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A nonlocal isoperimetric problem with dipolar repulsion

We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the nonlocal term asymptotically localizes and contributes to the perimeter term to leading order. We establish existence of generalized minimizers for all values of the dipolar strength, mass and regularization cutoff and give conditions for existence of classical minimizers. For subcritical dipolar strengths we prove that the limiting functional is a renormalized perimeter and that for small cutoff lengths all mass-constrained minimizers are disks. For critical dipolar strength, we identify the next-order $\Gamma$-limit when sending the cutoff length to zero and prove that with a slight modification of the dipolar kernel there exist masses for which classical minimizers are not disks.


Introduction
Understanding the emergence of spatial order from basic constitutive interactions is one of the most important problems in the natural sciences. For example, why do the ions of Na + and Cl − organize themselves into an alternating pattern arranged into a cubic crystal when precipitating from a supersaturated aqueous solution, something that can be readily observed in a simple tabletop experiment? Our present physical understanding is that this process is driven by the competition of repulsive electrostatic interactions between the like ions, attractive interactions between the opposite ions, and a hard-core repulsion at short distances, to minimize the total interaction energy (both quantum mechanical and thermal effects are also present, but are believed to be of secondary importance). In these terms, the fundamental problem was concisely articulated in 1967 by Uhlenbeck [60, p. 581]: "The basic difficulty lies perhaps in the fact that one does not really understand the existence of regular solids from the molecular forces." For ionic crystals in the much simpler periodic setting, the question goes back much further [10] and was resolved only very recently [7].
The above question becomes even more complicated for problems involving many-body effects. Perhaps the best known example is that of the spatial arrangement of protons and neutrons in nuclear matter, which is relevant both to the shape of ordinary atomic nuclei [29] and the exotic phases of matter in the crust of neutron stars [52]. The same problem is also ubiquitous in various hard and soft condensed matter systems in which mesoscopic phases form as a result of competing attractive and repulsive interactions operating on different scales [6,30,47,49,55]. The earliest model that captures the competition of short-range attractive forces and long-range repulsive forces in the case of the atomic nuclei was conceived in 1929 by Gamow [24] and further refined by Heisenberg [29] and von Weizsäcker [61]. It is now known as the liquid drop model of the atomic nucleus. In this model, the nucleus is treated as a drop of incompressible liquid held together by surface tension and subject to Coulombic repulsion by the uniformly distributed positive charge of the protons. For dense nuclear matter, the same model is considered within a large periodic box and is known to produce multiple morphologies referred to as nuclear "pasta" phases [52].
Mathematically, the liquid drop model belongs to a class of geometric variational problems in which one minimizes energies of the form (for a non-technical overview, see [12]) E(Ω) := P (Ω) + 1 2ˆΩˆΩ G(x − y) dx dy, (1.1) among measurable sets Ω ⊂ R n subject to a mass constraint |Ω| = m for m > 0. Here |Ω| denotes the n-dimensional Lebesgue measure of the set Ω, P (Ω) is the perimeter of Ω, which is a suitable generalization of the surface measure (for precise definitions, see below), and G(x) is some "repulsive" kernel. In the case of the liquid drop model in the whole space, one chooses the Newtonian potential G(x) := 1 4π|x| and minimizes over all Ω ⊂ R 3 . In a periodic setting, one should instead consider Ω to be a subset of a large three-dimensional torus, and G should be the periodic Green's function of the Laplacian (with uniform neutralizing background) [38]. Note that with different spatial dimensionalities and different choices of repulsive kernels the model is also relevant to a number of other physical situations [35,36,47,48]. In fact, the case of dipolar repulsion considered in this paper falls within the above framework as well.
In the mathematical literature, nonlocal isoperimetric problems in which perimeter competes with a nonlocal repulsive term seem to have been largely unnoticed until quite recently, with a notable exception of a paper by Otto on the dynamics of labyrinthine pattern formation in ferrofluids [51], and a paper by Rigot dealing mostly with regularity of minimizers of (1.1) [53]. Gamow's liquid drop model caught the attention of mathematicians after reappearing as the leading order asymptotic problem in the studies of the Ohta-Kawasaki energy by Choksi and Peletier [13,14]. Since then the problem has enjoyed a considerable attention. In the following, we review some of the results obtained so far (naturally, the list of references below is not meant to be comprehensive).
The first study of the problem associated with (1.1), in which Ω ⊂ R n and G(x) = 1 |x| α is a Riesz kernel with α ∈ (0, n) was carried out by Knüpfer and Muratov [35,36]. This setting includes the classical Gamow's model, for which n = 3 and α = 1. For a range of parameters covering the latter, their results establish existence and radial symmetry of minimizers for sufficiently small masses, and non-existence for sufficiently large masses. Radial symmetry for small masses was also independently established by Julin in R n for all n ≥ 3 in the case of G(x) = 1 |x| n−2 [32], and by Bonacini and Cristoferi for a range of Riesz kernels [8]. In bounded domains with a particular choice of boundary conditions, Cicalese and Spadaro proved that minimizers are close to balls in the vanishing mass limit, but cannot be exactly spherical unless the original domain is a ball [15]. A further generalization to all Riesz kernels and nonlocal perimeters is due to Figalli, Fusco, Maggi, Millot and Morini [19] in the spirit of the currently developing theory of nonlocal minimal surfaces (see, for example, [11]). Non-existence of minimizers for large masses for Gamow's model was also independently established by Lu and Otto [41], and Frank, Killip and Nam provided an explicit estimate for the mass beyond which minimizers do not exist [21]. It is currently an open problem whether the minimizers are balls whenever they exist in the case of Gamow's model.
On the other hand, minimizers always exist in the periodic setting. Alberti, Choksi and Otto [2] proved uniform distribution of mass, which is a popular first step for approaching the question of periodicity of minimizers. In various low volume fraction regimes, Choksi and Peletier [13] and Knüpfer, Muratov and Novaga [38] proved that the minimizers consist of small isolated droplets, whose shape is asymptotically determined by solutions of the whole space problem for certain masses a) Fig. 5.118. Modification of a bubble lattice in perpendicular fields. The initial "microcrystalline" bubble lattice (a) expands and "melts" under the action of a perpendicular field by the familiar process of the growing of large cells and the collapse of small cells. Remarkably, a certain class of small cells, the five-sided cells, remain metastable up to quite high fields (c-e). After their collapse the network expands rapidly (f ). Reducing the field again (g-h) does not lead to a renucleation of bubbles but to a folding of the remaining cell walls into a maze pattern at zero field (h), which appears again "crystallized", but in a completely different configuration compared to (a). (Sample: BiCuGa-modified yttrium iron garnet film of 5 µm thickness; courtesy W. Tolksdorf ) irregularities. The mentioned reproducibility applies to the nucleation pattern only; not to the subsequently developing coarser configurations.
The beautiful patterns developing out of bubble lattices in a perpendicular field parallel to the bubble magnetization have always attracted the attention of investigators [1057,883]. In such fields the bubbles expand, transforming the former matrix into a network of narrow domains in which each cell represents a former bubble domain. A magnetization cycle starting from a remanent bubble lattice state and ending at a band domain state as shown in Fig. 5.118 demonstrates configurational hysteresis phenomena that appear to be independent of crystal lattice defects or imperfections.
The process starts with the collapse of smaller cells and the growth of larger cells. Peculiar is the relative stability of small fivefold cells, so-called bubble traps [755] (see Sketch 5.20), which persist up to a critical field [slightly above b) Observation and Interpretation fication of a bubble lattice in perpendicular fields. The initial bubble lattice (a) expands and "melts" under the action of a by the familiar process of the growing of large cells and the col-. Remarkably, a certain class of small cells, the five-sided cells, up to quite high fields (c-e). After their collapse the network . Reducing the field again (g-h) does not lead to a renucleation folding of the remaining cell walls into a maze pattern at zero ears again "crystallized", but in a completely different configu-(a). (Sample: BiCuGa-modified yttrium iron garnet film of 5 µm W. Tolksdorf ) mentioned reproducibility applies to the nucleation pattern ubsequently developing coarser configurations. patterns developing out of bubble lattices in a perpendicular e bubble magnetization have always attracted the attention 057, 883]. In such fields the bubbles expand, transforming into a network of narrow domains in which each cell repbubble domain. A magnetization cycle starting from a rettice state and ending at a band domain state as shown in trates configurational hysteresis phenomena that appear to crystal lattice defects or imperfections. arts with the collapse of smaller cells and the growth of larger he relative stability of small fivefold cells, so-called bubble etch 5.20), which persist up to a critical field [slightly above c) Fig. 5.118. Modification of a bubble lattice in perpendicular fields. The initial "microcrystalline" bubble lattice (a) expands and "melts" under the action of a perpendicular field by the familiar process of the growing of large cells and the collapse of small cells. Remarkably, a certain class of small cells, the five-sided cells, remain metastable up to quite high fields (c-e). After their collapse the network expands rapidly (f ). Reducing the field again (g-h) does not lead to a renucleation of bubbles but to a folding of the remaining cell walls into a maze pattern at zero field (h), which appears again "crystallized", but in a completely different configuration compared to (a). (Sample: BiCuGa-modified yttrium iron garnet film of 5 µm thickness; courtesy W. Tolksdorf ) irregularities. The mentioned reproducibility applies to the nucleation pattern only; not to the subsequently developing coarser configurations.
The beautiful patterns developing out of bubble lattices in a perpendicular field parallel to the bubble magnetization have always attracted the attention of investigators [1057,883]. In such fields the bubbles expand, transforming the former matrix into a network of narrow domains in which each cell represents a former bubble domain. A magnetization cycle starting from a remanent bubble lattice state and ending at a band domain state as shown in Fig. 5.118 demonstrates configurational hysteresis phenomena that appear to be independent of crystal lattice defects or imperfections.
The process starts with the collapse of smaller cells and the growth of larger cells. Peculiar is the relative stability of small fivefold cells, so-called bubble traps [755] (see Sketch 5.20), which persist up to a critical field [slightly above Sketch 5.20. (as already mentioned, they are presently unknown, although conjectured to be balls; see also [22]). Similar results are also available for Gamow's model with screening and for the Ohta-Kawasaki energy, which can be understood as a diffuse interface approximation to Gamow's liquid drop model [14,26,27,48]. In particular, it is proved that in two dimensions the droplets become almost circular. In other regimes, Sternberg and Topaloglu identified stripes as the global minimizers in the case of the two-dimension torus [56], and Morini and Sternberg showed that such patterns also turn out to be minimizers in thin domains [45]. Another class of (anisotropic) nonlocal isoperimetric problems in dimensions n ≥ 2 in which minimizers are one-dimensional and periodic is given by Goldman and Runa [28], and by Daneri and Runa [18].
In this paper, we study a nonlocal isoperimetric problem described by (1.1) in two space dimensions, n = 2, in which the kernel G is of dipolar type, i.e., G(x) 1 |x| 3 . This is the interaction experienced at large distances by spins lying on the plane and oriented perpendicularly to it. Just like the classical Gamow's model, the model under consideration is relevant to the multidomain patterns observed in perpendicularly magnetized thin film ferromagnets [30], as well as ferroelectric films [57], Langmuir monolayers [5] and ferrofluid films subject to a strong perpendicular applied field [31,54] (for an experimental realization involving a magnetic garnet film, see Figure 1). Note, however, that setting G(x) = 1 |x| 3 would result in an ill-defined problem because of the divergence of the integral defining the nonlocal contribution at short scales. Furthermore, redefining the energy in the spirit of nonlocal minimal surfaces [11] to reduce the integral to that over Ω and Ω c , up to an additive constant, would not help, either, since the singularity of the kernel is still too strong. Therefore, a genuine regularization at short scale δ > 0 is necessary to make sense of the energy in (1.1) with this kind of kernel. This is a novel feature of the considered nonlocal isoperimetric problem compared to those studied previously.
More specifically, we study the following nonlocal isoperimetric problem, as proposed by Kent-Dobias and Bernoff [34]. We wish to minimize among all finite perimeter sets Ω ⊂ A m of fixed mass m > 0, i.e., with Here, α, β > 0 are fixed parameters, P (Ω) is the perimeter of a measurable set Ω in the sense of De Giorgi [4]: and g δ (r) := g(r/δ) is a cutoff function at scale δ > 0 that makes the integral well-defined. The specific choice is not essential, so for simplicity we work with g(r) := χ (1,∞) (r) throughout the rest of the paper. Note that by a rescaling we can always choose α = 1. We are interested in the regime in which δ is much smaller than the characteristic length scale of minimizers, expressing the physical condition that the regularization happens on the atomic scale, which is much smaller than the scale of the observed patterns. Ultimately, we wish to send the parameter δ to zero to obtain results that are insensitive to the short-scale cutoff. As we show below, for a meaningful limit as δ → 0 to exist at fixed value of m > 0 it is necessary to renormalize the strength β of the dipolar interaction. We set β := λ/| log δ| for some λ > 0 and restrict ourselves to the case δ < 1 2 . Then the nonlocal term can be rewritten, so that, up to an additive constant depending only on m, λ and δ, and with α = 1 the energy in (1.2) is equal to where χ Ω is the characteristic function of the set Ω. First, we will make sure to demonstrate that minimizers always exist in the generalized sense of consisting of finitely many components that are "infinitely far apart" from each other. In the subcritical regime, λ < 1, we then prove that the Γ-limit of the energy is given by (1 − λ)P (Ω), i.e., the nonlocal term localizes to leading order and renormalizes the perimeter as δ → 0. Moreover, we prove in Theorem 2.1 that the minimizers of E λ,δ exist also in the classical sense and are in fact disks for all δ 1. The strategy is to make use of sufficiently uniform regularity estimates for minimizers, first on the level of density estimates and then in terms of curvature, as well as stability of the disk with respect to perturbations of curvature.
On the other hand, it is easy to see that disks are no longer classical minimizers for δ 1 as soon as λ > 1, as it is more convenient to split a single disk into multiple components. In fact, by proving that (the components of generalized) minimizers cannot contain disks of radius larger than r(δ) > 0 with r(δ) → 0 as δ → 0 we see that generalized minimizers cannot be asymptotically well-behaved in this limit. As such, a more precise analysis likely requires further insight into the question whether minimizers are large collections of disks or exhibit stripe-like behavior (see Figure  1).
This naturally brings us to the critical case λ = 1, where in view of the above arguments a transition from classical, radial to non-trivial minimizers occurs for δ 1. In this case, we compute the next-order Γ-limit as δ → 0 by observing that the sequence | log δ|E 1,δ (Ω) for a fixed set Ω ⊂ R 2 of finite perimeter is monotone decreasing in δ. Suitably integrating by parts allows us to represent the limit in a closed form that in fact does not fall within the class of problems given by (1.2). While we cannot directly address the issue of minimizers for the limit, we are able to prove that after modifying the functional by reducing the repulsion at infinity there exist masses for which classical minimizers exist, but are not given by disks. The idea of the proof is to construct a long stripe with large mass whose energy per mass is lower than the energy per mass of disks, ruling out the optimality of any collection of disks.
The paper is organized as follows: In Section 2, we give the precise statements of our main results. In Section 3, we give various representations of the energies and their rescaling behaviors. Section 4 is dedicated to proving existence of generalized minimizers together with control over the number of their components. We also already start to look into the regularity properties of minimizers on a qualitative level. Section 5 contains a thorough discussion of the subcritical case λ < 1, culminating in the proof of Theorem 2.1. The critical case λ = 1, and in particular the existence of non-radial minimizers (Theorem 2.8) for the modified problem is dealt with in Section 6. Finally, Proposition 2.9 characterizing the behavior of generalized minimizers as δ → 0 in the supercritical case λ > 1 is presented in Section 7.

Main results
Our first result identifies the minimizers of E λ,δ over A m in the subcritical regime of small δ and 0 < λ < 1 as disks. The significance of the condition λ < 1 is that even though the nonlocal term renormalizes the perimeter in the limit δ → 0, we still retain control over the perimeter as is evident by the L 1 -Γ-limit of E λ,δ for δ → 0 being given by (1 − λ)P (Ω), see Proposition 5.3. This allows us to obtain density estimates for Ω and the reduced boundary ∂ * Ω (for the definition, see, e.g., [42,Chapter 15]) that are uniform in δ. As the localization also turns out to take place in the Euler-Lagrange equation of E λ,δ , we can control the curvature of minimizers, which together with a stability property of disks allows us to conclude that the minimizers are disks.
Remark 2.2. Note that we formulated the theorem for mass m = π in order to have a somewhat clean criterion for the smallness of δ in terms of λ. However, the rescaling given in Lemma 3.2 allows to obtain a similar result for all masses by replacing δ with π m δ and λ with |log( √ π m δ)| | log δ| λ. Also note that when m = π and λ approaches 1 for a fixed value of δ, condition (2.1) may still be satisfied, while condition (2.2) fails, indicating the possibility of existence of non-radial minimizers. Whether or not the latter indeed happens for minimizers of E λ,δ over A π is an open problem.
As the first-order Γ-limit in the critical case λ = 1 degenerates, we have to compute a higherorder Γ-limit to obtain a helpful limiting description. It turns out that the appropriate sequence to analyze is | log δ|E 1,δ , as suggested, for example, by the estimate of Proposition 5.1. The L 1 -Γ-limit is then given by  Theorem 2.3. For every m > 0, the Γ-limit of | log δ|E 1,δ restricted to A m , with respect to the L 1 -topology as δ → 0, is given by E 1,0 in the following sense: (i) (Lower bound) Let Ω, Ω δn ∈ A m such that |Ω δn ∆Ω| → 0 and δ n → 0 as n → ∞. Then it holds that (ii) (Upper bound) For every set Ω ∈ A m , a recovery sequence is given by the constant sequence, i.e., it holds that In particular, the Γ-limit coincides with the pointwise limit.
The limiting functional has previously been investigated numerically by Bernoff and Kent-Dobias [34] on the basis of a formal calculation. We are also able to rigorously justify their representation and extend it to sets with a C 2 -boundary.

Proposition 2.4.
If Ω ⊂ R 2 is a set of finite perimeter with a C 2 -boundary, which is decomposed into positively oriented closed Jordan curves γ i : [0, P i ] → R 2 of length P i parametrized by arc length for 1 ≤ i ≤ N , with some N ∈ N, then it holds that (2.7) When trying to prove a compactness result for the whole space corresponding to Theorem 2.3, one has to deal with the issue that the functionals E λ,δ have no a priori compactness due to their translational symmetry. Already for fixed parameters this issue manifests itself in minimizing sequences potentially breaking up into multiple pieces which the repulsive nonlocal term then pushes infinitely far apart from each other. In particular, we can in general only hope to prove that minimizers exist in a generalized sense: We say that a generalized minimizer of E over A m is a minimizer of the functional defined on the domain The main point of this definition is finiteness of the number of pieces, which relies on almostminimality of minimizing sequences and is not true for arbitrary sequences of sets with uniformly bounded perimeters. The existence of generalized minimizers for all parameters λ, δ, m > 0 is the main content of Section 4.
Coming back to augmenting Theorem 2.3 with a compactness statement, we see that we have the same issue of sequences of sets breaking up into potentially infinitely many parts in the limit δ → 0. Instead of dealing with the technicalities of formulating a framework that can handle this issue, we choose to only analyze the compactness properties of minimizers of E 1,δ , which in any case is the main consequence one would like to extract from such a statement. Proposition 2.6. For m > 0, generalized minimizers of E 1,δ over A m are compact in the sense that their number of components is uniformly bounded as δ → 0, and each component is compact in L 1 after translation. Additionally, the limiting collection of sets is a generalized minimizer of E 1,0 .
Finally, we turn to investigating the properties of minimizers of E δ,1 for δ 1. We note that we currently do not know what they are for m > 0 fixed and all δ > 0 sufficiently small. In particular, they may or may not be disks, depending on the values of m, provided that δ 1. Results of experiments such as the images shown in Figure 1 suggest that in some regimes the minimizers may indeed be disks, while in other regimes the situation may be more complex. Our ansatz-based computations, however, indicate that disks are always preferred over stripe-like constructions that are tractable analytically, both at relatively low mass and in the limit of large masses. One could hope that numerically optimized stripes such as the ones obtained by Bernoff and Kent-Dobias [34] could have lower energy than disks, but this does not seem to be the case. Also a linear stability analysis is unhelpful, as instability of a single disk only occurs at masses at which two disks of equal mass have strictly lower energy. In other words, it seems that splitting a domain into finitely many disks may always decrease energy, thus making stripes always energetically disadvantageous. In fact, one might be tempted to argue that the persistence of balls as generalized minimizers is a universal feature of systems described by (1.1).
In the following we demonstrate that the above view is too simplistic and that the conjecture about the universal optimality of balls in the context of (1.1) is false. Indeed, we will show that under appropriate conditions stripes produce better candidates for generalized minimizers than collections of disks in R 2 . In order to tilt the favor towards stripes, for l > 0 we consider the energy This is precisely the energy of two identical, but oppositely oriented dipolar patches lying in the parallel planes separated by distance l. Such a model is, for example, relevant to synthetic antiferromagnets, in which the antiparallel alignment of spins in adjacent layers is favored by antiferromagnetic exchange coupling through a spacer layer (see, e.g., [46]). Heuristically, the kernel behaves as in a dipolar layer on scales r l, while on large scales the kernel decays faster, thus reducing the long-range repulsion in the far field. Notice that we recover the original energy in the limit l → ∞.
From the point of view of Theorem 2.3, the modification is merely a continuous perturbation: Lemma 2.7. For all l > 0 and m > 0, the L 1 -Γ-limit of F 1,δ,l restricted to A m is given, as δ → 0, by for Ω ∈ A m . Also for F 1,δ,l the number of components of generalized minimizers over A m is uniformly bounded in δ, and generalized minimizers of F 1,δ,l converge to generalized minimizers of F 1,0,l in the same sense as in Proposition 2.6.
For this energy we can prove existence of non-radial minimizers. In combination with Lemma 2.7, this represents the first instance, to our knowledge, in which whole space minimizers of a problem belonging to the class in (1.1) are not radially symmetric.
Theorem 2.8. There exists a universal constant c > 0 with the following property: Let 0 < l− 2 e 2 < c. Then there exists M = M (l) > 0 such that for all masses m > M any generalized minimizer of F 1,0,l over A m has at least one component which is not a disk. In particular, there exist masses m > 0 at which classical minimizers exist and are non-radial.
We conclude by briefly indicating that generalized minimizers in the supercritical case need to exhibit strongly irregular behavior, in the sense that their components cannot contain disks of O(1)-radii in the limit δ → 0. Note that such a statement cannot distinguish between generalized minimizers consisting of a large number of disks or thin, fingered structures. The point is that for λ > 1 it is beneficial to increase the perimeter of sets as the nonlocal term overcompensates the local contribution. We exploit this in the following proposition by cutting out disks from the minimizer and placing them at infinity. We expect the resulting relation r ≥ Cδ λ−1 λ between the radius r of the disks and δ to be sharp in terms of the exponent λ−1 λ , as the rescaling properties of E λ,δ , see Lemma 3.2, imply E λ,δ δ λ−1 λ Ω = δ λ−1 λ E 1,δ 1/λ (Ω) for all sets Ω ⊂ R 2 of finite perimeter. Proposition 2.9. There exists a universal constant C > 0 such that for all λ > 1 and all δ > 0 sufficiently small depending only on λ the following holds: If |B r (0) \ Ω| = 0 for a set Ω ⊂ R 2 of finite perimeter and some r ≥ Cδ λ−1 λ , then there exists a set Ω ⊂ R 2 andr > 0 such that Notation: Within proofs, the symbols "c" and "C" denote universal constants that may change from line to line. Furthermore, for (a, b) ∈ R 2 we define (a, b) ⊥ := (−b, a) to be its counter-clockwise rotation by 90 degrees. Depending on the specific situation, it is helpful to write the energy in one way or another. For example, the nonlocal term can be interpreted as a kind of nonlocal perimeter via straightforward combinatoricŝ In terms of the potential we can also represent it as because we have´R 2 |z| 2 −2l 2 (|z| 2 +l 2 ) 5/2 dz = 0. For explicit calculations it is helpful to rewrite both the energy and the potential in terms of boundary integrals, using integration by parts. To this end, we define Φ δ : R → R solving Writing the Laplacian in polar coordinates allows to straightforwardly determine the solution to be Similarly, we define Φ δ,l : R → R as the unique solution of where ν is the outward-pointing unit normal. If additionally the set Ω is such that there exists C Ω > 0 such that for H 1 -almost every x ∈ ∂ * Ω, then the energy can be represented as Proof. We only need to rewrite the nonlocal term. By integrability of the kernel, for every x ∈ Ω we getˆΩ whereν is the outward unit normal to the set of finite perimeter Ω c ∩ (B R (x) \ B r (x)), see for example [42,Lemma 15.12]. The same lemma furthermore implies that for almost all r, R > 0 we have the equality of measures (3.12) We first remove the second term. As Φ δ,l (R) decays at infinity like 1 Similarly, for every r > 0 we have (3.14) Next, we argue that for almost all x ∈ Ω the third term vanishes in the limit r → 0 along a suitable sequence. To this end, note that Φ δ,l (r) blows up at the origin as 1 r . As i.e., the set Ω is up to a set of measure zero given by its points of L 2 -density one, we only have to prove that Indeed, otherwise there would exist ε > 0 and r > 0 such that for almost all 0 <r < r. Integrating over 0 <r <r for all 0 <r < r then gives and thus x cannot be a point of L 2 -density one of Ω.
In order to carry out the remaining limit, we note that lim r→0 [42,Corollary 6.5]. As a result, we get for almost all x ∈ Ω, where r n is a sequence chosen due to equation (3.16). Therefore, for almost all x ∈ Ω we obtainˆΩ Consequently, after adding and subtracting the appropriate integral over Ω c in definition Going back to the original double integral we see that Here we were able to apply Fubini's theorem, since the integrand on the right hand side satisfies due to the explicit representation (3.7), and thus is integrable over Ω × ∂ * Ω w.r.t. the product measure L 2 ⊗ H 1 .
Using a similar choice of small scale cut-offs as in (3.12) with x and y interchanged, we get where in the last step we used the fact that εΦ δ,l (ε) → 0. Under the mild regularity assumption (3.9) we can cover B 1 (y) by a collection of overlapping dyadic annuli (3.26) and the desired statement (3.10).

Rescaling
We first record the behavior of the energy under rescalings. Especially the inequality (3.28) below turns out to be of central importance in proving that generalized minimizers exist, as well as in controlling the number of pieces.
As we only require the rescaling for the critical Γ-limit E 1,0 after the proof of Theorem 2.3, we may as well use the characterization as the pointwise limit to see that The modification due to l < ∞ transforms this into (3.29). 13 We can also control how much the energy increases if we have α > 1. However, we postpone the proof to Section 5 as it is a straightforward adaptation of the proof of Proposition 5.1. As we only need this statement for l = ∞ we directly formulate it for E λ,δ . Lemma 3.3. As before, consider α Ω = Ω, but now let α > 1. Then we have 4 Existence of classical and generalized minimizers of E λ,δ and F λ,δ,l The main purpose of this section is to prove existence of generalized minimizers, together with some control over how many components a generalized minimizer has. This is encoded in the function f λ,δ,l : R + → R defined as The crucial observation is that any component of a generalized minimizer (that is not classical) must have at least a certain amount of mass and thus can have at most a quantitatively controlled number of components.
Before we can embark on proving existence of (generalized) minimizers, we first need a number of observations: We first give a rough bound of the perimeter in terms of the energy. It is sufficient for the purposes of existence of minimizers, but we will do better in Proposition 5.1 below for asymptotic statements. The second observation is continuity of f λ,δ,l . Note that throughout this section all the constants λ > 0, m > 0, 0 < δ < 1 2 and 0 < l ≤ ∞ are fixed, unless stated otherwise.
Proof. The first statement is a straightforward consequence of the observation The second statement follows by combining the isoperimetric inequality with the above estimate to get for masses 0 < m < C(δ, λ, l).
Taking the limit ε → 0 and inserting α = m m , we see that Asf is locally bounded from above by testing with disks, this estimate provides a locally uniform modulus of continuity after symmetrizing the expression in m and m.
We are now in a position to prove existence of minimizers as long as the infimal energy is non-negative. Lemma 4.4. Let m > 0 be such that f λ,δ,l (m) > 0. Then every minimizing sequence over A m is compact in L 1 after translation. In particular, minimizers exist, and any generalized minimizer must be classical.
Proof. The basic strategy is applying the concentration-compactness principle due to Lions [40], see also Struwe [58,Section 4.3]. We have to deal with three cases: compactness, vanishing and splitting. Let Ω n be a minimizing sequence. By approximation, we may suppose that Ω n are smooth open sets, see [42,Theorem 13.8]. Lemma 4.2 implies that (4.11) Step 1: Compactness.
In this case we know the following: Up to extracting a subsequence and translating the sets, for every Therefore, the sequence of measures χ Ωn L 2 is tight. Together with the bound on the perimeter and the corresponding compact embedding theorem for BV-functions [4,Corollary 3.49] this implies that there exists a subsequence (not relabeled) and a set Ω ∈ A m such that |Ω n ∆Ω| → 0. Furthermore, the perimeter is lower semi-continuous with respect to this topology. As a result, in order to see that Ω is a minimizer of F λ,δ,l we only have to prove that the quadratic form for some C(δ, l) > 0.
In this case, we have for all R > 0 that (4.14) By Lemma 4.2 we have To argue that the perimeter is large, we decompose the plane into the squares Q k := k+[0, 1)×[0, 1) for k ∈ Z 2 . The relative isoperimetric inequality implies that there exists a constant c > 0 such that for all k ∈ Z 2 and n ∈ N, where P Q k (Ω n ) denotes the perimeter of Ω n relative to Q k . The vanishing property (4.14) implies that for large n ∈ N we have |Ω n ∩ Q k | ≤ ε for all k ∈ Z 2 , which implies Consequently, from the bound (4.11) we get for ε small enough, which is a contradiction.
In this case, there exists 0 < θ < 1 such that for any ε > 0 there exists R > 0 and a sequence x n ∈ R 2 with the following property: For any R > R we have We choose πλ | log δ| m will be shown to bound the diameter of each connected component. Let Ω (1) n be the union of all connected components of Ω n that intersect B R (x n ) and let Ω (2) n be the union of all connected components intersecting B c R (x n ), see Figure 3 for a sketch. Let the remainder be Ω As we chose a minimizing sequence consisting of smooth sets, it is easy to see that any connected component Ω n of Ω n satisfies sup x,y∈ Ωn |x − y| ≤ 1 2 P ( Ω n ) by noting that the supremum is achieved on ∂ Ω n . Consequently, by Lemma 4.2 we get sup x,y∈ Ωn for n sufficiently large. Therefore, we get Ω In particular, Ω n and Ω (2) n are disjoint. We now exploit the fact that the nonlocal term is repulsive at sufficiently large distances. The representation dy dx (4.22) and the observation for two disjoint sets Ω 1 , Ω 2 ⊂ R 2 allow us to compute It is straightforward to see that the kernel in (4.24) is positive for |x − y| ≥ δ, so that we get as a result of inf x∈Ω (1) n ,y∈Ω n and Ω 2 = Ω n in equation (4.23) we see Postprocessing this inequality by subtracting 6πλ 5| log δ|l m, we see for some C λ,δ,l > 0. Applying the isoperimetric inequality to Ω n , we see n | ≤ ε is small enough. As a result, we obtain We can now go to the limit n → ∞ along some subsequence such that we have α 1,n → α 1 (ε) and α 2,n → α 2 (ε) with due to the estimate (4.19). Then estimate (4.31) turns into Therefore, taking the limit ε → 0 we see that which in view of the positivity of f λ,δ,l (m) implies θ = 0 or θ = 1. However, this is a contradiction to 0 < θ < 1, see the beginning of Step 3. This concludes the proof.
We finally turn to prove existence of generalized minimizers: Proof of Proposition 4.1. As in this proof λ, δ and l are fixed, we drop them from the notation in f λ,δ,l , F λ,δ,l and m 0 (λ, δ, l). According to equation ( Step 1: We have m 0 < ∞.
We argue by providing an upper bound for the minimal energy, using disks as test configurations.
Letting Ω := B 1 (0) and Ω := B α −1 (0), we only have to show that we can improve estimate (3.34) to a lower bound that blows up in the limit α → 0. Indeed, with m = |Ω| and m = | Ω| we have the right-hand side of which converges to λ 2| log δ|ˆB 1 (0)ˆB c Step 2: The functional F has a minimizer at mass m 0 and all generalized minimizers are classical.
We may re-use large parts of the proof of Lemma 4.4. The compactness and vanishing cases work exactly the same. We only have to rule out splitting, for which we follow the previous proof up to estimate (4.30) and apply similar arguments as for estimates (4.31) and (4.34) to obtain for some 0 < θ < 1. By the choice of m 0 and Lemma 4.
The obvious contradiction rules out the case of splitting, and concludes the proof of this step.
Let Ω be a minimizer of F over A m 0 . For α < 1, we test the infimum of f ( m), where m := α −2 m 0 , with the rescaling α −1 Ω and apply the strict inequality of Lemma 3.2 to get for all m > m 0 . This clearly implies I + = (0, m 0 ). A similar argument shows that f is monotone decreasing on (m 0 , ∞). All other listed statements have already been proven.
We yet again re-use the proof of Lemma 4.4, which allows us to rule out vanishing, and of course implies existence of a minimizer in the compactness case. However, we will not be able to rule out splitting, and instead provide a lower bound for the masses that may split off. As in Step 2 of this proof, we see that for some 0 < θ < 1. If we had θm ≤ m 0 , then we would have f (θm) ≥ 0, which in turn would imply f ((1 − θ)m) < 0. Monotonicity of f then would give which is a clear contradiction. Switching the roles of θ and 1 − θ in this argument we obtain (1 − θ)m > m 0 (λ, δ, l). Consider two smooth competitors Ω 1 for f (θm) and Ω 2 for f ((1 − θ)m). We may use Ω := Ω 1 ∪ (Re 1 + Ω 2 ) for R > 0 large enough such that Ω 1 ∩ (Re 1 + Ω 2 ) = ∅ as a competitor for f (m). As the nonlocal interaction between the two sets vanishes in the limit R → ∞, we obtain f (θm) + f ((1 − θ)m) ≥ f (m). Combining this with estimate (4.39) we get 43) and the inequalities are, in fact, equalities.
As for m ≤ 2m 0 any splitting would violate estimate (4.41) or estimate (4.42), classical minimizers exist and any generalized minimizer is classical.
Step 5: Existence of generalized minimizers for masses m > 2m 0 and the bound for the number of their components.

The insights from
Step 4 allow us to set up an induction argument: For k ∈ N assume that generalized minimizers exist for all masses m ≤ (k + 1)m 0 and that they can have at most k components. Note that by the result in Step 4 this assumption is true for k = 1. Let now m ∈ ((k + 1)m 0 , (k + 2)m 0 ]. Then equation (4.43) says that a generalized minimizer with mass m is given by combining the generalized minimizers at masses θm and (1 − θ)m which exist because estimates (4.41) and (4.42) imply To see that the generalized minimizer has at most k + 1 components we note that the same proof as for estimates (4.41) and (4.42) implies that any component Ω i must have |Ω i | > m 0 . Therefore, any generalized minimizer at mass m clearly has at most k + 1 components. Furthermore, for masses m > 2m 0 the condition km 0 < m ≤ (k + 1)m 0 implies k < m m 0 , which implies k ≤ m m 0 − 1. Taking into account the fact that minimizers must be classical for masses m ≤ m 0 due to Lemma 4.4 gives us the bound on the number of components of generalized minimizers.
Next, we briefly discuss the regularity properties of generalized minimizers of F λ,δ,l . Even though we only provide a qualitative statement here, it is one of the crucial ingredients when proving that minimizers are disks for λ < 1 and l = ∞, as it will allow us to set up an iteration to enlarge the scale at which the boundary of a minimizer is locally a graph with respect to some coordinate direction.
Proposition 4.5. Let (Ω i ) i∈N be a generalized minimizer of F λ,δ,l over A m . Then there exists a Lagrange multiplier µ ∈ R such that for all i ∈ N with |Ω i | > 0 the following holds: (ii) We have that Ω i has a C 2,α -boundary for all α ∈ (0, 1).
(iii) For all y ∈ ∂Ω i we have where κ i is the curvature of Ω i (positive for convex sets) and v δ,l,i is the potential of Ω i defined in equation (3.2).
Proof. As the components of generalized minimizers clearly are classical minimizers, we may drop the index i for the proof of regularity and the form of the Euler-Lagrange equation. We will only recall the dependence when proving that the Lagrange-multiplier is independent of i. In order to obtain C 1,α -regularity for some α ∈ (0, 1), one can directly apply [53, Théorème 1.4.9 or Théorème 5.1.3], which yields α = 1 2 . One could alternatively prove that Ω is a (Λ, r 0 )-minimizer of the perimeter functional according to [42,Chapter 21] for some Λ, r 0 > 0 and then apply the regularity theory presented therein to prove C 1,α -regularity for all α ∈ (0, 1 2 ). The mass constraint can be dealt with by rescaling competitor sets to the appropriate mass and making use of Lemmas 3.2 and 3.3.
The computation of the Euler-Lagrange equation for a single set Ω i : with some µ i ∈ R, is a standard exercise in view of Lipschitz continuity of v i . For the latter, write arguing by approximation. To obtain higher regularity, we appeal to the regularity theory for the prescribed mean curvature equation, see for example [25]. To see that the Lagrange multiplier does not depend on the component, let i, j ∈ N be such that Ω i , Ω j = ∅. For ε ∈ R, dilating Ω i with the factor 1 + ε and Ω j with the factor 1 − m i m j ε gives a viable competitor (Ω ε,i ) i∈N for (Ω i ) i∈N . We therefore get which immediately implies µ i = µ j .
To conclude this section, we point out that in a large range of parameters any minimizer has to be connected. This is not clear a priori as the kernel K δ,l is not necessarily strictly positive. In fact, if we have δ > √ 2l, then it is easy to see that K δ,l (r) < 0 for √ 2l < r < δ, so that at certain distances the kernel is even attractive. Lemma 4.6. If δ < √ 2πm in the case l = ∞, or δ < √ 2l in the case l < ∞, then the regular representatives constructed in Proposition 4.5 of any minimizer Ω of F λ,δ,l over A m are connected.

The subcritical case λ < 1
The main goal in this section is to prove that minimizers are disks for λ < 1 and all δ sufficiently small. While in the previous section rough estimates were acceptable, the main theme here will be obtaining estimates that are as uniform as possible. To this end, it is crucial to realize that the nonlocal term localizes to leading order in the sense that in the limit δ → 0 it approaches −λP (Ω). Therefore, treating it as a volume term as we did for example in Proposition 4.5 has no chance of being accurate. Instead, one has to split the nonlocal contribution into the leading-order local and a higher-order nonlocal part.
The first instance of this idea is contained in the following lower bound for the energy in terms of the perimeter, adapted from Knüpfer, Muratov and Nolte [37]. Not only is it the first step to obtain uniform regularity estimates and the lower bound for the Γ-limit E λ,δ Γ → (1 − λ)P , it will in Section 6 also be the starting point in proving the Γ-lim inf inequality of the Γ-convergence in the critical case λ = 1.
Proposition 5.1. Let λ > 0, 0 < δ < 1 2 , m > 0 and Ω ∈ A m . If P (Ω) ≤ πm δ we have the bound Proof. We derive an upper bound for the nonlocal term by splitting the inner integral in into two parts.
For some cut-off length R > 0 to be determined later, we claim to have the estimatê Note that the integral on the left hand side is continuous with respect to the L 2 -topology due estimate (4.13). By approximating χ Ω with smooth functions f n ∈ C ∞ c (R 2 ; [0, 1]) such that f n → χ Ω in L 2 ,´f n dx = m and´R 2 |∇f n | dx → P (Ω) it is sufficient to prove the analogous result for smooth functions. By [37, estimate (3.8)] we get Using polar coordinates it is easy to see that which concludes the proof of the claim after noticing |χ Step 2: Large scales.
In order to estimate the large scales, we make use of the representation (3.1) to seê Step 3: Balancing the contributions. Combining this inequality with the local perimeter term in the energy gives the first desired estimate. If, on the other hand, πm P (Ω) < δ, we instead choose which gives the second desired inequality.

24
The argument in the proof of Proposition 5.1 also allows us to finally give an efficient proof of Lemma 3.3.

Proof of Lemma 3.3.
Recalling the proof of Lemma 3.2, specifically the estimate (3.34), we have where the kernel g δ (|z|) is positive due to α > 1. Choosing R = αδ in Step 1 of the proof of Proposition 5.1 gives Throughout the rest of this section we assume that 0 < λ < 1 and 0 < δ < 1 2 . Combining the result of Proposition 5.1, the isoperimetric inequality and the result of Lemma 4.4 immediately yields existence of minimizers of E λ,δ over A π in the subcritical case for all δ sufficiently small.
then there exists a minimizer of E λ,δ over A π .
As already mentioned, Proposition 5.1 allows us to compute the Γ-limit of E λ,δ in the L 1topology. Note that we do not need a compactness statement since we will quantify the convergence of the minimizers to disks,see Lemma 5.7, and we will in the end even see that minimizers are disks for δ > 0. Proposition 5.3. Let m > 0. As δ → 0, the L 1 -Γ-limit of the functionals E λ,δ restricted to A m is given by E λ,0 (Ω) := (1 − λ)P (Ω) (5.14) for Ω ∈ A m .
Proof. Using the lower bound of Proposition 5.1, the Γ-lim inf-statement follows from lower semicontinuity of the perimeter: Let Ω δ ⊂ R 2 be such that |Ω δ ∆Ω| → 0 as δ → 0. Then we get noting that the term involving log P is bounded below by the isoperimetric inequality. Turning towards the Γ-lim sup-statement, it is well-known that any set Ω ⊂ R 2 of finite perimeter can be approximated with smooth sets Ω n ⊂ R 2 such that |Ω n ∆Ω| → 0 and P (Ω n ) → P (Ω) as n → ∞, i.e., smooth sets not satisfying the mass constraint are dense in energy with respect to the perimeter P . In order to enforce the constraint, one approximates Ω ∩ B R (0) for R > 0 large by smooth sets and uses [20,Lemma 4] to correct the mass of the approximation. Therefore it is sufficient to prove that lim δ→0 E λ,δ (Ω) = P (Ω) (5.16) 25 for smooth sets Ω ⊂ R 2 .
To this end we make use of the boundary representation of the energy (3.10) of Lemma 3.1. For x ∈ ∂Ω we focus on the inner integral We parametrize ∂Ω about x via γ : (−c, c) → R 2 such that γ(0) = x and |γ| = 1 for some c > 0. Expanding the various expressions, see identity (3.5), about t = 0 gives As a result, we get that the leading order contribution of the inner integral iŝ Inserting this expression into the representation (3.10) gives the desired statement.
Our next result shows that the minimizers of E λ,δ have small isoperimetric deficit whenever δ is sufficiently small depending only on λ.
Lemma 5.4. Any minimizer Ω of E λ,δ over A π satisfies Proof. The lower bound of Proposition 5.1 implies for a universal constant C > 0 as the map x → x log x is bounded from below. On the other hand we have by the construction in Proposition 5.3 for B 1 (0). As a result, we obtain  Figure 4: Sketch of the sets F, F 1 , F 2 discussed in Lemma 5.5. The length of the dashed line is 1 2 Σ. a) If the common boundary of F 1 and F 2 is short, it is beneficial to cut away F 2 and dilate F 1 to account for the lost mass. b) Similarly, if the hole F 2 has little extra boundary, one can lower the energy by filling the hole and dilating.
We now turn to the heart of the matter: obtaining uniform regularity estimates for the minimizers. As is usually the case for these problems, the first step is to prove uniform density estimates. To this end, we present two non-optimality criteria in the spirit of Knüpfer and Muratov [36,Lemma 4.2]. The idea is to cut away small appendages and fill small holes, see Figure 4.
Lemma 5.5. There exists a universal constant σ 0 > 0 with the following property: Let F ⊂ R 2 be a set of finite perimeter with P (F ) ≤ 3π 1 2 |F | 1 2 . Let the two sets F 1 and F 2 be such that |F 2 | ≤ σ 0 (1 − λ) 2 min(1, |F 1 |) and, up to sets of measure zero, we have where the symbol "∪" stands for the disjoint union. If then there exists a set G of finite perimeter such that |G| = |F | and E λ,δ (G) < E λ,δ (F ).
Proof. Throughout the proof we drop the indices of the energy, as λ and δ are fixed. Furthermore, we will only deal with the case F = F 1∪ F 2 in detail and highlight the changes for the other case at the end. Let m 1 := |F 1 |, m 2 := |F 2 |, m := |F | = m 1 + m 2 and γ := m 2 m 1 . We will consider the competitor G := lF 1 , where we choose l := √ 1 + γ in order to have |G| = |F |.
In a first step we compare E(G) and E(F 1 ). By scaling, see Lemma 3.2, and concavity of the square root we know that |x−y| 3 for measurable sets Ω ⊂ R 2 , we see that where we made use of assumption (5.28) and obtained by straightforward combinatorics. Thus we get and, together with estimate (5.29), The isoperimetric inequality gives due to the assumption |F 2 | ≤ σ 0 (1 − λ) 2 min(1, |F 1 |) ≤ 4 e 2 π , provided we choose σ 0 < 4 e 2 π . Plugging this into the a priori bound from Proposition 5.1 and again applying the isoperimetric inequality we get Using this and the assumption on P (F ) in inequality (5.32) we obtain due to m 2 m 1 ≤ σ 0 (1 − λ) 2 < 1 for σ < 1. Therefore, we finally get as long as we choose σ 0 < 1 9 . For the second case, we set G := lF 1 with l := 1 − m 2 m 1 , as well as γ := m 2 m 1 as before, but with m = m 1 − m 2 now. There are only three estimates that are significantly different than in the first case, namely estimates (5.29), (5.31) and (5.38). Dealing with the second estimate is a matter of adjusting the combinatorics. In order to obtain an analogue of inequality (5.29) we have to use Lemma 3.3 to get for a universal constant C > 0, as long as σ 0 > 0 is small enough to carry out the expansions. The additional term can be estimated via where we exploited that Σ ≤ 1 2 E(F 2 ) straightforwardly implies P (F 1 ) ≤ P (F ) − 1 2 P (F 2 ) ≤ P (F ). Therefore, the additional term is estimated against the same quantity as the contribution γE(F 1 ), see estimate (5.36).
The analogue of the third estimate (5.38) is given by This concludes the proof.
With this tool at hand we are able to prove uniform density bounds for minimizers.
Lemma 5.6. There exist universal constants σ, c, r 0 > 0 such that under the condition | log δ| −1 ≤ σ 1−λ λ the following holds: Let Ω ⊂ R 2 be a minimizer of the energy E λ,δ over A π . Then for every x ∈ ∂Ω and r ≤ (1 − λ)r 0 we have . as long as we have σ > 0 small enough universal. To prove the lower L 2 -density bound we re-use large portions of the proof of [36,Lemma 4.3], exploiting the first case in our non-optimality criterion. Let x ∈ ∂Ω, F r 1 := Ω \ B r (x) and F r 2 := Ω∩B r (x). As for r > 0 small enough an application of Lemma 5.5 requires |F r 2 | ≤ σ 0 (1−λ) 2 , we choose r ≤ (1 − λ)r 0 for a universal r 0 > 0. For the analogue of estimate [36, (4.10)], we additionally apply Proposition 5.1 to get where we used the same arguments involving estimates (5.34) and (5.35) to omit the logarithmic contribution.

29
The upper L 2 -density bound follows in the same way by using the second case of Lemma 5.5. The lower density bound for the length follows from those of the mass via the relative isoperimetric inequality, while the upper bound is a straightforward consequence of see [36, equation (4.11)] for the proof.
As a first application of the uniform density estimates, we briefly quantify the convergence of minimizers to disks in order to have a quantitative dependence of the required smallness of δ on λ in the final statement.
Lemma 5.7. There exist universal constants C, σ > 0 such that under the condition | log δ| −1 ≤ σ (1−λ) 2 λ the following holds: The regular representative of any minimizer Ω of E λ,δ over A π is simply connected and, after translation, satisfies where dist denotes the Hausdorff-distance of sets.
Proof. Let Ω be the regular representative obtained in Proposition 4.5. Observe that by Lemma 4.6 the set Ω is connected. As the set Ω ⊂ R 2 is bounded with a smooth boundary, there exists a simply connected set Ω such that Ω ⊂ Ω, ∂ Ω ⊂ ∂Ω is a Jordan curve, and such that ∂Ω \ ∂ Ω is either empty or itself a union of disjoint Jordan curves γ i for 1 ≤ i ≤ N and some N ∈ N. In order to prove that Ω is simply connected, we only have to rule out the latter case. Towards a contradiction, we assume that ∂Ω \ ∂ Ω is indeed a non-empty union of Jordan curves.
As we haver := | Ω| π ≥ 1, the isoperimetric inequality and Lemma 5.4 imply Furthermore, for F 2 := Ω\Ω we have ∂F 2 = ∂Ω\∂ Ω. As a result of estimate (5.50), the isoperimetric inequality implies with σ 0 as in Lemma 5.5, provided λ (1−λ) 2 | log δ| ≤ σ for σ > 0 universally small enough and where we bounded | Ω| by a universal constant due to P ( Ω) ≤ P (Ω) and Lemma 5.4. Similarly to estimate (5.35) we see that Therefore, we can apply the second case of Lemma 5.5 with F 1 = Ω and F = Ω to conclude that Ω cannot have been a minimizer, which contradicts the assumption that Ω is not simply connected.
As Ω is simply connected, we may apply Bonnesen's inequality [9,23,50] and Lemma 5.4 to Ω to get that after a suitable translation we have dist ∂Ω, S 1 ≤ C P 2 (Ω) − P 2 (B 1 (0)) ≤ Cλ which concludes the proof. 30 We now have all the necessary ingredients to make the qualitative regularity of Proposition 4.5 quantitative. The basic strategy is to improve the scale at which ∂Ω is a graph. We first give a precise meaning to this statement: Definition 5.8. We say that a set Ω has a uniform C 2 -boundary at scale r > 0 if the boundary is an oriented C 2 -manifold such that for every x ∈ ∂Ω there exists a rotation S ∈ SO(2) and a function h ∈ C 2 ([−r, r]) with the following properties: In this terminology, Proposition 4.5 states that a minimizer Ω δ of E λ,δ has a uniform C 2boundary at some small scale r(Ω δ ) > 0. In the next statement, Proposition 5.9, we present the machinery which will allow us to iteratively increase the scale of regularity. The idea is to expand the potential v δ , see definition (3.2), into a localizing part that approaches the curvature and into a larger scale remainder. As we have λ < 1, the curvature is still the leading order term in the Euler-Lagrange equation (4.46). Once we obtain convexity of the minimizers, the proposition will also allow us to identify the minimizers as disks in combination with a rigidity statement for the disk.
To this end, we make use of the representation (3.8) of the potential. As we will later argue by taking differences of the Euler-Lagrange equation in two different points, the constant term drops out, so that we may focus on the integral. For x ∈ ∂Ω we split it up into the local contribution ∇Φ δ (|y − x|) · ν(y) dH 1 (y) (5.55) from B r (x) and into a remainder we abbreviate with Step 1: Expand the integrand in (5.55).
Without loss of generality we may assume x = 0. To control the localized part we transform the integral in (3.8) into the graph coordinates y = (t, h(t)). Recall that we asked h(t) to satisfy h(0) = h (0) = 0 in Definition 5.8. Let T 1 , T 2 > 0 be such that y(−T 1 ), y(T 2 ) ∈ ∂Ω∩∂B r (0). As the Jacobian in the transformation formula of boundary integrals is the length of the tangent, we see that ν(y) dH 1 (y) transforms into (−h (t), 1) dt, so that we aim to compute |y(t)| (t, h(t)) · (−h (t), 1) dt. (5.57) Note that the scalar product above may be represented as as well as |h(t)| ≤ 1 2 r −1 t 2 and Here, we use the O-notation to denote estimates up to universal constants. Also note that we always have |y(t)| ≥ |t|, so that as a consequence we get 1 |y(t)| 2 = 1 |t| 2 1 + O r −2 t 2 (5.63) and 1 |y(t)| 3 = 1 |t| 3 1 + O r −2 t 2 . (5.64) Step 2: If r ≤ δ we have the following: There exist y 1 , y 2 ∈ ∂Ω ∩ B r (0) such that Using the explicit form of Φ δ in (3.5), we see In order to get a lower bound, we choose t 1 ∈ argmax s∈[−T 1 ,T 2 ] κ(y(s)) and estimate This is the desired lower bound of estimate (5.65). The upper bound follows similarly by choosing t 2 ∈ argmin s∈[−T 1 ,T 2 ] κ(y(s)).

(5.71)
Again the argument for the upper bound is similar.
Let y 1 and y 2 realize the maximum in osc κ = max y 1 ,y 2 ∈∂Ω (κ(y 1 ) − κ(y 2 )). We first deal with the more complicated case r > δ. Taking the difference of the Euler-Lagrange equation (4.46) in the two points we see from Step 3 that there existỹ 1 ,ỹ 2 ∈ ∂Ω such that we have if we have | log δ| −1 ≤ σ 1−λ λ for a universal σ > 0 small enough. A similar argument in the case r < δ concludes the proof.
We are now in a position to prove convexity of the minimizers. The strategy is to show that the | log δ| −1 factor in front of the remainder in estimate (5.54) can be used to iteratively improve the scale r > 0 of regularity all the way up to some uniform radius independent of δ. The resulting improved bound for the curvature can be combined with the non-optimality criterion in Lemma 5.5 to also improve the scale at which ∂Ω is a graph.
Proposition 5.10. There exist universal constants σ, C > 0 with the following properties: If Additionally, the set Ω is convex and has a uniform C 2 -boundary at scale σ(1 − λ).
Proof. The proof consists of five steps. In the first four steps, we assume that Ω has a uniform C 2 -boundary at scale 0 < r < 1 2 (1 − λ)r 0 , where r 0 > 0 is the universal constant given in Lemma 5.6. Recall that such a value of r exists by Proposition 4.5, but depends on Ω and the values of δ and λ.
Step 2: For any K > 0 and all σ sufficiently small depending only on K, the curvature satisfies κ ∞ ≤ 1−λ Kr . We apply the Gauss-Bonnet theorem to the Jordan curve ∂Ω, see Lemma 5.7, to obtain ∂Ω κ dH 1 = 2π. (5.77) The mean value theorem then implies that there exists x ∈ ∂Ω with 2 3 ≤ κ(x) ≤ 1 because by estimate (5.45) and by the isoperimetric inequality we have 3π ≥ P (Ω) ≥ 2π In particular, for any K > 0 to be chosen later we have where the second inequality holds due Proposition 5.9 and Step 1, the third is a result of | log δ| −1 ≤ σ (1−λ) 3 λ and the fourth follows from r ≤ σ(1 − λ), provided σ > 0 is small enough depending only on K.
Step 4: The set Ω has a uniform C 2 -boundary at scale 2r.
We can now iterate Proposition 5.9 and the result of Step 4 until we know that the minimizers have uniform C 2 -boundaries at scale r = (1 − λ)r 0 for a universal r 0 > 0. Then the estimate of Proposition 5.9 and the estimate (5.76) combine to for a universal C > 0. As there exists x ∈ ∂Ω such that κ(x) ≥ 2 3 , see Step 2, we have κ > 0 on ∂Ω under the assumption | log δ| −1 ≤ σ (1−λ) 5 λ . Therefore Ω is convex.
The final ingredient of the proof of Theorem 2.1 is a quantitative rigidity result in the C 1topology for the circle in terms of the oscillation of the curvature. We note that in general space dimensions this is the question of stability for the famous Aleksandrov's soap bubble theorem [3], stating that the only compact embedded hypersurfaces of constant mean curvature are spheres. As such the problem has attracted a significant amount of attention in recent years. We only single out a few contributions: Estimates controlling the distance to a circle in a C 1 -sense are [16,Theorem 1.1] and [17,Corollary 1.2]. However, these statements require control of (osc κ) α for some (explicitly given) exponent α < 1, which turn out to be suboptimal at least in two space dimensions. On the other hand, [17,Theorem 1.1] and [43,Theorem 4.3] provide a C 0 -estimate in terms of osc κ, which, however, is still not enough for our purposes. We therefore give our own version, the proof of which turns out to be an elementary calculation.
Lemma 5.11. Let P > 0 and let γ : [− 1 2 P, 1 2 P ] → R 2 be a C 2 -regular Jordan curve, parametrized by arc length and oriented such thatγ(t) = κ(t)γ ⊥ (t). Let R = P 2π be the radius of the circle whose length equals the length of γ. Then there exists t 0 ∈ [0, P ) and x 0 ∈ R 2 such that for all t ∈ [− 1 2 P, 1 2 P ] we have Proof. By the Gauss-Bonnet theorem, we have´P 2 − P 2 κ(γ(t)) dt = 2π. Therefore the mean value theorem implies that there exists t 1 ∈ [− 1 2 P, 1 2 P ] such that κ(t 1 ) = 2π P = R −1 . In particular, we obtain κ − R −1 ∞ ≤ osc κ. Without loss of generality we may assumeγ(0) = 0 1 , which will turn out to fix the choice t 0 = 0. We abbreviate and calculateḟ Therefore we see Integrating once more, we see that there exists a constant x 0 ∈ R 2 such that for all t ∈ − 1 2 P, 1 2 P . We are now in a position to prove that minimizers are disks.
6 The critical case λ = 1 We finally turn to analyzing the critical case. The first step is to prove that the pointwise limit of | log δ|E 1,δ exists and coincides with the L 1 -Γ-limit. To this end, we observe that the inequality used in Step 1 in the proof of Proposition 5.1 allows us to decouple the dependence of the energy | log δ|E 1,δ and the sets Ω δ on δ. In order to identify the limit, we re-use Step 1 of the proof of Lemma 3.1.
Proof of Theorem 2.3.
Step 1: Compactness for sequences of uniformly bounded sets with finite energy.
Step 2: Monotonicity for fixed sets.
We start with proving the lower bound. Let Ω δn ∈ A m for δ n → 0 as n → ∞ be such that there exists Ω ∈ A m with lim n→∞ |Ω δn ∆Ω| = 0. For everyδ > 0 we then have by lower semi-continuity of the perimeter and continuity of the nonlocal term, see estimate (4.13).
As a next step we re-derive the representation of the Γ-limit used by Bernoff and Kent-Dobias [34].
For the compactness part, we exploit the control over the number of pieces of a generalized minimizer given in Proposition 4.1.
Proof of Proposition 2.6. Step 1: A generalized minimizer of E 1,δ can have at most N (m) components.
According to Proposition 4.1, in order to prove that generalized minimizers of E 1,δ enjoy a uniform control over the number of their parts, we only have to see that m 0 (1, δ, ∞) = sup{m : inf Am E λ,δ > 0} is strictly positive uniformly in δ. To this end, we use the lower bound of Proposition 5.1 to get a universal constant c > 0 such that for all sets of finite perimeter Ω ⊂ R 2 we have | log δ|E 1,δ (Ω) ≥ P (Ω) log cP (Ω) m . (6.25)
Step 2: We have lim sup δ→0 N δ i=1 P (Ω i,δ ) ≤ C(m) for some 0 < C(m) < ∞, where (Ω i,δ ) i∈N is a generalized minimizer of E 1,δ over A m and N δ ≤ N (m) is the number of its components.
Step 4: The collection of sets (Ω i ) i∈N is a generalized minimizer of E 1,0 over A m .
The argument for the perimeter of a generalized minimizer can easily be adjusted using the bound | log δ|F 1,δ,l ( Ω) ≤ | log δ|E 1,δ, ( Ω) + 4πm 3l for all Ω ∈ A m . The rest of the argument carries over with the only difference that the application of Lemma 4.6 is immediate in the case l < ∞.
Finally, we come to proving existence of non-radial minimizers. The proof relies heavily on some tedious calculations, of which we delegate most to Mathematica 11.1.1.0 software.
Proof of Theorem 2.8.
Step 1: Compute the energy of disks.