Large mass rigidity for a liquid drop model in 2D with kernels of finite moments

Motivated by Gamow's liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter $P(E)$ is replaced by $P(E)-\gamma P_\varepsilon(E)$, with $0<\gamma<1$ and $P_\varepsilon$ a nonlocal energy such that $P_\varepsilon(E)\to P(E)$ as $\varepsilon$ vanishes. We prove that unit area minimizers are disks for $\varepsilon>0$ small enough. More precisely, we first show that in dimension $2$, minimizers are necessarily convex, provided that $\varepsilon$ is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their boundary is a small Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we prove that (in arbitrary dimension $n\geq 2$) the unit ball in $\mathbb{R}^n$ is the unique unit-volume minimizer of the problem among centered nearly spherical sets. As a consequence, up to translations, the unit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, sufficiently integrable kernel. In that formulation, our main result states that if the first moment of the kernel is smaller than an explicit threshold, there exists a critical mass $m_0$ such that for any $m>m_0$, the disk is the unique minimizer of area $m$ up to translations. This is in sharp contrast with the usual case of Riesz kernels, where the problem does not admit minimizers above a critical mass.


Introduction
Given a positive, radial, measurable kernel G : R n → [0, +∞) with finite first moment (that is, |x|G(x) ∈ L 1 (R n )), we consider the nonlocal perimeter functional P G (see e.g. [10,6]) defined on measurable sets E ⊆ R n by (1.1) Here 1 E denotes the indicator function of E. For ε > 0, we introduce the rescaled kernel G ε (x) := ε −(n+1) G(ε −1 x), x ∈ R n . As will be justified later, the first moment of G is fixed to an explicit dimensional constant (see (H.2)) so that P Gε (E) converges to P (E) as ε vanishes. Given γ ∈ (0, 1) and ε > 0, we study the minimization problem and (P γ,ε ) is equivalent to the problem (P γ,ε ) min P (F ) − γ P G (F ) : |F | = |B 1 |/ε , in the sense that E ε is a minimizer of (P γ,ε ) if and only if F ε := ε −1 E ε is a minimizer of (P γ,ε ). Then, if in addition we assume that G is integrable in R n , we may write thus (P γ,ε ) is in fact equivalent to where we have set m ε := ε −1 |B 1 | and G := 2γG. When n = 3 and G(x) = 1/(8π|x|), this is Gamow's liquid drop model (see [11] for a general overview); note however that in that case, the minimized functional cannot be rewritten as the difference between the perimeter and a nonlocal perimeter, since G is not integrable at infinity. As a prototypical model for various physical systems involving the competition between short-range attractive forces and long-range repulsive ones, generalizations of this model have gained increasing interest during the past decade, in particular generalizations in higher dimensions, where the Coulomb potential is replaced with Riesz potentials, that is, G(x) = |x| α−n , α ∈ (0, n). In particular, it was shown that for every Riesz kernel, in the small mass regime, the unique minimizer of the liquid drop model (G mε ) is the ball, up to translations (see [24,25,23,7,15]). Conversely, for α ∈ [n − 2, n), the problem admits no minimizer above a critical mass (see [7,24,25,28,16,18]; see also [17]). More general kernels of Riesz-type were studied e.g. in [9,33,30], where the unit ball is shown to be the unique minimizer in the small mass regime. Although the small mass regime has been extensively studied, the literature on large mass minimizers for Gamow-type problems is still sparse, since existence is rather unexpected in that case, and is usually only recovered by adding an extra attractive potential, such as in [1,2,21], or by adding a density to the perimeter, as in [3], where the authors show that if the density is a power-law growing sufficiently fast at infinity, then minimizers always exist, and are balls for large masses. It is worth mentioning that in the case of general kernels with compact support, the author of [36] shows that minimizers exist for all masses.
Between Riesz kernels, which are not integrable at infinity, where (G mε ) does not admit minimizers above a critical mass, and compactly supported kernels, where (G mε ) always admits minimizers, it is natural to wonder what happens with non-compactly supported but reasonably decaying kernels, such as Bessel kernels. These kernels behave as Riesz potentials near the origin, but decrease exponentially at infinity. They were suggested in [26] as a replacement for Riesz kernels for modeling diblock copolymers when long-range interactions are partially screened by fluctuations in the background nuclear fluid density. In [30] (see Section 1.2 therein), motivated by some model for cell motility, the authors suggest to study problem (G mε ) for rescalings of the kernel G, where G is the fundamental solution of Id + (−∆) s/2 G = δ 0 , for s ∈ (0, 2). As pointed out in [30,Rem. 1.3], for s ∈ (0, 1) their asymptotic rescalings correspond to the small mass regime, while for s ∈ (1, 2) they correspond to the large mass regime. The authors focus on the case s ∈ (0, 1). Our work actually addresses the case s ∈ (1, 2) (and with more general kernels).
The study of the liquid drop model in the large mass regime for integrable kernels with finite first moment (such as Bessel kernels or the ones just mentioned for s ∈ (1, 2)), which is equivalent to the study of (P γ,ε ) when ε is small, has been started by the second author in [35]. The existence of minimizers for any γ ∈ (0, 1) was established therein for ε small enough, as well as the convergence of minimizers to the unit ball as ε vanishes. It was conjectured there that the ball is actually the unique minimizer up to translations, for ε small enough. In this paper, we give a positive answer to this conjecture in dimension n = 2 under reasonable assumptions on the first moment of G and the second moment of ∇G, which are still satisfied by Bessel kernels. The conjecture remains open in higher dimensions.
Let us introduce the "critical energies" Although the paper mostly deals with the "subcritical case" γ < 1, we focus on the critical energies in Section 3.1, and show that they decrease by convexification. Finally, for any k ∈ N {0}, we denote by the k-th moment of the (k − 1)-th radial derivative of the kernel G, whenever it is well-defined. In the paper, starting from Section 2, we shall always implicitly assume that the kernel G satisfies the following general assumptions: (H.1) G is nonnegative and radial, that is, there exists a measurable function g : (0, +∞) → [0, +∞) such that G(x) = g(|x|) for almost every x ∈ R n ; (H.2) the first moment of G is finite and set to be where, for any ν ∈ S n−1 , the constant K 1,n is defined by Starting from Section 4, we may explicitly use the extra assumption: These assumptions are in particular satisfied by the Bessel kernels B κ,α , that is, fundamental solutions of the operators (I − κ∆) α/2 , for κ, α > 0 (see e.g. [35, §3.2] or [22] for their definition and properties).
Even though our main result is in dimension n = 2, where we prove that the unit disk is the only minimizer up to translations, provided that ε is small enough, note that the intermediate results of Section 3.4 and Section 4 are obtained in arbitrary dimension. That is, convex minimizers are nearly spherical sets, and the unit ball of R n is the unique minimizer among nearly spherical sets whenever ε is small enough.
Note also that the kernel G is assumed to be radial but not necessarily radially nonincreasing, as is often the case. Let us emphasize that, contrarily to the small mass regime for Riesz-type potentials, here the nonlocal perimeter term does not vanish in the limit but rather converges to a fraction of the standard perimeter.
We shall now state the main result of the paper.
In terms of Gamow's problem (G mε ), this means that if is the first moment of the kernel G defined as in (1.4)) and G is in addition integrable, then there exists a critical mass m * such that the only solutions of (G mε ) with m ε > m * are the disks of area m ε . In the particular case of the Bessel kernels G = B κ,α with κ, α > 0, such a critical mass exists whenever (see [35,Cor. 3.9 & Prop. 3.10]).
The existence of minimizers for small ε was shown in [35], where the second author also proved that they are necessarily connected whenever ε is small enough. The idea for proving the convexity of minimizers is to study the critical energy on the real line, and show that it decreases by convexification and by expansion of segments, so that, by a slicing argument, the critical energy of a connected set in dimension 2 decreases after convexification. As a consequence, since the perimeter of a connected set is also reduced by convexification, so is F γ,Gε . This slicing argument is specific to the dimension 2, where a line intersects a connected set if and only if it intersects its convex hull. This fails in higher dimension.
Note that this is not enough to conclude that minimizers are convex. Indeed, although for every minimizer E ε ⊂ R 2 with ε small enough, we have F γ,Gε (co(E ε )) F γ,Gε (E ε ), where co(E ε ) denotes the convex hull of E ε , the volume of co(E ε ) is larger than |B 1 | if E ε is not convex. However, using the fact that a minimizer E ε is already close to the unit ball by [35] and the convexity of co(E ε ), we prove that, if E ε is not convex, scaling down co(E ε ) to make its volume equal to |B 1 | strictly decreases the energy F γ,Gε , which contradicts the minimality of E ε .
The convexity of minimizers E ε allows us to improve the convergence of ∂E ε towards ∂B 1 as ε goes to 0, from the previously known Hausdorff convergence to Lipschitz convergence. We deduce that minimizers are nearly spherical sets, whose definition is given just below.
In dimension n = 2, we use the terminology "t-nearly circular set" for "t-nearly spherical set".
Theorem 2 is a direct consequence of Theorem 1 and the uniform convergence of minimizers already shown in [35], using the geometric fact that the normal vectors to the boundary of a convex set lying between two balls B r and B R , r < 1 < R converge to those of the unit sphere as r, R → 1 (see [34,19]).
We end the proof by showing that for ε and t small enough, any centered t-spherical minimizer of (P γ,ε ) is the unit ball. This last result is not specific to dimension n = 2.
Theorem 3 (Minimality of the unit ball among nearly spherical sets; see Proposition 4.3) Assume that γ ∈ (0, 1) and G satisfies (H.1) to (H.3). Then there exist t * = t * (n, G, γ) > 0 and ε 3 = ε 3 (n, G, γ) > 0 such that, for every t < t * , if E is a t-nearly spherical set, then we have and the inequality is strict if E = B 1 (in the sense that they differ by a set positive measure).
Theorem A is then an immediate consequence of Theorems 1 to 3. The proof of Theorem 3 relies on a bound (in our case, an upper bound) on the quantity P Gε (E t )− P Gε (B 1 ) for a centered t-nearly spherical set E t with ∂E t = (1 + tu(x)) : x ∈ S n−1 , in terms of the L 2 norms of u and ∇ τ u on the sphere. In the case of the local perimeter, this kind of control is well-known and is originally due to Fuglede (see [19,Th. 1.2]), who proved provided that t is small enough, depending only on n. Similar results were obtained for so-called fractional perimeters P s as well as for Riesz potentials in [15] (see Theorems 2.1 and 8.1 therein), where the quantities are bounded in terms of the L 2 norm and fractional Sobolev seminorms of u on the sphere. Our computations are inspired by the ones in [15], however, due to the general form of the kernel G, they are more involved and quite tricky at times. As a last introductory remark, let us comment on the constants t * , ε A , ε i , i ∈ {1, 2, 3} and their dependence in γ. As expected, the constants vanish as γ tends to 1, and the rate at which they vanish depends on the convergence rate of the quantity P (B 1 ) − P Gε (B 1 ) (quadratic in ε) and on the decay of the kernel G at infinity. Assuming that G(x) = O(|x| −(n+1+β) ) at infinity, by Remark 3.12 and Remark 4.4, as γ tends to 1, we have Outline of the paper. -The structure of the paper follows the strategy of the proof. In Section 2 we recall some useful results from [35] on minimizers of (P γ,ε ) and some facts on nonlocal perimeters. In Section 3, we show that 2D minimizers are convex and thus nearly circular sets for small ε, that is, Theorem 1 and Theorem 2, where the latter is a consequence of Proposition 3.11. Finally, Section 4 is dedicated to the proof of Theorem 3, which is a consequence of Proposition 4.3.

Notation
Operations on sets. -For any set E ⊆ R n , E c := R n E denotes its complement, co(E) its convex hull (that is, the intersection of all convex sets containing E), and |E| its Lebesgue measure, whenever E is measurable. We write E F for the symmetric difference of E and F , and E F for the union of E and F whenever they are disjoint.
Hausdorff measures. -We denote by H k the k-dimensional Hausdorff measure in R n . When integrating with respect to the measure H k in a variable x, we use the notation dH k x instead of the more standard but less compact dH k (x).
Balls and spheres. -We denote by B r (x) the open ball in R n of radius r centered at x. For brevity, we write B r when x is the origin. The volume of B 1 is ω n := |B 1 | = π n/2 /Γ (1 + n/2), and the area of the unit sphere S n−1 is H n−1 (S n−1 ) = nω n , which we also write |S n−1 | for simplicity.
Sets of finite perimeter. -We denote by BV(R n ) the space of functions with bounded variation in R n . For any f ∈ BV(R n ) we let |Df | be its total variation measure, and set [f ] BV(R n ) := R n |Df |. For a set of finite perimeter E in R n , we let 1 E ∈ BV(R n ) be its characteristic function (i.e., 1 E (x) = 1 if x ∈ E and 0 otherwise), and define its perimeter by P n (E) := R n |D 1 E |. When there can be no confusion, we may drop the superscript and simply write P (E) for the perimeter functional in R n . We denote by µ E := D 1 E the Gauss-Green measure associated with the set of finite perimeter E, and by ν E (x) the outer unit normal of ∂ * E at x, where ∂ * E stands for the reduced boundary of E. We refer to e.g. [13,Chap. 5] or [29] for further details on functions of bounded variations and sets of finite perimeter.
For a general nonnegative radial kernel K with finite first moment, we have the following control of P K by the perimeter, as an immediate consequence of [35,Prop. 3.1] and of the second expression of the nonlocal perimeter given by (1.1).
Proposition 2.1. -Let K : R n → [0, +∞) be a kernel satisfying the same assumptions (H.1) and (H.2) as G, except that the value of its first moment is not prescribed. Then, for every set of finite perimeter E in R n , we have P K (E) K 1,n I 1 K P (E). In particular, for the kernels G ε , we have We also have the following convergence result, which is a consequence of [12] and our choice of I 1 G .
Proposition 2.2. -For any set of finite perimeter E in R n , we have We will use the following computation obtained in [35,Lem. 3.5], which clarifies the behavior of the nonlocal perimeter under scaling. where P Gε (E) is defined by Let us remark that in [35], G is assumed to be in addition integrable in R n , but Lemma 2.3 can be deduced by approximating G with maps G k : x → χ k (|x|)G(x) ∈ L 1 (R n ). Indeed, let χ k ∈ C ∞ (R + , [0, 1]) be cutoff functions with χ k (r) = 0 for r 1/k, χ k (r) = 1 for r 2/k, so that I 1 In order to study the minimality of the unit ball among nearly spherical sets, we will use the following Bourgain-Brezis-Mironescu-type result (see [8]) for approximating the H 1 seminorm on the sphere by nonlocal seminorms. When n = 2, we assume in addition that g is such that the family (η ε ) ε>0 satisfies Then for any u ∈ H 1 (S n−1 ), we have where q η (ε) vanishes as ε goes to 0, and depends only on n and G. In addition, for any u ∈ H 1 (S n−1 ), Proof. -One easily checks that assumptions (H.1) and (H.2) ensure that the family (η ε ) ε>0 is a (n − 1)-dimensional approximation of identity, up to multiplication by the constant K 2,n−1 = 1/(n − 1), i.e., loc (0, +∞) and the functions t → t n g(t) and t → t n+1 g (t) are integrable on (0, +∞). In addition, integrating the function (t n+1 g(t)) between r and R, we have the relation Since t n g(t) and t n+1 g (t) are integrable on (0, +∞), this implies that r n+1 g(r) has a limit in 0 + and at infinity. By the integrability of t n g(t) on (0, +∞), these limits are necessarily 0. In particular, r n+1 g(r) vanishes uniformly on (R, 2) as ε → 0, for every R ∈ (0, 2).
Eventually, gathering results from [35, Th. A & B] (see also Theorem 4.16 therein), we have existence and convergence results for minimizers of (P γ,ε ). We also know that minimizers are connected for small ε. Here connectedness is to be understood in a measure-theoretic sense for sets of finite perimeter, often referred to as indecomposability, as defined below (see [4]).
Definition 2.6. -We say that a set of finite perimeter E is decomposable if there exist two sets of finite perimeter E 1 and E 2 such that E = E 1 E 2 , |E 1 | > 0, |E 2 | > 0 and P (E) = P (E 1 ) + P (E 2 ). Naturally, we say that a set of finite perimeter is indecomposable if it is not decomposable.
Let us remark that by [4,Th. 2], the notion of connectedness and indecomposability coincide whenever E is an open set of finite perimeter such that H n−1 (∂E) = H n−1 (∂ * E).
Proof. -In [35], the kernel G is assumed to be integrable in R n . However, it is actually only required for the two following reasons: first, to be able to write (1.2) and obtain the equivalence with the Gamow-type minimization problem (G mε ); second, by this equivalence, to deduce that minimizers of (P γ,ε ) are so-called quasi-minimizers of the perimeter, and thus are (non-uniformly in ε) C 1,1/2 -regular outside a "small" singular set. Here, we do not need the equivalence with (G mε ) nor the a priori regularity of minimizers. In the end, apart from the C 1,1/2 partial regularity of minimizers, all the conclusions of [35, Th. A & B] follow. More precisely, there exists ε 0 = ε 0 (n, G, γ) such that, for any ε such that 0 < ε < ε 0 , (P γ,ε ) admits a minimizer. In addition, any such minimizer E ε is indecomposable and, up to a translation and Lebesgue-negligible set, it satisfies where δ : (0, +∞) → (0, 1/4) is a function depending only on n, G, and γ vanishing in 0 + . To conclude, there remains to show that in dimension n = 2, E ε is equivalent to a connected set with (2.4), that is, to link the indecomposability of E ε with the topological notion of connectedness. It is not a trivial question, at least without (weak) regularity results on minimizers. However, [4,Th. 8] shows that in dimension 2, Since Remark 2.8. -In higher dimensions n 3, one could show as well that any minimizer of (P γ,ε ) is equivalent to a connected set for small ε, without assuming that G ∈ L 1 (R n ). Indeed, proceeding e.g. as in [32,Lem. 5.6], it is possible to obtain uniform (with respect to ε) density estimates for minimizers, and with those to deduce that any minimizer is equivalent to an open set E ε such that ∂E ε = spt µ ε . The indecomposability of E ε then implies connectedness by [4].
and that ε 0 is actually chosen so that δ(ε) C(n) for every ε such that 0 < ε < ε 0 , for some dimensional constant C(n). A computation shows that P ( and we can choose Hence, the closer γ is to 1, the smaller ε 0 is.

3.
Minimizers are nearly circular sets in dimension 2 3.1. Decrease of the critical energy by convexification. -We can recover the nonlocal perimeter of a measurable set E ⊆ R n by integrating the 1-dimensional nonlocal perimeter of all 1-dimensional slices of E in a given direction, and averaging over all the directions.
for every measurable set J ⊆ R.
Proof. -By the change of variable y = x + rσ with fixed x and Fubini's theorem, we have where we have used the definition of ρ ε for the last equality. Then, for σ fixed, let us make the change of variable Finally, using Fubini's theorem and making the change of variable t = s + r, where s is fixed, we obtain, by definition of E σ,y and P 1 ρε , This concludes the proof.
Hence by Proposition 2.1, we have Similarly, as a straightforward consequence of [5,Th. 3.103] (see also Crofton's formula [31, §3.16] or [14,Th. 3.2.26]), for any set of finite perimeter E ⊆ R n , E σ,y is a one-dimensional set of finite perimeter for H n−1 -almost every σ and y, and we have is the standard perimeter in dimension 1. Hence, we have the following representation of the critical energy E Gε .
We give a simple expression of the one-dimensional critical energy. of a segment (a, b) ⊆ R.
In particular E 1 ρε ((a, b)) decreases as the interval grows. In addition, Proof. -By a change of variable and Fubini's theorem, for any a ∈ R, we have Similarly, P 1 ρε ((b, +∞)) = 1, and since P 1 ((−∞, a)) = P 1 ((b, +∞)) = 1, (3.3) follows. Next, for every a, b ∈ R such that a < b. We also have For a general set of finite perimeter in R, we have the following expression of the critical energy.
Let us point out that E c may have up to two unbounded connected components, and their critical energy is 0 by Lemma 3.4. As a consequence of Lemmas 3.4 and 3.5, the 1D critical energy decreases by convexification and the energy of a nonempty open segment is a decreasing function of its length.
by Lemma 3.4 and Remark 3.2, so the result holds true. If E does not have finite perimeter, then E 1 ρε (E) = +∞, and the result holds true as well. Thus, let us assume that E is a bounded set of finite perimeter. In particular, up to a negligible set, E is the disjoint union of k open intervals with k 1 since L 1 (E) > 0. In addition, since E is bounded, E c has two unbounded components C 1 = (−∞, a) and C k+1 = (b, +∞) (up to renumbering). By Lemmas 3.4 and 3.5, we have with −∞ a 1 < b 1 < · · · < a k < b k +∞, so that, setting b 0 := −∞ and a k+1 := +∞, the connected components of E c are given by Beware that C 0 and C k are either empty or unbounded. If a 1 = −∞, C 0 = ∅, and if b k = +∞, C k = ∅, in which cases it is an abuse to consider them connected components of E c , but they do not contribute to the terms in (3.6). Omitting the integrand ρ ε (s − t) ds dt for the sake of readability, let us write Note that this holds even if a 1 = −∞ or b k = +∞. Let us define and similarly Notice that R k = L 0 = 0. We observe that by definition of L i , R i and (3.7), (3.8), Ci {t ∈ E c : t>ai+1} and similarly, Ci {t ∈ E c : t<bi} The two previous equations hold even if C 0 , C k are empty or unbounded. Inserting (3.10) and (3.11) into (3.9) yields which concludes the proof.
We easily deduce from Lemma 3.5 and the slicing decomposition of the critical energy stated in Corollary 3.3 that in dimension n = 2, the critical energy of a connected set decreases by convexification.
Proof. -First, recall that a bounded convex set is a set of finite perimeter, since it is Lebesgue-equivalent to an open set with Lipschitz boundary, thus co(E) is a set of finite perimeter. Then, by Corollary 3.3 we have where E σ,t := s ∈ R : tσ ⊥ + sσ ∈ E , and F σ,t := s ∈ R : tσ ⊥ + sσ ∈ co(E) .
Since E is connected, for every σ ∈ S 1 and t ∈ R, the slice F σ,t is empty if and only if E σ,t is empty (this is the argument which is valid in dimension 2 only). In addition, since E and co(E) are bounded sets of finite perimeter in R 2 , for H 1 -almost every σ ∈ S 1 and L 1 -almost every t ∈ R, E σ,t and F σ,t are bounded sets of finite perimeter in R, and for every nonempty slice, F σ,t is an interval s.t. E σ,t ⊆ F σ,t (since F σ,t is a slice of a convex set). Hence, by Lemmas 3.4 and 3.5, for H 1 -almost every σ ∈ S 1 and L 1 -almost every t ∈ R, there holds In view of (3.12), this concludes the proof.
Lemma 3.8. -There exists ε 1 = ε 1 (G, γ) > 0 such that the following holds. If E ⊆ R 2 is a convex set such that with 0 < ε < ε 1 , and where δ is the function given by Theorem 2.7, then As a consequence, we obtain that minimizers of (P γ,ε ) are necessarily convex for small ε in dimension n = 2, that is Theorem 1: Proof of Theorem 1. -Let E ε ⊆ R 2 be a minimizer of (P γ,ε ) with 0 < ε < ε 1 , where ε 1 is to be chosen later, smaller than ε 0 (G, γ) and ε 1 (G, γ) given respectively by Theorem 2.7 and Lemma 3.8. Thus, up to a Lebesgue-negligible set and a translation, E ε is connected and satisfies By contradiction, let us assume that E ε is not convex, so that |co(E ε )| > |B 1 |. We have . This contradicts the minimality of E ε , whence E ε is convex.
3.3. Proof of Lemma 3.8. -In order to prove that F γ,ε (co(E ε )) (strictly) decreases by scaling down co(E ε ) by t ε for small ε, we need to estimate the term P Gε (t ε co(E ε )) appearing when applying Lemma 2.3. While it is not so difficult to see that for a fixed set E with C 1 boundary, P Gε (E) converges to P (E) as ε vanishes, here we do not have uniform regularity estimates on minimizers of (P γ,ε ). The lack of such uniform regularity estimates as ε goes to 0 is the main obstacle for proving that the unit ball is, up to translations, the unique minimizer for ε small enough. With sufficient regularity, the arguments of Section 4 would yield the result. However, here we can make use of the convexity of co(E ε ) and the fact that it lies between two disks whose radii are close to 1 to prove the following. Lemma 3.9. -For any α ∈ (0, 1), there exist positive constants r = r(α) and ε 2 = ε 2 (G, γ, α) such that the following holds. If E ⊆ R 2 is a convex set such that with 0 < ε < ε 2 , and where the function δ is given by Theorem 2.7, then we have Proof. -We proceed in two steps.
-We show that the normal vector ν E (x) converges uniformly for H 1 -almost every x ∈ ∂E to ν B1 (x) as ε vanishes, in the sense that , for H 1 -almost every x ∈ ∂E.
Then we can estimate P Gε (E) when E ⊆ R 2 is a convex set lying between the disks B 1−δ(ε) and B 1+δ(ε) , for any ε small enough.
Lemma 3.10. -For any τ > 0, there exists ε 3 = ε 3 (G, γ, τ ) > 0 such that the following holds. If E ⊆ R 2 is a convex set such that with 0 < ε < ε 3 , where the δ function is the one from Theorem 2.7, then we have Proof. -Let E be as in the statement of the lemma, with 0 < ε < ε 3 (G, γ, τ ), where ε 3 is to be fixed later. Recall that since E is a bounded convex set, it is a set of finite perimeter and its topological boundary is H 1 -equivalent to ∂ * E. We proceed in two steps.
Step 1. Upper bound. -First, note that by convexity of E, we have for H 1 -almost every x ∈ ∂E, and every y ∈ E, and, defining for H 1 -almost every x ∈ ∂E, Then, letting g ε = ε −3 g(ε −1 ·), by a change of variable and the coarea formula, we find where we also used the definition of K 1,2 for the third equality, and (H.2) for the last one.
We are now in position to prove Lemma 3.8.

With (3.18) and Proposition 2.1, this leads to
Hence, for every t ∈ (1/2, 1), by our choice of τ , it follows This proves the lemma.

3.4.
Convex minimizers are nearly spherical sets. -In arbitrary dimension, it is classical to improve Hausdorff convergence of the boundary of minimizers to Lipschitz convergence once convexity is established. Here, we show that convex minimizers of (P γ,ε ) are centered t(ε)-nearly spherical sets (see Definition 1), up to a translation, where the function t vanishes in 0 + .
Proposition 3.11 (Convex minimizers are nearly spherical sets). -There exists ε 4 = ε 4 (n, G, γ) > 0 such that the following holds. If E ε ⊆ R n is a convex minimizer of (P γ,ε ) with 0 < ε < ε 4 , then, up to a translation, we have is the function of Theorem 2.7 vanishing in 0 + , and C , C > 0 only depend on n.
Proof. -In the proof, we write · ∞ for · L ∞ (S n−1 ) . Let E ε be a convex minimizer (P γ,ε ) with 0 < ε < ε 4 , where ε 4 is to be fixed later. If ε 4 < ε 0 , where ε 0 is given by Theorem 2.7, up to a translation and a negligible set, E ε lies between the balls B 1−δ(ε) and B 1+δ(ε) . By convexity, this implies that the set E ε itself (without the addition or the subtraction of a negligible set) satisfies, up to a translation, provided that ε 4 is small enough depending only on n, G and γ. Hence, up to translating E ε by y ε , we may assume that it satisfies with C := C(n) + 1 and is centered, that is, J.É.P. -M., 2022, tome 9 By convexity of E ε , for every x ∈ S n−1 , there is a unique point of intersection p ε (x) = t ε (x)x of {tx : t > 0} and ∂E ε , and by (3.19), |p ε (x) − x| C δ(ε). The map p ε : and u ε ∞ C δ(ε). In addition, the fact that E ε is convex implies u ε ∈ Lip(S n−1 ). Moreover, for any x ∈ S n−1 , the distance between the normal vector of ∂E ε (which exists for H n−1 -almost every x ∈ ∂E ε ) at p ε (x) and the vector x (which is the normal vector of S n−1 at x) is controlled by p ε − Id ∞ = u ε ∞ , in view of Step 1 of the proof of Lemma 3.9 (or simply by [34,Cor. 1]), which gives a control of ∇ τ u ε ∞ by u ∞ . More precisely, by [19, Ineq. ( * * )], we have where we used the inequality u ε ∞ C δ(ε) 1/2 for the last inequality, provided that ε 4 (n, G, γ) is chosen small, and defined C := 6 √ C . This concludes the proof.
Observe that Theorem 2 is an immediate corollary of Theorem 1 and Proposition 3.11.
In the end, ε 1 and ε 2 can be chosen to be

Minimality of the unit ball among nearly spherical sets
This section is devoted to the proof of the minimality of the unit ball of R n among t-nearly spherical sets, for small t and ε.

4.1.
A Fuglede-type result for the nonlocal perimeter. -For a centered nearly spherical set E t , we wish to estimate the quantity F γ,ε (E t ) − F γ,ε (B 1 ) from below. Thus, we need an estimate of P (E t ) − P (B 1 ) from below, and an estimate of P ε (E t ) − P (B 1 ) from above. For the former we use the following lower bound.
Lemma 4.1. -There exist positive constants t = t(n) and C = C(n) such that the following holds. For any centered t-nearly spherical set E t with 0 < t < t described by the function u, we have This is an improvement of the original lower bound of Fuglede [19,Th. 1.2]. It can be found in [20, Proof of Th. 3.1]. We establish a similar Taylor expansion for the nonlocal perimeter P Gε . Let us point out that similarly to fractional perimeters, fractional Sobolev-type seminorms associated with the kernel G ε (and its derivatives) naturally appear in the expansion. However, these converge to the H 1 seminorm as ε vanishes, so we choose here to control P Gε (E t ) − P Gε (B 1 ) directly in terms of the H 1 seminorm of u on S n−1 . There exist positive constants ε 5 = ε 5 (n, G) and t 0 = t 0 (n) such that the following holds. If E t is a centered t-nearly spherical set with 0 < t < t 0 described by the function u, then for any ε such that 0 < ε < ε 5 , we have where C = C(n, G), q η is given by Lemma 2.4, and q η (ε) → 0 as ε → 0.
When ε vanishes, the quadratic terms from the Taylor expansion of P Gε (E t ) − P Gε (B 1 ) compensate exactly those of P (B 1 ) − P (E t ), so that the constant γ must be smaller than one, and the expansion needs to be pushed to the third order to obtain a useful estimate.
Proof. -In the proof, unless stated otherwise, C denotes a positive constant depending only on n and possibly changing from line to line. For the sake of brevity we write B for the open unit ball in R n , ∂B for the unit sphere S n−1 , and · ∞ for · L ∞ (S n−1 ) . Let 0 < ε < ε 5 , where ε 5 = ε 5 (n, G) is to be fixed later.
Let t 0 = t 0 (n) < 1/8 to be fixed later, and let E t be a centered t-nearly spherical set such that ∂E t = (1 + tu(x))x : x ∈ S n−1 with 0 < t < t 0 . We proceed in three steps.
Step 1. -We rewrite P Gε (E t ) in a more convenient form, introducing two terms that we will control from above in the next steps. Using polar coordinates, we have By symmetry of G, we see that Changing variables, we deduce This concludes Step 1.
Step 2. Estimation of ϕ ε (t). -Note that by Lemma 2.4 and Remark 2.5, we have Then let us write: On the one hand, since u ∞ 1, on the domain of integration we have Thus where η ε is defined as in Lemma 2.4 so that g ε (r) = η ε (r)/(2r 2 ). In view of Lemma 2.4 and Remark 2.5 again, it follows (4.5) for any ε small enough (depending only on n and G).
Let us now bound the term I 2 (t). Integrating on a line (recall that g is absolutely continuous in (0, +∞) by (H.3)), and using the inequality (1 + ta) n−1 (1 + tb) n−1 2 2(n−1) for any a, b such that 0 |a|, |b| u ∞ 1, since t < t 0 < 1, we find Observe that in (4.6), if the term (x−y) inside ∇G ε were not perturbed by st(ax−by), using the inequality on the domain of integration, we would have which we could estimate in terms of ∇ τ u 2 L 2 (S n−1 ) by applying directly Lemma 2.4 with t → t|g (t)| in place of g. Here, to deal with this perturbation, we apply the technical Lemma A.2, by showing that the right-hand side of (4.6) is bounded by a term of the form is a small perturbation of (x − y), and k ε is a kernel defined further below. Let us remark that, since G is radial, we have ∇G ε (x) = g ε (|x|)x/|x|, thus Then notice that, on the domain of integration, we have (4.10) since t < t 0 < 1/4. By (4.7), (4.10), and the fact that t < t 0 < 1/8, on the domain of integration we deduce Hence, with (4.9), it follows where we have set k(r) := r|g (r)| and k ε (r) := ε −(n+1) k(ε −1 r) for all r > 0 and ε > 0. For x, y ∈ ∂B fixed, making the changes of variables a = u(y) + r(u(x) − u(y)), and b = u(x) + ρ(u(y) − u(x)) in (4.6) yields, with (4.8) and (4.11), dr dρ dH n−1 x dH n−1 y ds.
-By Fubini's theorem, we have Notice that for any rotation R ∈ SO(n), |rx − ρ(R −1 y)| = |r(Rx) − y|, so that, for any σ ∈ ∂B, making the change of variable y = R −1 y with R a rotation mapping x to σ, by symmetry of G, we find Hence, the above integral does not depend on x ∈ ∂B, and by averaging over ∂B, we obtain ∂B G ε (rx − ρy) dH n−1 Inserting this into (4.13) and using Fubini's theorem once again yields Then, by Proposition 2.1, we deduce Since t < t 0 < 1/8 and u ∞ 1, we have thus, by (A.1) of Lemma A.1, from (4.14) it follows This concludes Step 3.

4.2.
Minimality of the unit ball. -In order to take advantage of Lemma 4.2, one should have that for a centered t-nearly spherical set E t such that ∂E t = (1 + tu(x))x : x ∈ S n−1 , the quantity from (4.1) controls u 2 H 1 (S n−1 ) for small t and ε. This is the purpose of Lemma A.1 in appendix.
With Lemma 4.2 and Fuglede's result for the local perimeter, we deduce a lower bound for F γ,Gε (E t ) − F γ,Gε (B 1 ).
Then there exist positive constants t * and ε 3 depending only on n, G and γ such that the following holds. If E t is a centered t-nearly spherical set such that ∂E t = (1 + tu(x))x : x ∈ S n−1 with 0 < t < t * , then for every ε such that 0 < ε < ε 3 , we have Proof. -Assume 0 < ε < ε 3 , and 0 < t < t * , where t * = t * (n) and ε 3 = ε 3 (n, G, γ) will be fixed later. If E t is a centered t-nearly spherical set and t * < t 1 as well, where t 1 = t 1 (n) is given by Lemma A.1, we have in particular Then, choosing t * < t where t = t(n) is given by Lemma 4.1, we have for some positive constant C = C(n). Then, assuming that t * < t 0 (n) as well, and ε < ε 5 (n, G), where t 0 and ε 5 are given by Lemma 4.2, we find for some positive constant C = C (n, G), with q η (ε) → 0 as ε → 0. From (4. 16) and (4.17), it follows Eventually, choosing ε 3 , t * small enough depending only on n, G and γ, for any ε and t such that 0 < ε < ε 3 and 0 < t < t * , we have which proves the result.

Appendix. Additional computations for Section 4
In the following lemma, we establish some general inequalities on functions u : S n−1 → R describing centered nearly-spherical sets. For this, we need to recall a few basic facts and notation on spherical harmonics. For k 0, we denote by S k the subspace of spherical harmonics of degree k (i.e., restrictions to S n−1 of polynomials of degree k in R n ), which is a finite-dimensional vector space of degree d(k). Let (Y i k ) i∈{1,...,d(k)} be an orthonormal basis of S k for the standard scalar product of L 2 (S n−1 ). When there can be no confusion, we write Y k for a generic vector of the basis of S k . It is well-known that the family (Y i k ) i∈{1,...,d(k)} k∈N is a Hilbert basis of L 2 (S n−1 ) which diagonalizes the Laplace-Beltrami operator on the sphere, and the eigenvalue associated with Y i k is l k := l k = k(k + n − 2), for all i ∈ {1, . . . , d(k)}. We recall that d(0) = 1, d(1) = n, and that the Y i 1 may be chosen colinear to Lemma A.1. -There exist positive constants t 1 = t 1 (n) and C = C(n) such that the following holds. If E t is a centered t-nearly spherical set such that then we have Thus, writing we deduce (A.1), choosing e.g. t 1 = 1 and C depending only on n. There remains to show (A.2). We decompose u in spherical harmonics Recall that d(0) = 1, Y 1 0 is constant, d(1) = n, and that Y i 1 is colinear to x → x i . Since 1 = n − 1 and k 2n for k 2, it follows On one hand, so that by (A.1), we have up to choosing t 1 = t 1 (n) small enough, and using t < t 1 and u L ∞ (S n−1 ) 1. On the other hand, the barycenter condition Using the binomial formula again and Y i 1 (x) = |B 1 | −1/2 x i , we obtain, for i ∈ {1, . . . , n}, Next, using u L ∞ (S n−1 ) 1, Cauchy-Schwarz inequality, and n|B 1 | = |S n−1 |, we get Whence, choosing t 1 even smaller, but still depending only on n, we may assume Gathering (A.4) to (A.6), we find which proves (A.2) and concludes the proof. We establish a technical lemma to control terms of the form by the H 1 (S n−1 ) norm of u, where Φ tu (x, y, r, ρ) is a small perturbation of (x − y), and k ε are suitable rescalings of a nonnegative kernel.