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On an isoperimetric problem with a competing non-local term. I. The planar case

This paper is concerned with a study of the classical isoperimetric problem modified by an addition of a non-local repulsive term. We characterize existence, non-existence and radial symmetry of the minimizers as a function of mass in the situation where the non-local term is generated by a kernel given by an inverse power of the distance. We prove that minimizers of this problem exist for sufficiently small masses and are given by disks with prescribed mass below a certain threshold, when the interfacial term in the energy is dominant. At the same time, we prove that minimizers fail to exist for sufficiently large masses due to the tendency of the low energy configuration to split into smaller pieces when the non-local term in the energy is dominant. In the latter regime, we also establish linear scaling of energy with mass suggesting that for large masses low energy configurations consist of many roughly equal size pieces far apart. In the case of slowly decaying kernels we give a complete characterization of the minimizers.


Introduction
The isoperimetric problem is a classical problem in the calculus of variations, one formulation of which seeks to find a set of the smallest perimeter enclosing a prescribed volume. By the famous result of De Giorgi, in the Euclidean space R n the solution of this problem is well known to be a ball [12]. In this paper we are interested in the question how the solution of the isoperimetric problem is affected by an addition of a repulsive long-range force. Specifically, for n ≥ 2 we wish to study the variational problem associated with the energy functional E[u] := R n |∇u| dx + R n R n u(x)u(y) |x − y| α dx dy, (1.1) where u is the characteristic function of a subset of R n with finite perimeter and mass m > 0, and α ∈ (0, n) is a parameter. More precisely, we look for minimizers of the energy E[u] over all u ∈ A, where A := u ∈ BV (R n , {0, 1}) : The choice of the non-local term in (1.1) is motivated by the form appearing in a number of physical problems. In particular, the nonlocal term with α = 1 in either two or three space dimensions arises naturally due to Coulombic forces (electrostatic repulsion in the three-dimensional ambient space), in which case the characteristic function u may be associated with the uniform charge density over a subset of either the three-dimensional space or the two-dimensional plane [7,8,13,21,32,33,41]. The case α = 1 in three dimensions also arises in the studies of models of diblock copolymer melts and related polymer, as well as other, systems (see e.g. [11,17,22,25,27,29,34,40]). More generally, the non-local term is chosen in this form to have the following four properties [28]: a) The non-local term is invariant with respect to translations and rotations.
b) The non-local term is repulsive.
c) The non-local term is scale-free.
d) The non-local term scales with length faster faster than volume.
Indeed, the kernel in the non-local term depends only on the distance between two points. The repulsive nature of the non-local term is due to the fact that for α > 0 the kernel is positive and monotonically decreasing with distance. It is also important to note that the quadratic form in L 2 (R n ) generated by the kernel is positive-definite. We point out that when the non-local term has opposite sign (attractive long-range forces), the minimizers of the considered variational problem are still balls, since the non-local term in (2.1) increases with respect to Schwarz symmetrization [26]. Furthermore, the non-local term is scale-free, in the sense that dilations only result in the appearance of a multiplicative factor in front of the non-local term. The scale-free nature of the non-local term allows one to reduce the number of free parameters of the problem to a single one, which we choose to be the mass m. Indeed, in this case a coefficient in front of the non-local term may be eliminated via a rescaling in space (the coefficient in front of the interfacial term in the energy can be eliminated by rescaling the energy). Finally, the scaling property is ensured by the condition α < n. Notice that the non-local term is always infinite when α ≥ n and if the interior of the support of u is non-empty.
We note that a suitably regularized version of the non-local term in the energy in (2.1) with α ∈ (n, n + 1) and a negative sign in front leads to a non-local isoperimetric problem, in which the non-local term gives a generalized notion of the perimeter [5]. In fact, when α → n + 1 from below, the latter Γ-converges, after a suitable rescaling, to the usual perimeter [2] (see also [6]). In our problem, on the other hand, the non-local term has a very different effect. It acts as the square of the negative Sobolev norm of u and, therefore, favors rapid oscillations of u. This leads to a competition between the perimeter and the non-local term that can give rise to the appearance of non-trivial energy minimizing patterns in bounded domains [9, 10, 24, 28-30, 36, 38]. Our whole space problem, in turn, appears as a limit problem in the studies of Γ-convergence of the functional in (2.1) in the presence of a small coefficient multiplying the perimeter term [9] (see also [10,18,19,30] for a related problem).
Despite the apparent simplicity of the model, for the problem under consideration even the basic question of existence of minimizers is not completely straightforward. While in the surface energy-dominated regime (small masses, m ≪ 1) one would naturally expect the minimizers to exist and be in some sense approximations to balls, for non-local energy-dominated regime (large masses, m ≫ 1) the energy may be lowered by splitting a given configuration into several pieces and moving them far apart. In this situation the minimizers may fail to exist. Our goal is to address these questions analytically.
In this paper, we present a detailed analysis of existence vs. non-existence of minimizers of the considered variational problem in the particular case of two space dimensions. We chose to treat the n = 2 case separately, because in two space dimensions many technicalities simplify substantially, allowing one to concentrate on the issues associated with non-locality and making the analysis more transparent (the general case will be treated elsewhere [23]). Furthermore, in two dimensions the obtained results appear to be optimal, the estimates are readily made explicit, and the obtained results are applicable to a number of physical systems, including high-T c superconductors, magnetic bubble materials and ferroelectrics [13,21,32,41]. What we prove in the following sections is that for n = 2 and 0 < α < 2 the basic picture presented above is correct: the minimizer of the considered problem exists for small enough m and does not exist for large enough m. Note that the considered problem is different from the one studied in [37], where the non-local term has a compactly supported kernel and minimizers exist for all masses. Moreover, we prove that for m sufficiently small the minimizer is precisely a single disk.
The main ideas of the proofs are as follows. Existence of minimizers for small masses is proved by showing that the members of a minimizing sequence can be chosen to be connected. Non-existence is proved by showing that for large masses the minimizers must be long and slender, so it is always possible to reduce the energy by cutting the set in two and moving the pieces far apart. The fact that the minimizer at small masses is a ball is proved by exploiting the good stability properties of the minimizers of the usual isoperimetric problem.
We note that the intricate case of intermediate masses remains largely open. In particular, an interesting open question is weather the minimizer of the considered problem is, in fact, a ball whenever it exists. We prove that this is indeed the case for α sufficiently small. Another interesting open question is about the structure of the set of values of m for which the minimizer exists, in particular whether it is an interval. Again, in the case of small α we prove the latter to be the case and compute the precise threshold value of m separating the existence and non-existence regimes. In the full generality, however, these questions are currently out of reach for the methods of the present paper, since in our analysis we mainly employ the properties of the problem in the regimes dominated by either of the two terms in the energy. New tools that deal with the joint effect of the local and the non-local terms need to be developed to further address the finer properties of the minimizers in the considered problem in the regime of intermediate masses. One step in that direction is the precise scaling of minimal energy obtained by us for large masses, using an interpolation inequality relating the interfacial and the non-local parts of the energy.
Our paper is organized as follows. In Sec. 2, we collect all the main results of our paper. In Sec. 3, we present some background results and the results of explicit computations for several configurations. In Sec. 4, we prove existence of minimizers for sufficiently small masses. In Sec. 5, we prove the optimal scaling of the minimal energy for large masses. In Sec. 6, we prove non-existence of minimizers for large masses. In Sec. 7, we prove that minimizers for sufficiently small masses are disks. Finally, in Sec. 8, we prove that for sufficiently small α minimizers exist if and only if they are disks and if and only if their mass is less or equal than an explicit threshold value.

Statement of results
Throughout the rest of this paper we always assume that n = 2 and α ∈ (0, 2). The considered variational problem is then equivalent to minimizing where Ω is a set of finite perimeter in the plane, |Ω| denotes the Lebesgue measure of Ω, i.e., |Ω| = H 2 (Ω) and |∂Ω| denotes the perimeter of Ω, i.e., |∂Ω| = H 1 (∂Ω) (for definitions, see e.g. [3]). We note that, when dealing with minimizers of E in (2.1), we can always assume that ∂Ω is a collection of C 2,β curves, for some β ∈ (0, 1). This is due to the fact that minimizers of E are quasi-minimizers of the perimeter and, therefore, the standard regularity theory of minimal surfaces applies to them. By a straightforward cutting argument, we also show that minimizers are connected (but not necessarily simply-connected).
The proposition below collects some basic properties of minimizers: Let Ω be a minimizer of E. Then (i) The boundary ∂Ω of Ω is of class C 2,β , for some β ∈ (0, 1), with the regularity constants depending only on m and α.
(ii) Ω is bounded and connected. Moreover, Ω contains at most finitely many holes.
(iii) The Euler-Lagrange equation for (2.1) is where κ(x) and v(x) are the curvature and the non-local potential at the point x ∈ ∂Ω, respectively, and µ ∈ R is the Lagrange multiplier due to the mass constraint (the sign of the curvature is chosen to be positive for convex sets).
Note that further C 3,β regularity can be inferred when α < 1, since v ∈ C 1,β (R 2 ) for some β ∈ (0, 1) in that case. We now state our main results. Concerning the existence of minimizers, as was noted already in the introduction the question is not straightforward, since the minimizing sequences for (2.1) may consist of disconnected pieces moving off to infinity away from each other. What we can establish, however, is that in the regime of small masses, i.e., when the perimeter is the dominant term in the energy, minimizers of E do indeed exist. Our next result gives a complete characterization of the minimizers for sufficiently small masses.
Since in the case of small masses the energy is dominated by the perimeter, it is expected that the minimizer should be close to a disk. However, our result is stronger, stating that even for small (but positive) masses the minimizers is precisely a disk with mass m. At the same time, our next theorem shows that the minimizer (global or local) cannot be a disk if m is sufficiently large. Note that these kinds of results have been derived for a number of related problems [10,20,21,29,31,38,42].
Theorem 2.4 (Global and local instability of a disk). Let Ω = B R (0) with R = (m/π) 1/2 . There are two constants m c1 < m c2 given in (3.13) and (3.23) such that the following holds: (ii) Ω is not a local minimizer (with respect to arbitrarily small perturbations of the boundary) if m > m c2 = m c2 (α) > 0.
Note that by Theorem 2.4 disks cease to be global minimizers before they undergo shape instability. While Theorem 2.4 only shows that a minimizer cannot be given by a disk with mass m if m is sufficiently large, the next theorem encompasses a more general result: For sufficiently large masses the energy (2.1) does not have a minimizer. This point was conjectured in [9,Remark 4.2] for the case n = 3 and α = 1. In fact, a more general non-existence result holds for all α < 2 in dimensions n > 2 as well [23].
Theorem 2.5 (Non-existence). There exists m 2 = m 2 (α) > 0 such that there is no minimizer of E for all m > m 2 .
The non-attainability of the minimal energy for large masses established by Theorem 2.5 is related to the fact that for m ≫ 1 it is advantageous for the mass to escape to infinity. One could imagine that in this case the minimizing sequence consists asymptotically of disconnected sets of approximately equal masses m i ∼ 1 (for a related result in two dimensions, see [9,18,30]). In particular, the minimal energy should scale linearly with the mass m for large m. The next theorem supports this picture. In other words, for sufficiently small values of α the constants in Theorems 2.2-2.5 obey m 0 = m 1 = m 2 = m c1 . Thus, our results support a recent conjecture of [10] in the present setting for sufficienly slowly decaying kernels. We note that the arguments in the proof of Theorem 2.7 also imply that as soon as it is known that for a given value of α the minimizers of E can only be disks, we have m 0 = m 1 = m 2 = m c1 without the need to assume that α is small. The latter follows from a global result in Lemma 3.6(ii) concerning minimizers of E among sets consisting of two arbitrary disks.

Preliminaries
In this section, we present the necessary background for the proofs of the main theorems, as well as the results of some precise ansatz-based calculations. We first outline the proof of the regularity of ∂Ω for minimizers: Let Ω be a minimizer of E. Then ∂Ω is of class C 2,β , for some β ∈ (0, 1), with the regularity constants depending only on m and α.
Proof. Regularity of ∂Ω follows from the fact that every minimizer of E is a quasiminizer of the interfacial energy. More precisely, we claim that for any set of finite perimeter for some universal c > 0 1 . Indeed, assuming that (3.1) holds, we can immediately apply [37, Theorem 1.4.9] to conclude the uniform C 1,β regularity of ∂Ω for any minimizer Ω of E. Furthermore, the boundary satisfies the weak form of the Euler-Lagrange equation (2.2). Noting that clearly v ∈ C 0,β (R 2 ) for some β ∈ (0, 1) and since ∂Ω is locally a graph of a C 1,β function, the regularity assertion of the lemma follows by further application of the regularity theory for graphs [16] (for details of the argument, see e.g. the last paragraph of Sec. 2 in [39], as well as [15]). It hence remains to show (3.1). By a direct computation, for any R > 0 we have The statement then follows from the minimizing property of Ω by choosing R = (2 − α) 1/2 m 1/2 in (3.2).

Lemma 3.2.
Let Ω be a minimizer of E. Then Ω is connected.
Proof. By uniform Hölder estimates of Lemma 3.1, we have Ω ⊂ B R 0 (0) for some R 0 > 0 (where R 0 depends on the configuration). Therefore, if Ω (1) is a connected component and Ω\Ω (1) = ∅, then for any R > 0 we can consider a set Ω ′ obtained by translating Ω (1) outside B R 0 +R (0). The energy of the obtained set is then But for R sufficiently large, this inequality contradicts the minimizing property of Ω.
Let Ω be a minimizer of E.
Then Ω contains at most finitely many holes.
Proof. By Lemma 3.1 we can cover ∂Ω by finitely many balls of equal radius, which depends only on α and m, such that in each ball ∂Ω is a graph of a C 1 function. Therefore, since the perimeter of Ω is bounded, ∂Ω breaks into a finite collection of simple closed curves.
We now turn to the exact computations related to sets enclosed by ellipses. In the lemma below, we obtain an expression for the energy of such a set.

Lemma 3.4.
Let Ω e be a set enclosed by an ellipse of eccentricity e. Then where E(x) is the complete elliptic integral of the second kind, 2 F 1 (a, b; c; z) is the hypergeometric function, and R = (m/π) 1/2 .
Proof. We consider the region Ω e enclosed by an ellipse whose semi-axes are a and b. Since the area of the ellipse is πab = πR 2 , we have a = R/ 4 √ 1 − e 2 and b = R 4 √ 1 − e 2 . We also recall that the perimeter of the ellipse is given by the well-known expression where E(x) is the complete elliptic integral of the second kind [1]. To compute the non-local part E nl of the energy, we pass to the Fourier space. In terms of the Fourier transform of u = χ Ωe , the characteristic function of Ω e , the nonlocal part of the energy is given by (see e.g. [26]) is the Fourier transform of the kernel in the non-local term and Γ(x) is the Gammafunction [1]. To proceed, we note that if x = (x 1 , x 2 ), the rescaling x 1 → x 1 /a and , after a suitable translation, into the characteristic function of B R (0). Therefore, with q = (q 1 , q 2 ) one can write explicitly upon integration where J 1 (x) is the Bessel function of the first kind [1]. Performing another change of variables: , and then introducing polar coordinates q 1 = s cos t,q 2 = s sin t, upon integration in s we obtain Finally, performing the integration in t, we obtain is the hypergeometric function [1]. Combining this formula with (3.5), we obtain the result.
Setting e = 0, we also obtain the precise formula for the energy of a single ball of mass m = πR 2 .
The result in Lemma 3.4 enables us to give the proof of Theorem 2.4 concerning the failure of minimality (either global or local) of Ω = B R (0) for R = (m/π) 1/2 large enough. The proof relies on the explicit formula for the energy of elliptical domains (balls, in particular) obtained in Lemma 3.4 and Corollary 3.5. The proof of Theorem 2.4 follows directly from the next two lemmas. Note that it is easy to see that m c1 < m c2 where m c1 and m c2 are defined in (3.13) and (3.23). (3.13) (ii) The disk of area m has lower energy than any two non-overlapping disks of the same total area if m ≤ m c1 .
Proof. We first compare the energy of Ω = B R (0) with that of , and e 1 is the unit vector along the x 1 -axis. It is easy to see from (3.12) that if m > m c1 , where m c1 is defined in (3.13) and r is sufficiently large, contradicting minimality of Ω. We now show that it is energetically advantageous to replace a set Ω consisting of two non-overlapping disks of mass tm and (1 − t)m, with arbitrary t ∈ (0, 1), by a single disk of mass m whenever m ≤ m c1 . Indeed, by positivity of the kernel in the non-local part of the energy and (3.12) we have E(Ω) > 2 √ π m 1/2F (t, m/m c1 ), wherẽ The statement of the lemma is equivalent to showing that F (t, µ) :=F (t, µ) −F (0, µ) ≥ 0 for all µ ≤ 1 and, by symmetry, for all t ∈ [0, 1/2]. We claim that for fixed t the minimum of F as a function of µ is attained when µ = 1. Indeed, differentiating this expression with respect to µ, we find that Clearly, F 1 (t) is strictly convex and in view of the fact that F 1 (0) = 0, we have F 1 (t) < 0 and, hence, ∂F/∂µ < 0 for all t ∈ (0, 1/2]. Therefore, it is sufficient to show that F (t) ≥ 0 for µ = 1. We now write To prove that this function is non-negative for all α ∈ (0, 2) and t ∈ [0, 1/2] is a tedious exercise in calculus (the reader can readily verify this fact by plotting F as a function of two variables, α and t). Below we sketch the analytical argument. Introduce a cutoff parameter t 0 = 1/8. It is then not difficult to see that for all t ∈ [t 0 , 1/2] we have Then, by inspection we find that F (t, 1) ≥ 0 for all t ∈ [t 0 , 1/2] and all α ∈ (0, 2). We now turn to t ∈ (0, t 0 ). Here we have the following estimates: Again, by inspection these estimates imply that F (t, 1) ≥ 0 for all t ∈ (0, t 0 ) and all α ≤ 1.
To cover the range α > 1, we use a different estimate for F 1 : After some further manipulations, one can obtain a lower bound (3.23) Then the disk of area m is not a local minimizer of E (with respect to arbitrarily small perturbations of the boundary) if m > m c2 .
Proof. We expand the expression in (3.4) for the energy of Ω e as in Lemma 3.4 in the power series in e at e = 0: for sufficiently small e if m > m c2 , where m c2 is defined in (3.23). Under this condition the energy decreases upon arbitrarily small distortion of a disk into an ellipse.
Lastly, we also need to characterize the non-local potential generated by a unit ball and, specifically, its behavior near the boundary, in order to prove Theorem 2.3.

27)
where Γ(x) is the Gamma-function and (3.28) Proof. The proof is by an explicit computation. Introducing the Fourier transformv and using (3.8) and (3.9) with e = 0, we obtain whereû q is the Fourier transform of the characteristic function of the unit ball centered at the origin. Inverting the Fourier transform and integrating over the directions of q, with z = |q| we arrive at where J n (x) are the Bessel functions of the first kind. But the right-hand side of (3.31) coincides with the right-hand side of (3.26). Finally, the expansion in (3.27) is an immediate consequence of (3.26).

Existence of minimizers for small masses
We now prove the existence result in Theorem 2.2. The strategy of the proof is to suitably localize the minimizing sequence for E in (2.1). Existence of minimizers then follows by the usual compactness and lower-semicontinuity results for functions of bounded variation [3].
with Ω k ⊂ R 2 and |Ω k | = m, be a minimizing sequence for E. Without loss of generality, we may assume that each Ω k consists of N k < ∞ disjoint open connected components Ω |, and that the interfaces ∂Ω k are smooth. As a first step, we use a ball of radius R = (m/π) 1/2 as a test function to obtain an upper bound for the minimal energy. By comparing the energy of Ω with the energy of the ball B R (0), we may assume that for some C > 0 depending only on α (for the precise constant, see (3.12)). Suppose now that N k > 1 and, hence, |Ω (i) k | ≤ m/2 for all 2 ≤ i ≤ N k . By the isoperimetric inequality and by positivity of the non-local term in the energy, we have Squaring both sides of (4.2) and combining it with (4.1), after some algebraic manipulations we obtain for some C > 0 depending only on α.
On the other hand, consider a set Ω ′ k obtained by erasing Ω from Ω k and then dilating the resulting set by λ k = m/(m − |Ω , we replace the set Ω k by Ω ′ k in the minimizing sequence and repeat the above process. Then, after finitely many steps either the set Ω k is connected, or E(Ω ′ k ) ≥ E(Ω k ). In the latter case we can write and, therefore, Applying again the isoperimetric inequality on the left-hand side of (4.5) and using the fact that by (4.1) we have E(Ω k ) ≤ Cm 1/2 for some C > 0 depending only on α and m ≤ 1, we then conclude that in this case for some c > 0 depending only on α.
It is easy to see that for sufficiently small m the two inequalities in (4.3) and (4.6) are incompatible. Thus, given a minimizing sequence {Ω k } ∞ k=1 , for sufficiently small m it is always possible to construct another minimizing sequence {Ω ′ k } ∞ k=1 , in which each set Ω ′ k is connected. By a suitable translation, one can further assume that the origin belongs to each of Ω ′ k . In turn, since the perimeters of Ω ′ k are uniformly bounded above, we have Ω ′ k ⋐ B R (0) for some large enough R > 0. Therefore, introducing the characteristic functions u k ∈ BV (B R (0); {0, 1}) of Ω ′ k , we get that the functions u k are equibounded in BV (B R (0); {0, 1}). So up to extraction of a subsequence u k → u ∈ BV (B R (0); {0, 1}) strongly in L 1 (B R (0)) and u k ⇀ u in BV (B R (0); {0, 1}), with the limit independent of R. In particular, B R (0) u dx = m. Since the perimeter is lower-semicontinuous, and the non-local term is continuous with respect to the above convergence (the latter follows immediately from (3.2)), we conclude that the set Ω = {u = 1} is a minimizer.

Scaling of the minimal energy
We now consider the opposite extreme, in which the non-local term favors splitting of the set Ω into smaller disconnected sets. The corresponding scaling of the minimal energy is described by Theorem 2.6 whose proof is an immediate consequence of the following three lemmas. We note that the main point of Theorem 2.6 is the ansatz-free lower bound for large m which matches the upper bound from an ansatz consisting of a collection of equal size balls far apart. We also note that we only need to prove the bounds in Theorem 2.6 for sufficiently large masses. Indeed, by the isoperimetric inequality and by the positivity of the non-local term we have E(Ω) ≥ |∂Ω| ≥ 2 √ π m 1/2 , so E(Ω) is uniformly bounded away from zero whenever m ≥ c, for any c > 0. We begin with an ansatz-based upper bound.
Lemma 5.1. For every m ≥ 1 there exists Ω such that E(Ω) ≤ Cm for some C > 0 depending only on α.
Proof. The proof is by an explicit construction. We take where N = ⌊(m/π)⌋, e 1 is the unit vector along the x 1 direction, r = π −1/2 (m − πN ) 1/2 , and R > 2, i.e., we take Ω to be a linear chain of non-overlapping unit balls (except for the last one, whose radius is chosen to accommodate the mass constraint). Then by (3.12) we have So the assertion of the lemma follows by choosing R = m 1/α + 2.
We now turn to the ansatz-free lower bound. Here we present a simpler proof, which does not rely on Fourier techniques and the properties of special functions. The result is obtained from the following interpolation inequality: for some universal C > 0, which is valid for any u ∈ BV (R 2 ) ∩ L ∞ (R 2 ). Indeed, for any admissible set Ω, let u be the characteristic function of Ω. Applying (5.3), we then have For the proof of (5.3), we only need to take into account the non-local interaction on intermediate length scales of order R > 0, which will be determined later. With the change of variables z = y − x, we have Using the fact that R ≤ |z| ≤ 2R and that |B 2R (0)\B R (0)| = 3πR 2 , we hence get for some universal C, C ′ > 0 (recall that α ∈ (0, 2)), where we argued by approximating u with smooth functions, noting that by an argument similar to the one used in (3.2) the non-local term is continuous in the L 1 -topology. Therefore for some universal C > 0. Estimate (5.3) then follows by minimizing the right hand side of (5.7) in R, i.e. we choose (5.8) This concludes the proof of (5.3).
The following lemma strengthens the lower bound for configurations which satisfy the linear scaling of the energy:

Non-existence of minimizers for large masses
We now present the proof of Theorem 2.5. We begin with a basic estimate (from above and below) of the diameter of a minimizer Ω of E.
Lemma 6.1. Let m ≥ 1, let Ω be a minimizer of E and let d := diam(Ω). Then for some c, C > 0 depending only on α.
Proof. We first recall that by Proposition 2.1 the set Ω is regular and connected. Therefore, we have 2d ≤ |∂Ω| ≤ E(Ω) ≤ Cm for some C > 0 depending only on α, in view of Lemma 5.1.
On the other hand, which yields the second inequality.
Let us note that as an immediate consequence of Lemma 6.1 we get non-existence of minimizers for large masses when α < 1. We call this regime far field-dominated, as opposed to the opposite regime (α ≥ 1), which we call near-field dominated. We will see again in Sec. 7 that this distinction also plays a role for minimizers at small masses. Proof. By Lemma 5.1 and 6.1 every minimizer Ω has to satisfy E ≤ Cm and E ≥ diam(Ω) ≥ cm 1/α . For α < 1 and sufficiently large m, both inequalities cannot be satisfied at the same time.
We now turn to completing the proof of Theorem 2.5, which in view of Corollary 6.2 amounts to the proof of the following proposition. Proof. We argue by contradiction. Let Ω be a minimizer of E for some m ≥ 1. Introducing d := diam(Ω), let x 1 , x 2 ∈ ∂Ω be such that |x 1 − x 2 | = d. For every s ∈ (0, d), define T (s) to be the line perpendicular to x 1 − x 2 and located at distance s from x 1 . The line T (s) cuts the set Ω into two non-empty parts. We define the part of Ω which is closer to x 1 as Ω s . We also define V (s) := |Ω s | and A(s) := |Ω ∩ T (s)|. Note that A ∈ L ∞ (0, d) (since the diameter of Ω is bounded), and by Cavalieri's principle we have Also, without loss of generality we may assume that V (d/2) ≤ m/2. In particular, this implies that for all s ∈ (0, d/2), Now, consider a new set Ω ′ = (T R Ω s ) ∪ (Ω\Ω s ), where T R denotes a translation by distance R > 0 along the vector −−→ x 2 x 1 , i.e., the set Ω ′ is obtained by cutting Ω with T (s) and moving the resulting pieces distance R apart. We have Therefore, from the minimizing property of Ω we obtain Integrating this expression from s ∈ (0, d/2) to d/2, we then conclude that In particular, by Lemma 6.1 for some c > 0 depending only on α, i.e., V (s) becomes uniformly small for s ∈ (0, d/4] and m ≫ 1. Let us now show that the latter is impossible. We consider a different set Ω ′′ obtained by erasing Ω s from Ω and dilating the resulting set Ω\Ω s by a factor λ s = m/(m − V (s)) > 1 to make Ω ′′ admissible. By the minimizing property of Ω and positivity of the kernel in the non-local term, we have where we argued as in (4.4). Therefore, by isoperimetric inequality and Lemma 5.1 we have

Shape of minimizers for small masses
We now turn to the proof of Theorem 2.3. Here it is convenient first to rescale length in such a way that the rescaled set Ω has a fixed mass. Let us define a positive parameter Then the renormalized energy is related to the original energy as where Ω ε is obtained by dilating the set Ω by a factor of (m/π) −1/2 . We note that by virtue of Theorem 2.2, the minimizers of E ε exists for all ε ≤ ε 1 (α), where ε 1 is related to m 1 via (7.1). Furthermore, the regularity result in Proposition 2.1, with constants depending on ε and α, holds for the minimizers of E ε . Expressed in terms of the rescaled problem, Theorem 2.3 takes the following form: Proposition 7.1. There exists ε 0 = ε 0 (α) > 0 such that for all ε ≤ ε 0 the minimizer of E ε is a unit disk.
The proof proceeds differently for the far field-dominated (α < 1) and near field-dominated (α ≥ 1) regimes. But before we turn to the proof, let us establish a number of basic properties of the minimizers of E ε that we will need in our analysis. Recall that for any set of finite perimeter in R 2 , the isoperimetric deficit is given by We begin with a basic estimate of the isoperimetric deficit of minimizers.
Let Ω be a minimizer of E ε , and let D(Ω) be the isoperimetric deficit of Ω.
Then for some C > 0 depending only on α we have Proof. The proof is obtained by testing E ε with a unit disk. The assertion follows immediately from the minimizing property of Ω, positivity of the non-local term and the fact that by (3.12), We next establish that for small values of ε the minimizers are necessarily convex and, hence, simply connected.
The next lemma is key to the analysis of the small ε regime and is based on a Bonnesentype inequality for convex sets with small isoperimetric deficit [4] (for a review, see [35]). In view of Lemma 7.2, the latter is the case for the minimizers of E ε , when ε is sufficiently small. We will use a version of the result that was proved by Fuglede [14], which connects the isoperimetric deficit to the spherical deviation of the set Ω from a unit ball centered at the barycenter of Ω, to prove this lemma.
Proof. When ε is sufficiently small, the minimizer Ω of E exists, has small isoperimetric deficit by Lemma 7. We can now proceed to the conclusion of the proof of Theorem 2.3. We start with the far field-dominated case.
Proof. If ε is sufficiently small, there exists a minimizer Ω of E. Furthermore, the set Ω satisfies the conclusions of Lemma 7.4. Since Ω is a minimizer, we have E(Ω) ≤ E(B 1 (x 0 )), where x 0 is the barycenter of Ω, which is equivalent to (7.13) Let u and u B be the characteristic functions of Ω and B 1 (x 0 ), respectively, and let v B be as in (3.25). Then, since the non-local kernel is positive-definite (as can be seen from (3.7) and (3.8)), and since R 2 (u B − u) dx = 0, we have where v 0 is given by (3.28). Thus for some universal C > 0. On the other hand, by Lemma 3.8, we have |v B − v 0 | ≤ Cδ in B 1+δ (0)\B 1−δ (0), for some C > 0 depending only on α. Combining this inequality with by (7.11), (7.13) and (7.15), we get for some universal c > 0 and some C > 0 depending only α. Therefore, as long as ε is small enough, we have D(Ω) = 0, implying that Ω = B 1 (x 0 ).
We note that the above proof fails in the near field-dominated regime, α ≥ 1, since in this case v B fails to be in C 1 (R 2 ), as can be seen from (3.27) (in fact, the radial derivative of v B gets singular at ∂B 1 (0)). Therefore, a more delicate analysis of the contribution of the deviation of Ω from a ball to the non-local part of the energy is necessary. In fact, we need to prove some cancellations in the difference of the two nonlocal energies (related to the minimizer and the corresponding ball of the same area) to obtain an analog of (7.16). For this, we will make a more detailed use of the Euler-Lagrange equation.
It remains to prove the following proposition.
Proof. The main point here is to obtain the inequality in the right-hand side of (7.16) from (7.13). The conclusion then follows as in the proof of Proposition 7.5. We begin by writing where and i.e., x 1 (x) is the center of a ball whose center is shifted from x 0 in the direction of x in such a way that x ∈ ∂B 1 (x 1 (x)). Introducing polar coordinates as in Lemma 7.4(ii), we have (with a slight abuse of notation) Let us estimate the term in (7.22) first. In the following we will only explicitly consider the case α > 1, the case α = 1 is treated analogously. In view of (3.27), with s = r − 1 we have (7.23) for some C > 0 depending only on α, where However, the integral in the second line of (7.23) is identically zero, so we have II = O(δ 2 ). We now turn to estimating (7.21), which can be written as where d(s, s ′ , θ, θ ′ ) is the distance between the points with polar coordinates (1 + s, θ) and (1 + s ′ , θ ′ ), and ρ B (θ, θ ′ ) solves (7.26) and for each θ simply describes the polar graph r(θ ′ ) = 1 + ρ B (·, θ ′ ) of a circle shifted by ρ(θ) in the direction of θ from the origin. Clearly, ρ B (θ, ·) ∈ C ∞ (R) for sufficiently small δ, and furthermore for all θ, θ ′ ∈ R we have for some universal C > 0. In addition, for small enough δ we have d(s, s ′ , θ, θ ′ ) ≥ c|θ − θ ′ |, (7.28) for some universal c > 0. Combining all the information above, we can write (7.29) for some C, C ′ , C ′′ > 0 depending only on α, where here and below the subscript θ denotes a derivative with respect to θ. So, in order to conclude that I = O(δ 2 ) as well, it remains to show that for some C > 0 depending only on α.
To obtain (7.30), we write the Euler-Lagrange equation for ρ(θ) in polar coordinates. Using the well-known formula for the curvature in polar coordinates, we can write (7.7) in the form where v = v(1 + ρ(θ), θ). In fact, by continuity of ρ(θ) there exists θ * such thatv = v(1 + ρ(θ * ), θ * ). Now, by [14,Lemma 2.2], for sufficiently small δ we also have with some universal C > 0. Therefore, subtracting 1 from both sides of (7.31), after a straightforward calculation we obtain On the other hand, arguing as in (7.29), we have for some C, C ′ > 0 depending only on α. In particular, since the same estimate holds for θ = θ * , we have for some C > 0 depending only on α.

Complete characterization in the case of small α
In this section, we present the proof Theorem 2.7. The proof is a slight modification of the proof of Proposition 7.5, and so we find it more convenient to work with the energy in (7.2) (but now without the smallness assumption on ε). The proof also requires a refinement of the non-existence result from Sec. 6.
In terms of E ε , the result we wish to obtain is a consequence of the following proposition.
Proposition 8.1. There exists α 0 > 0 such that for all α ≤ α 0 the minimizer of E ε , if it exists, is given by The proof follows from a sequence of lemmas.
Let Ω ⊂ R 2 be a set of finite perimeter, and let |Ω| = π. Then where the constant C > 0 depends only on d := diam(Ω).
Our next lemma establishes non-existence of minimizers of E ε for sufficiently large ε uniformly in α (as long as α ≤ α 0 < 1 for some fixed α 0 ). Lemma 8.3. For every α 0 ∈ (0, 1) there exists ε 2 > 0 (depending only on α 0 ) such that for every α ∈ (0, α 0 ] there is no minimizer of E ε for any ε > ε 2 . Proof. We prove the statement for the original energy E, which amounts to existence of m 2 = m 2 (α 0 ) > 0 such that there is no minimizer of E for all m > m 2 and α ∈ (0, α 0 ]. By Lemma 5.1, for a minimizer Ω of E we have E(Ω) ≤ Cm for all m ≥ 1, where the dependence of the constant C > 0 on α is via E(B 1 (0)). By continuous dependence of E(B 1 (0)) on α ∈ [0, α 0 ] (see (3.12)), we can, in fact, choose C ≥ 1 to depend only on α 0 . Therefore, arguing as in the proof of Lemma 6.1, we have We next prove that minimizers of E ε must have small isoperimetric deficit for sufficiently small α.
The result in Lemma 8.4 implies that we can use the same ideas as in Sec. 7 (in the far field-dominated case), replacing ε with α and taking advantage of the smallness of α, to prove radial symmetry of minimizers. In particular, we have the analog of Lemma 7.3: Lemma 8.5. There exists α 0 ∈ (0, 1) such that for every α ∈ (0, α 0 ] any minimizer Ω of E ε is convex. Similarly, the analog of Lemma 7.4 is the following: Lemma 8.6. There exists α 0 ∈ (0, 1) such that for every α ∈ (0, α 0 ] any minimizer Ω of E ε satisfies B 1−δ (x 0 ) ⊂ Ω ⊂ B 1+δ (x 0 ), where x 0 is the barycenter of Ω, for some δ ≤ C D(Ω), with some universal C > 0.
Proof of Proposition 8.1. We argue as in the proof of Proposition 7.5. Repeating the steps of that proof with the help of Lemmas 8.4, 8.5, and 8.6, we obtain for some universal c, C > 0, where the second inequality in (8.3) follows from the fact that the potential v B given by (3.25) obeys for some universal C > 0, provided that α 0 is sufficiently small. The proof is then completed by observing that (8.3) implies D(Ω) = 0 for α 0 sufficiently small. . Then E(Ω k ) → e(m) := inf |Ω|=m E(Ω) as k → ∞, and by an approximation argument we may assume that all sets Ω k consist of N k < ∞ disjoint open connected components. In fact, Ω k can be chosen so that N k is independent of k. Indeed, by Theorem 2.3 we can lower the energy by replacing all the connected components whose mass is less than m 0 with balls of the same mass translated sufficiently far apart (as in the proof of Lemma 3.2). Then, if more than one ball is present in the resulting set, by Lemma 3.6(ii) we can further lower the energy by merging these balls, two at a time, and translating the resulting balls further apart.
In view of the above argument we may assume that Ω k = N j=1 Ω (j) k with N k ≤ N for some N ≤ 1 + (m/m 0 ) and the sets Ω (j) k are connected (some of Ω (j) k are empty if there are less than N connected components). After taking a subsequence, we may assume that for k | → µ j for some constants e j ≥ 0 and µ j ≥ 0 as k → ∞, and, furthermore, by compactness each set Ω (j) k "converges" to a set Ω (j) as k → ∞ after a suitable translation. More precisely, if u (j) k are the characteristic functions of Ω (j) k translated to contain the origin, then u is the characteristic function of Ω (j) . Furthermore, since the sets Ω Indeed, if Ω j ⊂ R 2 are such that |Ω j | = µ j and E(Ω j ) < e(µ j ) + δ for some δ > 0, we can construct a set Ω ′ with |Ω ′ | = m and E(Ω ′ ) < N j=1 e(µ j ) + 2δ by taking Ω ′ to be a union of Ω j translated sufficiently far apart. The result then follows by arbitrariness of δ. At the same time, we have N j=1 E(Ω (j) ) ≤ e(m). We now claim that E(Ω (j) ) = e(µ j ). Indeed, clearly E(Ω (j) ) ≥ e(µ j ) for all 1 ≤ j ≤ N . On the other hand, by (8.5) and ( Again, since e(µ j ) ≤ E(Ω (j) ) for each 1 ≤ j ≤ N , we get e(µ j ) = E(Ω (j) ) as well. Thus, each set Ω (j) is a minimizer of E with prescribed mass µ j . Therefore, by Proposition 8.1 for each 1 ≤ j ≤ N the set Ω (j) is either a ball or is empty. Then, repeating the argument at the beginning of the proof, with the help of Lemma 3.6(ii) we conclude that E(B R (0)) ≤ N j=1 E(Ω (j) ), where R = (m/π) 1/2 , and, hence, B R (0) is a minimizer by (8.6).
Remark 8.7. It is easy to see that to the leading order in α the non-local part of the energy in (8.1) is generated by the kernel g(x − y) ≃ ε ln |x − y| −1 , which appears in the studies of the sharp interface version of the Ohta-Kawasaki energy in two dimensions [9,18,30,43]. In this respect the result of Proposition 8.1 is closely related to the rigidity result obtained in [30,Proposition 3.5].
Finally, let us point out that the fact that the minimizers in Theorem 2.7 exist if and only if m ≤ m c1 , where m c1 is given by (3.13), does not rely on smallness of α and would remain valid as long as minimizers of E are disks whenever they exist. This can also be seen from the following general result, which says, essentially that any set of finite perimeter can be replaced by a set with lower energy consisting of a union of finitely many disjoint sets, each of which is a minimizer of E.
If Ω is a minimizer of E, there is nothing to prove. So assume that it is not. Without loss of generality, we may assume that Ω consists of N < ∞ disjoint open connected components, denoted by Ω i , and that Ω has smooth boundary. By positivity and decay of the kernel in the non-local part of the energy, we have where Ω i are minimizers of E with mass m i , whenever such a minimizer exists. The strict inequality in (8.10) follows from the fact that for N = 1 the set Ω is not a minimizer, while for N > 1 the energy can be reduced by spreading different connected components sufficiently far apart (as, e.g., in the proof of Lemma 5.1). Suppose that Ω i = Ω i . If the minimizer of E exists for mass m = m i , we replace Ω i with Ω ′ i = Ω i . By the minimizing property of Ω i , we then have E(Ω i ) = inf m=m i E. Alternatively, if the minimum of E is not attained at m = m i , there exists δ 0 > 0 such that if Ω ′ i has mass m i and E(Ω ′ i ) < inf m=m i E + δ 0 , then Ω ′ i is disconnected. Indeed, if not, there exists a minimizing sequence consisting of Ω k ⊂ R 2 with |Ω k | = m i and each Ω k connected. Then by the argument in the proof of Theorem 2.2 the minimum of E is attained, contradicting our assumption.
We can, therefore, replace all sets Ω i = Ω i for those values of m i at which the minimum of E with m = m i is not attained with disconnected sets Ω ′ i such that for some δ ∈ (0, δ 0 ) to be specified later. Observe that by Theorem 2.2 for all those we have m i > m 1 = m 1 (α) > 0. Therefore, the number of the components Ω i = Ω i corresponding to m i at which the minimum of E at m = m i is not attained is bounded above in terms of |Ω|. We now apply to each such component the algorithm in the proof of Theorem 2.2 to lower energy by erasing the smallest connected component of Ω ′ i and rescaling the resulting set back to mass m i . In view of the fact that δ < δ 0 , this process must terminate before only one connected component remains. Then, arguing as in the proof of Theorem 2.2, we conclude that the mass of each remaining connected component of Ω ′ i is bounded below by some c > 0 depending only on E(Ω) and α.
We are now able to choose δ > 0 sufficiently small and construct a new set Ω ′ with |Ω ′ | = |Ω| and E(Ω ′ ) < E(Ω) by taking the union of all connected components of the sets Ω ′ i constructed above, suitably translated to be sufficiently far apart. In this process the mass of each connected component of Ω ′ that is distinct from a minimizer is bounded above by max Ω i =Ω i |Ω i | − c. Repeatedly applying this process, we then find that after finitely many iterations all connected components are minimizers.