On the Shape of Small Liquid Drops Minimizing Nonlocal Energies

We study the equilibrium shape of liquid drops minimizing the fractional perimeter under the action of a potential energy. We prove, with a quantitative estimate, that the small volume minimizers are convex and uniformly close to a ball.


Introduction
Given s ∈ (0, 1) and m > 0, we study the following minimization problem: min E s,g (E) := P s (E) + where P s (E) is the s-fractional perimeter of a measurable set E ⊂ R N defined as and E g(x) dx is the potential energy of E associated with a locally Lipschitz and coercive function g : R N → R. The notion of s-fractional perimeter was introduced by Caffarelli, Roquejoffre, and Savin in [4], who were motivated by the classical phase field model when long-range correlations exist.After their celebrated work, variational problems involving the s-fractional perimeter have been studied by many authors as an analogy of the classical ones and our paper is in this line of research.
Our research is motivated by the classical minimization problem for the equilibrium shape of liquid drops and crystals.The classical problem can be formulated as follows: min E(E) := where f : R N → [0, ∞) is a convex, positively 1-homogeneous function, ν E is the measuretheoretic unit normal to E, and ∂ * E is the reduced boundary (see the definition of ν E and ∂ * E in [18]).Notice that, if f is the Euclidean norm, then the first term in E is exactly the classical perimeter of a set E ⊂ R N in the sense of De Giorgi.Almgren originally proposed the following question on the shape of minimizers for Problem (1.3).This question is mentioned in [19].
Question: If the potential g is convex, then is it true that the minimizer for Problem (1.3) is convex (at least connected) and unique?The geometric properties of minimizers for liquid drop and crystal models associated with Problem (1.3) have been studied for decades by many authors, for instance, Almgren, Avron, Baer, De Philippis, Figalli, Finn, Goldman, Gonzalez, Maggi, Massari, McCann, Tamanini, Taylor, Zia, and so on [11,13,1,12,10,2,7].Partial answers to Question are given by some of these authors.
In particular, Figalli and Maggi in [10] extensively studied the shape of minimizers for Problem (1.3) in the situation where the contribution coming from the potential term E → E g(x) dx can be negligible compared with the perimeter term associated with f .They proved that the minimizers of E are uniformly close to a Wulff shape when the volume is sufficiently small.In addition, if N = 2, they showed that the minimizers are convex and, if N ≥ 3, assuming a stronger regularity on the 1-homogeneous function f and the potential g of the energy E, they also proved the convexity of minimizers with a quantitative estimate for the second fundamental form on the boundaries of minimizers.We remark that De Philippis and Goldman in [7] proved that, if the function f is smooth and uniformly elliptic, and the potential g is convex and coercive, then any minimizer for Problem (1.3) without volume constraint is convex in any dimension.Moreover, under the same assumptions on f and g as above, they showed that, in dimension 2, any minimizer for Problem (1.3) (with volume constraint in this case) is convex and unique.On the other hand, Indrei in [14] proved the non-existence of minimizers for Problem (1.3) in dimension 2 if the potential g is merely convex and satisfies g(0) = 0.Moreover, Indrei and Karakhanyan in [15] recently showed the sharp quantitative stability for Problem (1.3) if the function f = 1 on S N −1 and the potential g is radial and non-decreasing and g(0) = 0, which implies that the unique minimizer for Problem (1.3) is a ball.
To the best of our knowledge, the shape of minimizers for the nonlocal version of Problem (1.3), that is, Problem (1.1) is not well-understood and there are few references on the problem.For instance, Cesaroni and Novaga in [6] proved the existence and regularity of minimizers for Problem (1.1) in the case that the potential function is either coercive or Z N -periodic.Our goal in this paper is to study the shape of minimizers for Problem (1.1) and to establish a nonlocal version of some of the results shown in [10].Precisely, we prove Theorem 1.1.Let s ∈ (0, 1).Assume that the potential g satisfies (H1) and (H2) (see Section 2 for the assumptions).Then, there exist constants m 0 = m 0 (N, s, g) > 0 and R 0 = R 0 (N, g) > 0 such that, for any m ∈ (0, m 0 ), every minimizer E m of E s,g with |E m | = m satisfies the following: there exists a point for some r 0 ∈ (0, Cm s 2 /(2N 2 ) ) and some constant C > 0 depending only on N, s, and g, where we set Moreover, we have that ∂E m is of class C 2,α with α < s and ∂E m converges to ∂B 1 in C 2 as m ↓ 0, by proper scaling and translation.In particular, E m is convex for small m ∈ (0, m 0 ).
From a geometric point of view, the existence of minimizers for Problem (1.3) is related to the existence of a smooth hypersurface M on which the following prescribed mean curvature equation holds: where then we can obtain the following Euler-Lagrange equation: As a nonlocal analogy of the classical problem, the existence of minimizers of our energy E s,g is related to the problem of finding a set E with a smooth boundary such that the following prescribed "fractional" mean curvature equation holds on ∂E: Here H s,∂E (x) is the s-fractional mean curvature on ∂E at x, which is defined as where "P.V." means the Cauchy principle value.Indeed, assuming that E m is a minimizer of E s,g among sets with volume m > 0 and ∂E m is smooth, we can show that the following Euler-Lagrange equation holds true: where λ m is a Lagrange multiplier.As we mentioned above, the authors in [6] proved the existence of bounded minimizers with smooth boundary for Problem (1.1) and this implies the existence of compact hypersurfaces M ⊂ R N on which the equation (1.8) holds true for any m > 0. In this context, our main theorem may imply the existence of a compact, smooth hypersurface M m ⊂ R N with non-negative mean curvature such that the equation (1.8) holds on M m for small m > 0 and M m , by properly rescaling, converges to the sphere in R N as m ↓ 0. The proof of Theorem 1.1 is divided in several steps.The former part of Theorem 1.1 (the uniform closeness of minimizers to the Euclidean ball) will be proved in Section 3 and the latter part of Theorem 1.1 (the regularity and convexity of minimizers) will be proved in Section 5.
The organization of our paper is as follows: in Section 2 we fix some notation and we introduce the mathematical setting of our problem.
In Section 3 we first establish properties of minimizers of (2.1) and consider a suitable rescaling for them.Then, we prove the uniform closeness of the rescaled minimizers to the unit Euclidean ball for small volumes in Theorem 3.7.The strategy follows some arguments in [6,8,10].Precisely, we derive suitable integro-differential inequalities for estimating the volume of the difference between rescaled minimizers and balls centred in the origin.A similar argument can be found in [6,Proposition 3.2], where the authors, applying a nonlocal version of the so-called Almgren's Lemma (see [10,Lemma 3]), prove the boundedness of minimizers of (2.1) for a fixed value of the volume.The first main difference with the approach in [6] is that our estimates not only aim at obtaining boundedness of minimizer, but also at closeness in L ∞ to the unit ball (for small volumes).The second one is that our estimates are uniform with respect to the volume of minimizers.To obtain the uniformity with respect to the volume, we adapt ideas in [10] coming from the classical case.In particular, we do not employ a nonlocal Almgren's Lemma such as [6, Lemma 3.1], but we made use of a suitable dilation.Indeed, with our approach, the constants involved in our estimates depend only on the volume of the minimizers rather than on the minimizers themselves.
In Section 4, with the uniform proximity result in our hands, we can reformulate Problem (2.1) into a variational unconstrained problem, which becomes crucial for establishing regularity.
In Section 5, we obtain the regularity and convexity of minimizers for Problem (1.1), employing several results on the regularity of the so-called "almost minimizers" or "Λminimizers", where Λ is independent of minimizers, of the s-fractional perimeter and the bootstrap argument for integro-differential equations shown in [3].As a consequence, by applying the regularity result for the "Λ-minimizers" shown in [9,Corollary 3.6] and from the uniform closeness of the minimizers to balls that we prove in Section 3, we obtain the convergence of the minimizers to the Euclidean ball in C 2 -topology, which implies in particular the convexity of minimizers.

Notation and Setting of the Problem
We denote the Euclidean norm of x ∈ R N by |x|, the Euclidean ball of radius r > 0 and centre x ∈ R N by B r (x) and the Euclidean ball of volume m > 0 and centre x ∈ R N by B m (x).We also set B r := B r (0) and We denote the Lebesgue measure of R N by L N and the class of all Lebesgue measurable sets of R N by M N .For E ∈ M N we also set |E| := L N (E) and we denote the topological boundary of E ∈ M N by ∂E.H N −1 denotes the (N − 1)−dimensional Hausdorff measure of R N .
Let E m be a minimizer of the constrained problem (1.1) with volume m, that is Moreover, we recall the definition of (1.4) for every m > 0, so that B m = B σ(m) (the ball of volume m is equal to the ball of radius σ(m)).
Without loss of generality (thanks to the structure of the volume constrained problem in (2.1)) in the whole work we assume inf R N g = g(0) = 0.
(2.2) Indeed, fixed any x ∈ R N such that g(x) = inf R N g, by replacing the potential g with ĝ, defined by ĝ(x) := g(x + x) − inf R N g, on the one hand we observe that ĝ satisfies (H1), (H2) and (2.1) on the other one we note that the minimizers of E s,ĝ with volume m are the same minimizers of E s,g with volume m (upon a translation via x).
Remark 2.1.We observe that E m is well-defined, since the problem (1.1) admits solution if the potential g is measurable, bounded from below and satisfies assumption (H1).This can be proved via the Direct Method as done in [6,Proposition 5.3].
As far as uniqueness is concerned, to the best of our knowledge, there are no complete results available in the literature.Even in the classical case (see Problem (1.3)) a general answer has not been provided yet, even though there are some partial results.Precisely, from [7,Remark 1.6] it follows that for large volumes, minimizers of (1.3) are unique when the potential g is convex and coercive.
We also mention that in [19,Theorem 1.1] uniqueness (up to translations) is established in 2D, for any volume but among convex competitors, when the potential g is convex and the set where it vanishes is bounded but non-empty.
For every m > 0, we let be the family of all the minimizers of E s,g with volume constraint |E i m | = m.We observe that S m = ∅ because there always exists a minimizer of E s,g for every m > 0. In the sequel, by a little abuse of notation, we will drop the dependence on i ∈ I m in E i m when it is not essential to highlight it (and also in other quantities possibly depending on i ∈ I m ).

Uniform
Furthermore, x attains the infimum in the definition of A(E) if and only if x + x attains the infimum in the definition of A(λE + x).
We are now ready to establish properties of minimizers of (1.1).Proposition 3.4 (Properties of Minimizers).Let s ∈ (0, 1) and assume that g satisfies (H1), (H2) and (2.2).Let S m := {E i m } i∈Im be as in (2.3) and σ(m) be as in (1.4) for every m > 0.Then, there exists C(N, s) > 0 s.t. for every m > 0 and i ∈ I m A for some Furthermore, if we let for every m > 0.
Proof.This proof follows the strategy adopted in part of [10,Theorem 9] combined with [6,Proposition 3.2].Let m > 0 and i ∈ I m .By the non-negativity of g (recall (2.2)) and the minimality of Then, ) is actually a minimum.This observation joint with Theorem 3.3 and (3.1) implies (3.2) for some x i m ∈ R N .Thanks to the local boundedness of g (see (H2)), from (3.2) we can find a constant m 0 (N, s, g) > 0 such that for all m ≤ m 0 and i ∈ I m ; which (since for all m ≤ m 0 and i ∈ I m .Since (H1) holds, we observe that {g = 0} is contained in a compact subset of R N .Then, (3.4) implies (3.3) for some R 0 = R 0 (N, g) > 0 (possibly choosing m 0 = m 0 (N, s, g) > 0 smaller).
We now claim that (3.4) holds.
We define d m := sup i∈Im dist(x i m , {g = 0}) for every m ∈ (0, m 0 ] and we consider the set We first observe that the fact that σ(m) converges to zero as m → 0 implies, whenever 0 is an accumulation point for (0,

by definition of d m and the triangular inequality).
We now observe that by the fractional isoperimetric inequality and (3.8) it immediately follows Then, by (3.9), (3.11) and the non-negativity of g (see (2.2)): g, Therefore, since g is continuous and g(0) = 0 (see (2.2) and (H2)): whenever 0 is an accumulation point for I m 0 .From (3.12) and the coercivity of g (see (H1)) and the non-negativity of g (see (2.2)) we then deduce that lim m→0 m∈Im 0 whenever 0 is an accumulation point for I m 0 .Therefore, (3.4) is proved by combining (3.10) and (3.13). Since In order to prove the uniform closeness of the rescaled minimizers E m to the Euclidean ball for small volumes (see Theorem 3.7), we need two technical lemmas.

+∞) be absolutely continuous and monotone non-decreasing
such that f (0) = 0.Then, if the integral in (3.15) is finite.
Proof.We prove only (i) since the proof of (ii) is analogous.By Fubini's Theorem: We also used that the function (b, +∞) ∋ t → (t − a) We prove (i).
We highlight that in this proof the constant c in (3.16) cannot vary from line to line: it has to be considered fixed.Now we argue in a similar fashion as in [8,Lemma 4.1].We define the auxiliary function h : [a, +∞) → R as for every ̺ ≥ a.Our goal is now to prove that We define Letting n go to +∞, we obtain h(R * ) ≥ lim .The proof of (i) is complete.The proof of (ii) is similar to the proof of (i) and thus is left to the reader.
Proof.The proof is in the spirit of the one of [10, Theorem 5].
We choose m 0 = m 0 (N, s, g) smaller than the constant m 0 in the statement of Proposition 3.4 so that σ(m 0 ) < 1, (3.22) and we choose R 0 = R 0 (N, g) and x m as in the statement of Proposition 3.4 (so that |x m | ≤ R 0 whenever m ∈ (0, m 0 ), as in Proposition 3.4).
For convenience, we put E := E m and we change our focus from the variable m to ε := σ(m) s .Therefore, applying Proposition 3.4, we deduce that there exists ε 0 (N, s, g) > 0 such that 0 < ε ≤ ε 0 (N, s, g) and We claim (possibly choosing a smaller ε 0 (N, s, g) or, equivalently, a smaller m 0 (N, s, g)) that there exists a constant C(N, s, g) such that for some r 0 = r 0 (ε, N, s, g) ≤ C(N, s, g)ε s/N .From (3.23) (in alternative we can simply use the minimality, testing E against B 1 ) we deduce (3.24) STEP 1: proof of the inclusion E ⊂ B 1+r 0 .
For every r > 0 we define the non-negative and monotone non-increasing absolutely continuous function u(r) := |E \ B r |.Thanks to the monotonicity and the hypotheses we observe that for every r ≥ 1 u(r) ≤ u(1) ≤ C(N, s, g)ε. (3.25) If the set {r > 1 : u(r) > 0} is empty, there is nothing to prove, so we assume it is not.We let We consider the following competitor (with volume equal to |E| = |B 1 |) for the volume constrained problem, After a computation, thanks to (3.25), we observe that λ is arbitrarily close to 1 (upon choosing ε 0 (N, s, g) small enough).In particular (upon choosing ε 0 (N, s, g) small enough) we get 0 < λ − 1 ≤ C(N)u(r) (3.28) and, since r < 2, F ⊆ B λr ⊆ B 3 .At this point we make use of the minimality of E (see (3.7)) to write Now, we recall [8, Lemma 2.1] and making use of the isoperimetric inequality for P s and the fact that E\Br E∩Br We then estimate (3.30) using the fact that g is locally Lipschitz (see ((H2))) and nonnegative (see (2.2)) together with (3.25), (3.28), (3.22) and the fact that r < 2 (possibly choosing a smaller ε 0 (N, s, g)): Now we deal with (3.31).Arguing exactly as in [6, equation ( 15)], we find E\Br E∩Br Gathering all the previous inequalities together, we find that for every 1 < r < min{2, r 1 }, Now, recalling (3.25), upon choosing ε 0 (N, s, g) smaller, we can rewrite (3.32) as Now we integrate both sides of (3.33) with respect to the variable r and apply Lemma 3.5 part (i) to obtain that for some c = c(N, s, g) > 0 for every 1 < ̺ < min{2, r 1 }.Recalling (3.25), upon choosing C(N, s, g) larger and ε 0 (N, s, g) smaller, we can assume Without loss of generality we can also assume that Thanks to the previous step, upon choosing m 0 = m 0 (N, s, g), we can assume (3.38) For every r > 0 we define the non-negative and monotone non-decreasing absolutely continuous function w(r) := |B r \ E|.Thanks to the monotonicity and the hypotheses we observe that for every 0 ≤ r ≤ 1 w(r) ≤ w(1) ≤ C(N, s, g)ε. (3.39) We observe that 0 ∈ {r ∈ [0, 1) : w(r) = 0} and we let Our goal is to prove that 1 − r 2 ≤ C(N, s, g)ε s/N .(3.41) Without loss of generality we suppose that r 2 < 1, otherwise (3.41) trivialises itself, and we fix r > 0 such that r 2 < r < 1.By definition of r 2 we have that We consider the following competitor (with volume equal to After a computation, thanks to (3.39), we observe that µ is arbitrarily close to 1 (upon choosing ε 0 (N, s, g) small enough).In particular (upon choosing ε 0 (N, s, g) small enough) we get 0 < 1 − µ ≤ C(N)w(r) (3.42) and, since r < 1, F ⊆ B µr ⊆ B 1 .At this point we make use of the minimality of E (see (3.7)) to write Now, we recall [8, Lemma 2.1] and making use of the isoperimetric inequality for P s , the invariance under complementation of P s and the fact that We first observe that (3.43) is negative.We then estimate (3.44) using the fact that g is locally Lipschitz (see ((H2))) and non-negative (see (2.2)) together with (3.38), (3.42), (3.22) and the fact that r < 2: Now we deal with (3.45).Following th scheme of the proof of [6, equation ( 15)], we write Gathering all the previous inequalities together, we find that for every r s.t.r 2 < r < 1,

C(N, s, g)(w(r))
(3.47) Now we integrate both sides of (3.47) with respect to the variable r and apply Lemma 3.5 part (ii) obtain that for some c = c(N, s, g) > 0 2c

Unconstrained Minimization Problem
Now we show that our minimization problem (1.1) is equivalent to the rescaled and unconstrained problem min E µ s,gm (E) : for µ > 0 sufficiently large.Here we recall that g m is defined by (3.6), that is, for x ∈ R N where σ(m) is defined by (1.4) and x m is given in Proposition 3.4.for every µ ≥ µ 0 , where g m is defined by (3.6).
Proof.We revisit the proof of [6,Lemma 3.4] making use of the extra information given by Theorem 3.7 and tracking carefully all the constants.In this proof c = c(N, s, g) > 0, c ′ = c ′ (N, s, g) > 0 are fixed constants defined by, where R 0 = R 0 (N, s, g) is as in the statement of Proposition 3.4. Let for all λ > 0 with |λ − 1| ≤ ε 1 . Let We fix m 0 = m 0 (N, s, g) ∈ (0, 1) as in the statement of Theorem 3.7 (or smaller) and m such that 0 < m ≤ m 0 (N, s, g).
We fix µ ≥ µ 0 and we assume by contradiction that ∃G µ ⊆ B 3 of finite s-perimeter such that Then, since E m solves (3.7), Which, implies In particular, The combination between (4.3) and (4.4) leads to: Applying exactly the same argument in the proof of [6,Lemma 3.4], we obtain: and By (4.2), (4.5), (4.6), (4.7) we deduce that that is the desired contradiction.

Regularity and Convexity of Minimizers
In this section, our goal is to show the following lemma: Lemma 5.1.Let s ∈ (0, 1) and assume that g satisfies ( H1), ( H2) and (2.2).For every m > 0 let E m be a minimizer of E s,g with |E m | = m and let E m be as in (3.5).Then, there exists m 0 > 0, which depends only on N, s and g, such that, for any m ∈ (0, m 0 ) the boundary of We remark that Lemma 5.1 actually implies that the latter part of Theorem 1.1 is valid.Therefore, combining Lemma 5.1 with Theorem 3.7, we conclude the proof of our main theorem, that is, Theorem 1.1.
Note that the similar concept of the almost minimality for P s , which is more general, was also given by Caputo and Guillen [5].See [5] for the details.
Next we recall the regularity result of fractional minimal cones in R 2 by Savin and Valdinoci [20].

Theorem 5.4 ([20]).
Assume that E ⊂ R 2 is an s-fractional minimal cone, namely, E is a minimizer of P s in any ball and satisfies that E = t E for any t > 0. Then E is a half-plane.
In particular, by combining the blow-up and blow-down arguments in [4], one may obtain that s-fractional minimal surfaces in R 2 are fully C 1,α -regular for any α ∈ (0, s).

Corollary 5.5 ([20]). If E is an s-fractional minimal set in an open
Originally, the regularity of nonlocal (fractional) minimal surfaces, which are defined by the boundaries of sets minimizing the fractional perimeter, was obtained by Caffarelli, Roquejoffre, and Savin [4].Precisely, they proved that every fractional minimal surface is locally C 1,α outside of a closed singular set of Hausdorff dimension at most N − 2.Moreover, thanks to Corollary 5.5, it follows that the singular set of fractional minimal surfaces has Hausdorff dimension at most N − 3. Now we prove that the rescaled minimizers for our problem are (Λ, δ)−minimizers of P s where Λ is independent of the minimizers.Lemma 5.6 ((Λ, δ)-Minimality or Almost Minimality).Let s ∈ (0, 1) and δ ∈ (0, 1).Assume that g satisfies (H1), (H2) and (2.2).For every m > 0, let E m be a minimizer of E s,g with |E m | = m and let E m be as in (3.5).Then there exist m 0 = m 0 (N, s, g) > 0 and Λ = Λ(N, s, g) such that, for every m ∈ (0, m 0 ), E m is a (Λ, δ)−minimizer of P s in the sense of Definition 5.2.
Proof.We fix δ ∈ (0, 1).From Proposition 4.1, we know that E m is also a solution to the problem (4.1), for some large constant µ > 0 independent of m and E m ⊂ B 3/2 .Now, letting F ⊂ R N , x ∈ ∂ E m , and r ∈ (0, δ) with E m △ F ⊂ B r (x), from the minimality of E m , we have that ) since F ⊂ B 3 .Hence, from (5.3), we can compute as follows: Moreover, from the fact that Therefore, by Proposition 3.4 and the definition of g m , we have where R > 0 is a constant independent of m.Hence, from (5.4) and (5.5), we obtain where m 0 > 0 is as in Proposition 4.1.This implies that E m is an almost minimizer or (Λ, δ)-minimizer of P s in the sense of (5.1) with Λ := σ(m 0 ) s g L ∞ (B R (0)) + µ < ∞, which is independent of minimizers.
As a consequence of Lemma 5.6 and the aforementioned regularity results for the almost minimizers of the s-fractional perimeter, we obtain Lemma 5.7 (C 1,α -regularity for Minimizers of E s,gm ).Let s ∈ (0, 1) and assume that g satisfies (H1), (H2) and (2.2).For every m > 0 let E m be a minimizer of E s,g with |E m | = m and let E m be as in (3.5).There exists Proof.The idea is to apply Theorem 5.3 to our problem.To this aim, it is sufficient to show that every minimizer E m is indeed a (Λ, δ)−minimizer in the sense of (5.1) for some Λ > 0 depending on N, s and g only, which is guaranteed by Lemma 5.6.
Therefore, from Theorem 5.3 (see also [9,Corollary 3.5]) and the regularity of minimal cones in [20], we obtain that the boundary of E m is of class C 1,α for some α ∈ (0, 1) outside of a closed set of Hausdorff dimension at most N − 3.
From the uniform closeness of minimizers to the Euclidean ball stated in Theorem 3.7, we can immediately obtain the Hausdorff convergence of the minimizers to the unit ball.
Then, with the Hausdorff convergence of the minimizers at our disposal, we can exploit the regularity criterion (5.2) and the smoothness of the limit set B 1 via the argument in [18,Theorem 26.6] in order to obtain the following result (see also [9, Theorem 3.3 and Corollary 3.6]).Lemma 5.8.Let s ∈ (0, 1).Assume that g satisfies (H1), (H2) and (2.2).For any m > 0, let E m be a minimizer of E s,g with |E m | = m and let E m be as in (3.5).Then, there exist a constant m 0 > 0 and a bounded sequence of functions {ϕ m } m∈(0, m 0 ) ⊂ C 1,α (∂B 1 ) with some α ∈ (0, 1) (independent of m) such that, for any m ∈ (0, m 0 ), (5.6) Proof.First of all, we recall that, from Theorem 3.7, there exist positive constants m 0 = m 0 (N, s, g) and C = C(N, s, g) such that, if m ∈ (0, m 0 ), E m satisfies the inclusion for some r 0 ∈ (0, Cm s 2 /(2N 2 ) ).Thus, (5.7) implies that E m converges to B 1 (0) as m ↓ 0 in L 1 -sense and moreover, by definition, ∂ E m converges to ∂B 1 (0) in the Hausdorff distance.Now, as we observe in Lemma 5.6 and Lemma 5.7, we obtain that E m is also an almost minimizer of P s and that ∂ E m is of class C 1,α with some α ∈ (0, 1) for m ∈ (0, m 0 ) outside of a closed set of Hausdorff dimension at most N − 3.Although the singular sets can appear on each ∂ E m , by combining the convergence of {∂ E m } m∈(0, m 0 ) to ∂B 1 in the Hausdorff distance with the regularity criterion (5.2), we can show that there exists a constant m ′ 0 > 0 such that, for each m ∈ (0, m ′ 0 ), there exists a function Therefore, the representation of ∂ E m in (5.6) holds.We refer to [9,Corollary 3.6] for the similar argument.Finally, we show the convergence of the functions {ϕ m } m∈(0, m ′ 0 ) in C 1 sense as in (5.6) by using the argument in [18,Theorem 26.6].Let z m ∈ ∂ E m be any point and let R ∈ (0, 1).By the Hausdorff convergence of {∂ E m } m∈(0, m ′ 0 ) , we can choose a subsequence (denoted by the same indices) such that z m → z 0 ∈ ∂B 1 (0) as m ↓ 0. We define the cylinder Then, from the regularity of each ∂ E m and ∂B 1 (0), for each m ∈ (0, m ′ 0 ), we can choose a C 1,α -function From the regularity criterion (5.2) and the same argument as in, for instance, [18,Theorem 26.3], we further obtain the Hölder continuity of the function ∇ ′ u m , that is, for any y ′ , z ′ ∈ B ′ R (z ′ 0 ) and some constant C > 0 independent of m, where ∇ ′ is the gradient in R N −1 and α ∈ (0, 1) is as in Lemma 5.7.Moreover, thanks to the convergence E m → B 1 (0) in L 1 -sense, we have Thus, we obtain for any ξ ∈ C ∞ c (B ′ R (z 0 )).From (5.8), we have that {∇ ′ u m } m∈(0, m ′ 0 ) is equi-continuous and bounded in C 0 (B ′ R ), and thus, by the Ascoli-Arzelá theorem, we can extract a subsequence (denoted by the same indices) such that ∇ ′ u m → v as m ↓ 0 uniformly in B ′ R (z ′ 0 ).From (5.9), the limit v coincides with ∇ ′ u 0 .Therefore, we obtain that the graph functions {u m } m∈(0, m ′ 0 ) of ∂ E m converge to the graph function u 0 of ∂B 1 in C 1 sense.From the choice of ϕ m , this implies that ϕ m C 1 (∂B 1 ) converges to 0. Therefore, we conclude the proof of (5.6).Now we improve the regularity of the boundary of minimizers of E s,gm by employing the regularity result for solutions to integro-differential equations via the bootstrap argument.This result was obtained by Barrios,Figalli,and Valdinoci [3,Theorem 1.6].They proved the following regularity theorem on the solutions to integro-differential equations.For simplicity, we do not describe all of the theorem here.See [3,Theorem 1.6] for the full statement.
Proof.We may assume that 0 ∈ ∂ E m by translation.By Lemma 5.8, we can represent ∂ E m ∩ (B ′ r (0) × (−r, r)) ⊂ R N −1 × R for r > 0 as a graph of a C 1,α -function u for some α ∈ (0, 1) by choosing the proper coordinate.From the fact that E m is a minimizer of E s,gm with volume constraint and by employing the computation shown in [3], we may obtain the following Euler-Lagrange equation in the viscosity sense: r, where L r satisfies (A1) and (A2) in Theorem 5.9, spt L r (x ′ , •) ⊂ B ′ r ′ /2 (0) for any x ′ ∈ B ′ r ′ /2 (0), G is some smooth function (see [3,Section 3] for the precise expression), and λ m is a Lagrange multiplier.Then, since the potential g is locally Lipschitz, we now apply Theorem 5.9 several times, if necessary, to conclude that the regularity of u can be improved up to C 2,β with β ∈ (0, s).From the compactness of ∂ E m and by a simple covering argument, we obtain the C 2,β -regularity of ∂ E m for any β ∈ (0, s) with the following estimate , where C > 0 is a constant depending only on N, s, and g C 0,1 (B 2R 0 ) and R 0 is as in Proposition 3.4.
Finally, we are ready to prove Lemma 5.1, that is, the main lemma in this section.
Proof of Lemma 5.1.Taking into account all the arguments in Proposition 3.4, Theorem 3.7, Lemma 5.8, and Lemma 5.10, we obtain that there exist a constant m 0 = m 0 (N, s, g) > 0 and a sequence of functions {ϕ m } m∈(0, m 0 ) ⊂ C 2,α (∂B 1 (0)) for any α ∈ (0, s) such that where C 1 > 0 is a constant depending only on N, s, and g.Moreover, from the proof of Lemma 5.10 and (5.10), we can obtain for any m ∈ (0, m 0 ), by choosing m 0 > 0 small if necessary, where C 2 > 0 is a constant depending only on N, s, and g and λ m is a Lagrange multiplier as in Lemma 5.
where {Φ t } |t|<1 is one-parameter diffeomorphism associated with T ∈ C ∞ c (R N ; R N ), that is, Φ t (x) := x+tT (x) for any x ∈ R N and |t| < 1.Then, substituting the identity function Id : x → x with T (by using a suitable cut-off function or by approximation), we obtain, from (5.12), (5.13) From Theorem 3.7 and the assumption on g, we obtain for any m ∈ (0, m 0 ) where C = C(N, s, g) > 0 is a constant and R 0 is given in Proposition 3.4 depending only on N, s, and g.Then, from (5.13) and (5.14) and by the minimality of E m , we obtain for any m ∈ (0, m 0 ).Therefore, from (5.11) and (5.15), we obtain for any m ∈ (0, m 0 ) and some constant C 3 > 0 independent of m.
Remark 5.11.In the proof of Lemma 5.1, we have observed the convexity of minimizers for small volumes, assuming that g is locally Lipschitz, coercive, and inf R N g = g(0) = 0.
In addition, if we further assume that g is convex and radially symmetric, that is, there exists a convex function G : [0, ∞) → R such that g(x) = G(|x|) for any x ∈ R N , then we can show that the unique minimizer of E s,gm with volume |B 1 | for m > 0 is the ball up to negligible sets.Indeed, from the isoperimetric inequality of P s and the symmetric rearrangement (see, for instance, [16]), we have P s (E * ) ≤ P s (E),  12, we observe that if the function (0, +∞) ∋ m → λ m were continuous in a neighbourhood of zero, this would imply that it would be also surjective when restricted to a neighbourhood of zero (thanks to (5.19)).As a consequence, there would exist λ > 0 such that we would be able to solve a fractional version of the H-bubble problem, that is: we might find a hypersurface ∂E satisfying the equation for every λ > λ.
Regarding a sufficient condition for the continuity of the Lagrange multiplier λ m , we remark that, if for m 0 > 0 sufficiently small a set E m 0 with |E m 0 | = m 0 > 0 is the unique minimizer of E s,g among sets with volume m 0 , then the Lagrange multiplier λ m is continuous in a neighbourhood of zero( with respect to m > 0).Indeed, to prove this we first take any sequence {m i } i ⊂ (0, ∞) converging to a fixed m 0 ∈ (0, ∞).We observe, from the minimality of E m i , that sup i P s (E m i ) < ∞.Then, from the compactness of P s , the uniqueness of E m 0 , and the coercivity of g, we deduce that E m i → E m 0 in L 1 -sense.Thus, from the minimality of E m 0 , the uniform boundedness of E m i and by using the dominated convergence theorem, we have In the same way, we can also obtain that lim inf i→∞ E s,g [m i ] ≥ E s,g [m 0 ].From this, we derive the continuity of E s,g [m] at m 0 .Therefore, from the minimality of E m i and E m 0 and from (5.21), we obtain that P s is continuous at m 0 .Finally, from the assumption on g and the uniform boundedness of E m i , we can also obtain that the potential term in (5.20) is continuous at m 0 , that is,
By definition of h and thanks to (3.21), we finally observe that R * = a + (2u(a)) s N N sc

Corollary 5 . 12 .
E * g m (x) dx ≤ E g m (x) dx for any measurable set E ⊂ R N , where we denote by E * the open ball of radius equal to |E| 1/N |B 1 | −1/N centred at the origin.Under the assumptions of Lemma 5.1, for every m > 0 the following Euler-Lagrange equation holds: H s,∂Em + g = λ m on ∂E m , (5.18) where λ m is a Lagrange multiplier associated to the volume constraint such that lim m→0 λ m = +∞.(5.19) Proof.By making the same computation as in (5.13), we can show that the identity Nmλ m = (N − s)P s (E m ) + Em ∇g(x) • x dx + N Em g(x) dx (5.20) holds for m > 0 and that λ m → +∞ as m ↓ 0.

Remark 5 .
13 (H s -bubble problem).The classical H-bubble problem, which was originally raised by S.-T.Yau[21, Problem 59], is formulated as follows: given a function H : R N → R, the question is to find an immersed hypersurface M ⊂ R N such that its mean curvature at p ∈ M is equal to H(p).(1.1) is related to a nonlocal version of this geometric problem.Using the same notation as in Lemma 5.1 and Corollary 5.
of E m i , we can also obtain the continuity of P s at m 0 , that is,P s (E m i ) − −− → i→∞ P s (E m 0 ).Indeed, from the minimality of E m i , setting F m i := m i m 0 1/N E m 0 , we have E s,g [m i ] := inf{E s,g (E) | |E| = m i } ≤ E s,g (F m i )for any i ∈ N. Thus, from the minimality of E m 0 , we obtain that lim supi→∞ E s,g [m i ] ≤ E s,g [m 0 ].
Proximity to the Unit Ball and we observe that it is an interval and that * := inf I ∈ a, a + (2u(a)) n } n∈N ⊆ [a, R * ] converging to R * , such that h(R n ) ≥ u(R n ) for every n ∈ N. So,combining (3.16) and (3.19), we get 10. Now we derive the upper bound of | λ m | from the minimality of E m .Since E m is a minimizer of E s,gm with | E m | = |B 1 |, by considering the Euler-Lagrange equation with volume constraint, we have s (Φ t ( E m )) + Φt( Em)