Cyrill B Muratov
email: muratov@njit.edu

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

Berardo Ruffini
email: berardo.ruffini@umontpellier.fr

On Equilibrium Shape of Charged Flat Drops

Equilibrium shapes of two-dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed.

Introduction

In this paper we are interested in a geometric variational problem that arises in the studies of electrified liquids (for a recent overview, see, e.g., [START_REF] De La | The fluid dynamics of Taylor cones[END_REF]). Consider an insulating fluid between two parallel plates, also made from an insulating material. From a small opening in one of the plates an immiscible, electrically conducting incompressible liquid is introduced and is allowed to fill a region whose lateral dimensions greatly exceed the distance between the plates. Because of its essentially two-dimensional character, we call the resulting pancake-shaped drop of the conducting liquid a flat drop. A voltage is then applied through the opening, imparting an electric charge to the liquid. One would like to understand the equilibrium shape attained by the drop, either at constant voltage or constant total charge, and either at constant or variable volume. Physically, this problem is interesting because the electric field produced by the charges tends to destabilize the liquid interface [START_REF] De La | The fluid dynamics of Taylor cones[END_REF][START_REF] Fontelos | Symmetry-breaking bifurcations of charged drops[END_REF][START_REF] Landau | Course of Theoretical Physics[END_REF][START_REF] Taylor | Disintegration of water drops in an electric field[END_REF][START_REF] Zeleny | Instability of electrified liquid surfaces[END_REF], an effect that is used extensively in a variety of applications such as mass spectrometry [START_REF] Gaskell | Electrospray: Principles and practice[END_REF], electrospinning [START_REF] Barrero | Micro-and nanoparticles via capillary flows[END_REF] and microfluidics [START_REF] Castro-Hernandez | Breakup length of AC electrified jets in a microfluidic flowfocusing junction[END_REF]. Mathematically, this problem is interesting because the interplay between the capillary and Coulombic forces produces a competition that makes prediction of the energy minimizing configurations a highly non-trivial task.

The equilibrium configurations of electrified conducting liquids may be studied with the help of a basic variational model that goes back all the way to Lord Rayleigh [START_REF] Rayleigh | On the equilibrium of liquid conducting masses charged with electricity[END_REF]. Let K ⊂ R 3 be a compact set with smooth boundary. If σ is the surface tension, is the relative permittivity of the surrounding medium and 0 is the permittivity of vacuum, then the energy of a charged, perfectly conducting liquid drop occupying K, with volume V and the total charge Q, is given by [START_REF] Landau | Course of Theoretical Physics[END_REF]

E Q 3d (K) = σH 2 (∂K) + Q 2 2C(K) , (1.1)
1
where H s denotes the s-dimensional Hausdorff measure, with H 2 (∂K) representing the surface area of the liquid, and

1 C(K) := 1 4π 0 inf µ(K)=1 K K dµ(x)dµ(y) |x -y| , (1.2)
is the inverse electric capacitance of K in the SI units. Here µ is a probability measure supported on K, and the infimum in (1.2) is attained by a unique measure that concentrates on ∂K [START_REF] Landkof | Foundations of modern potential theory[END_REF]. If instead the voltage U on the drop is prescribed, then the total energy (including that of the charge reservoir) is

E U 3d (K) := σH 2 (∂K) - 1 2 C(K)U 2 . (1.3)
For flat drops, this model can be reformulated as follows. Let d be the distance between the plates and assume that K = Ω × [0, d], where Ω ⊂ R 2 is a compact set with smooth boundary which represents the shape of the flat drop. Note that this means that we are assuming the contact angle θ = π 2 , corresponding to a flat meniscus. For a general contact angle θ ∈ [0, π] the set K may be similarly defined in terms of Ω, with the shape of the meniscus determined by balancing the capillary forces locally. In this case the meniscus cross-section will have the shape of a circular arc of length = π-2θ 2 cos θ d, with = d if θ = π 2 . Then, to the leading order in d L, where L is the characteristic dimension of the drop, the energy E Q 3d (K) is given by

E Q 2d (Ω) := σ H 1 (∂Ω) + Q 2 8π 0 I 1 (Ω), (1.4)
where we introduced the Riesz capacitary energy of a set:

I α (Ω) := inf Ω Ω dµ(x) dµ(y) |x -y| α : µ(Ω) = 1 , (1.5)
which is more generally defined for all Borel sets Ω ⊂ R N and α ∈ (0, N ). Similarly, E U 3d (K) becomes

E U 2d (Ω) := σ H 1 (∂Ω) - 2π 0 U 2 I 1 (Ω) . (1.6)
The volume of the flat drop is, to the leading order, equal to V = |Ω|d, where |Ω| denotes the area of Ω. Hence, fixing the volume of K amounts to fixing |Ω|. We note that the infimum in the definition of I 1 (Ω) is attained by a unique measure supported on Ω, and the value of I 1 (Ω) is proportional to the inverse of the Riesz capacity of Ω [START_REF] Landkof | Foundations of modern potential theory[END_REF]. Also, for fixed volume there is no shape-dependent contribution to the energy from the liquid-solid interface to the leading order. We point out that the same type of mathematical models is relevant to systems which give rise to electric charges concentrating on planar regions, such as high-T c superconductors [START_REF] Emery | Frustrated electronic phase-separation and high-temperature superconductors[END_REF], Langmuir monolayers [START_REF] Andelman | Phase transitions in Langmuir monolayers of polar molecules[END_REF] and, possibly, graphene [START_REF] Samaddar | Charge puddles in graphene near the Dirac point[END_REF]. In addition, it should be possible to rigorously obtain the corresponding two-dimensional expressions for the energy from the original three-dimensional energy in the limit d → 0 via Γ-convergence for a suitable notion of convergence of sets [START_REF] Braides | Γ-convergence for beginners[END_REF].

We are now in a position to introduce the dimensionless versions of the energies that we will analyze. Setting

E Q λ (Ω) := E Q 2d (LΩ) σ L , λ := Q 2 8π 0 σ L 2 , (1.7)
we obtain a one-parameter family of energy functionals

E Q λ (Ω) = H 1 (∂Ω) + λI 1 (Ω), (1.8)
with λ > 0 a fixed dimensionless parameter, defined on all compact subsets of R 2 . Similarly, setting

E U λ (Ω) := E U 2d (LΩ) σ L , λ := 2π 0 U 2 σ , (1.9)
we obtain a one-parameter family of energy functionals

E U λ (Ω) = H 1 (∂Ω) - λ I 1 (Ω) , (1.10)
with λ > 0 a fixed dimensionless parameter. These energy functionals and their global minimizers with or without constraints are the main subject of the present paper.

There has recently been a great interest in non-local variational problems involving energies like those in (1.8) and (1.10). In particular, the non-local isoperimetric problems governed by (1.1), or (1.8), or their generalizations involving Riesz energy, in which the measure µ is fixed to be the uniform measure, was recently treated in [11, 23, 25, 26, 35-37, 41, 44] (see also [START_REF] Alberti | Uniform energy distribution for an isoperimetric problem with long-range interactions[END_REF][START_REF] Choksi | Small volume fraction limit of the diblock copolymer problem: I. Sharp interface functional[END_REF][START_REF] Goldman | The Γ-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density[END_REF][START_REF] Goldman | The Γ-limit of the two-dimensional Ohta-Kawasaki energy. Droplet arrangement via the renormalized energy[END_REF][START_REF] Knüpfer | Low density phases in a uniformly charged liquid[END_REF] for closely related problems on large bounded domains; these lists are by far not meant to be exhaustive). One of the motivations for those studies comes from the fact that the three-dimensional problem with uniform charge density corresponds to the famous model of the atomic nucleus introduced by Gamow [START_REF] Gamow | Mass defect curve and nuclear constitution[END_REF] and is used to explain the stability of nuclear matter and nuclear fission [START_REF] Bohr | The mechanism of nuclear fission[END_REF][START_REF] Weizsäcker | Zur Theorie der Kernmassen[END_REF]. It is now known that in the case of the energy in (1.8) with I 1 evaluated, using dµ = |Ω| -1 dx, the unique minimizer with |Ω| = m fixed is a ball as long as m ≤ m 0 λ -1 , where m 0 > 0 is a universal constant. On the other hand, there is no minimizer with |Ω| = m as soon as m > m 2 λ -1 , where m 2 ≥ m 0 is another universal constant [START_REF] Knüpfer | On an isoperimetric problem with a competing non-local term I. The planar case[END_REF]. These conclusions have been extended to the case of higher-dimensional problems and more general Riesz kernels [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF][START_REF] Julin | Isoperimetric problem with a Coulombic repulsive term[END_REF][START_REF] Knüpfer | On an isoperimetric problem with a competing non-local term II. The general case[END_REF][START_REF] Lu | Nonexistence of minimizer for Thomas-Fermi-Dirac-von Weizsäcker model[END_REF]. The basic intuition behind these results is that when the value of m is sufficiently small, the perimeter term dominates the energy, forcing the minimizer to coincide with the one for the classical isoperimetric problem. On the other hand, when the value of m is sufficiently large, the repulsive term becomes dominant, forcing the charges to move off to infinity. The observations above motivate the conjecture, first stated in [START_REF] Choksi | Small volume fraction limit of the diblock copolymer problem: II. Diffuse interface functional[END_REF], that minimizers of the problem with uniform charge density are balls, unless it is better to split one ball into to two equal size balls. Whether or not this conjecture is true in its full generality is a very difficult question, but it was answered in the affirmative in the case of two-dimensional problems involving Riesz kernels with a sufficiently small exponent α [START_REF] Knüpfer | On an isoperimetric problem with a competing non-local term I. The planar case[END_REF][START_REF] Muratov | On an isoperimetric problem with a competing non-local term: quantitative results[END_REF]. A weaker version of this result in higher dimensions, namely, that minimizers of the above problem with α sufficiently small are balls whenever they exist, without specifying the threshold for existence, was established in [START_REF] Bonacini | Local and global minimality results for a nonlocal isoperimetric problem on R N[END_REF].

One might expect that the capacitary problem in (1.1) should exhibit a similar behavior. Studies of instability of balls with respect to small smooth perturbations in three space dimensions go back to Lord Rayleigh [START_REF] Rayleigh | On the equilibrium of liquid conducting masses charged with electricity[END_REF], and emergence of regular non-symmetric critical points out of the bifurcation at the critical charge has been established much more recently in [START_REF] Fontelos | Symmetry-breaking bifurcations of charged drops[END_REF] (for some numerical work, see also [START_REF] Basaran | Axisymmetric shapes and stability of isolated charged drops[END_REF][START_REF] Burton | Simulations of Coulombic fission of charged inviscid drops[END_REF][START_REF] Garzon | Numerical simulations of electrostatically driven jets from nonviscous droplets[END_REF]). Yet, a striking result from a recent paper [START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF] shows that the problem of minimizing the energy in (1.1) never admits global minimizers with |Ω| = m. Even worse, this problem was shown never to admit any regular local minimizers, understood in any reasonable sense [START_REF] Muratov | On well-posedness of variational models of charged drops[END_REF]. This severe illposedness of the three-dimensional capacitary problem has to do with an incompatibility between the perimeter term, which sees two-dimensional sets, and the Coulombic term, which sees sets of positive Newtonian capacity. The latter, however, could concentrate on sets of positive s-dimensional Hausdorff measure with 1 < s < 2 [START_REF] Landkof | Foundations of modern potential theory[END_REF]. Thus, there is a gap between the Hausdorff dimension of the sets that are negligible with respect to the perimeter and those negligible with respect to the Coulombic energy, which prevents the perimeter from imparting classical regularity to the minimizers. Note that existence may be restored, if one looks for minimizers in more regular classes, such as, for example, the class of convex compact sets [START_REF] Goldman | On minimizers of an isoperimetric problem with long-range interactions and convexity constraint[END_REF]. Also, well-posedness under confinement in three dimensions can be restored by including entropic effects into the model, resulting in a finite screening length [START_REF] Muratov | On well-posedness of variational models of charged drops[END_REF].

Coming back to our main problem of interest, which consists of minimizing E Q λ (Ω) in (1.8) among all compact sets Ω ⊂ R 2 with |Ω| = m fixed, one can see that the reduced dimensionality of the problem makes it a borderline case between the ill-posed capacitary problem in three-dimensions and the well-posed uniform charge density problem in two dimensions. Indeed, in this case the perimeter controls sets of zero Riesz capacity for α = 1 in two space dimensions. Therefore, the question of existence vs. non-existence of minimizers in this setting becomes particularly delicate. In what follows, we will show that the non-local isoperimetric problem associated with (1.8) remains on the "good side" of the critical value of α = 1, exhibiting existence of minimizers for small values of m and non-existence for large values of m.

Remarkably, we are able to give a complete explicit solution to this problem, both with and without the constraint on |Ω|. In the absence of the constraint, the energy E Q λ is uniquely minimized by balls of radius R = 1 2 √ λ. With the constraint |Ω| = m the unique minimizers of E Q λ are balls when λ ≤ 4m/π, while there are no minimizers when λ > 4m/π. We note that balls loose their minimizing property before it becomes energetically favorable to split one ball into two well separated balls with equal area and charge, which happens for λ > 4m √ 2/π, or before deforming a ball into an elliptical domain of the same area lowers energy, which happens for λ > 12m/π. The nature of the instability for λ > 4m/π is similar to the one for the three-dimensional capacitary problem [START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF]: one can lower the energy by splitting off a large number of small balls, putting the right amount of charge on them and sending them off to infinity. This catastrophic loss of stability suggests that the ill-posedness exhibited by the three-dimensional problem [START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF][START_REF] Muratov | On well-posedness of variational models of charged drops[END_REF] begins to manifest itself in our two-dimensional problem at a critical value of charge. Lastly, we also provide a complete solution to the problem of minimizing E U λ from (1.10). Without the constraint on |Ω|, this energy has minimizers if and only if λ = π 2 , and the only minimizers are balls (of any radius). If also |Ω| = m, then balls of area m are the only minimizers when λ ≤ π 2 , and there are no minimizers for λ > π 2 .

The strategy of the proof of our main theorems is to take advantage of the fact that without the volume constraint the energy E Q λ of a set Ω decreases if we convexify each connected component of Ω. Then we exploit the fact that the perimeter is linear for the Minkowski sum [START_REF] Gardner | The Brunn-Minkowski inequality[END_REF], while the non-local term I 1 is sublinear [START_REF] Novaga | Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian[END_REF]. This, together with some classical, simple results in the theory of convex bodies, allows us to transform each convex component of a set into a ball. Eventually we show that by suitably merging several balls into one ball we decrease the energy. Let us stress that such a technique is very special and applies only to the two-dimensional case and only to the capacitary energy I 1 . It is indeed an open problem whether I α is a subadditive operator with respect to the Minkowski summation for α = 1 (and for α = N -2 in R N with N ≥ 3). For the constrained problem, we combine the previous result with the well-known maximizing property of balls with respect to the Riesz capacitary energy [START_REF] Pólya | Inequalities for the capacity of a condenser[END_REF][START_REF] Pólya | Isoperimetric Inequalities in Mathematical Physics[END_REF] to prove existence for small charges. Finally, the non-existence for large charges for the constrained problem is obtained by constructing a test configuration with many balls, whose energy in the limit coincides with that of the unconstrained problem.

Let us also point out that while our paper is concerned exclusively with global energy minimizers, existence of local minimizers or critical points of the energy is naturally an interesting question, both physically and mathematically. Care, however, is needed in defining the appropriate notion of solution in view of the a priori lack of regularity and potentially severe ill-posedness of the considered variational problems [START_REF] Muratov | On well-posedness of variational models of charged drops[END_REF]. In particular, one needs to take into consideration the possibility of either local minimizers or critical points consisting of sets with many small holes ("froth"). Some new quantitative insights into the capacitary energy of sets of finite perimeter would be needed to deal with those issues, which goes beyond the scope of the present work.

The rest of the paper is organized as follows: in Section 2, we give the precise definitions of the variational problems to be studied and state our main results, in Section 3 we discuss some preliminary results, and in Section 4 we present the proofs of the main theorems.

Setting and statements of the main results

In this section we state the main results of our paper. Viewing E Q λ in (1.8) as a map from B(R 2 ) to [0, +∞], where B(R 2 ) is the family of all Borel subsets of R 2 , we start by defining the appropriate admissible classes in the domain of

E Q λ . Given two sets A, B ⊂ R 2 we introduce the equivalence relation A ∼ B if H 1 (A \ B) = H 1 (B \ A) = 0. We then set A := {Ω ⊂ R 2 : Ω compact, |Ω| > 0, H 1 (∂Ω) < ∞}/ ∼ . (2.1)
Notice that the equivalence relation ∼ in the definition of the class A will be needed in order to have uniqueness of minimizers of E Q λ over A. The definition above ensures that 0 < E Q λ (Ω) < ∞ whenever Ω ∈ A. We note that physically the class A corresponds to flat drops with fixed charge and variable volume, which can be achieved by placing a porous membrane, permeable to the liquid but impermeable to the charge-carrying ions, into the opening supplying the liquid. Similarly, for m > 0 we define

A m := {Ω ∈ A : |Ω| = m}. (2.2)
Note that the definitions of both the energy and the associated admissible classes A and A m involve the Hausdorff measure H 1 of the topological boundary. Unfortunately, H 1 fails to be lower semicontinuous even with respect to Hausdorff convergence of compact sets, making it difficult to apply the direct method of the calculus of variations to establish existence of minimizers of E Q λ . One possibility to proceed might be to replace H 1 (∂Ω) with P (Ω), where

P (Ω) := sup Ω ∇ • φ dx : φ ∈ C ∞ c (R 2 , R 2 ), φ L ∞ (R 2 ) ≤ 1 , (2.3)
is the perimeter functional (which coincides with H 1 (∂Ω) for sets with smooth boundary [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]), and work in the class of sets of finite perimeter. This, however, would bring us to an ill-posed problem: since I 1 is a capacitary term, it is not well defined on sets which are defined up to sets of Lebesgue measure zero [START_REF] Landkof | Foundations of modern potential theory[END_REF]. This means, for example, that it is possible to find a compact set consisting of a disjoint union of a closed ball of area m and a closed set of Lebesgue measure zero but positive Riesz capacity (by [START_REF] Landkof | Foundations of modern potential theory[END_REF]Theorem 3.13], for the latter one can take any compact set K ⊂ R 2 such that 0 < H s (K) < ∞ for some 1 < s < 2). Putting all the charge on the latter and then translating and dilating it off to infinity, one recovers the perimeter of the ball of area m as the infimum of the energy. Thus the infimum would clearly not be attained, and this is the reason why we restrict ourselves to the classes A and A m . Notice that, for all Ω ∈ A, we have P (Ω) ≤ H 1 (∂Ω). Indeed, following [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] we have that P (Ω) = H 1 (∂ M Ω) where

∂ M Ω := x ∈ R 2 : lim r→0 |Ω ∩ B r (x)| πr 2 ∈ {0, 1} ⊆ ∂Ω. (2.4)
However, P (Ω) = H 1 (∂Ω) as soon as Ω has Lipschitz boundary.

In this paper we do not rely on the direct method of the calculus of variations. Instead, we directly show that balls are minimizers for our problem. We also point out that by the results of [START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF], the infimum of the energy over A m is never attained, if I 1 is replaced by I α with α ∈ (0, 1), contrary to the case of α = 1 considered here.

We now state our first result concerning minimizers of E Q λ over A.

Theorem 1. Let λ > 0 and let R λ = 1 2 √ λ. Then the closed ball B R λ is the unique (up to translations) minimizer of E Q λ over A.
We note that the optimal radius R λ can be determined by minimizing over balls, for which the energy is explicitly (see Lemma 3)

E Q λ (B R ) = 2πR + λπ 2R ≥ 2π √ λ = E Q λ (B R λ ), (2.5)
with the lower bound in the right-hand side attained only if R = R λ . We also note that because of the characterization of sets of finite perimeter in the plane [START_REF] Ambrosio | Connected components of sets of finite perimeter and applications to image processing[END_REF], the statement of Theorem 1 remains true when the minimization is carried out among all sets of finite perimeter, provided that the set Ω is identified with its measure-theoretic closure. Alternatively, the statement holds when minimizing E Q λ among convex compact sets. We caution the reader, however, that not every minimizing sequence for E Q λ (after suitable translations) has a ball of radius R λ as the limit.

Remark 1. For λ > 0 the infimum of E Q λ over A is achieved by a sequence of N balls of radius R λ /N running off to infinity. There are also minimizing sequences consisting of N balls with N → ∞. Indeed, taking

Ω ε = N i=1 B R λ /N (x i ), with |x i -x j | R λ
, in view of Theorem 1 and Lemma 2 we have

E Q λ (Ω ε ) ≤ N E Q λ/N 2 (B R λ /N ) + ε = E Q λ (B R λ ) + ε = inf Ω∈A E Q λ (Ω) + ε, (2.6)
for an arbitrary ε > 0. Thus, the energy can be equally minimized by a fine "mist" of droplets moving off to infinity by distributing the charge proportionally to their perimeter.

As a byproduct of the proof of Theorem 1 we can show that balls minimize I 1 (that is, maximize the capacity) among sets of fixed perimeter, thus generalizing a result in [START_REF] Novaga | Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian[END_REF]Corollary 3.2].

Corollary 1. The closed ball B R is the unique (up to translations) minimizer of I 1 among the sets Ω ∈ A with P (Ω) = 2πR. The ball B R is also the unique minimizer of I 1 among the sets Ω ∈ A with H 1 (∂Ω) = 2πR.

Turning now to the minimizers of E Q λ over A m , we have the following result.

Theorem 2. Let λ > 0, m > 0, let R = m/π, and define

λ Q 0 := 4m π . (2.7)
Then:

(i) The closed ball B R is the unique (up to translation) minimizer of E Q λ over A m , if λ ≤ λ Q 0 . (ii) There is no minimizer of E Q λ over A m , if λ > λ Q 0 .
One should contrast the result in Theorem 2 with the expectation in the case of nonlocal isoperimetric problems with uniform charge that balls are the unique minimizers, unless it becomes advantageous to split a ball into two equal size balls [START_REF] Choksi | Small volume fraction limit of the diblock copolymer problem: II. Diffuse interface functional[END_REF][START_REF] Knüpfer | On an isoperimetric problem with a competing non-local term I. The planar case[END_REF][START_REF] Muratov | On an isoperimetric problem with a competing non-local term: quantitative results[END_REF]. In our problem the latter would happen at the value of λ at which two balls of area m/2 infinitely far apart and carrying equal charge have the same total energy as one ball of area m and the same total charge. By Lemma 4, this is the case if λ > λ Q c1 , where

λ Q c1 := 4m √ 2 π . (2.8)
Comparing this value with λ Q 0 , one sees, however, that balls become unstable before it becomes energetically favorable to split them into two equal balls. Furthermore, by Lemma 4 the instability with respect to elongations occurs at even higher values of λ > λ Q c2 , where

λ Q c2 := 12m π .
(2.9)

An inspection of the proof of Theorem 1 shows that the nature of the instability of balls at λ = λ Q 0 is, in fact, very different from the two just described: as soon as λ > λ Q 0 , it becomes energetically favorable to split off many small balls and move them far apart. Those balls will have a small total area, but carry a finite fraction of the total charge. This nonlinear instability is closely related to the ill-posedness of the three-dimensional capacitary problem [START_REF] Goldman | Existence and stability for a non-local isoperimetric model of charged liquid drops[END_REF][START_REF] Muratov | On well-posedness of variational models of charged drops[END_REF].

Observe that, as with minimizing E Q λ over A, the infimum over A m for λ ≤ λ Q 0 may also be achieved by a sequence of many balls moving off to infinity, as in Remark 1. By (2.5), in this case we have

inf Ω∈Am E Q λ (Ω) = √ 4πm + λπ π 4m ∀λ ≤ λ Q 0 . (2.10)
On the other hand, when the infimum is not attained, we can still characterize the optimal scaling of the energy in terms of λ.

Theorem 3. Let m > 0 and let λ > λ Q 0 , where

λ Q 0 is defined in (2.7). Then inf Ω∈Am E Q λ (Ω) = 2π √ λ.
(2.11)

Thus, we have inf Ω∈Am E Q λ (Ω) ∼ max √ m, √ λ .
Note that in physical terms this result means that the minimal energy is linear in charge for all sufficiently large charges. In this case the infimum of the energy is achieved by a union of a ball carrying the fraction λ Q 0 /λ of the total charge whose area converges to m and many small balls with vanishing total area and carrying the fraction (λ -λ Q 0 )/λ of the total charge, far apart. We conclude by turning our attention to the problem associated with the energy E U λ from (1.10). One key difference between E U λ and E Q λ is that the former is not bounded from below a priori. Consider the energy of a ball of radius R, which equals

E U λ (B R ) = 2πR - 2λR π , (2.12)
and observe a marked difference between this expression and the one in (2.5), which has a unique minimum as a function of R. Here, instead, the energy is a one-homogeneous function of R vanishing at λ = λ U 0 , where λ U 0 := π 2 . At λ = λ U 0 any ball is a minimizer of the energy (among balls), while for λ < λ U 0 the infimum of the energy is attained by a sequence of shrinking balls, and for λ > λ U 0 the energy is not bounded from below. This picture remains valid also among all sets in A.

Theorem 4. Let λ > 0 and define

λ U 0 := π 2 .
(2.13)

Then:

(i) If λ < λ U 0 , we have inf Ω∈A E U λ (Ω) = 0,
(iii) If λ > λ U 0 , we have inf Ω∈A E U λ (Ω) = -∞.
Similarly, in the case of minimizing over A m we have the following result.

Theorem 5. Let λ > 0, m > 0, let R = m/π, and let λ U 0 be defined in (2.13). Then:

(i) If λ ≤ λ U 0 , then B R is the unique minimizer (up to translations) of E U λ over A m . (ii) If λ > λ U 0 , then inf Ω∈A E U λ (Ω) = -∞.
In some sense, the problem associated with E U λ is simpler, because both the perimeter and the capacitary term are one-homogeneous functions with respect to dilations, making the problem very degenerate.

Preliminary results

3.1.

Basic facts about the capacitary energy. We start by collecting several results about existence and uniqueness of equilibrium measures on admissible sets.

Lemma 1.

Let Ω ∈ A. Then there exists a unique probability measure µ over R 2 supported on Ω such that

I 1 (Ω) = Ω Ω dµ(x)dµ(y) |x -y| . (3.1)
Furthermore, µ(∂Ω) = 0, and we have dµ(x) = ρ(x)dx for some ρ ∈ L 1 (Ω) satisfying 0 < ρ(x) ≤ C/dist (x, ∂Ω) for some constant C > 0 and all x ∈ int(Ω).

Proof. For the existence and uniqueness of the equilibrium measure µ we refer to [40, pp. 131-133]. By [40, Theorems 3.13 and 3.14], we have µ(Σ) = 0 for any set Σ such that H 1 (Σ) < ∞, so that in particular µ(∂Ω) = 0.

To show that the equilibrium measure µ is absolutely continuous with respect to the Lebesgue measure in the plane and has a strictly positive density, we recall that the Riesz potential v : R 2 → [0, ∞] associated with Ω:

v(x) := Ω dµ(y) |x -y| , (3.2)
is equal to I 1 (Ω) in Ω up to a set of zero 1-Riesz capacity and satisfies v(x) ≤ I 1 (Ω) for all x ∈ R 2 [40, Theorem 2.6 and page 137]. Therefore, by [START_REF] Landkof | Foundations of modern potential theory[END_REF]Theorem 3.13] we have v(x) = I 1 (Ω) for a.e. x ∈ Ω.

Let now φ ∈ C ∞ c (R 2 ) and ψ ∈ C ∞ (R 2 ) ∩ L 1 (R 2 ) be such that (-∆) 1/2 φ = ψ, which is equivalent to [19, Lemma 3.2 and Proposition 3.3] 1 4π R 2 2φ(x) -φ(x -y) -φ(x + y) |y| 3 dy = ψ(x) ∀x ∈ R 2 . (3.3)
Multiplying (3.2) by ψ and integrating over R 2 , from Fubini-Tonelli theorem we obtain

1 4π R 2 R 2 v(x)(2φ(x) -φ(x -y) -φ(x + y)) |y| 3 dx dy = 2π Ω φ(x) dµ(x), (3.4)
where we also took into account that φ

(x) = 1 2π R 2 ψ(y)
|x-y| dy for every x ∈ R 2 [42, Lemma 1.3]. Note that (3.4) is the distributional form of the equation (-∆) 1/2 v = 2πµ. Finally, observe that for any φ with support in the interior of Ω we have

1 4π R 2 R 2 v(x)(2φ(x) -φ(x -y) -φ(x + y)) |y| 3 dx dy = lim r→0 1 4π R 2 \Br(0) R 2 v(x)(2φ(x) -φ(x -y) -φ(x + y)) |y| 3 dx dy = lim r→0 1 4π R 2 \Br(0) R 2 φ(x)(2v(x) -v(x -y) -v(x + y)) |y| 3 dx dy = 1 4π Ω φ(x) R 2 2v(x) -v(x -y) -v(x + y) |y| 3 dy dx =: Ω φ(x)ρ(x) dx, (3.5)
where we took into account that the inner integrand in the last line of (3.5) is zero for a.e. y ∈ B r (0) with r := dist(supp φ, ∂Ω). Thus dµ(x) = ρ(x)dx, and the rest of the claims follows from the definition of ρ, noting that the integrand in the last line of (3.5) is strictly positive on a set of positive measure and is bounded above by 2I 1 (Ω)/|y| 3 for all y ∈ R 2 \B r (0).

We next state a result about the capacitary energies of unions of disjoint sets from A.

Lemma 2.

Let Ω 1 ∈ A and Ω 2 ∈ A be two disjoint sets, and let θ ∈ [0, 1]. Then

I 1 (Ω 1 ∪ Ω 2 ) ≤ θ 2 I 1 (Ω 1 ) + (1 -θ) 2 I 2 (Ω 2 ) + 2θ(1 -θ) dist (Ω 1 , Ω 2 ) . (3.6)
Moreover, there exists θ ∈ (0, 1) such that

I 1 (Ω 1 ∪ Ω 2 ) > θ2 I 1 (Ω 1 ) + (1 -θ) 2 I 2 (Ω 2 ). (3.7)
Proof. Let θ ∈ [0, 1], and let µ 1 and µ 2 be the equilibrium measures for I 1 (Ω 1 ) and I 1 (Ω 2 ), respectively. Then μ = θµ 1 + (1 -θ)µ 2 is a probability measure supported on Ω 1 ∪ Ω 2 .

Hence

I 1 (Ω 1 ∪ Ω 2 ) ≤ Ω 1 ∪Ω 2 Ω 1 ∪Ω 2 dμ(x) dμ(y) |x -y| = θ 2 Ω 1 Ω 1 dµ 1 (x) dµ 1 (y) |x -y| +(1 -θ) 2 Ω 2 Ω 2 dµ 2 (x) dµ 2 (y) |x -y| + 2θ(1 -θ) Ω 2 Ω 1 dµ 1 (x) dµ 2 (y) |x -y| ≤ θ 2 I 1 (Ω 1 ) + (1 -θ) 2 I 1 (Ω 2 ) + 2θ(1 -θ) dist (Ω 1 , Ω 2 )
, which gives (3.6). Let now µ be the equilibrium measure for I 1 (Ω 1 ∪ Ω 2 ). Observe that by Lemma 1 we have µ(Ω i ) > 0 for i = 1, 2, in view of the fact that the sets Ω i have non-empty interiors. Define the probability measures µ i := µ -1 (Ω i )µ| Ω i , and let θ := µ(Ω 1 ) ∈ (0, 1). Then it holds

I 1 (Ω 1 ∪ Ω 2 ) = Ω 1 ∪Ω 2 Ω 1 ∪Ω 2 dµ(x) dµ(y) |x -y| = µ 2 (Ω 1
)

Ω 1 Ω 1 dµ 1 (x) dµ 1 (y) |x -y| +µ 2 (Ω 2 ) Ω 2 Ω 2 dµ 2 (x) dµ 2 (y) |x -y| + 2µ(Ω 1 )µ(Ω 2 ) Ω 2 Ω 1 dµ 1 (x) dµ 2 (y) |x -y| > θ2 I 1 (Ω 1 ) + (1 -θ) 2 I 1 (Ω 2 ) , which gives (3.7).
The first result in Lemma 2 basically says that the capacitary energy of a union of two admissible sets may be estimated from above by the self-energies of the components with an arbitrary partition of charges, plus an interaction energy in terms of the (positive) distance between these sets. Similarly, the second result says that the capacitary energy may be estimated from below by the sum of the capacitary energies for a suitable distribution of charges.

3.2. Spherical and elliptical domains. We continue by summarizing some well-known facts about the Riesz capacity of balls and elliptical sets.

Lemma 3.

Let where K(m) and E(m) are the complete elliptic integrals of the first and second kind, respectively [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]. In particular, when a = b we have I 1 (Ω e ) = π 2a . The equilibrium measure µ for I 1 (Ω e ) is given by

Ω e = (x, y) : x 2 a 2 + y 2 b 2 ≤ 1 , (3.8
dµ(x, y) = 1 2π a 2 b 2 -b 2 x 2 -a 2 y 2 dx dy. (3.10)
In addition, we have

I 1 (Ω e ) ≤ π 5 4|Ω e |H 1 (∂Ω e ) 1/3 . (3.11)
Proof. Most of the formulas in this lemma are well-known. For the reader's convenience, we present some details regarding the Riesz capacitary energy of an elliptical set. The value of I 1 (Ω e ) and the expression for the equilibrium measure µ follow by passing to a suitable limit from the corresponding expressions for the ellipsoid, which can be found by an explicit solution of the Laplace's equation in three dimensions in ellipsoidal coordinates [START_REF] Landau | Course of Theoretical Physics[END_REF]. The formula in (3.11) is motivated by [START_REF] Dudko | Rate constant for diffusion-influenced ligand binding to receptors of arbitrary shape on a cell surface[END_REF], where it was shown that

4|Ω e |H 1 (∂Ω e )I 3 1 (Ω e ) π 5 1/3 = 1 + O e 8 , (3.12)
and the inequality can be seen numerically for any particular value of e ∈ (0, 1). To obtain the upper bound on the whole interval is not straightforward, however, since many cancellations give rise to (3.12). The proof of (3.11) amounts to showing that the function

f (x) := 16 √ 1 -x K 3 (x)E(x) π 4 , (3.13)
is bounded above by 1 for all x ∈ (0, 1). For this, we differentiate f to obtain, using the identities associated with the complete elliptic integrals [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]:

f (x) = 8K 2 (x)(3E 2 (x) -(2 -x)E(x)K(x) -(1 -x)K 2 (x)) π 4 x √ 1 -x . (3.14)
Therefore, our proof is concluded, once we prove that the function

g(x) := 3E(x) -(2 -x)K(x) -(1 -x) K 2 (x) E(x) , (3.15)
is negative for all x ∈ (0, 1). This is done by using the estimates

K(x) ≥ K 15 (x), E(x) ≤ E 15 (x), (3.16)
where K 15 (x) and E 15 (x) are the 15th-order Taylor polynomials for K(x) and E(x), respectively, obtained by truncating the power series representations of K(x) and E(x). Substituting (3.16) into (3.15), we find that g(x) ≤ x 4 P 27 (x)/Q 15 (x), where P 27 (x) and Q 15 (x) are two explicit polynomials of degrees 27 and 15, respectively, with P 27 (0) > 0 and Q 15 (0) < 0. The proof is concluded by verifying that both P 27 and Q 15 do not change sign in [0, 1]. All these tedious computations were carried out using computer algebra software Mathematica 10.4.

We now observe that since for a given Ω ∈ A m it is always possible to find e ∈ [0, 1) such that Ω e ∈ A m and H 1 (∂Ω e ) = H 1 (∂Ω), by comparing the energy of Ω with that of Ω e we obtain the following result from (3.11).

Corollary 2. Let λ > 0, m > 0 and let Ω ∈ A m be a minimizer of E Q λ over A m . Then

E Q λ (Ω) ≤ H 1 (∂Ω) + λ π 5 4mH 1 (∂Ω) 1/3 .
(3.17)

Note that numerically the right-hand side of (3.11) gives a remarkably accurate value of I 1 (Ω e ) for practically all values of e [START_REF] Dudko | Rate constant for diffusion-influenced ligand binding to receptors of arbitrary shape on a cell surface[END_REF]. For example, when e = 0.7, it predicts the value of I 1 (Ω e ) with the relative error of 5 × 10 -5 . Even at e = 0.9999 the relative error of the leading order expression is within about 25%. Let us also point out that the leading order term in (3.11) was found to be a very good approximation for the values of I 1 for sets of varied simple shapes, not limited to elliptical sets [START_REF] Dudko | Rate constant for diffusion-influenced ligand binding to receptors of arbitrary shape on a cell surface[END_REF]. In fact, if (3.17) were an equality, one would immediately conclude from the isoperimetric inequality that balls are the only minimizers of E Q λ over A m for all λ ≤ λ Q c2 , where λ Q c2 is given by (2.9). Recalling Lemma 2, the computations of Lemma 3 allow us to make conclusions about instabilities of balls. Lemma 4. Let m > 0, let R = m/π and let λ Q c1 and λ Q c2 be as in (2.8) and (2.9). Then (i) If λ > λ Q c1 , a ball with area m is unstable with respect to splitting, i.e.,

E Q λ (B R ) > E Q λ (B R/ √ 2 (0) ∪ B R/ √ 2 (x 0 ) for all x 0 ∈ R 2 with |x 0 | R. (ii) If λ > λ Q c1
, a ball with area m is unstable with respect to elongations, i.e., there exists an elliptical set

Ω e such that E Q λ (B R ) > E Q λ (Ω e ).
Proof. The first assertion follows by a straightforward computation, using Lemma 3 and Lemma 2. For the second, we use Corollary 2, minimizing the right-hand side of (3.17) with respect to H 1 (∂Ω) ≥ 2πR.

We now show that a single ball has lower energy than a disjoint union of balls with the same total perimeter. Lemma 5. Let Ω = N i=1 B r i (x i ) be a union of a finite number of disjoint closed balls, and let r := N i=1 r i . Then

E Q λ (B r ) ≤ E Q λ (Ω) , (3.18)
with equality if and only if

Ω = B r (x) for some x ∈ R 2 .
Proof. If Ω is a ball then there is nothing to prove. Assume that Ω = B r 1 (x 1 ) ∪ B r 2 (x 2 ) is the union of two disjoint balls. We show that the ball B r , with r 1 + r 2 = r, has always less energy. Indeed, for every θ ∈ [0, 1] it holds

1 r 1 + r 2 ≤ θ 2 r 1 + (1 -θ) 2 r 2 , (3.19)
which is readily verified by noticing that θ = r 1 /(r 1 + r 2 ) minimizes the right-hand side of (3.19), and equality holds for θ = θ. Recalling that I 1 (B r ) = π/(2r), inequality (3.19) can be rephrased as

I 1 (B r ) ≤ θ 2 I 1 (B r 1 (x 1 )) + (1 -θ) 2 I 1 (B r 2 (x 2 )) ∀θ ∈ [0, 1]. (3.20)
On the other hand, since H 1 (∂B r ) = H 1 (∂B r 1 (x 1 )) + H 1 (∂B r 2 (x 2 )), from (3.20) and (3.7) it follows that

E Q λ (B r ) ≤ E Q λ θ2 (B r 1 (x 1 )) + E Q λ(1-θ) 2 (B r 2 (x 2 )) < E Q λ (B r 1 (x 1 ) ∪ B r 2 (x 2 )). (3.21)
Assume now that Ω consists of more than two disjoint balls. In this case we write Ω = Ω ∪ Ω , where Ω := B r 1 (x 1 ) ∪ B r 2 (x 2 ) and Ω := Ω \ Ω = ∅. Again, from Lemma 2 we know that there exists θ ∈ (0, 1) and δ > 0 such that

I 1 (Ω) ≥ θ2 I 1 (Ω ) + (1 -θ) 2 I 1 (Ω ) + δ. (3.22)
Moreover, from the previous discussion it follows that I 1 (Ω ) ≥ I 1 (B r(x)), where r := r 1 + r 2 , for any x ∈ R 2 , so that we get

I 1 (Ω) ≥ θ2 I 1 (B r(x)) + (1 -θ) 2 I 1 (Ω ) + δ. (3.23)
Recalling (3.6), we then obtain the estimate

I 1 (Ω) ≥ θ2 I 1 (B r(x)) + (1 -θ) 2 I 1 (Ω ) + δ ≥ I 1 (B r(x) ∪ Ω ) + δ - 2 θ(1 -θ) dist(B r(x), Ω ) . (3.24) If we choose x ∈ R 2 such that dist(B r(x), Ω ) > (2δ) -1 , we then get I 1 (Ω) > I 1 (B r(x) ∪ Ω ). Recalling that H 1 (∂Ω) = H 1 (∂B r(x) ∪ ∂Ω ), this implies that E Q λ (Ω) > E Q λ (B r(x) ∪ Ω ) . (3.25)
The proof is then easily concluded by means of an inductive argument.

Our next lemma is a version of a well-known Brunn-Minkowski inequality for the Newtonian capacity of convex bodies in R 3 [START_REF] Borell | Capacitary inequalities of the Brunn-Minkowski type[END_REF][START_REF] Caffarelli | On the Case of Equality in the Brunn Minkowski Inequality for Capacity[END_REF][START_REF] Colesanti | The Brunn-Minkowski inequality for p-capacity of convex bodies[END_REF]. Lemma 6. For any pair of convex bodies Ω 1 , Ω 2 in R 2 and for any λ ∈ [0, 1], it holds

C 1 (λΩ 1 + (1 -λ)Ω 2 ) ≥ λC 1 (Ω 1 ) + (1 -λ)C 1 (Ω 2 ), (3.26)
where C 1 (Ω) := I -1 1 (Ω) is the 1-Riesz capacity of Ω. Moreover the equality holds if and only if Ω 1 is homothetic to Ω 2 .

Proof. The inequality has been proved in [START_REF] Novaga | Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian[END_REF]Theorem 1.1 (ii)] and can also be seen as the limit case of the corresponding inequality for Newtonian capacity of thin cylindrical domains in R 3 [START_REF] Borell | Capacitary inequalities of the Brunn-Minkowski type[END_REF].

To get the characterization of the equality case, we introduce u : R 3 → R, which is the harmonic extension of v in (3.2) to R 3 \(Ω × {0}). Notice that up to a constant factor the function u is the capacitary potential of a convex set Ω × {0} ⊂ R 3 , and the level sets {x ∈ R 3 : u(x) > t} are regular convex bodies in R 3 for all t ∈ (0, 1) (see [45, Theorem 1.1 (i)]). With this observation, the final part of the proof of [16, Theorem 1, pp. 476-478] applies without modifications.

The following is the cornerstone result for the proof of Theorem 1. Its proof, in a different setting, was developed in [START_REF] Bucur | Optimal convex shapes for concave functionals[END_REF].

Lemma 7. Let Ω ∈ A be a convex set. Then E Q λ (Ω) ≥ E Q λ (B Ω )
, where B Ω is a ball with the same perimeter as Ω, with equality if and only if Ω is a closed ball.

Proof. By a result of Hadwiger [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF]Theorem 3.3.2], we know that there exists a sequence of Minkowski rotation means of Ω which converges to a ball in the Hausdorff metric. Namely, there exists a sequence of rotations {T n } n∈N such that the sets

Ω n := n k=1 T k (Ω) n (3.27)
converge in the Hausdorff metric to a ball B Ω . We note that by Lemma 6 we have

I 1 (Ω n ) = 1 C 1 (Ω n ) ≤ 1 C 1 (Ω) = I 1 (Ω). (3.28)
Suppose that the equality holds in (3.28) for any n ∈ N, so that Ω n is homothetic to Ω.

Since Ω n → B Ω in Hausdorff metric, we get that Ω = B Ω . Since the perimeter is linear with respect to the Minkowski sum of convex sets in R 2 , we have H 1 (∂B Ω ) = H 1 (∂Ω). Gathering the information above, we get

H 1 (∂B Ω ) + λI 1 (B Ω ) = lim n→∞ H 1 (∂Ω n ) + λI 1 (Ω n ) ≤ H 1 (∂Ω) + λI 1 (Ω),
with equality if and only if Ω = B Ω . This concludes the proof.

Indecomposable components. Given a set Ω ∈ A, let Ω

M be its measure theoretic closure, namely:

Ω M := x ∈ R 2 : lim inf r→0 |Ω ∩ B r (x)| πr 2 > 0 . (3.29)
Since P (Ω M ) = P (Ω) = H 1 (∂ M Ω) ≤ H 1 (∂Ω) < +∞, the set Ω M is a set of finite perimeter [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]. Therefore, following [START_REF] Ambrosio | Connected components of sets of finite perimeter and applications to image processing[END_REF], for a set Ω ∈ A we can write Ω M = ( i Ω i ) ∪ Σ, with H 1 (Σ) = 0, where the sets Ω i ∈ A are the so-called indecomposable components of Ω M (finitely many or a countable family). In particular, the sets Ω i admit unique representatives that are connected and satisfy the following properties:

• H 1 (Ω i ∩ Ω j ) = 0 for i = j, • |Ω M | = i |Ω i |, • P (Ω) = P (Ω M ) = i P (Ω i ),
• Ω i = int(Ω i ). Moreover, each set Ω i is indecomposable in the sense that it cannot be further decomposed as above. We refer to these representatives of Ω i as the connected components of Ω ∈ A. Note that such a definition is necessary, since we are dealing with equivalence classes of subsets of R 2 for which the usual notion of connectedness is not well defined (consider, for instance, the union of two touching closed balls, which is equivalent to the same set minus the contact point). We point out, however, that this notion coincides with the standard notion of connected components in the following sense: if Ω has a regular boundary, for instance, Lipschitz continuous, then the components Ω i are (the closure of) the usual connected components of the interior of Ω.

Observe that H 1 (∂ co(Ω i )) ≤ H 1 (∂Ω i ) for every i, where co(Ω i ) denotes the convex envelope of Ω i . This follows from the fact that the outer boundary of a connected component can be parametrized by a Jordan curve of finite length (see [START_REF] Ambrosio | Connected components of sets of finite perimeter and applications to image processing[END_REF]Section 8]). In addition, since ∂Ω is negligible with respect to the equilibrium measure for I 1 (Ω) by Lemma 1, we have I 1 (Ω) = I 1 (Ω M ). Lastly, we have |Ω| = |Ω M | and int(Ω) = int(Ω M ).

We conclude this section by showing that any admissible set may be replaced by an admissible set with lower energy that consists of only a finite number of connected components. In particular, this result shows that minimizers of E Q λ over A are connected whenever they exist. We note that replacing an admissible set by a set with finitely many connected components as in the lemma below imparts a certain degree of regularity. Namely, our procedure removes the parts of the topological boundary of an admissible set that have postive H 1 measure, but do not belong to the measure theoretic closure (e.g., a line segment attached to a ball). Lemma 8. Let Ω ∈ A. Then there exists a set Ω ∈ A such that Ω has a finite number of connected components, and

E Q λ (Ω ) ≤ E Q λ (Ω). (3.30)
Moreover, the inequality in (3.30) is strict whenever Ω has more than one connected component.

Proof. Without loss of generality, we may assume that Ω M has at least two indecomposable components. Let Ω M = ( i∈I Ω i ) ∪ Σ be the decomposition of Ω M into indecomposable components. Let 0 < ε < 1/2 to be fixed later, and let N ≥ 2, depending on ε, be such that 1 -δ := N i=1 µ(Ω i ) ≥ 1 -ε, where µ is the equilibrium measure of Ω. If I is finite, we let N be the number of indecomposable components of Ω.

For i ∈ I, we let ξ i be the barycenter of Ω i and set

Ω N (t) := N i=1 (Ω i -ξ i + ite 1 ) , (3.31)
where e 1 denotes the unit vector in the first coordinate direction. Let also f t i (x) := x -ξ i + ite 1 and let µ t i := (f t i ) µ| Ω i be the pushforward of µ i := µ| Ω i through the map f t i . By a straightforward computation, we get

I 1 (Ω) -I 1 (Ω N (t)) ≥ 1 - 1 (1 -ε) 2 N i=1 dµ i (x) dµ i (y) |x -y| + 2 N -1 i=1 N j=i+1 dµ i (x) dµ j (y) |x -y| - 2 (1 -ε) 2 N -1 i=1 N j=i+1 dµ t i (x) dµ t j (y) |x -y| . (3.32)
Since Ω ∈ A, there exists R > 0 such that diam(Ω) < R. Moreover if t > 2R we have that whenever i = j, for any (x, y) ∈ (-

ξ i + Ω i + ite 1 ) × (-ξ j + Ω j + jte 1 ) it holds |x -y| ≥ t -2R. Let also M (Ω) := 2 N -1 i=1 N j=i+1 dµ i (x) dµ j (y) > 0. ( 3.33)
The last inequality follows from Lemma 1. With these choices, we get

I 1 (Ω) -I 1 (Ω N (t)) ≥ -8ε N i=1 dµ i (x) dµ i (y) |x -y| + 1 R - 4 t -2R M (Ω) ≥ -8εI 1 (Ω) + 1 R - 4 t -2R M (Ω) ≥ M (Ω) 4R , (3.34)
where the last inequality follows if we choose t ≥ 10R and ε small enough depending on Ω. Since H 1 (∂Ω) ≥ H 1 (∂Ω N (t)), we get that Ω := Ω N (t) is the desired set.

Proofs of the main results

We turn now to the proofs of Theorems 1-5.

Proof of Theorem 1. Let Ω ∈ A, hence E Q λ (Ω) < +∞. We first show that there exists a closed ball B such that

E Q λ (Ω) ≥ E Q λ (B), (4.1
) and equality holds if and only if Ω itself is a ball.

Thanks to Lemma 8 we can suppose that the number of the connected components Ω i of Ω is finite. If Ω has only one connected component, then we can convexify it and use Lemma 7. So we suppose that Ω has at least two connected components.

Let co(Ω i ) be the convex envelope of Ω i , and denote by µ the equilibrium measure of Ω and µ i its restriction to Ω i . Note that µ i (Ω i ) > 0 for all i by Lemma 1. Then by Lemma 7 we get

E Q λ (Ω) = i H 1 (∂Ω i ) + λ i dµ i (x) dµ i (y) |x -y| + λ i =j dµ i (x) dµ j (y) |x -y| ≥ i H 1 (∂ co(Ω i )) + λ i (µ i (Ω i )) 2 I 1 (Ω i ) + λM (Ω) diam(Ω) ≥ i H 1 (∂ co(Ω i )) + λ i (µ i (Ω i )) 2 I 1 (co(Ω i )) + λM (Ω) diam(Ω) ≥ i H 1 (∂B i )) + λ i (µ i (Ω i )) 2 I 1 (B i ) + λM (Ω) diam(Ω) . (4.2)
where M (Ω) := i =j µ(Ω i )µ(Ω j ) > 0 and B i are balls with mutual distance greater than a number n to be fixed later, and such that H 1 (∂B i ) = H 1 (∂Ω i ). In the first inequality we used the fact that (µ i (Ω i )) -1 µ i is admissible in the definition of I 1 (Ω i ), while for the second one we exploited the inequality I 1 (Ω i ) ≥ I 1 (co(Ω i )). Let now µ i be the optimal measure for B i . Then the measure i µ(Ω i ) µ i is a probability measure on ∪ i B i . Thus

I 1 (∪ i B i ) ≤ i µ 2 (Ω i ) d µ i (x) d µ i (y) |x -y| + i =j µ(Ω i )µ(Ω j ) d µ i (x) d µ j (y) |x -y| ≤ i µ 2 (Ω i ) I 1 (B i ) + i =j µ(Ω i )µ(Ω j ) n . (4.3)
By (4.2) and ( 4.3) we get

E Q λ (Ω) ≥ E Q λ (∪ i B i ) + λM (Ω) diam(Ω) - λM (Ω) n ≥ E Q λ (∪ i B i ), (4.4)
where the last inequality holds if we choose n big enough, depending on Ω.

To conclude, we only have to show that

E Q λ (∪ i B i ) ≥ E Q λ (B) (4.5)
for a ball B such that H 1 (∂B) = i H 1 (∂B i ), where equality holds only if the union of balls in the left-hand side of (4.5) is indeed a unique ball. But this is exactly the statement of Lemma 5. Once we know that the minimizer of E Q λ in A is a ball, we conclude by optimizing on the radius of all possible balls.

Proof of Corollary 1. The proof is contained in that of Theorem 1, also noting that at every step of the proof the perimeters of the considered sets cannot increase. Therefore, if the perimeter of the final ball B is strictly less than H 1 (∂Ω), we can further decrease I 1 by dilating that ball to ensure that its perimeter equals H 1 (∂Ω).

Proof of Theorem 2. We first prove (i). Notice that if λ = λ Q 0 , by Theorem 1 we know that the ball B R is a global minimizer of E Q λ in A. In particular

E Q λ Q 0 (Ω) ≥ E Q λ Q 0 (B R ), with
equality if and only if Ω is a closed ball. Now, since I 1 is maximized by balls under volume constraint, see [46, p. 14] and [47, Sec. VII.7.3, p. 157], and since λ Q 0 -λ > 0, we get

E Q λ (Ω) = E Q λ Q 0 (Ω) -(λ Q 0 -λ)I 1 (Ω) ≥ E Q λ Q 0 (B R ) -(λ Q 0 -λ)I 1 (Ω) ≥ E Q λ Q 0 (B R ) -(λ Q 0 -λ)I 1 (B R ) = E Q λ (B R ), (4.6)
with equality if and only if Ω is a ball; thus B R is a minimizer. Moreover all other minimizers are translates of B R . Indeed every minimizer has B R as its interior (up to a translation), and the boundary of the minimizer differs from ∂B R by a set of zero H 1 measure.

We now prove (ii). We have to show that there is no minimizer of E Q λ over A m for λ > λ Q 0 . We argue by contradiction and assume that there is a minimizer Ω ∈ A m of E Q λ . In view of the statement of Theorem 1, we have E Q λ (Ω) > 2π √ λ. We now consider a competitor set Ω = B R (0) ∪ N i=1 B r/N (x i ) , where N ∈ N, r > 0 is to be determined, R = m π -r 2 N , and x i ∈ R 2 with |x i -x j | R for any i = j. As in Lemma 2, the energy of Ω may be estimated from above by partitioning the total charge between B R (0) and B r/N (x i ). Suppose that the fraction of the total charge on each ball B r/N (x i ) is equal to θ/N for some θ ∈ (0, 1). We chose r = 1 2 θ √ λ and θ in such a way that the charge fraction 1 -θ on B R (0) is optimal for a ball of radius R according to Theorem 1: Then we get

λ(1 -θ) 2 = 4 R 2 =λ Q 0 - θ 2 λ N , (4.7)
which for λ > λ Q 0 has a unique solution for θ whenever N is sufficiently large. On the other hand, by Lemma 2 with our choices of the charge partitioning we have, for any ε > 0 and x i sufficiently far apart:

E Q λ ( Ω) ≤ E Q λ(1-θ) 2 (B R 0 (0)) + N i=1 E Q λθ 2 /N 2 (B r/N (x i )) + ε = 2π(1 -θ) √ λ + N i=1 2πθ √ λ N + ε = 2π √ λ + ε. (4.8)
In view of arbitrariness of ε, this contradicts the minimality of

E Q λ (Ω) > 2π √ λ.
Proof of Theorem 3. Just use as a competitor the set Ω constructed in the proof of Theorem 2, point (ii), to show that for any ε > 0 there exists a set whose energy is lower than 2π √ λ + ε. By Theorem 1 its energy is greater than 2π √ λ, thus we conclude.

Proof of Theorem 4. By Corollary 1 we know that for all Ω ∈ A we have E U λ (Ω) ≥ E U λ (B Ω ), where B Ω is a ball with the same perimeter as Ω, with equality if and only if Ω is a ball. As the energy of a ball of radius R is given in (2.12), this immediately implies that if λ = λ U 0 , then the energy of any ball is zero. In particular, statement (ii) follows. If λ > λ U 0 , by (2.12) the energy of a ball of radius R diverges to -∞ as R → +∞, proving statement (iii). To prove (i), let λ < λ U 0 and recall that I -1 1 is minimized by balls under volume constraint. Then we have that, for any Ω ∈ A, it holds

E U λ (Ω) = E U λ U 0 (Ω) + (λ U 0 -λ)I -1 1 (Ω) ≥ E U λ U 0 (B R ) + (λ U 0 -λ)I -1 1 (B R ) = E U λ (B R ) = (λ U 0 -λ) 2R π > 0, (4.9)
where B R is a ball with the same volume as Ω. Therefore, the infimum of E U λ is zero, and it is not attained.

Proof of Theorem 5. By (4.9) and Theorem 4 (ii) we immediately obtain (i). Let us prove (ii). For n ∈ N and d > 0 sufficiently large, we let r n := m/(πn) and Since λ > π 2 , the latter diverges to -∞ as n → ∞.

  and the infimum is not attained.(ii) If λ = λ U 0 , then inf Ω∈A E U λ (Ω) = 0, and the infimum is attained by any closed ball and no other set.

) be an ellipse with eccentricity e = 1 -

 1 (b/a)2 , where a and b are the major and minor semi-axes, respectively. ThenH 1 (∂Ω e ) = 4aE(e 2 )and I 1 (Ω e ) = a -1 K(e 2 ), (3.9)

Ω n,d := n i=1 B 2 √ πm 1

 i=121 rn (x i ) , (4.10) where the centers x i ∈ R 2 are chosen in such a way that |x i -x j | ≥ d for i = j. Thus |Ω n,d | = m, and by Lemma 2 its asymptotic energy as d → ∞ can be estimated as lim sup d→∞ E U λ (Ω n,d )≤n E U λ (B rn ) =

Acknowledgments

The authors wish to thank M. Goldman, J. Lamboley and V. Moroz for helpful discussions on the subject. The work of CBM was supported, in part, by NSF via Grants No. DMS-1313687 and DMS-1614948. MN and BR were partially supported by GNAMPA of INdAM, and by the University of Pisa via grant PRA-2015-0017.