and Γ ε := ∂E ∩ H - 3ε . -We let ∆V := |E| -|E ε |, ∆P := P (E) -P (E ε ) and ∆I 0 := I 0 (E ε ) -I 0 (E).
We point out some simple remarks:

-Thanks to Theorem 3.1 we have that the optimal measure f satisfies f ∈ L p (∂E) for some p = p(E) > 2.

-If E is a convex body then γ ε is bounded away from π 2 and |x 3ε

1 -x ε 1 | ∼ |x 3ε 2 -x ε 2 | ∼ ε.
-The quantities ∆V , ∆P and ∆I 0 are nonnegative by definition.

-All the constants involved up to now depend only on the Lipschitz character of ∂E. In particular, if E n is a sequence of convex bodies converging to a convex body E, then these constants depend only on the geometry of E. Before stating the main result of this section, we prove two simple regularity lemmata. Proof. Let x 0 ∈ ∂E be fixed. Since E is convex, there exist R > 0 and a convex function

u : I → R such that ∂E ∩ B R (x 0 ) = {(t, u(t)) : t ∈ I} for some interval I ⊂ R. Furthermore, u L ∞ 1. Let x ∈ ∂E ∩ B R (x 0 )
. Without loss of generality, we can assume that x = 0 = (0, u(0)). By convexity of u, up to adding a linear function, we can further assume that u ≥ 0 in I. Thanks to the Lipschitz bound on u, for x = (t, u(t)) ∈ ∂E ∩ B R (x 0 ), we have

|x| = (t 2 + |u(t)| 2 ) 1/2 ∼ t. (4.3) Let now ε > 0. For δ > 0, let -1 t δ 1 < 0 < t δ 2 1 such that x i δ = (t δ i , u(t δ i
)) for i = 1, 2 (see the notation above). By (4.3), there exists λ > 0 depending only on the Lipschitz character of u, such that |t λε i | ≥ ε. Without loss of generality, we can now assume that u(-ε) ≤ u(ε). In particular, considering the ∆V associated to λε, we have that (see Figure 2)

∆V ≥ 2εu(ε) - 2ε(u(ε) -u(-ε)) 2 - ε -ε u(t) dt = ε(u(ε) + u(-ε)) - ε -ε u(t) dt .
Calling ψ(z) := This proves that there exists ε 1 > 0 such that for 0

≤ ε ≤ ε 1 , ψ(ε) ε 2+β .
Putting this back into (4.4) and dividing by ε, we get

u(-ε) + u(ε) ε 1+β + ε -1 ψ(ε) ε 1+β .
In other words, we have proven that u is differentiable in zero with u (0) = 0 and that for |ε| small enough, |u(ε) -u(0) -u (0)ε| ε 1+β . Since the point zero was arbitrarily chosen, this yields that u is differentiable everywhere and that for t, s ∈ I with |t -s| ≤ ε 1 ,

|u(t) -u(s) -u (s)(t -s)| |t -s| β+1 ,
which is equivalent to the C 1,β regularity of ∂E 3 . Lemma 4.3. Suppose that the minimizer E for (4.1) has boundary of class C 1,β , for some 0 < β < 1. Then, for every

x 0 ∈ ∂E, x ∈ Σ ε and y ∈ B R (x 0 ), γ ε (x, y) ε β + |x -y| β . (4.5)
Proof. Without loss of generality, we can assume that x 0 = 0. As in the proof of Lemma 4.2, since E is convex and of class C 1,β , in the ball B R (0), for a small enough R, ∂E is a graph over its tangent of a C Let ξ ε be the angle between L ε and the horizontal line (see Figure 3). Since γ ε (x, y) = γε ± ξ ε , (4.5) holds provided that we can show

ξ ε ε β . (4.7) Let t ε 1 , t ε 2 ∼ ε be such that x ε 1 = (-t ε 1 , u(-t ε 1 )) and x ε 2 = (t ε 2 , u(t ε 2
)). We see that ξ ε is maximal if u(-t ε 1 ) = 0, and then

t ε 1 = ε. In that case, tan ξ ε = u(t ε 2 ) ε+t ε 2 . Since u(t ε 2 ) ε 1+β , and t 2 ε ε, we obtain ξ ε ∼ tan ξ ε ε 1+β ε = ε β ,
proving (4.7). This concludes the proof of (4.5).

We pass now to the main result of this section.

Theorem 4.4. Every minimizer of (4.1) is C 1,1 . Moreover, for every Q 0 and every Q ≤ Q 0 , the C 1,1 character of ∂E depends only on Q 0 , the Lipschitz character of ∂E and f L p (∂E) .

Proof. Let E be a minimizer of (4.1), x 0 ∈ ∂E be fixed and let ε ≤ ε 0 . With the above notation in force, we begin by observing that using E ε as a competitor, by minimality of E for (4.1), and by means of the estimate

||E| -1| -||E ε | -1| ≥ -||E| -|E ε || = -∆V,
we get

Q 2 ∆I 0 ≥ ∆P -Λ∆V. (4.8)
We are thus going to estimate ∆I 0 , ∆P and ∆V in terms of ε and γ ε . This will give us a quantitative decay estimate for γ ε . This in turn, in light of (4.9) below and Lemma 4.2, will provide the desired regularity of E.

Step 4). Therefore, we obtain

∆V ∼ ε 2 cos γ ε sin γ ε ∼ ε 2 γ ε .
Step 2 (Perimeter estimate): Since the triangle with vertices

x 0 , x ε 1 , x ε 2 is contained inside E ∩ H + ε , it holds ∆P = P (E) -P (E ε ) ≥ 2ε (1 -cos γ ε ) εγ 2 ε . (4.10)
Step 3 (Non-local energy estimate): We now estimate ∆I 0 . Since µ ε is a competitor for I 0 (E ε ), recalling that Π is the identity outside Σ ε , we have

∆I 0 = I 0 (E ε ) -I 0 (E) ≤ ∂Eε×∂Eε f ε (x)f ε (y) log 1 |x -y| - ∂E×∂E f (x)f (y) log 1 |x -y| = ∂E×∂E f (x)f (y) log 1 |Π(x) -Π(y)| - ∂E×∂E f (x)f (y) log 1 |x -y| = ∂E×∂E f (x)f (y) log |x -y| |Π(x) -Π(y)| .
∆I 0 ≤ Σ 3ε ×Σ 3ε f (x)f (y) log |x -y| |Π(x) -Π(y)| + 2 Σε Γε f (x)f (y) log |x -y| |Π(x) -y| =: I 1 + 2I 2 .
We first estimate I 1 :

I 1 = Σ 3ε ×Σ 3ε f (x)f (y) log 1 + |x -y| -|Π(x) -Π(y)| |Π(x) -Π(y)| ≤ Σ 3ε ×Σ 3ε f (x)f (y) |x -y| -|Π(x) -Π(y)| |Π(x) -Π(y)| .
Since for any x, y ∈ Σ 3ε we have (with equality if x, y ∈ Σ ε ),

cos(γ ε (x, y))|x -y| ≤ |Π(x) -Π(y)|,
we get Using then Hölder's inequality (recall that f ∈ L p (∂E) for some p > 2) to get

I 1 ≤ Σ 3ε ×Σ 3ε f (x)f (y) 1 cos(γ ε (x, y)) -1 Σ 3ε ×Σ 3ε γ 2 ε (x, y)f (x)f (y). ( 4
Σ 3ε f ≤ Σ 3ε f p 1/p H 1 (Σ 3ε ) p-1 p ε p-1 p , (4.12)
and γ ε (x, y) 1, we obtain

I 1 ε 2 p-1 p . ( 4

.13)

We can now estimate I 2 :

I 2 = Σε Γε f (x)f (y) log 1 + |x -y| -|Π(x) -y| |Π(x) -y| ≤ Σε Γε f (x)f (y) |x -y| -|Π(x) -y| |Π(x) -y| .
Denote by z the projection of Π(x) on the line containing x and y. Then, since the projection is a 1-Lipschitz function, it holds |z -y| ≤ |Π(x) -y|. Thus,

|x -y| -|y -Π(x)| = |x -z| + |z -y| -|y -Π(x)| ≤ |x -z|.

Arguing as in

Step 1, we get |x -

Π(x)| ≤ |x ε 2 -x ε 2 | εγ ε .
Furthermore, the angle zΠ(x)x equals γ ε (x, y) (see Figure 5), so that Therefore,

|x -y| -|y -Π(x)| ≤ |x -z| = |x -Π(x)| sin(γ ε (x, y)) εγ ε γ ε (x, y).
I 2 εγ ε Σε Γε f (x)f (y)γ ε (x, y) |y -x| . (4.14)
There exists M > 0 which depends only on the Lipschitz character of ∂E such that for

x ∈ Σ ε and y ∈ Γ ε ∩ B M (x 0 ), |y -x| ≥ min i=1,2 |y -x ε i |. Let Γ N ε := Γ ε ∩ B M (x 0 ) and Γ F ε := Γ ε ∩ B c M (x 0
). We then have

I 2 εγ ε Σε×Γ N ε f (x)f (y)γ ε (x, y) min i |y -x ε i | + Σε×Γ F ε f (x)f (y)γ ε (x, y) =: I N 2 + I F 2 .
We begin by estimating I F 2 . Since γ ε (x, y) 1, using Hölder's inequality we find

I F 2 εγ ε Γε f Σε f ≤ εγ ε f L p H 1 (Γ ε ) 1-1 p f L p H 1 (Σ ε ) 1-1 p εγ ε H 1 (Σ ε ) 1-1 p ε 2-1 p γ ε . (4.15)
We can now estimate I N 2 . Recall that

I N 2 := εγ ε Σε×Γ N ε f (x)f (y)γ ε (x, y) min i |y -x ε i | . (4.16)
As before, we use γ ε (x, y) 1 together with Hölder's inequality applied twice to get

Σε×Γ N ε f (x)f (y)γ ε (x, y) min i |y -x ε i | ε 1-1/p Γ N ε 1 min i |y -x ε i | p/(p-1) (p-1)/p
.

Since E is convex, its boundary can be locally parametrized by Lipschitz functions so that, if M is small enough (depending only on the Lipschitz regularity of ∂E), then for y ∈ Γ N ε , min i (y, xε i ) ∼ min i |y -xε i | (where (x, y) denotes the geodesic distance on ∂E). From this we get 1) . From this we conclude that

Γ N ε 1 min i |y -x ε i | p/(p-1) ε -1/(p-
I N 2 γ ε ε 2-2 p .
(4.17)

Step 4 (C 1,β regularity): We now prove that E has boundary of class C 1,β . To this aim, we can assume that ∆V ∆P . Indeed, if ∆V ∆P , thanks to (4.9) and (4.10), we would get γ ε ε and thus ∆V ε 3 , which by Lemma 4.2 would already ensure the C 1,1 regularity of ∂E. Using (4.8), (4.10), (4.13), (4.15) and (4.17), we get

Q 2 (ε 1-2 p + γ ε (ε 1-1 p + ε 1-2 p )) γ 2 ε . (4.18)
Now since ε

1-1 p ε 1-2
p , this reduces further to

Q 2 (ε 1-2 p + γ ε ε 1-2 p ) γ 2 ε . (4.19)
We can now distinguish two cases. Either

Q 2 ε 2( 1 2 -1 p )
γ 2 ε and then γ ε Qε

( 1 2 -1 p ) or Q 2 γ ε ε 1-2 p γ 2 ε and then γ ε Q 2 ε 1-2 p .
Thus in both cases, since p > 2, we find γ ε Qε β for some β > 0 and we can conclude, by means of (4.9) and Lemma 4.2, that ∂E is C 1,β .

Step 5 (C 1,1 regularity): Thanks to Lemma 3.6, we get that f ∈ L ∞ with f L ∞ depending only on the Lipschitz character of ∂E and on f L p . Using this new information, we can improve (4.13), (4.15) and (4.17) to

I 1 ε 2 , I F 2 γ ε ε 2 , and I N 2 γ ε ε 2 | log ε|. (4.20)
Arguing as in Step 4, we find γ ε Qε 1/2 and thus ∂E is of class C 1,1/2 . In order to get higher regularity, we need to get a better estimate on γ ε (x, y). Going back to (4.11) and using (4.5) with β = 1/2, we find the improved estimate

I 1 ε 3 . (4.21)
If we also use (4.5) in (4.16), we obtain

I N 2 εγ ε Σε×Γ N ε ε 1/2 + |x -y| 1/2 min i |y -xε i | εγ ε Σε×Γ N ε ε 1/2 + min i {|x -xε i | 1/2 + |y -xε i | 1/2 } min i |y -xε i | εγ ε Σε×Γ N ε ε 1/2 + min i |y -xε i | 1/2 min i |y -xε i | ε 2 γ ε Γ N ε ε 1/2 min i |y -xε i | + 1 min i |y -xε i | 1/2 ε 2 γ ε (ε 1/2 | log ε| + 1) ε 2 γ ε .
As in the beginning of Step 4, we can assume that ∆V ∆P , so that by (4.8) and (4.10) we have Q 2 ∆I 0 ∆P εγ 2 ε . By the previous estimate for I N 2 , (4.21) and the second inequality in (4.20) we eventually get

Q 2 ε 2 γ ε ∼ Q 2 (ε 3 + ε 2 γ ε ) εγ 2 ε ,
which leads to γ ε Q 2 ε. By using again Lemma 4.2, the proof is concluded.

Minimality of the ball for N = 2 and Q small

We now use the regularity result obtained in Section 4 to prove that for small charges, the only minimizers of F Q,0 in dimension two are balls.

Theorem 5.1. Let N = 2 and α = 0. There exists Q 0 > 0 such that for Q < Q 0 , up to translations, the only minimizer of (1.1) is the ball.

Proof. Let E Q be a minimizer of F Q,0 and let B be a ball of measure one. By minimality of E Q , we have

P (E Q ) -P (B) ≤ Q 2 (I 0 (B) -I 0 (E Q )) ≤ Q 2 I 0 (B).
(

Using the quantitative isoperimetric inequality (see [START_REF] Fusco | The sharp quantitative isoperimetric inequality[END_REF]), we infer

|E Q ∆B| 2 P (E Q ) -P (B) ≤ Q 2 I 0 (B).
This implies that E Q converges to B in L 1 as Q → 0. From the convexity of E Q , this implies the convergence also in the Hausdorff metric. Since the sets E Q are all uniformly bounded and of fixed volume, they are uniformly Lipschitz. Theorem 4.4 then implies that ∂E Q are C 1,1 -regular sets with C 1,1 norm uniformly bounded. Therefore, thanks to the Arzelà-Ascoli's Theorem, we can write

∂E Q = {(1 + ϕ Q (x))x : x ∈ ∂B} ,
with ϕ Q C 1,β converging to 0 as Q → 0 for every β < 1. From Lemma 3.6 we infer that the optimal measures µ Q for E Q are uniformly C 0,β and in particular are uniformly bounded. Using now [11, Prop 6.3], we get that for small enough Q, µ Q

2

L ∞ (P (E Q ) -P (B)) I 0 (B) -I 0 (E Q ) Putting this into (5.1), we then obtain

P (E Q ) -P (B) Q 2 (P (E Q ) -P (B))
from which we deduce that for Q small enough, P (E Q ) = P (B). Since, up to translations, the ball is the unique solution of the isoperimetric problem, this implies E Q = B.

Asymptotic behavior as Q → +∞

In this section we characterize the limit shape of (suitably rescaled) minimizers of F Q,α , with α ∈ [0, 1], as the charge Q tends to +∞. For this, we fix a sequence Q n → +∞.

6.1. The case α ∈ [0, 1). For n ∈ N, we let V n := Q - 2N (N -1) 1+(N -1)α n (so that V n → 0 as n → +∞) and A n,α := E ⊂ R N convex body, |E| = V n , F n,α (E) := V -N -2 N -1 n P (E) + I α (E) for E ∈ A n,α .
It is straightforward to check that if E is a minimizer of (1.1), then the rescaled set

E := Q - 2(N -1) 1+(N -1)α n E is a minimizer of F n,α in the class A n,α .
We begin with a compactness result for a sequence of sets of equibounded energy. Proposition 6.1. Let α ∈ [0, 1) and let E n ∈ A n,α be such that

sup n F n,α (E n ) < +∞.
Then, up to extracting a subsequence and up to rigid motions, the sets E n converge in the Hausdorff topology to the segment [0, L] × {0} N -1 , for some L ∈ (0, +∞).

Proof. The bound on I α (E n ) directly implies with (2.2) (or (2.3) in the case α = 0) that the diameter of E n is uniformly bounded from below.

Let us show that the diameter of E n is also uniformly bounded from above. Arguing as in Theorem 2.3, let R n = N i=1 [0, λ n i ] be the parallelepipeds given by Lemma 2.2, and assume without loss of generality that

λ n 1 ≥ λ n 2 ≥ • • • ≥ λ n N .
In the case α > 0, (2.1) directly gives the bound while for α = 0, we get using (2.1) and (2.3), that |I 0 (R n )| is uniformly bounded, from which the bound on the diameter follows, using once again (2.1). Moreover, from (2.2) and (2.3), we obtain that

λ n i ∼ V 1 N -1 n
(where the constants depend on F n,α (E n )), for i = 2, . . . , N . The convex bodies E n are therefore compact in the Hausdorff topology and any limit set is a non-trivial segment of length L ∈ (0, +∞).

In the proof of the Γ-convergence result we will use the following result. Lemma 6.2. Let 0 < γ < β with β ≥ 1, V > 0 and L > 0, then

min L 0 f γ : L 0 f β = V, f concave and f ≥ 0 = (β + 1) γ/β γ + 1 L 1-γ β V γ/β . (6.1)
Proof. For L, V > 0, let

M (L, V ) := min L 0 f γ : L 0 f β = V, f concave and f ≥ 0 .
Let us now prove (6.1). By scaling, we can assume that L = V = 1. Thanks to the concavity and positivity constraints, existence of a minimizer for (6.1) follows. Let f be such a minimizer. Let us prove that we can assume that f is non-increasing. Notice first that by definition, there holds

M (1, 1) = 1 0 f γ .
Up to a rearrangement, we can assume that f is symmetric around the point 1/2, so that f is non-increasing in [1/2, 1] and

1 1/2 f γ = 1 2 M (1, 1) = M (1/2, 1/2).
Letting finally for x ∈ [0, 1], f (x) := f ( 1 2 (x + 1 2 )), we have that f is non-increasing, admissible for (6.1) and

1 0 f γ = 2 1 1/2 f γ = M (1, 1)
, so that f is also a minimizer for (6.1). Assume now that f is not affine in (0, 1). Then there is x > 0 such that for all 0

< x ≤ x f (x) > f (0) -(f (0) -f (1))x. Let f := λ -(λ -f (1))x with λ > f (0) chosen so that 1 0 f β-1 f = 1 0 f β . (6.2)
Now, let g := f -f . Since f + g = f is concave, for every 0 ≤ t ≤ 1, f + tg is a concave function. For δ ∈ R, let f t,δ := f + t(g + δ(1 -x)). Let finally δ t be such that

1 0 f β t,δt = 1 0 f β .
Thanks to (6.2) and since β ≥ 1, |δ t | = O(t). Since f t,δt is concave, by the minimality of f we get

1 0 f γ t,δt - 1 0 f γ ≥ 0.
Dividing by t and taking the limit as t goes to zero, we obtain

1 0 f γ-1 g ≥ 0.
Let z ∈ (0, 1) be the unique point such that f (z) = f (z) (so that f (x) > f (x) for x < z and f (x) < f (x) for x > z). We then have,

0 ≤ 1 0 f β-1 f -f f β-γ = z 0 f β-1 f -f f β-γ + 1 z f β-1 f -f f β-γ < 1 f β-γ (z) z 0 f β-1 ( f -f ) + 1 z f β-1 ( f -f ) = 1 f β-γ (z) 1 0 f β-1 ( f -f ),
which contradicts (6.2).

We are left to study the case when f is linear. Assume that f (1) > 0 and let

δ := 1 0 f β-1 1 0 xf β-1 > 1,
so that in particular, 1 0 f β-1 (1 -δx) = 0. Up to adjusting the volume as in the previous case, for t > 0 small enough, f + t(1 -δx) is admissible. From this, arguing as above, we find that

1 0 f γ-1 (1 -δx) ≥ 0.
By splitting the integral around the point z = δ -1 ∈ (0, 1) and proceeding as above, we get again a contradiction. As a consequence, we obtain that f (x) = λ(1 -x), with λ = (β + 1) 1/β so that the volume constraint is satisfied. This concludes the proof of (6.1).

We now prove the following Γ-convergence result. Theorem 6.3. For α ∈ [0, 1), the functionals F n,α Γ-converge in the Hausdorff topology, as n → +∞, to the functional

F α (E) :=                  C N L 1 N -1 + I α ([0, 1]) L α if E [0, L] × {0} N -1 and α > 0 C N L 1 N -1 + I 0 ([0, 1]) -log L if E [0, L] × {0} N -1 and α = 0 +∞ otherwise,
where E F means that E = F up to a rigid motion, and

C N := ω 1/(N -1) N -1
N (N -2)/(N -1) with ω N the volume of the ball of radius one in R N (so that for N = 2 we have C 2 = 2).

Proof. By Proposition 6.1 we know that the Γ-limit is +∞ on the sets which are not segments.

Let us first prove the Γ-limsup inequality. Given L ∈ (0, +∞), we are going to construct E n symmetric with respect to the hyperplane

{0} × R N -1 . For t ∈ [0, L/2], we let r(t) := N Vn ω N -1 L 1/(N -1)

-2t

L and then

E n ∩ R + × R N -1 := t, B N -1 r(t) : t ∈ [0, L/2] ,
where B N -1 r(t) is the ball of radius r(t) in R N -1 . With this definition, |E n | = V n , so that E n ∈ A n,α . We then compute

P (E n ) = 2 L/2 0 H N -2 (S N -2 )r(t) N -2 1 + |r | 2 = 2(N -1) ω N -1 N V n ω N -1 L N -2 N -1 L/2 0 1 - 2t L N -2 1 + c N L 2 V n L 2 N -1 1/2 = C N V N -2 N -1 n L 1 N -1 + o V N -2 N -1 n .
Letting µ α be the optimal measure for I α ([-L/2, L/2]), we then have

F n,α (E n ) ≤ C n L 1 N -1 + I α ([0, L]) + o(1),
which gives the Γ-limsup inequality.

We now turn to the the Γ-liminf inequality. Let E n ∈ A n,α be such that

E n → [0, L] × {0} N -1 in the Hausdorff topology. Since I α is continuous under Hausdorff convergence, it is enough proving that lim inf n→+∞ V -N -2 N -1 n P (E n ) ≥ C N L 1 N -1 . (6.3)
Let L n := diam(E n ). By Hausdorff convergence, we have that L n → L. Moreover, up to a rotation and a translation, we can assume that [0, L n ] × {0} N -1 ⊂ E n . For N = 2, we directly obtain P (E n ) ≥ 2L n which gives (6.3). We thus assume from now on that N ≥ 3. Let E n be the set obtained from E n after a Schwarz symmetrization around the axis R × {0} N -1 . By Brunn's principle [START_REF] Brunn | Über Ovale und Eiflächen[END_REF], E n is still a convex set with

P (E n ) ≥ P ( E n ) and |E n | = | E n |.
We thus have that

E n = t∈[0,Ln] {t} × B N -1 r(t)
for an appropriate function r(t), and, by Fubini's Theorem,

L N 0 r(t) N -1 = V n ω N -1
.

By the Coarea Formula [2, Th. 2.93], we then get

P ( E n ) ≥ H N -2 (S N -2 ) Ln 0 r(t) N -2 1 + |r (t)| 2 ≥ H N -2 (S N -2 ) Ln 0 r(t) N -2 .
Applying then Lemma 6.2 with γ = N -2 and β = N -1, we obtain (6.3).

Remark 6.4. For α ∈ [0, 1) and N ≥ 2, it is easy to optimize F α in L and obtain the values L N,α given in Theorem 1.4.

From Proposition 6.1, Theorem 6.3 and the uniqueness of the minimizers for F α , we directly obtain the following asymptotic result for minimizers of (1.1). Corollary 6.5. Let α ∈ [0, 1) and N ≥ 2. Then, up to rescalings and rigid motions, every sequence E n of minimizers of (1.1) converges in the Hausdorff topology to [0, L N,α ]× {0} N -1 .

6.2.

The case N = 2, 3 and α = 1. In the case α ≥ 1, the energy I α is infinite on segments and thus a Γ-limit of the same type as the one obtained in Theorem 6.3 cannot be expected. Nevertheless in the Coulombic case N = 3, α = 1 we can use a dual formulation of the non-local part of the energy to obtain the Γ-limit. As a by-product, we can also treat the case N = 2, α = 1.

For N = 2, 3 and n ∈ N, we let

A n,1 := E ⊂ R 3 convex body, |E| = Q -2(N -1) n (log Q n ) -(N -1) , F n,1 (E) := Q 2(N -2) n (log Q n ) N -2 P (E) + I 1 (E) log Q n , for E ∈ A n,1 .
As before, if E is a minimizer of (1.1), then the rescaled set

E := Q - 2(N -1) N n (log Q n ) -(N -1) N E is a minimizer of F n,1 in A n,1 . Let C ε := [0, 1] × B ε ⊂ R 3 be
a narrow cylinder of radius ε > 0 (where B ε denotes a two-dimensional ball of radius ε). We begin by proving the following estimate on I 1 (C ε ): Proposition 6.6. It holds

lim ε→0 I 1 (C ε ) | log ε| = 2 . (6.4)
As a consequence, for every L > 0,

lim ε→0 I 1 ([0, L] × B ε ) | log ε| = 2 L . (6.5)
Proof. Estimate (6.4) is well-known (see for instance [START_REF] Maxwell | On the Electrical Capacity of a long narrow Cylinder, and of a Disk of sensible Thickness[END_REF]). We include here a proof for the reader's convenience.

To show that

lim ε→0 | log ε| -1 I 1 (C ε ) ≤ 2,
we use µ ε :=

I 1 (C ε ) ≤ 1 π 2 ε 4 Cε×Cε dxdy |x -y| = 1 π 2 ε 4 Cε Cε+y dz |z| dy ≤ 1 πε 2 1/2 -1/2 Bε 1 (z 2 1 + |(z 2 , z 3 )| 2 ) 1/2 = 4 ε 2 1/2 0 ε 0 r (z 2 1 + r 2 ) 1/2 = 4 ε 2 1/2 0 z 2 1 + ε 2 -z 1 = 4 ε 2 1 8 1 + 4ε 2 - 1 8 + ε 2 2 log 1 2ε + 1 + 1 4ε 2 = 2| log ε| + o(| log ε|).
In order to show the opposite inequality, we recall the following definition of capacity of a set E:

Cap(E) := min R 3 |∇φ| 2 : χ E ≤ φ, φ ∈ H 1 0 (R 3 )
Then , if E is compact, we have [START_REF] Landkof | Foundations of modern potential theory[END_REF][START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF]]

I 1 (E) = 4π Cap(E)
.

Thus (6.4) will be proved once we show that

Cap(C ε )| log ε| ≤ 2π + o(1). (6.6)
For this, let λ > 0 and µ > 0 to be fixed later and let and we find (6.6). Estimate (6.5) then follows by scaling.

f λ (x ) :=        1 for |x | ≤ ε 1 - log(|x |/ε) log(λ/ε) for ε ≤ |x | ≤ λ 0 for |x | ≥ λ and ρ µ (z) :=                    0 for z ≤ -µ z + µ µ for -µ ≤ z ≤ 0 1 for 0 ≤ z ≤ 1 1 - z -1 µ for 1 ≤ z ≤ 1 + µ 0 for z ≥ 1 + µ.
As a simple corollary we get the two dimensional result This gives together with (6.4) the corresponding lower bound.

We can now prove a compactness result analogous to Proposition 6.1.

Proposition 6.8. Let E n ∈ A n,1 be such that sup n F n,1 (E n ) < +∞. Then, up to extracting a subsequence and up to rigid motions, the sets E n converge in the Hausdorff topology to a segment [0, L] × {0} N , for some L ∈ (0, +∞).

Proof. We reason as in the proof of Proposition 6.1. Since the case N = 2 is easier, we focus on N = 3. Let R n = 3 i=1 [0, λ i,n ] be given by Lemma 2.2 and let us assume without loss of generality that i → λ i,n is decreasing. Then (2.1) applied with V = Q -4 n (log Q n ) -2 , directly yields an upper bound on λ 1,n (and thus on diam(E n )).

We now show that the diameter of E n is also uniformly bounded from below. Unfortunately, (2.2) does not give the right bound and we need to refine it using (6.4). As in Proposition 6.1, the energy bound I 1 (E n ) log Q n , directly implies that λ 1,n 1 log Q n , from which, using (2.1) and

:= Q -2 n log Q n we get λ 1,n log Q n λ 1,n I 1 (E n ) ∼ λ 1,n I 1 (R n ) = I 1 3 i=1 0, λ i,n λ 1,n I 1 (C εn ) ∼ | log ε n | ∼ log Q n , which implies λ 1,n 1 ,
and gives a lower bound on the diameter of E n . Arguing as in the proof of (2.2), we then get

λ 3,n ≤ λ 2,n Q -2 n (log Q n ) -1 . (6.8)
It follows that the sets E n are compact in the Hausdorff topology, and any limit set is a segment of length L ∈ (0, +∞).

Arguing as in Theorem 6.3, we obtain the following result. Theorem 6.9. The functionals F n,1 , Γ-converge in the Hausdorff topology, to the functional

F 1 (E) :=    C N L 1 N -1 + 4 L if E [0, L] × {0} N -1 +∞ otherwise,
where C N is defined as in Theorem 6.3.

Proof. Since the case N = 2 is easier, we focus on N = 3. The compactness and lower bound for the perimeter are obtained exactly as in Theorem 6.3. For the upper bound, for L > 0 and n ∈ N, we define E n as in the proof of Theorem 6.3, by first letting V n := Q -4 n (log Q n ) -2 (recall that N = 3) and then for t ∈ [0, L/2], r(t) := 3Vn πL 1/2 1 -2t L and

E n ∩ R + × R 2 := t∈[0, L 2 ] {t} × B 2 r(t)
where B 2 r(t) is the ball of radius r(t) in R 2 . As in the proof of Theorem 6.3, we have

lim n→+∞ Q 2 n log Q n P (E n ) = C 3 L 1 2 .
Remark 6.10. As before, optimizing F 1 with respect to L, one easily obtains the values of L N,1 given in Theorem 1.4.

Remark 6.11. By analogy with results obtained in the setting of minimal Riesz energy point configurations [START_REF] Hardin | Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds[END_REF][START_REF] Martínez-Finkelshtein | Asymptotics for minimal discrete Riesz energy on curves in R d[END_REF], we believe that for every N ≥ 2, α > 1 and L > 0, (6.5) can be generalized to

lim ε→0 I α ([0, L] × [0, ε] N -1 ) ε 1-α = C α L α , (6.9)
for some constant C α depending only on α. This result would permit to extend Theorem 6.9 beyond α = 1. Let us point out that showing that the right-hand side of (6.9) is bigger than the left-hand side can be easily obtained by plugging in the uniform measure as a test measure. However, we are not able to prove the reverse inequality.

  4. C 1,1 -regularity of minimizers for N = 2 and α = 0

Figure 1 .

 1 Figure 1.

Lemma 4 . 2 .

 42 Let 0 < β ≤ 1 and C, ε 0 > 0 be given. Then, every convex body E such that for every x 0 ∈ ∂E and every ε ≤ ε 0 ,∆V ≤ Cε 2+β (4.2)is C 1,β with C 1,β norm depending only on the Lipschitz character of ∂E, ε 0 and C.

z

  -z u(t)dt, from (4.2) we obtainεψ (ε) -ψ(ε) ε 2+β .(4.4)

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3.

Figure 4 . 1 , x ε 2 , xε 1 and x ε 1 , x ε 2 , xε 2 .

 412 Figure 4. ∆V is contained in the union of the triangles of vertices x ε 1 , x ε 2 , xε 1 and x ε 1 , x ε 2 , xε 2 .

. 11 )Figure 5 .

 115 Figure 5. The angle zΠ(x)x equals γ(x, y).

  On the other hand, since |y -x| ≥ 2ε (indeed |x -x 0 | ≤ ε and |y -x 0 | ≥ 3ε), we have |y -Π(x)| ≥ |y -x| -|x -Π(x)| |y -x| -ε |y -x|.

  We finally let φ(x , z):= f λ (x )ρ µ (z). Since ρ µ , f λ ≤ 1 and |ρ µ | ≤ µ -1 , by definition of Cap(C ε ), we have Cap(C ε ) ≤ We now choose λ := | log ε| -1 ε and µ := | log λ| -1 = (log | log ε|) -1 so that log(λ/ε) = | log ε| + log | log ε|, µ → 0 and µ λ so that µ log(λ/ε) + λ 2 µ = o(| log ε| -1 )

Corollary 6 . 7 .I 1 ( 2 . ( 6 . 7 )

 671267 It holdslim ε→0 [0, 1] × [0, ε]) | log ε| = Proof.The upper bound is obtained as above by testing withµ ε := ε -1 χ [0,1]×[0,ε] . By identifying [0, 1] × [0, ε] with [0, 1] × [0, ε] × {0} ⊂ C ε we get that I 1 ([0, 1] × [0, ε]) ≥ I 1 (C ε ).

  1,β function u. Up to a rotation, we can further assume that this tangent is horizontal so that for some interval I ⊂ R, we have ∂E∩B

R (0) = {(t, u(t)) : t ∈ I}. In particular, if x = (t, u(t)) ∈ ∂E∩B R (0), |u(t)| |t| 1+β and |u (t)| |t| β . 3 indeed, for |s-t| ≤ ε1, |u (t)-u (s)| ≤ |t-s| -1 (|u(t)-u(s)-u (s)(t-s)|+|u(s)-u(t)-u (t)(s-t)|) |t -s| β

  1 πε 2 χ Cε as a test measure in the definition of I 1 (C ε ). Then, noting that for every y ∈ C ε ,

	Cε+y	dz |z|	≤	[-1/2,1/2]×Bε	dz |z|	,
	we obtain

  3 i=1 λ i,n ∼ Q -4 n (log Q n ) -2 , we get

	λ 2,n Q -2 n .
	In particular, it follows that
	λ 2,n λ 1,n	log Q n n Q 2	.
	By Proposition 6.6, letting ε n

Let µ n be the optimal measure for I 1 (E n ), and let ε n := 3Vn πL 1/2 . For L > δ > 0, there holds [-L-δ 2 , L-δ 2 ] × B 2 εn ⊂ E n so that by (6.5),

Letting δ → 0 + , we obtain the upper bound.

We are left to prove the lower bound for the non-local part of the energy. Let E n be be a sequence of convex sets such that

. We can assume that sup n F n,1 (E n ) < +∞, since otherwise there is nothing to prove. Let δ > 0. Up to a rotation and a translation, we can assume that [0,

3 ) be such that

Up to a rotation of axis R × {0} 2 , we can assume that x 1 = (a, n 1 , 0) for some n 1 ≥ 0. Let finally x 2 be such that

On the other hand, by convexity, the tetrahedron T with vertices 0, x 1 , x 2 and (L -δ, 0, 0) is contained in E n . We thus have

we also have

Arguing as in the proof of (2.2), we get from the energy bound, (L -δ)

, and thus

, where the constants involved might depend on L. We therefore have

where the last inequality follows from (6.5). Letting δ → 0, we conclude the proof.