Volume constrained minimizers of the fractional perimeter with a potential energy

We consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume inte- gral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove existence and regularity of minimizers under suitable assumptions on the potential energy, which cover the periodic case. In the small volume regime we show that minimizers are close to balls, with a quantitative estimate.


Introduction
Let s ∈ (0, 1) and let E ⊂ R N be a measurable set, the fractional perimeter P s (E) of E is defined as the squared H s/2 -seminorm of the characteristic function of E, i.e.
This notion has been introduced in [14,3] and has been widely studied in the last years (see [8,7] and references therein).
It is well known that balls are the unique minimizers of the fractional perimeter among sets with the same volume. Indeed, the following fractional isoperimetric inequality holds for sets of finite volume (see [3,8]): where B is the ball of radius 1, and equality holds if and only if E is a ball. The isoperimetric inequality (2) can also be localized in bounded sets with Lipschitz boundary (see [7,Lemma 2.5]).
In this paper we are interested in existence and properties of minimizers of the following isoperimetric problem In particular we will provide regularity properties of minimizers under the assumption that g : R N → R is locally Lipschitz continuous and bounded from above, see Corollary 3.5, whereas the existence of a solution of the isoperimetric problem is obtained for g periodic, see Theorem 5.1, or g coercive, that is see Proposition 5.3. Our main result is the following.
Theorem 1.1. Assume that g is locally Lipschitz and either coercive or Z Nperiodic. Then for any s ∈ (0, 1) and m > 0 there exists a bounded minimizer E of (3). Moreover, ∂E is of class C 2,α for any α < s outside of a closed singular set S of Hausdorff dimension at most N − 3.
Existence of such minimizers is related to the problem of finding compact solutions to the geometric equation where H s denotes the s-mean curvature at a point x ∈ ∂E (see [3,1]) , that is, Indeed if E is a critical point of the functional and ∂E is of class C 1,α for some α > s, then it is easy to prove that E solves the prescribed fractional curvature problem (5). Note that in general there is no existence for minimizers of the problem (6), due to the lack of compactness. As a corollary of our main result, we get that if E is a minimizer of (3), then there exists a constant µ m , depending on m, such that where S is the singular set in Theorem 1.1.
We will also show in Proposition 4.1 that, in the small volume regime, the contribution of the volume term E g(x)dx becomes irrelevant, and the minimizers converge, after appropriate rescalings, to a ball. Note that if g is close to a constant, it is known that solutions to (5) are necessarily compact and close to balls in the Hausdorff distance (see [6]).

Notation and basic estimates
Given a set E ⊂ R N , we denote as E c its complement, that is, E c = R N \ E. We denote by B r the ball of center 0 and radius r, whereas B(x, r) is the ball centered at x and with radius r. We also let ω N = |B 1 |.
Given E, F two sets in R N , the symmetric difference of E and F is defined We recall the following computation, that will be useful in the sequel (see It is possible to define the nonlocal perimeter of E in a bounded set Ω as follows: Finally we recall the following formula (see [7,Lemma 2.4]). Given two disjoint bounded open sets Ω 1 , Ω 2 , then there holds

Regularity of minimizers
In this section we shall assume that g is locally Lipschitz continuous and bounded from above (10) and we will prove regularity of minimizers. We start with a nonlocal version of the so-called Almgren's Lemma (see [9, Lemma 2.3]). Lemma 3.1. Let s ∈ (0, 1) and let E ⊂ R N be a measurable set with P s (E) < +∞. Let x 0 ∈ R n and r > 0 be such that Then there exist positive constants k 0 , C, depending on E, such that for any k ∈ (−k 0 , k 0 ) there exists a measurable set F with P s (F ) < +∞, satisfying the following properties Notice that such a vector field T necessarily exists since otherwise we would have which, by the relative isoperimetric inequality, would contradict (11). For t ∈ (−1, 1), we define the maps Φ t (x) = x + tT (x). It is easy to see that Φ t is a diffeomorphism of R N for t sufficiently small, moreover the Jacobian of Φ t is given by JΦ t (x) = 1 + t div T (x) + o(t).
By construction E∆Φ t (E) ⊂⊂ B(x 0 , r), moreover For k sufficiently small, we then let F : is such that |F | = |E| + k, so that Properties 1 and 2 are verified. We now compute Using the regularity of T , we get that there exists a constant C (depending on T ) such that Substituting this estimate in the expression for P s (Φ t (E)) above, we obtain that where C depends on T, N, s. This shows that the set F also satisfies Property 3, and the proof is concluded.
Using this lemma we get boundedness of minimizers.
In particular, there exists R, depending on E, such that E ⊆ B R , up to a suitable translation.
Proof. Let E be a minimizer of (3). For r ≥ 0 we define Then f (r) is a nonincreasing function and by the coarea formula, we have We claim that there exists R, such that f (r) = 0 for r ≥ R. Let us assume by contradiction that f (r) > 0 for any r > 0. Without loss of generality we can also assume that E ∩ B 1 = ∅ and E c ∩ B 1 = ∅. Moreover, we fix R 0 ≥ 1 such that f (r) < k 0 for any r ≥ R 0 , where k 0 is as in Lemma 3.1. Then by Lemma 3.1 for any r ≥ R 0 there exists a set F such that: By the first two properties in Lemma 3.1, we have that |G| = |E|. Therefore, by minimality of E and recalling (7), we get By Property 3 in Lemma 3.1 we get that Notice that by the construction in Lemma 3.1, using the locally Lipschitz regularity of g, we have also Using the coarea formula and recalling that E \ B r = F \ B r , we get Substituting (13), (14), (15) in (12), we eventually obtain for some C ′ > 0. Hence, by the isoperimetric inequality (2) we get Recalling that f (r) is decreasing to 0 as r → +∞, we can choose R 1 > R 0 such that for all r ≥ R 1 . Therefore, for r ≥ R 1 we obtain that f satisfies the inequality We integrate (16) on (R, +∞), with R > R 1 , and we exchange the order of integration to get We now compute Using again the fact that f is decreasing to 0, we can choose R sufficiently large such that Substituting this inequality in (17), we get that f satisfies the integrodifferential inequality for all R ≥ R 2 , with R 2 sufficiently large.
Proceeding now exactly as in [7, Lemma 4.1], from (18) we can conclude that there exists R such that f (r) = 0 for every r ≥ R.
Once we have boundedness of minimizers, we can obtain regularity. We will use the following result about regularity of local almost minimizers of the fractional perimeter, proved in a more general setting in [5, Thm 1.1, Thm 1.2]. Moreover, in [5] it is proved that the singular set has Hausdorff dimension at most N − 2, improved to N − 3 in [13, Corollary 2]. Theorem 3.3. Let s ∈ (0, 1), δ > 0, Ω an open set. Let E be a nonlocal almost minimal set. This means that for any x 0 ∈ ∂E, for any r < min(δ, d(x 0 , ∂Ω)) and for any measurable set F with E∆F ⊂ B(x 0 , r), the following holds P s (E, Ω) ≤ P s (F, Ω) + Kr N .
Then E has boundary of class C 1 outside of a closed singular set S of Hausdorff dimension at most N − 3.
We start showing that any solution to the isoperimetric problem (3) is actually also a local minimizer for a suitably defined unconstrained problem.
Lemma 3.4. Let (10) hold. Let E be a minimizer of (3) with |E| = m. Then there exists R > 0 and µ 0 , depending on E, such that E ⊆ B R/2 and E is a solution to Proof. First of all, without loss of generality, for simplicity we let m = 1. Let E be a minimizer of F among sets F with |F | = 1. Then, by Proposition 3.2 there exist R depending on E, N, s and g ∞ such that E ⊆ B R/2 .
We argue by contradiction and we assume there exists a sequence µ n → +∞ and F n ⊆ B R such that We observe that ||F n | − 1| > 0, since otherwise we would get a contradiction to the previous inequality by minimality of E among sets of volume 1.
From now on we assume µ n > g L ∞ (BR) for every n. We observe that Using this computation and minimality of F n , say (19), we get that there exists C indipendent of n such that In particular this implies that |F n | → 1 as n → +∞. Let λ n = |F n | −1/N . Then, by the computation above, λ n → 1 as n → +∞. We defineF n = λ n F n . So, by definition |F n | = 1 and, by minimality of E, we get where K g (R) is the Lipschitz constant of g in B R . So, using both (19) and (20), we obtain that So, we divide both sides by ||F n | − 1| = |λ N n − 1|λ −N n and we obtain, recalling that P s (F n ) ≤ C, So, in particular, recalling that λ n → 1 as n → +∞, we get that µ n ≤ C for some constant depending on R, g L ∞ (BR) , K g (R), N, s, in contradiction with the assumption that µ n → +∞.
Finally we will use the bootstrap argument in [2,Theorem 5] and the Lipschitz regularity of g to improve the regularity of ∂E from C 1 to C 2,α for any α < s. Corollary 3.5. Assume (10). Let E be a minimizer of (3). Then ∂E is of class C 2,α for every α < s, up to a closed singular set S of Hausdorff dimension at most N − 3.
Proof. Observe that Lemma 3.4 implies that E is a nonlocal almost minimal set in B R . Take δ < R/2, Ω = B R and K = ( g L ∞ (BR) + µ 0 )ω N . Then for any x 0 ∈ ∂E, for any r < δ and for any measurable set F with E∆F ⊂ B(x 0 , r), the following holds Therefore , so we can apply Theorem 3.3 and conclude that ∂E is of class C 1 , up to a closed singular set S of Hausdorff dimension at most N − 3. Eventually we use the bootstrap argument in [2, Theorem 1.5] and the Lipschitz regularity of g to improve the regularity of ∂E from C 1 to C 2,α for any α < s.

Asymptotics of minimizers for small volumes
In this section we discuss the asymptotic behavior of minimizers of (3) in the small volume regime. We will prove in particular that the volume term becomes irrelevant for small volumes.
First of all observe that if E is a minimizer of (3) with mass constraint |E| = m, then E λ = λE is a minimizer of among all sets of volume |E| = λ N m.
We show that minimizers of (3), properly rescaled, tend to a ball as the volume goes to zero. Proposition 4.1. Assume that g ∈ L ∞ . Then for ε ∈ (0, 1) let E ε be a minimizer of (3) with volume constraint |E ε | = ε N ω N , and letẼ ε = ε −1 E ε . Then, as ε → 0, the setsẼ ε converge in the L 1 -topology, up to translations, to the unit ball B, and in particular there holds where the constant C depends only on N, s.
Proof. Note that by the observation aboveẼ ε is a minimizer of the functional F ε −1 , defined in (21), among sets of volume ω N . By minimality ofẼ ε we then get, for every x ∈ R N , Recalling the quantitative isoperimetric inequality for the fractional perimeter (see [8, where C(N, s) depends only on N, s, we then get from which we obtain (22).
Remark 4.2. The result in Proposition 4.1 also holds if g belongs to L ∞ loc and is coercive. Indeed, the proof is the same once we show that the points x in (22) can be chosen in a fixed compact set, independent of ε, and this can be easily proved reasoning as in Proposition 5.3.

Existence result
We now prove existence of minimizers under suitable assumptions on the function g.

Periodic case
The first case we consider is the case in which g is Z N periodic. The construction of a minimizer to (3) follows the same strategy as in the proof of [7, Theorem 7.2], which is based on a concentrated compactness type argument.
Theorem 5.1. Assume (10) and that g is a Z N periodic function. Then, for every m > 0 there exists a bounded minimizer E to (3).
Proof. Without loss of generality we shall assume that m = 1/2, since the argument is the same for all values of m > 0.
We recall a technical Lemma proved in [10,Lemma 4.2].
Lemma 5.2. Let C > 0 and let {x i } i∈N be a non-increasing sequence of positive numbers such that Then there exists k 0 ∈ N such that, for all k ≥ k 0 there holds Let now E n be a minimizing sequence for (3), that is, In particular, since the function g is bounded, we have where C does not depend on n. For n ∈ N, we also let {Q i,n } i∈N be a partition of R N into disjoint unit cubes such that the quantities x i,n = |E n ∩ Q i,n | are non-increasing in i. In particular, there holds Recalling the fractional isoperimetric inequality (2), which can be also localized in Lipschitz domains (see [7,Lemma 2.5]), we also have for some constants C, C ′ > 0. By Lemma 5.2 we then obtain that for some c > 0 and for all k ∈ N. By a diagonal argument, up to extracting a subsequence, we can assume that x i,n → α i as n → ∞, for some α i ∈ [0, 1/2]. By (23) and (24) we then get Fix now z i,n ∈ Q i,n . Up to extracting a further subsequence, we can suppose that d(z i,n , z j,n ) → c ij ∈ [0, +∞], and that there exists G i ⊆ R N such that for every i ∈ N. We say that i ∼ j if c ij < +∞ and we denote by [i] the equivalence class of i. Notice that G i equals G j up to a translation, if i ∼ j.
To prove it, we fix M ∈ N and R > 0. Let Q R = [−R, R] N . We take different equivalence classes i 1 , . . . , i M and we notice that if i k = i j then the set z i k ,n +Q R is moving far apart from the set z ij ,n + Q R , and so we have lim n→+∞ zi k ,n +QR zi j ,n +QR dx dy |x − y| N +s = 0.
By (26), the lower semicontinuity of the perimeter and (9), we obtain By sending first R → +∞ and then M → +∞, this yields (27). Now we claim that Indeed, for every i ∈ N and R > 0 we have If j is such that j ∼ i and c ij ≤ R 2 , possibly increasing R we have Q j,n − z i,n ⊂ Q R for all n ∈ N, so that and so Letting R → +∞ we then have Putting together (27) and (29) we then get This means in particular that each set G i is a minimizer of F among sets of volume equal to |G i |, hence it is bounded thanks to Proposition 3.2. Assume now that at least two of the sets G i 's have positive volume, and let F := ∪ [i] (G i + w i ), where the vectors w i ∈ Z N are chosen in such a way that the sets (G i + w i ) are pairwise disjoint. Then, by Z N periodicity of g, and by (7) we get thus leading to a contradiction. It follows that there existsī such that |Gī| = 1/2, so that Gī is a (bounded) minimizer of the functional F .

Coercive case
We now assume that g is coercive.
Proposition 5.3. Assume that g is a measurable function, bounded from above, and coercive. Then for every m > 0 there exists a minimizer to (3).
Proof. The argument is the same as for local perimeter functionals (see [9,Lemma 6]). First of all observe that, up to adding a constant, we can assume that g ≤ 0.
Let E n be a minimizing sequence, then For R > 0, we compute Since by assumption sup R N \BR g → −∞ as R → +∞, this implies that sup n |E n \ B R | → 0 as R → +∞.
By (31), (32) and the compact embedding of H s/2 into L 1 , there exists a set E with |E| = m such that, up to a subsequence, E n → E in L 1 . By the lower semicontinuity of P s wth respect to the L 1 convergence, it follows that E is a minimizer of (3).