Second order expansion for the nonlocal perimeter functional

The seminal results of Bourgain, Brezis, Mironescu and D\'avila show that the classical perimeter can be approximated by a family of nonlocal perimeter functionals. We consider a corresponding second order expansion for the nonlocal perimeter functional. In a special case, the considered family of energies is also relevant for a variational model for thin ferromagnetic films. We derive the Gamma--limit of these functionals. We also show existence for minimizers with prescribed volume fraction. For small volume fraction, the unique, up to translation, minimizer of the limit energy is given by the ball. The analysis is based on a systematic exploitation of the associated symmetrized autocorrelation function.


Introduction and statement of main results
The seminal results of Bourgain, Brezis, Mironescu [8] and Dávila [18] show that the classical perimeter can be approximated by a family of nonlocal perimeter functionals. In a slightly reformulated setting this implies that for any function u ∈ BV (T , {0, 1}) we have for a family of kernels K : R d → [0, ∞), K ∈ L 1 (B 1 ), K (z) = K (|z|) and which asymptotically have a singularity at the origin as → 0 (see (3)-(4) below). Here, T is the flat torus with sidelength > 0, Furthermore, M is defined by for > 0 and where ω d is the volume of the unit ball in R d . In this paper we consider the corresponding second order expansion for the nonlocal perimeter functional, i.e. we consider the family of energies We assume that the kernels satisfy K K 0 for 0 with In particular, this includes the following kernels K (2) (r) = r −d+ (6) K (3) (r) = r −q χ ( ,∞) (r) for q > d.

E
(1) is a sharp interface version of a problem from micromagnetism for thin plates with perpendicular anisotropy [26,29,34,32]. In this setting, {u = 1} and {u = 0} are the magnetic domains where the magnetization points either upward or downward. The interfacial energy penalizes interfaces between the magnetic domains, while the nonlocal term describes the dipolar magnetostatic interaction energy between the domains. For E (2) , the nonlocal term in the energy (2) is the so-called fractional perimeter P s (Ω) with s = 1 − of the set Ω := spt u (see e.g. [10,11,9]), and the functional is the second order asymptotic expansion of the fractional perimeter as s 1. The third energy E (3) shows that our model also includes some cases where the nonlocal term in the energy is more singular in terms of its scaling than the perimeter.
In the first two cases (5)- (6), the nonlocal term and the perimeter term in (2) both formally have the same scaling for = 0. However, due to the non-integrability of the kernel r −d we have M → ∞, and both terms in (2) are infinite for = 0. In our analysis we show that the singularity of these two terms cancels in the leading order, and we calculate the remainder which describes the limiting behaviour of (2) for → 0. Our main result establishes compactness and Γ-convergence for the family of energies (2): (ii) (Γ-convergence) E Γ → E 0 in the L 1 -topology, where for u ∈ BV (T ; {0, 1}) and E 0 (u) = +∞ otherwise.
In fact, statement (i) of Theorem 1.1 is a consequence of the fact that we have a uniform BV estimate for our family of energies (2) as stated in Proposition 3.1. The Γ-convergence result demonstrates that the functional E 0 , given in (8), is the second order of the asymptotic expansion for the nonlocal perimeter functional and quantifies asymptotically the error in the approximation (1).
A key observation for our analysis is that both interfacial energy and nonlocal energy can be expressed solely in terms of the autocorrelation function As a result the energies E for ≥ 0 can be written in terms of the autocorrelation functions for ≥ 0, cf. Proposition 2.6 for the radially symmetrized expression. We note that the new formulated energy has a simpler structure: it is linear in the space of autocorrelation functions. This leads to easier proofs for the Γconvergence for more general kernels compared with the existing literature, e.g. [32,12]. We note that the method of autocorrelation function can also be used for a simpler proof of Dávila's result [18], see Appendix A. We have chosen a periodic setting, but our methods can also be used to derive corresponding results for the corresponding full space problem.
While autocorrelation functions are natural and often used tools in physics and stochastic geometry, they seem not have been used in the context of nonlocal isoperimetric problems before. We note that the idea of linearization of the problem by a change of configuration space also appears in the formulation of quantum field theory as well as in the theory of optimal transport. In both cases the problem gets linear but the configuration space becomes very complex.
We next consider the minimization problem for energies E with prescribed volume fraction, i.e. for some fixed λ ∈ [0, 1] we assume We first note that for any ≥ 0, the minimization problem has a solution: The proof for Proposition 1.2 is based on the uniform BV -bound in Proposition 3.1. Next we note that the energy E (χ Ω ) only depends on the boundary ∂Ω for any set Ω ⊂ R d . In fact, this can be seen from which follows from the symmetry property for the autocorrelation function (Proposition 2.3(v)). In particular, the minimal energy for prescribed volume fraction is symmetric with respect to λ = 1 2 ∀ ≥ 0. We thus expect that minimizers for E in the case of the equal volume fraction λ = 1 2 are equally distributed stripes.
In the following, we discuss properties of minimizers of the limit energy for the special choice (5) or (6) of the kernel, i.e. K 0 (r) = r −d , and the corresponding energy is denoted by E (1) 0 . We first prove that when the volume is sufficiently small, the unique minimizer (up to translation) is given by a single ball: the unique, up to translation, minimizer u of E (1) 0 in BV (T ; {0, 1}) with constraint (9) is a single ball. Theorem 1.3 actually holds for a slightly larger class of kernels, cf. Proposition 3.3. The proof relies on the sharp quantitative isoperimetric inequality in [20] and a careful study of the limit energy using the autocorrelation function. The optimality of the ball as minimizers has been shown for related models with interfacial term and competing nonlocal energy such as the Ohta-Kawasaki energy [27,28,24,7,19,33]. The difference in our model in comparison to these previous results is that the nonlocal energy has the same critical scaling as the perimeter, which requires a particularly careful analysis. We note that Theorem 1.3 is concerned with the case of small volume instead of small volume fraction, as the smallness assumption for λ is not uniform in .
To study the minimizers for small volume fraction, we calculate the energy E (1) 0 for periodic stripes and lattice balls. For this, we allow that the centers of the balls are arranged in an arbitrary Bravais lattice which requires a slightly more general formulation of our energy (cf. Definition 4.1). In particular, we show that in 2d when the volume fraction is almost equal, equally distributed stripes have strictly smaller energy than all lattice ball configurations: Proposition 1.4 (Ball vs. stripe patterns). Let d = 2 and K 0 (r) = r −2 . Then there is c 0 > 0 such that if |λ − 1 2 | < c 0 , then the energy of equally distributed stripes is strictly smaller than the energy of periodic ball configurations.
The precise statement and the proofs are given in Section 4.
Previous literature and related models. We note that isoperimetric problems have been extensively researched in the mathematical community. A prototype model which has been investigated is the sharp-interface Ohta-Kawasaki energy, where the system is described by a sharp interface term together with a nonlocal Coulomb interaction term (or more general Riesz interaction term), see e.g. [2,3,13,14,15,17,22,23,25,31]. However, our model is different from these models, as the nonlocal term asymptotically has the same (or higher) scaling as the interfacial energy.
When u ∈ BV (R 2 ; {0, 1}) and K is defined in (5), the energy has been studied by Muratov and Simon in [32], where the authors show among other results the Γ-convergence of the energy. The Γ-convergence for the energies (6) is studied by Cesaroni and Novaga [12]. Our model includes the models considered in both [32] and [12] (in the periodic setting). We generalize some results of these papers, however, with a different strategy of proofs and in a more general setting. Moreover, we characterize the minimizers of the limit energy when the volume is small, and show that in 2d stripes have strictly smaller energy than lattice of balls if the volume fraction is close to 1/2. These have not been considered in [32,12].
Structure of paper. In Section 2 we introduce the autocorrelation functions, collect and prove some of their basic properties. We write the energy in terms of the autocorrelation function and use it to explore some properties of the energy. We also derive different formulations of the energy. In Section 3, we give the proof of Theorem 1.1 and Theorem 1.3. The stripes and balls configurations are studied in Section 4.
Notation. Throughout the paper unless specified we denote by C a positive constant depending only on d. By T := R d /( Z) d we denote the flat torus in R d with side length > 0. We repeatedly identify functions on T with T -periodic functions on R d using the canonical projection Π : R d → T . Similarly, any set Ω ⊂ T can be identified with its periodic extension onto R d . Let M be the space of signed Radon measures of bounded variation on T . For µ ∈ M we write µ := |µ|(T ) for its total variation. For any function of bounded variation u ∈ BV (T ), we analogously write ∇u = |∇u|(T ). By the structure theorem for BV functions we have ∇u = σ|∇u| for some σ : T → S d−1 . By a slight abuse of notation we sometimes write Given a measurable set Ω ⊂ T (or Ω ⊂ R d ), the Lebesgue measure of Ω is denoted by |Ω|. If Ω has finite perimeter, the perimeter of Ω is denoted by P (Ω) := ∇χ Ω . Furthermore, ω d is the volume of the unit ball in R d and

Autocorrelation functions and energy
In this section we give various different formulations for our energy, in particular in terms of the autocorrelation function.
2.1. Autocorrelation functions. In this section we introduce the autocorrelation function in our setting and show some of properties of the autocorrelation function. At the end of the section, we compute explicitly the autocorrelation function for a single ball and stripes. The autocorrelation function of a characteristic function is defined as follows: Its radially symmetrized version c u : [0, ∞) → [0, ∞) is given by We can also write the autocorrelation function (10) as C u = 1 |T | u * Iu, where Iu(x) := u(−x). We note that both C u and c u are dimensionless. Moreover, they are invariant under translation and reflection with respect to the origin.
Autocorrelation functions are widely used in other fields such as physics and stochastic geometry: One common use of these functions is to study spectral properties of observed patterns in theoretical physics (e.g. astrophysics). In the field of stochastic geometry, the autocorrelation function is also called covariogram or set covariance (cf. e.g. [30] and [4]). A common question in this field is to reconstruct a set from its autocorrelation function. One conjecture is e.g. whether the autocorrelation function for a convex set can characterize the set uniquely up to translation and reflection [30].
We note that the definition of autocorrelation function can be extended to more general configurations: Remark 2.2 (General periodic configurations). Let Λ ⊂ R d be a periodicity cell and assume that u : R d → R is Λ-periodic, i.e. u(x + Λ) = u(x). Then the autocorrelation function C u can be defined by Alternatively, one can define it by replacing T by the periodicity cell in (10), and it is easy to see that the two definitions coincide. In fact, the definition (11) is also well-defined for more general functions such as almost periodic functions, cf. eg. [16]. The symmetrized autocorrelation function is defined accordingly. Equation (11) shows in particular that C u (and c u ) do not depend on the choice of periodicity cell.
Some properties of the autocorrelation function are derived in e.g. [21,Prop. 11]. However, otherwise we have not found many results about analytical properties of the autocorrelation function in the mathematical literature. We hence collect some basic properties about C u in the following proposition: and Proof. The periodicity and symmetry of C u follows directly from the periodicity of u and the change of variable y = x − z.
) and takes the maximum at z = 0. By the triangle inequality and since ũ L ∞ , u L ∞ ≤ 1, for any z ∈ R d we have The identity (iii) follows directly from Fubini and since u is periodic. Estimate (iv) is given in [21,Prop.5].
We note that (13) with g := ∂ i u and f := ∂ j u defines a bilinear operator Passing to the limit in the weak formulation of (13) one gets ∂ ij C u ∈ M and moreover |T |∂ ij C u = −∂ j u * I∂ i u.
(v): Follows directly from the definition.
We note some of the assertions in Proposition 2.3 can be easily generalized to the case of u ∈ BV (T ) in which case an additional norm u L ∞ has to be added on the right hand side.
In the following we derive properties of the symmetrized autocorrelation. Similarly, these properties extend to general functions u ∈ BV (T ). However, in the general case we have to replace ∇u by D s u in (v) where D s u is the jump part of the derivative of u.
(v): By (iv) and [21,Prop. 11] (noting that the identity also holds for periodic functions) the derivative c (0) exists and we have For the last identity, we have used that To conclude we note that by Integrating by parts and since Thus the series (15) is absolutely convergent. Then where J α are the Bessel functions of the first kind. The decay estimate for (17) is absolutely and uniformly convergent for all r > 0 by (16)) The decay estimate for (18) then follows analogously as for (17). This completes the proof for (vi).

2.2.
Properties of the energies. In this section, we express our energies in terms of the autocorrelation function and use this expression to explore their properties. We first note that by Proposition 2.4 our energies can be expressed in terms of the symmetrized autocorrelation function c u : Proposition 2.6 (Energy in terms of c u ). Let d ≥ 1. Let ≥ 0 and suppose that K , K 0 satisfy (3)-(4). Then for any u ∈ BV (T ; {0, 1}) we have Proof. We first note that the integral in (19) is well-defined in R ∪ {±∞}: Thus the integrand is nonnegative in (0, 1) and negative in (1, ∞). This combined with our assumptions (3)-(4) on the kernels K and that c u ∈ C 0,1 ((0, ∞)) gives that the integral (19) is welldefined in R for > 0, and is well-defined in R ∪ {+∞} for = 0. To show the uniform lower bound we observe that where in the last inequality we have used the assumption (4). In view of Proposition 2.4(i)(ii) and using polar coordinates we have Then (19) directly follows from Proposition 2.4(iv).
The truncation at r = 1 is arbitrary. More generally, we have: Although the energies are linear in terms of the autocorrelation functions, they are not linear in u. Indeed, they are sublinear in the following sense: Then the interaction energy is non-negative and for Ω i := spt u i , i = 1, 2 we have Proof. Let u := u 1 + u 2 . By assumption, we have c u (0) = c u 1 (0) + c u 2 (0) and c u (0) = c u 1 (0) + c u 2 (0). Together with u = u 1 + u 2 the assertion follows from the energy expression (19) and Using the symmetry of the sum and substituting the supports of u i , we obtain the desired expression for I .
The space of functions with finite limit energy is quite complicated: the condition u ∈ BV (T , {0, 1}) is in general only a necessary but not sufficient condition for a configuration to have finite energy. Below we construct a BV function with infinite energy E 0 using the nonnegativity of the interaction energy in Lemma 2.8: On the other hand, from Lemma 2.8 and the estimate of the energy for a single disk (cf. Lemma 4.4(i) below) we have We end this section by a remark on the finite energy configurations for E becomes increasingly smaller as q increases. Indeed, as q getting larger the (symmetrized) autocorrelation function c u needs to converge faster to its affine approximation at 0 in order to get a finite energy (cf. Proposition 2.6). It follows from the explicit computation of c u (cf. Lemma 2.5) that, if Ω is a ball and u = χ Ω then E (3) 0 (u) < ∞ for q < d + 2 and infinite for q ≥ d + 2.
If Ω consists of multiple stripes, then E

Equivalent formulations of the energy.
In this section, we give some further geometric representations of the limit energy E 0 . These representations are in particular helpful for the calculation of the energy for specific configurations. We note that the assumption K 0 L 1 (R d ) = ∞ is used in the proof of Lemma 2.11 below. Hence, the results do not hold for the approximating energies E . Throughout the section F, G : R + → R + are defined by Note that for K 0 (r) = r −d we have F (r) = σ d−1 r −1 and G(r) = σ d−1 | ln r|.
We start with an auxiliary lemma: Proof. (i): With the representation (19) of the energy, we have where we also used (3)-(4). Arguing by contradiction we may assume that there exist δ 0 > 0 and c > 0 such that Since (21) and (20) as well as the definition of F we get Since F (δ 0 ) < ∞ by the assumption (4) of the kernel K 0 , from (22) one has that lim r→0 ln F (r) < ∞. This is a contradiction, because lim r→0 F (r) ≥ Analogously as for (i) and using that lim r→0 G(r) ≥ 1 0 K 0 (r)r d−1 dr = ∞, one then gets a sequenceδ j → 0 which satisfies (ii).
With Lemma 2.11 at hand we can express the energy using geometric quantities of Ω := spt u: where ν is the measure theoretic unit outer normal of Ω at x. We use the notation z := y − x. Then we have (ii) Letδ j → 0 be any sequence as in Lemma 2.11(ii). Assume that where P j := {(x, y) ∈ ∂Ω × ∂Ω per :δ j ≤ |x − y| ≤ R j } with Ω per ⊂ R d being the periodic copies of Ω.

Proof. (i):
We start with the representation (19) of E 0 in Proposition 2.6. We note that F (R) → 0 as R → ∞ which follows from (4). Integrating by parts both integrals in (19) (for the first integral of (19) we integrate by parts from δ j to 1, where the sequence δ j is from Lemma 2.11(i), and then let δ j → 0) yields Here we have used that the integrand for I 2 is nonnegative and thus the limit exists. We note that To calculate I 2 and I 3 we first express c u (r) using the geometric quantities. Using ∇u(x) = −ν(x)H d−1 ∂ * Ω and with Ω r (x) := Ω−x r , for a.e. r > 0 we have In the limit as r → 0 we get Inserting (25) and (26) into the integrand of I 2 and by Fubini we get Observing that for each ν ∈ S d−1 fixed, f (K) := − S d−1 ∩K ω · ν dω achieves its maximum when K = H − (x), we have Similarly, inserting (25) into the integrand of I 3 we have The desired identity then follows from (24), (27) and (27) with ω = y−x |y−x| ∈ S d−1 with y ∈ B 1 (x) and r = |y − x| and by a change from polar coordinates to Cartesian coordinates.
(ii): We choose a sequenceδ j → 0 such that the convergence in Lemma 2.11(ii) holds. Starting point is the representation (24) of the energy. Integrating by parts and using that G(1) = 0 we have By our choice of sequence, the second term on the right hand side vanishes as j → ∞. By assumption as well as Proposition 2.4(vi)(v) the term related to the evaluation at R j vanishes as j → ∞. Adding the expressions for shortrange and long-range interaction together we obtain the claimed equivalent form of the energy. Proof. (i): We use the expression of the energy E (u) in terms of the autocorrelation function in Proposition 2.6. By the assumptions (3)-(4) as well as K K 0 , there are C 0 , C 1 > 0 and a measurable set A ⊂ (0, 1), which depends on K 0 , such that for sufficiently small > 0 we have Using (28) as well as where we have used Proposition 2.4(ii) and (28) for the last estimate. Recalling the expression for c u (0) in Proposition 2.4(iv), we thus obtain the desired estimate from the above inequality.
(ii): By (i) we have ∇u ≤ C. By the compactness of the BV functions, there is a subsequence of u (which we still denote by u ) and u ∈ BV (T , {0, 1}), such that u → u in L 1 (T ) and ∇u ≤ lim inf →0 ∇u .
To show the convergence of ∇u , by Proposition 2.4 it suffices to show c u (0) → c u (0). Assume c u j (0) → α for some α ∈ R along a subsequence. By the lower semi-continuity of the BV-norm we have α ≤ c u (0). It remains to show that α ≥ c u (0): By Fatou's lemma and (28) for any δ ∈ (0, 1), On the other hand, by Proposition 2.4 Taking the difference of the above two inequalities yields As a consequence of Proposition 3.1, we get the existence of minimizers for E with prescribed volume fraction: Proof for Proposition 1.2. Let ≥ 0 be fixed. It follows from Lemma 2.5 (ii) that E (u) < +∞ if u is a stripe configuration, and furthermore E (u) ≥ −C for all u ∈ BV (T ; {0, 1}) (cf. Proposition 2.6). Hence the infimum of E (u) exists and is finite. Let {u k } be a minimizing sequence for E . Then by Proposition 3.1(ii) up to a subsequence u k → u in L 1 (T ) and ∇u k → ∇u for some u ∈ BV (T ; {0, 1}). In particular u satisfies (9). Hence c u k → c u pointwisely and c u k (0) → c u k (0) by Proposition 2.4(ii) and (v). Applying Fatou's lemma (in (0, 1)) and dominated convergence theorem (in (1, ∞)) to the autocorrelation function formulation of E in Proposition 2.6, we obtain that E (u) ≤ lim inf k→∞ E (u k ). This implies that u is a minimizer.
At the end of the section we establish the Γ-convergence of E : Proof. We use the representation of E from Proposition 2.6, i.e.
(i): Liminf inequality. We need to show that for any sequence u → u in L 1 and sup E (u ) ≤ C < ∞ we have By Proposition 3.1(i) the sequence u is uniformly bounded in BV (T ; {0, 1}). This together with the L 1 convergence of u and Proposition 3.1(ii) yields that u ∈ BV (T ; {0, 1}) and ∇u → ∇u . Then by Proposition 2.4 we have c u → c u uniformly and c u (0) → c u (0). Moreover, the integrands of the energy are nonnegative for r ∈ (0, 1). Thus by Fatou's lemma (applied to the integral over (0, 1)) and the dominated convergence theorem (applied to the integral over (1, ∞)) one has which yields (29).

3.2.
Proof of Theorem 1.3. In this section, we give the proof of Theorem 1.3, i.e., we show that if the volume u L 1 (T ) = λ|T | is sufficiently small depending on the dimension, then the minimal energy is attained by a single ball. We give the proof for a class of energies which in particular includes the energy E (1) 0 in Theorem 1.3. More precisely, we assume that there is some We note that the assumption (30) holds for K (30) together with (33) ensures that the energy of a single ball is finite. Then Theorem 1.3 is a consequence of the next proposition: Proof. The proof is based on a contradiction argument. Similar proofs have been made using the sharp isoperimetric deficit (cf. [20]) e.g. in [27,28,24,7,19,33]. We adapt these arguments to the autocorrelation function formulation of our model. The novelty in our proof is that the nonlocal term has the same scaling as the perimeter term.
The isoperimetric deficit of the bounded Borel set Ω ⊂ R d is defined by where B ⊂ R d is the ball with |B| = |Ω|. The sharp estimate on the isoperimetric deficit in [20] entails that where δ := min{1, D(Ω)}, noting that the left hand side of (31) is called the Fraenkel asymmetry of Ω.
Let η > 0. By assumption (4) we have Since c u (r) − c u (0) − rc u (0) ≥ 0 and K 0 ≥ 0, for η ∈ (0, ρ) we have Using By Lemma 2.5 and since ≥ 4ρ, we have |T ||c u 0 (r)| ≤ Crρ d−3 for r ≤ ρ 2 and |T |( Using (33) we then get By an application of Proposition 2.4(ii) we get where we have used the estimate of the isoperimetric deficit (31) and the definition of δ in the last inequality. Combining (32), (34) and (35) and recalling −c u 0 (0) = Inserting the assumption (30) to the above inequality and by (4), we obtain With the choice η := δρ, the above inequality gives X η ≤ C(1+δ) M X η +C d,K δρ. Taking M = 4C and recalling that δ ≤ 1 we thus get X η ≤ C d,K ρ. Since X η → ∞ as η → 0, then necessarily we have ρ ≥ η ≥ C for some C = C(d, K) > 0, and hence λ|T | = ω d ρ d ≥ c > 0 for some constant c = c(d, K) > 0. We thus get a contradiction if λ|T | ≤ m 0 for some m 0 = m 0 (d, K) sufficiently small. Remark 3.4. Proposition 3.3 is for small volume configurations instead of the volume fraction, as the smallness condition on λ is not independent of . Actually it follows from Lemma 4.4 below that when the volume fraction λ is sufficiently small, periodic balls have smaller energy than a single ball in T for sufficiently large .

Stripes and balls configurations
In this section we explicitly compute the limit energy for stripes and lattice balls for given volume fraction. Throughout this section we assume the kernel is the prototypical K (1) 0 (r) = r −d . In the previous sections we have used the technical assumption that the configurations are T -periodic for some arbitrary periodicity . To compare the energy for configurations with fixed volume fraction, it is natural to consider more general configurations. We hence write exists. Then we set As noted in Remark 2.2, this energy is independent of the lattice Λ where it arises from. In particular, the definition (36) is consistent with and generalizes the definition of the energy E 0 in Theorem 1.1(ii). We also note that we analogously get a more general definition of the energies E . However, these will not be used in this article.
Here u is any function in BV loc (R d ; {0, 1}) such that C u is well-defined. One can show that A λ is convex. Even though the limit energy is a linear functional in terms of autocorrelation functions, the problem still seems to be very hard since it is difficult to characterize the space A λ .
In the rest of the section we calculate and compare the energy for stripe and ball configurations with the prototypical kernel K 0 (r) = r −d (and the corresponding energy is denoted by E (1) 0 ). We recall the generalized harmonic number H q with index q ≥ 0 is given by Then the following holds: (i) The function u := χ Ω is T a -periodic and where H q is the harmonic number (cf. (37)). (ii) Among configurations with prescribed λ ∈ (0, 1) the minimal energy is Proof. For the calculation we reduce the problem to the calculation of the interaction energy between two parallel slices. Note that the configuration is T a -periodic.
where we used that the inner integral in the first line above is independent of x and y 1 − x 1 = ρ. An explicit calculation yields By Proposition 2.12(ii) with I(0) := 0 and I(−ρ) := I(ρ) for ρ < 0 we get The infinite sum in the first line above is taken in the p.v. sense . Inserting the formula for I(ρ) and since P (Ω) = 2a d−1 , we arrive at Using d 0 = λa and the Euler's formula Π ∞ n=1 (ii): The results follows by a standard calculation.
As a competitor we next consider a lattice of balls, arranged on a Bravais lattice. We need to fix some notations: For a set of linearly independent vectors v i ∈ R d , 1 ≤ i ≤ d we consider the Bravais lattice By |Λ| we denote the volume of the periodicity cell, i.e.
The energy for lattice balls can then be formulated in terms of the Appell series F 4 , which in turn can be expressed using the Pochhammer symbols (a) n , given by (a) n := 1 for n = 0 and (a) n := a(a + 1) · · · (a + n − 1) for n ≥ 1. We set With these notations, we have where H q is defined in (37) and where where the Appell series H d is defined in (39).
(ii) Let Λ 0 be a fixed lattice with |Λ 0 | = 1 and let e B,Λ 0 (λ) be the minimal energy among lattices of the form Λ = aΛ 0 for a > 0 and for prescribed volume fraction λ = |Bρ| |Λ| . Then for all λ ≤ λ 0 , where λ 0 = λ 0 (Λ 0 ) is the largest volume fraction which can be realized by balls in the lattice. The radius of the optimal ball configuration is given by Proof. (i): We introduce the full-space (radially-symmetrized) autocorrelation function for the single ballũ := χ Bρ bỹ Using Lemma 2.8, we decompose the energy as Here, E self is the self-interaction energy of a single ball, i.e.
Furthermore, the interaction energy of a single ball B ρ with other copies is Computation of E self (ρ): By definition we havec u (r) = 0 for r ≥ 2ρ. Using Remark 2.7 and integration by parts we have Sincec u (r) = |Λ|c u (r) for 0 ≤ r ≤ 2ρ, in view of the formula in Lemma 2.5(i) and with the change of variables r = ρ Using the formulac u (0) = − Estimate of I int (Λ, ρ): For the estimate of I int (Λ, ρ) we note that by (50) in Appendix B we have Together with (43) and (42), this yields (i).
(ii): Using λ = |Bρ| |Λ| , we can further express E self in terms of λ and ρ as E self (ρ) |Λ| We note that I Λ = 1 a 2d+1 I Λ 0 , where Using this and λ = |Bρ| a d and ω d ρ d = λa d , the averaged interaction energy can be rewritten in terms of ρ and λ as From Lemma B.1 and thatρ |e| ≤ 1 2 , which follows from (40), we have that 1 |e| d+1 . Thus the self-interaction energy in (44) is of leading order in λ for λ 1 for fixed lattice Λ 0 . By minimization (44) in ρ we obtain Since expression (45) is of lower order in λ, (ii) follows by a standard argument.
(ii) if d = 2 and |λ − 1 2 | < δ 0 , then e S (λ) < e B,Λ 0 (λ). Proof. (i): Comparing the leading order (in λ) terms of e S (λ) and e B,Λ 0 (λ) (which is independent of Λ 0 ) and using that (ii): For large volume fraction the interaction energy is no more of lower order. Moreover, noting that balls have smaller self-interaction energy than stripes, we have to estimate the lower bound of interaction energy for balls. When d = 2, by Lemma 4.3(ii) and since H 1 Now we estimate the lower bound for the interaction energy among Bravais lattices in R 2 . Given any Bravais lattice Λ ⊂ R 2 with a 2 = |Λ| and Λ 0 := 1 a Λ, from (50) in Lemma B.1 and using that λa 2 = πρ 2 we have Thus combining with the self-interaction energy in Lemma 4.4 we have that for u = q∈Λ B ρ (q) with 0 < ρ ≤ 1 2 min p,q∈Λ |p − q| and λ = |Bρ| |Λ| = 1 2 , By Rankin [35], ζ(Λ 0 ) attains the minimum at the triangle lattice H nor : 2 )) among all 2d Bravais lattices with volume 1, and furthermore from its explicit expression we have that ζ(H nor ) > 8. Thus (46) together with the standard minimization gives ).
Since the energy functionals are continuous in λ , then there is δ 0 > 0 independent of Λ 0 such that if λ ∈ ( 1 2 − δ 0 , 1 2 + δ 0 ) stripes have strict smaller energy than any 2d lattice balls with the volume fraction λ. Remark 4.6 (Triangular lattice in 2d). We conjecture that in 2d for sufficiently small volume fraction λ, the triangle lattice H nor has the smallest energy among all lattice of balls. Indeed, we recall from Lemma 4.4(i) that the energy consists of the self-interaction energy and the interaction energy. In view of (44), the self-interaction energy is independent of the lattice Λ 0 . For the interaction energy, cf. (45), we expect that I Λ 0 achieves its minimum for the triangle lattice H nor if λ is sufficiently small. To prove such a result one would (at least) need to extend the methods in [6] to the case of non-integrable potentials and we leave it for the future work.

Appendix A. Connection to the results by Dávila
We start by recalling the classical result by Dávila [18]. Let K : R d → R + be a family of nonnegative radially symmetric kernels, which satisfy Let Ω ⊂ R d be a bounded set with finite perimeter. Then Dávila showed that for u := χ Ω one has Using the autocorrelation function, one can give a simpler proof for (48) With this at hand, (48) directly follows from writing the above equation in terms of u (cf. Proposition 2.4(iv) and (v) in the non-periodic setting). We note that the decay of K at infinity is not needed when Ω is bounded. Furthermore, instead of K L 1 (R d ) it suffices to normalize the LHS of (48) by K L 1 (Br 0 ) for arbitrary r 0 > 0. Now we comment on our assumptions on the kernels K as well as the convergence in (1). Since in this paper we are interested in the second order asymptotic expansion for the perimeter, we assume that K K 0 as 0 for some measurable function K 0 : R d → R + with K 0 L 1 (R d ) = ∞, i.e. (3). Moreover, in the periodic setting one needs to consider the interaction of Ω with its periodic copies, consequently the autocorrelation functioñ c u (r) −c u (0) does not decay as → ∞ but remains uniformly bounded. Thus in view of (49) we assume ∞ 1 K 0 (r)r d−2 dr < ∞ (cf. (4)) such that the energy functional is well-defined. Under these assumptions (1) holds true, which is an analogue of (48) in the periodic setting and whose proof is a simple modification of that for (48). We also mention that the monotone convergence of K is used when we prove the Γ-convergence of E (cf. Theorem 3.2) to avoid the concentration effect. It might be possible to weaken this assumption, but this was not our aim in this paper.

Appendix B. Interaction energy of two balls
In this section we provide the explicit expression for the full-space interaction energy between two.
Lemma B.1 (Interaction energy between two balls in full space). Let q ∈ R d . Then for 0 < ρ ≤ |q| 2 we have where H d (t) is given in (39). In particular, for C = C d > 0 we have Proof. By scaling and rotation invariance, we may assume that ρ = 1, λ := |q| ≥ 2 and q = |q|e 1 . We note that the Fourier transform for χ B 1 (0) is where J ν is the Bessel function of first kind. Using that B 1 (0) =: B 1 and B 1 (q) = B 1 + q are disjoint, we have By Plancherel and with the change of variable h → h |ξ| we have where the second equation of (52) follows from the and where .