Nonlocal minimal surfaces: interior regularity, quantitative estimates and boundary stickiness

We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present at least a sketch of the proofs of these results, in a way that aims to be as elementary and self contained as possible.


Introduction
The study of surfaces which minimize the perimeter is a classical topic in analysis and geometry and probably one of the oldest problems in the mathematical literature: according to the first book of Virgil's Aeneid, Dido, the legendary queen of Carthage, needed to study problems of geometric minimization in order to found her reign in 814 B.C. (in spite of the great mathematical talent of Dido and of her vivid geometric intuition, Aeneas broke his betrothal with her after a short time to sail the Mediterranean towards the coasts of Italy, but this is another story).
The first problem in the study of these surfaces of minimal perimeter (minimal surfaces, for short) lies in proving that minimizers do exist. Indeed "nice" sets, for which one can compute the perimeter using an intuitive notion known from elementary school, turn out to be a "non compact" family (roughly speaking, for instance, an "ugly" set can be approximated by a sequence of "nice" sets, thus the limit point of the sequence may end up outside the family). To overcome this difficulty, a classical tool of the Calculus of Variations is to look for minimizers in a wider family of candidates: this larger family has to be chosen to satisfy the desired compactness property to ensure the existence of a minimum, and then the regularity of the minimal candidate can be (hopefully) proved a posteriori.
To this end, one needs to set up an appropriate notion of perimeter for the sets in the enlarged family of candidates, since no intuitive notion of perimeter is available, in principle, in this generality. The classical approach of Caccioppoli (see e.g. [Cac27]) to this question lies in the observation that if Ω and E are 1 smooth sets and ν is the external normal of E, then, for any vector field T ∈ C 1 0 (Ω, R n ) with |T (x)| 1 for any x ∈ Ω, we have that T · ν |T | |ν| 1.
Consequently, the perimeter of E in Ω, i.e. the measure of the boundary of E inside Ω (that is, the (n − 1)-dimensional Hausdorff measure of ∂E in Ω), satisfies the inequality (1.1) Per (E, Ω) = H n−1 (∂E) ∩ Ω ∂E T · ν dH n−1 = E div T (x) dx, for every vector field T ∈ C 1 0 (Ω, R n ) with T L ∞ (R n ,R n ) 1, where the Divergence Theorem has been used in the last identity.
Viceversa, if E is a smooth set, its normal vector can be extended near ∂E, and then to the whole of R n , to a vector field ν * ∈ C 1 (R n , R n ), with |ν * (x)| 1 for any x ∈ R n . Then, if η ∈ C ∞ 0 (Ω, [0, 1]), with η = 1 in an ε-neighborhood of Ω, one can take T := ην * and find that T ∈ C 1 0 (Ω, R n ), |T (x)| 1 for any x ∈ R n and E div T (x) dx = ∂E T · ν dH n−1 = ∂E ην * · ν dH n−1 = ∂E η dH n−1 By taking ε as small as we wish and recalling (1.1), we obtain that (1.2) Per (E, Ω) = sup While (1.2) was obtained for smooth sets E, the classical approach for minimal surfaces is in fact to take (1.2) as definition of perimeter of a (not necessarily smooth) set E in Ω. The class of sets obtained in this way indeed has the necessary compactness property (and the associated functional has the desired lower semicontinuity properties) to give the existence of minimizers: that is, one finds (at least) one set E ⊆ R n satisfying (1.3) Per (E, Ω) Per (F, Ω) for any F ⊆ R n such that F coincides with E in a neighborhood of Ω c . The boundary of this minimal set E satisfies, a posteriori, a bunch of additional regularity properties -just to recall the principal ones: We refer to [Giu77] for complete statements and proofs (in particular, the claim in (1.4) here corresponds to Theorem 10.11 in [Giu77], the claim in (1.5) here to Theorem 11.8 there, and the claim in (1.6) here to Theorem 16.4 there).
A natural problem that is closely related to these regularity results is the complete description of classical minimal surfaces in the whole of the space which are also graphs in some direction (the socalled minimal graphs). These questions, that go under the name of Bernstein's problem, have, in the classical case, the following positive answer: If n 8 and E is a minimal graph, then E is a halfspace; (1.7) The statement in (1.7) is sharp, since there exist (1. 8) examples of minimal graphs in dimension 9 and higher that are not halfspaces.
We refer to Theorems 17.8 and 17.10 in [Giu77] for further details on the claims in (1.7) and (1.8), respectively.
It is also worth recalling that (1.9) surfaces minimizing perimeters have zero mean curvature, see e.g. Chapter 10 in [Giu77].
Recently, and especially in light of the seminal paper [CRS10], some attention has been devoted to a variation of the classical notion of perimeters which takes into account also long-range interactions between sets, as well as the corresponding minimization problem. This type of nonlocal minimal surfaces arises naturally, for instance, in the study of fractals [Vis91], cellular automata [Imb09,CS10] and phase transitions [SV12] (see also [BV16] for a detailed introduction to the topic).
A simple idea for defining a notion of nonlocal perimeter may be described as follows. First of all, such nonlocal perimeter should compute the interaction I of all the points of E against all the points of the complement of E, which we denote by E c .
On the other hand, if we want to localize these contributions inside the domain Ω, it is convenient to split E into E ∩ Ω and E \ Ω, as well as the set E c into E c ∩ Ω and E c \ Ω, and so consider the four possibilities of interaction between E and E c given by Among these interactions, we observe that the latter one only depends on the configuration of the set outside Ω, and so I(E \ Ω, E c \ Ω) = I(F \ Ω, F c \ Ω) for any F ⊆ R n such that F \Ω = E \Ω. Therefore, in a minimization process with fixed data outside Ω, the term I(E \ Ω, E c \ Ω) does not change the minimizers. It is therefore natural to omit this term in the energy functional (and, as a matter of fact, omitting this term may turn out to be important from the mathematical point of view, since this term may provide an infinite contribution to the energy). For this reason, the nonlocal perimeter considered in [CRS10] is given by the sum of the first three terms in (1.10), namely one defines As for the interaction I(·, ·), of course some freedom is possible, and basically any interaction for which Per s (E, Ω) is finite, say, for smooth sets E makes perfect sense. A natural choice performed in [CRS10] is to take the interaction as a weighted Lebesgue measure, where the weight is translation invariant, isotropic and homogeneous: more precisely, for any disjoint sets S 1 and S 2 , one defines 2 (1.11) I(S 1 , S 2 ) := S 1 ×S 2 dx dy |x − y| n+2s , with s ∈ 0, 1 2 . With this choice of the fractional parameter s, one sees that as long as pσ = 2s, that is the fractional perimeter of a set coincides (up to normalization constants) to a fractional Sobolev norm of the corresponding characteristic function (see e.g. [DNPV12] for a simple introduction to fractional Sobolev spaces).
Moreover, for any fixed y ∈ R n , Also, for any fixed x ∈ R n , div y ν(x) |x − y| n+2s−2 = (n + 2s − 2) Accordingly, by the Divergence Theorem 3 (1.13) That is, which suggests that the fractional perimeter is a weighted measure of the variation of the normal vector around the boundary of a set. As a matter of fact, as s 1/2, the s-perimeter recovers the classical perimeter from many point of views (a sketchy discussion about this will be given in Appendix A).
Also, in Appendix C, we briefly discuss the second variation of the s-perimeter on surfaces of vanishing nonlocal mean curvature and we show that graphs with vanishing nonlocal mean curvature cannot have horizontal normals.
Let us now recall (among the others) an elementary, but useful, application of this notion of fractional perimeter in the framework of digital image reconstruction. Suppose that we have a black and white digitalized image, say a bitmap, in which each pixel is either colored in black or in white. We call E the "black set" and we are interested in measuring its perimeter (the reason for that may be, for instance, that noises or impurities could be distinguished by having "more perimeter" than the "real" picture, since they may present irregular or fractal boundaries). In doing that, we need to be able to compute such perimeter with a very good precision. Of course, numerical errors could affect the computation, since the digital process replaced the real picture by a pixel representation of it, but we would like that our computation becomes more and more reliable if the resolution of the image is sufficiently high, i.e. if the size of the pixels is sufficiently small.
Unfortunately, we see that, in general, an accurate computation of the perimeter is not possible, not even for simple sets, since the numerical error produced by the pixel may not become negligible, even when the pixels are small. To observe this phenomenon (see e.g. [CSV16]) we can consider a grid of square pixels of small side ε and a black square E of side 1, with the black square rotated by 45 degrees with respect to the orientation of the pixels. Now, the digitalization of the square will produce a numerical error, since, say, the pixels that intersect the square are taken as black, and so each side of the square is replaced by a "sawtooth" curve (see Figure 1).
Notice that the length of each of these sawtooth curves is √ 2 (independently on how small each teeth is, that is independently on the size of ε). As a consequence, the perimeter of the digitalized image is 4 √ 2, instead of 4, which was the original perimeter of the square. This shows the rather unpleasant fact that the perimeter may be poorly approximated numerically, even in case of high precision digitalization processes. It is a rather remarkable fact that fractional perimeters do not present the same inconvenience and indeed the numerical error in computing the fractional perimeter becomes small when the pixels are small enough. Indeed, the number of pixels which intersect the sides of the original square is O(ε −1 ) (recall that the side of the square is 1 and the side of each pixel is of size ε). Also, the s-perimeter of each pixel is O(ε 2−2s ) (since this is the natural scale factor of the interaction in (1.11), with n = 2). Then, the numerical error in the fractional perimeter comes from the contributions of all these pixels 4 and it is therefore O(ε −1 ) · O(ε 2−2s ) = O(ε 1−2s ), which tends to zero for small ε, thus showing that the nonlocal perimeters are more efficient than classical ones in this type of digitalization process.
Thus, given its mathematical interest and its importance in concrete applications, it is desirable to reach a better understanding of the surfaces which minimize the s-perimeter (that one can call s-minimal surfaces). To start with, let us remark that an analogue of (1.9) holds true, in the sense that s-minimal surfaces have vanishing s-mean curvature in a sense that we now briefly describe. Given a set E with smooth boundary and p ∈ ∂E, we define The expression 5 in (1.14) is intended in the principal value sense, namely the singularity is taken in an averaged limit, such as For simplicity, we omit the principal value from the notation. It is also useful to recall (1.12) and to remark that H s E can be computed as a weighted boundary integral of the normal, namely (1.15) This quantity H s E is what we call the nonlocal mean curvature of E at the point p, and the name is justified by the following observation: Lemma 1.1. If E is a set with smooth boundary that minimizes the s-perimeter in Ω, then H s E (p) = 0 for any p ∈ (∂E) ∩ Ω.
The proof of Lemma 1.1 will be given in Section 2. We refer to [CRS10] for a version of Lemma 1.1 that holds true (in the viscosity sense) without assuming that the set has smooth boundary. See also [AV14] for further comments on this notion of nonlocal mean curvature.
Let us now briefly discuss the fractional analogue of the regularity results in (1.4) and (1.5). At the moment, a complete regularity theory in the fractional case is still not available. At best, one can 5 The definition of fractional mean curvature in (1.14) may look a bit awkward at a first glance. To make it appear more friendly, we point out that the classical mean curvature of ∂E at a point x ∈ ∂E, up to normalizing constants, can be computed via the average procedure To see this, up to rigid motions, one can assume that x is the origin and E, in a small neighborhood of the origin, is the subgraph of a function u : R n−1 → R with u(0) = 0 and ∇u(0) = 0. Then, we have that obtain regularity results either in low dimension or when s is sufficiently close to 1 2 (see [SV13,CV13] and also [BFV14] for higher regularity results): namely, the analogue of (1.4) is: Theorem 1.2 (Interior regularity results for s-minimal surfaces -I). Let E ⊂ R n be a minimizer for the s-perimeter in Ω. Assume that • either n = 2, • or n 7 and 1 2 − s ε * , for some ε * > 0 sufficiently small. Then, (∂E) ∩ Ω is smooth.
Similarly, a fractional analogue of (1.5) is known, by now, only when s is sufficiently close to 1 2 : Theorem 1.3 (Interior regularity results for s-minimal surfaces -II). Let E ⊂ R n be a minimizer for the s-perimeter in Ω. Assume that n 8 and 1 2 − s ε n , for some ε n > 0 sufficiently small. Then, (∂E) ∩ Ω \ Σ is smooth, being Σ a closed set of Hausdorff dimension at most n − 8. In [CSV16], several quantitative regularity estimates for local minimizers are given (as a matter of fact, these estimates are valid in a much more general setting, but, for simplicity, we focus here on the most basic statements and proofs). For instance, minimizers of the s-perimeter have locally finite perimeter (that is, classical perimeter, not only fractional perimeter), as stated in the next result: Theorem 1.4. Let E ⊂ R n be a minimizer for the s-perimeter in B R . Then for a suitable constant C > 0.
We stress that Theorem 1.4 presents several novelties with respect to the existing literature. First of all, it provides a scaling invariant regularity estimate that goes beyond the natural scaling of the s-perimeter, that is valid in any dimension and without any topological restriction on the s-minimal surface (analogous results for the classical perimeter are not known in this generality). Also, in spite of the fact that, for the sake of simplicity, we state and prove Theorem 1.4 only in the case of minimizers of the s-perimeter, more general versions of this result hold true for stable solutions and for more general interaction kernels (even for kernels without any regularizing effect). This type of results also leads to new compactness and existence theorems, see [CSV16] for full details on this topic.
As a matter of fact, we stress that the analogue of Theorem 1.4 for stable surfaces which are critical points of the classical perimeter is only known, up to now, for two-dimensional surfaces that are simply connected and immersed in R 3 (hence, this is a case in which the nonlocal theory can go beyond the local one). Now, we briefly discuss the fractional analogue of the Bernstein's problem. Let us start by pointing out that, by combining (1.4) and (1.7), we have an "abstract" version of the Bernstein's problem, which states that if E is a minimal graph in R n+1 and the minimal surfaces in R n are smooth, then E is a halfspace.
Of course, for the way we have written (1.4) and (1.7), this abstract statement seems only to say that 8 = 7+1: nevertheless this abstract version of the Bernstein's problem is very useful in the classical case, since it admits a nice fractional counterpart, which is: Theorem 1.5 (Bernstein result for s-minimal surfaces -I). If E is an s-minimal graph in R n+1 and the s-minimal surfaces in R n are smooth, then E is a halfspace.
This result was proved in [FV]. By combining it with Theorem 1.2 (using the notation N := n + 1), we obtain: Theorem 1.6 (Bernstein result for s-minimal surfaces -II). Let E ⊂ R N be an s-minimal graph. Assume that • either N = 3, • or N 8 and 1 2 − s ε * , for some ε * > 0 sufficiently small. Then, E is a halfspace. This is, at the moment, the fractional counterpart of (1.7) (we stress, however, that any improvement in the fractional regularity theory would give for free an improvement in the fractional Bernstein's problem, via Theorem 1.5).
We remark again that, differently from the claim in (1.8), it is not known if the statement in Theorem 1.6 is sharp, since there are no known examples of s-minimal graphs other than the hyperplanes.
It is worth recalling that, by a blow-down procedure, one can deduce from Theorem 1.2 that global s-minimal surfaces are hyperplanes, as stated in the following result: Theorem 1.7 (Flatness of s-minimal surfaces). Let E ⊂ R n be a minimizer for the s-perimeter in any domain of R n . Assume that • either n = 2, • or n 7 and 1 2 − s ε * , for some ε * > 0 sufficiently small. Then, E is a halfspace.
Of course, a very interesting spin-off of the regularity theory in Theorem 1.7 lies in finding quantitative flatness estimates: namely, if we know that a set E is an s-minimizer in a large domain, can we say that it is sufficiently close to be a halfspace, and if so, how close, and in which sense?
This question has been recently addressed in [CSV16]. As a matter of fact, the results in [CSV16] are richer than the ones we present here, and they are valid for a very general class of interaction kernels and of perimeters of nonlocal type. Nevertheless we think it is interesting to give a flavor of them even in their simpler form, to underline their connection with the regularity theory that we discussed till now.
In this setting, we present here the following result when n = 2 (see indeed [CSV16] for more general statements): Theorem 1.8. Let R 2. Let E ⊂ R 2 be a minimizer for the s-perimeter in B R . Then there exists a halfplane h such that We stress that Theorem 1.8 may be seen as a quantitative version of Theorem 1.7 when n = 2: indeed if E ⊂ R n is a minimizer for the s-perimeter in any domain of R n we can send R +∞ in (1.16) and obtain that E is a halfplane.
We observe that, till now, we have presented and discussed a series of results which are somehow in accordance, as much as possible, with the classical case. Now we present something with striking difference from the classical case. The minimizers of the classical perimeter in a convex domain reach continuously the boundary data (see e.g. Theorem 15.9 in [Giu77]). Quite surprisingly, the minimizers of the fractional perimeter have the tendency to stick at the boundary. This phenomenon has been discovered in [DSV], where several explicit stickiness examples have been given (see also [BLV16] for other examples in more general settings).
Roughly speaking, the stickiness phenomenon may be described as follows. We know from Lemma 1.1 that nonlocal minimal surfaces in a domain Ω need to adjust their shape in order to make the nonlocal minimal curvature vanish inside Ω. This is a rather strong condition, since the nonlocal minimal curvature "sees" the set all over the space. As a consequence, in many cases in which the boundary data are "not favorable" for this condition to hold, the nonlocal minimal surfaces may prefer to modify their shape by sticking at the boundary, where the condition is not prescribed, in order to compensate the values of the nonlocal mean curvature inside Ω.
In many cases, for instance, the nonlocal minimal set may even prefer to "disappear", i.e. its contribution inside Ω becomes empty and its boundary sticks completely to the boundary of Ω. In concrete cases, the fact that the nonlocal minimal set disappears may be induced by a suitable choice of the data outside Ω or by an appropriate choice of the fractional parameter. As a prototype example of these two phenomena, we recall here the following results given in [DSV]: Theorem 1.9 (Stickiness for small data). For any δ > 0, let Then, there exists δ o > 0, depending on s and n, such that for any δ ∈ (0, δ o ] we have that Theorem 1.10 (Stickiness for small s). As s → 0 + , the s-minimal set in B 1 ⊂ R 2 that agrees with a sector outside B 1 sticks to the sector. More precisely: let E s be the s-minimizer among all the sets E such that Then, there exists s o > 0 such that for any s ∈ (0, s o ] we have that E s = Σ.
We stress the sharp difference between the local and the nonlocal cases exposed in Theorems 1.9 and 1.10: indeed, in the local framework, in both cases the minimal surface is a segment inside the ball B 1 , while in the nonlocal case it coincides with a piece of the circumference ∂B 1 .
The stickiness phenomenon of nonlocal minimal surfaces may also be caused by a sufficiently high oscillation of the data outside Ω. This concept is exposed in the following result: (1.17) It is worth to remark that the stickiness phenomenon in Theorem 1.11 becomes "more and more visible" as the oscillation of the data increase, since, referring to (1.17), we have that hence the sticked portion of E M on ∂Ω becomes, proportionally to M , larger and larger when M → +∞. Also, the exponent 1+2s 2+2s in (1.17) is optimal, see again [DSV]. The stickiness phenomenon detected in Theorem 1.11 is described in Figure 3.
We believe that the stickiness phenomenon is rather common among nonlocal minimal surfaces. Indeed, it may occur even under small modifications of boundary data for which the nonlocal minimal surfaces cut the boundary in a transversal way. A typical, and rather striking, example of this situation happens for perturbation of halfplanes in R 2 . That is, an arbitrarily small perturbation of the data corresponding to halfplanes is sufficient for the stickiness phenomenon to occur. Of course, the smaller the perturbation, the smaller the stickiness: nevertheless, small perturbations are enough to cause the fact that the boundary data of nonlocal minimal surfaces are not attained in a continuous way, and indeed they may exhibit jumps (notice that this lack of boundary regularity for s-minimal surfaces is rather surprising, especially after the interior regularity results discussed in Theorem 1.2 and 1.3 and it shows that the boundary behavior of the halfplanes is rather unstable).
A detailed result goes as follows: Theorem 1.12 (Stickiness arising from perturbation of halfplanes). There exists δ 0 > 0 such that for any δ in(0, δ 0 ] the following statement holds true. Let Ω := (−1, 1) × R. Let also Let E be an s-minimal set in Ω among all the sets which coincide with F outside Ω. Then, for a suitable γ > 1.
The result of Theorem 1.12 is depicted in Figure 4. Let us briefly give some further comments on the stickiness phenomena discussed above. First of all, we would like to convince the reader (as well as ourselves) that this type of behaviors indeed occurs in the nonlocal case.
To this end, let us make an investigation to find how the s-minimal set E α in Ω : When α = 0, then E α = C α is the halfplane, so the interesting case is when α = 0; say, up to symmetries, α > 0. Now, we know how an investigation works: we need to place all the usual suspects in a row and try to find the culprit.
The line of suspect is on Figure 5 (remember that we have to find the s-minimal set among them). Some of the suspects resemble our prejudices on how the culprit should look like. For instance, for what we saw on TV, we have the prejudice that serial killers always wear black gloves and raincoats.
Similarly, for what we learnt from the hyperplanes, we may have the prejudice that s-minimal surfaces meet the boundary data in a smooth fashion (this prejudice will turn out to be wrong, as we will see).
In this sense, the usual suspects number 1 and 2 in Figure 5 are the ones who look like the serial killers.
Then, we have the regular guys with some strange hobbies, we know from TV that they are also quite plausible candidates for being guilty; in our analogy, these are the usual suspects number 3 and 4, which meet the boundary data in a Lipschitz or Hölder fashion (and one may also observe that number 3 is the minimal set in the local case).
Then, we have the candidates which look above suspicion, the ones to which nobody ever consider to be guilty, usually the postman or the butler. In our analogy, these are the suspects number 5 and 6, which are discontinuous at the boundary. Now, we know from TV how we should proceed: if a suspect has a strong and verified alibi, we can rule him or her out of the list. In our case, an alibi can be offered by the necessary condition for s-minimality given in Lemma 1.1. Indeed, if one of our suspects E does not satisfy that H s E = 0 along (∂E) ∩ Ω, then E cannot be s-minimal and we can cross out E from our list of suspects (E has an alibi!). Now, it is easily seen that all the suspects number 1, 2, 3, 4 and 5 have an alibi: indeed, from Figure 6 we see that H s E (p) = 0, since the set E occupies (in measure, weighted by the kernel in (1.14)) more than a halfplane 6 passing through p: in Figure 6 the point p is the big dot and the halfplane is marked by the line passing through it, so a quick inspection confirms that the alibis of number 1, 2, 3, 4 and 5 6 Indeed, in view of (1.14), we know that an s-minimal set, seen from any point of the boundary, satisfies a perfect balance between the weighted measure of the set itself and the weighted measure of its complement (here, weighted is intended with respect to the kernel in (1.14)). Since a halfplane also satisfies such perfect balance when seen from any point of its boundary (due to odd symmetry), one can say that a set is s-minimal when, at any point of its boundary, the weighted contributions of the set and its complement produce the same result as the ones of a hyperplane passing through On the other hand, the alibi of number 6 doesn't hold water. Indeed, near p, the set E is confined below the horizontal line, but at infinity the set E go well beyond such line: these effects might compensate each other and produce a vanishing mean curvature.
So, having ruled out all the suspects but number 6, we have only to remember what the old investigators have taught us (e.g., "When you have eliminated the impossible, whatever remains, however improbable, must be the truth"), to find that the only possible (though, in principle, rather improbable) culprit is number 6.
Of course, once that we know that the butler did it, i.e. that number 6 is s-minimal, it is our duty to prove it beyond any reasonable doubt. Many pieces of evidence, and a complete proof, is given in [DSV] (where indeed the more general version given in Theorem 1.12 is established). Here, we provide some ideas towards the proof of Theorem 1.12 in Section 5.
This set of notes is organized as follows. In Section 2 we present the proof of Lemma 1.1. Sections 3 and 4 are devoted to the proofs of the quantitative estimates in Theorems 1.4 and 1.8, respectively. Then, Section 5 is dedicated to a sketch of the proof of Theorem 1.12. We also provide Appendix A to discuss briefly the asymptotics of the s-perimeter as s 1/2 and as s 0 and Appendix B to discuss the asymptotic expansion of the nonlocal mean curvature as s 0. Finally, in Appendix C we discuss the second variation of the fractional perimeter functional. such point. This geometric trick often allows us to "subtract the tangent halfplane" from a set without modifying its fractional curvature (and this is often convenient to observe cancellations). Proof of Lemma 1.1. We consider a diffeomorphism T ε (x) := x + εv(x), with v ∈ C ∞ 0 (Ω, R n ) and we take E ε := T ε (E). By minimality, we know that Per s (E ε , Ω) Per s (E, Ω) for every ε ∈ (−ε 0 , ε 0 ), with ε 0 > 0 sufficiently small, hence Suppose, for simplicity, that I(E \ Ω, E c \ Ω) < +∞, so that we can write Moreover, if we use the notation X := T −1 ε (x), we have that Similarly, if Y := T −1 ε (y), we find that As a consequence, .
and so, interchanging the names of the variables, Consequently, Now, using the Divergence Theorem and changing the names of the variables we have that Accordingly, we find that Comparing with (2.1), we see that ∂E v(y) · ν(y) H s E (y) dH n−1 (y) = 0 and so, since v is an arbitrary vector field supported in Ω, the desired result follows.
3. Proof of Theorem 1.4 The basic idea goes as follows. One uses the appropriate combination of two general facts: on the one hand, one can perturb a given set by a smooth flow and compare the energy at time t with the one at time −t, thus obtaining a second order estimate; on the other hand, the nonlocal interaction always charges a mass on points that are sufficiently close, thus providing a natural measure for the discrepancy between the original set and its flow. One can appropriately combine these two facts with the minimality (or more generally, the stability) property of a set. Indeed, by choosing as smooth flow a translation near the origin, the above arguments lead to an integral estimate of the discrepancy between the set and its translations, which in turn implies a perimeter estimate.
We now give the details of the proof of Theorem 1.4. To do this, we fix R 1, a direction v ∈ S n−1 , a function ϕ ∈ C ∞ 0 (B 9/10 ) with ϕ = 1 in B 3/4 , and a small scalar quantity t ∈ − 1 100 , 1 100 , and we consider the diffeomorphism Φ t ∈ C ∞ 0 (B 9/10 ) given by Φ t (x) : We also define E t := Φ t (E). We have the following useful auxiliary estimates (that will be used in the proofs of both Theorem 1.4 and Theorem 1.8): Lemma 3.1. Let E be a minimizer for the s-perimeter in B R . Then
Proof. First we observe that , for some C > 0. This is indeed a general estimate, which does not use minimality, and which follows by changing variable in the integrals of the fractional perimeter (and noticing that the linear term in t simplifies). We provide some details of the proof of (3.6) for the facility of the reader. To this aim, we observe that Moreover, if, for any ξ, η ∈ R n , we set we have that g is bounded and A). Accordingly, for any A, B ∈ {B R , B c R , R n }, a change of variables x := Φ t (X) and y := Φ t (Y ) gives that for a suitable scalar functiong. Then, replacing t with −t and summing up, the linear term in t simplifies and we obtain This, choosing A and B appropriately, establishes (3.6).
On the other hand, the s-minimality of E gives that Per s (E, B R ) Per s (E ∪ B R , B R ), which, in turn, is bounded from above by the interaction between B R and B c R , namely I(B R , B c R ), which is a constant (only depending on n and s) times R n−2s , due to scale invariance of the fractional perimeter. That is, we have that Per s (E, B R ) CR n−2s , for some C > 0, and then we can make the right hand side of (3.6) uniform in E and obtain (3.2), up to renaming C > 0. The next step is to charge mass in a ball. Namely, one defines E ∪ t := E ∪ E t and E ∩ t := E ∩ E t . By counting the interactions of the different sets, one sees that To check this, one observes indeed that the set E t \ E interacts with E \ E t in the computations of Per s (E, B R ) and Per s (E t , B R ), while these two sets do not interact in the computations of Per s (E ∪ t , B R ) and Per s (E ∩ t , B R ) (the interactions of the other sets simplify). This proves (3.7). We remark that, again, formula (3.7) is a general fact and is not based on minimality. Changing t with −t, we also obtain from (3.7) that . This and (3.7) give that thanks to the s-minimality of E. In particular, This and (3.2) imply (3.3). Now, the interaction kernel is bounded away from zero in B R/2 , and so This is again a general fact, not depending on minimality. By plugging this into (3.3), we conclude that and so, again up to renaming C, Now, we recall (3.1) and we observe that E t ∩ B R/2 = (E + tv) ∩ B R/2 . Hence, the estimate in (3.8) becomes Since this is valid for any v ∈ S n−1 , we may also switch the sign of v and obtain that From (3.9) and (3.10) we obtain (3.4). Now we observe that, for any sets A and B, Indeed, this formula is clearly true if x ∈ B, since in this case the right hand side is nonpositive. The formula is also true if x ∈ A \ B, since in this case the left hand side is 1 and the right hand side is less or equal than 1. It remains to consider the case in which x ∈ A ∪ B. In this case, χ A (x) = 0, hence the right hand side is nonpositive, which gives that (3.11) holds true. By (3.11), This and (3.10) give that CR n−2s−2 2 t min which is (3.5). This ends the proof of Lemma 3.1.
With the preliminary work done in Lemma 3.1 (to be used here with R = 1), we can now complete the proof of Theorem 1.4. To this end, we observe that (3.12) for some C > 0.
Also, we observe that, for any a, b ∈ R, Indeed, up to exchanging a and b, we may suppose that a b; thus which proves (3.13). Using (3.5), (3.12) and (3.13), we obtain that up to renaming C. Dividing by t and sending t 0 (up to subsequences), one finds that for any v ∈ S n−1 , in the bounded variation sense. Since the direction v is arbitrary, this proves that This proves Theorem 1.4 with R = 1, and the general case follows from scaling.

Proof of Theorem 1.8
In this part, we will make use of some integral geometric formulas which compute the perimeter of a set by averaging the number of intersections of straight lines with the boundary of a set.
For this, we recall the notation of the positive and negative part of a function u, namely Notice that u ± 0, that |u| = u + + u − and that u = u + − u − . Also, if v ∈ ∂B 1 and p ∈ R n , we define v ⊥ := {y ∈ R n s.t. y · v = 0} and p + Rv That is, v ⊥ is the orthogonal linear space to v and p + Rv is the line passing through p with direction v. Now, given a Caccioppoli set E ⊆ R n with exterior normal ν (and reduced boundary denoted by ∂ * E), and v ∈ ∂B 1 , we set (4.1) with the sup taken over all smooth φ supported in the segment B 1 ∩ (y + Rv) with image in [0, 1]. We have (see e.g. Proposition 4.4 in [CSV16]) that one can compute the directional derivative in the sense of bounded variation by the formula and we also have that I v,± (y) is the number of points x that lie in B 1 ∩ (∂ * E) ∩ (y + Rv) and such that ∓v · ν(x) > 0. That is, the quantity I v,+ (y) (resp., I v,− (y)) counts the number of intersections in the ball B 1 between the line y + Rv and the (reduced) boundary of E that occur at points x in which v · ν(x) is negative (resp., positive). In particular, Furthermore, the vanishing of I v,+ (y) (resp., I v,− (y)) is related to the fact that, moving along the segment B 1 ∩ (y + Rv), one can only exit (resp., enter) the set E, according to the following result: which gives the desired result.
Then, there exists a horizontal halfplane h ⊂ R 2 such that (4.10) for some C > 0.
Proof. Given v ∈ ∂B 1 , we take into account the sets of y ∈ v ⊥ which give a positive contribution to I v,± (y). For this, we define From (4.3), we know that if y ∈ B ± (v), then I v,± (y) 1. As a consequence of this and of (4.4), we have that I v,± (y) dH 1 (y) H 1 (B ± (v)).
That is, we can denote by G E the set of vertical coordinates y for which the portion in B 1 of the horizontal line passing through y lies in E and, similarly, by G E c the set of vertical coordinates y for which the portion in B 1 of the horizontal line passing through y lies in E c and we obtain that G E ∪ G E c exhaust the whole of (−1, 1), up to a set of size at most 2µ. We also remark that G E lies below G E c : indeed, by (4.11), we have that vertical lines can only exit the set E (possibly with the exception of a small set of size µ). The situation is depicted in Figure 8. Hence, if we take h to be a horizontal halfplane which separates G E and G E c , we obtain (4.10).
With this, we can now complete the proof of Theorem 1.8. The main tool for this goal is Lemma 4.3. In order to apply it, we need to check that (4.8) and (4.9) are satisfied. To this end, we argue as follows. First of all, fixed a large R > 2, we consider, as in Section 3, a diffeomorphism Φ t such that Φ t (x) = x for any x ∈ R n \ B 9R/10 , and Φ t (x) = x + tv for any x ∈ B 3R/4 , and we set E t := Φ t (E). From (3.5) (recall that here n = 2), we have that for some C > 0. Thus, dividing by t and sending t 0, That is, recalling (4.5), We also observe that E has finite perimeter in B 1 , thanks to Theorem 1.4, and so we can make use of Lemma 4.2. In particular, by (4.6), after a rotation of coordinates, we may assume that Φ + (e 1 ) = Φ − (e 1 ). Hence (4.13) says that (4.14) max {Φ + (e 1 ), Φ − (e 1 )} = min {Φ + (e 1 ), Φ − (e 1 )} C R s . Also, up to a change of orientation, we may suppose that Φ − (e 2 ) Φ + (e 2 ), hence in this case (4.13) says that Φ − (e 2 ) C R s . From this and (4.14), we see that (4.8) and (4.9) are satisfied (with µ = C/R s ) and so by Lemma 4.3 we conclude that This completes the proof of Theorem 1.8: as a matter of fact, the result proven is even stronger, since it says that, after removing horizontal and vertical slabs of size C/R s , we have that ∂E in B 1 is a graph of oscillation bounded by C/R s , see Figure 8 (in fact, more general statements and proofs can be find in [CSV16]).

Sketch of the proof of Theorem 1.12
The core of the proof of Theorem 1.12 consists in constructing a suitable barrier that can be slided "from below" and which exhibits the desired stickiness phenomenon: if this is possible, since the sminimal surface cannot touch the barrier, it has to stay above the barrier and stick at the boundary as well.
Such barrier is constructed in [DSV] in an iterative way, that we now try to describe.
Step 1. Let us start by looking at the subgraph of the function y = x + , given 0. Then, at all the boundary points X = (x, y) with positive abscissa x > 0, the fractional mean curvature is at most for some c > 0. The full computation is given in Lemma 5.1 of [DSV], but we can give a heuristic justification of it, by saying that for small X the boundary point gets close to the origin, where there is a corner and the curvature blows up (with a negative sign, since there is "more than a hyperplane" contained in the set), see Figure 9. Also, the power 2s in (5.1) follows by scaling.
In addition, if is close to 0, this first barrier is close to a ninety degree angle, while if is large it is close to a flat line, and these considerations are also in agreement with (5.1).
Step 2. Having understood in Step 1 what happens for the "angles", now we would like to "shift iteratively in a smooth way from one slope to another", see Figure 10.
The detailed statement is given in Proposition 5.3 in [DSV], but the idea is as follows. For any K ∈ N, K 1, one looks at the subgraph of a nonnegative function v K such that •  • for any x K − q K , for some K K and q K ∈ 0, 1 K , • at all the boundary points X = (x, y) with positive abscissa x > 0, the fractional mean curvature is at most − c K |X| 2s , for some c > 0.
Step 3. If K is sufficiently large in Step 2, the final slope is almost horizontal. In this case, one can smoothly glue such barrier with a power like function like x 1 2 +s+ε 0 . Here, ε 0 is any fixed positive exponent (the power γ in the statement of Theorem 1.12 is related to ε 0 , since γ := 2+ε 0 1−2s ). The details of the barrier constructed in this way are given in Proposition 6.3 of [DSV]. In this case, one can still control the fractional mean curvature at all the boundary points X = (x, y) with positive abscissa x > 0, but the estimate is of the type either |X| −2s , for small |X|, or |X| − 1 2 −s+ε 0 , for large |X|. A sketch of such barrier is given in Figure 11. Figure 11. Description of Step 3.
Step 4. Now we use the barrier of Step 3 to construct a compactly supported object. The idea is to take such barrier, to reflect it and to glue it at a "horizontal level", see Figure 12. We remark that such barrier has a vertical portion at the origin and one can control its fractional mean curvature from above with a negative quantity for the boundary points X = (x, y) with positive, but not too large, abscissa.
Of course, this type of estimate cannot hold at the maximal point of the barrier, where "more than a hyperplane" is contained in the complement of the set, and therefore the fractional mean curvature is positive (the precise quantitative estimate is given in Proposition 7.1. of [DSV]).
Step 5. Nevertheless, we can now compensate this error in the fractional mean curvature near the maximal point of the barrier by adding two suitably large domains on the sides of the barriers, see Figure 13. The barrier constructed in this way is described in details in Proposition 7.3 of [DSV] and its basic feature is to possess a vertical portion near the origin and to possess negative fractional mean curvature.
By keeping good track of the quantitative estimates on the bumps of the barriers and on their fractional mean curvatures, one can now scale the latter barrier and slide it from below, in order to prove Theorem 1.12. The full details are given in Section 8 of [DSV].
Appendix A. A sketchy discussion on the asymptotics of the s-perimeter In this appendix, we would like to emphasize the fact that, as s 1/2, the s-perimeter recovers (under different perspectives) the classical perimeter, while, as s 0, the nonlocal features become predominant and the problem produces the Lebesgue measure ???-or, better to say, convex combinations of Lebesgue measures by interpolation parameters of nonlocal type.
First of all, we show that if E is a bounded set with smooth boundary, then where we denoted by κ n the n-dimensional volume of the n-dimensional unit ball. For further convenience, we also use the notation ω n := H n−1 (S n−1 ).
Notice that, by polar coordinates, We point out that formula (A.1) is indeed a simple version of more general approximation results, for which we refer to [BBM02,Dáv02,ADPM11,Pon04,CV11] and to [CV13] for the regularity results that can be achieved by approximation methods. See also [Dip14] for further comments and examples.
The proof of (A.1) can be performed by different methods; here we give a simple argument which uses formula (1.13). To this aim, we fix x ∈ ∂E and δ > 0. If y ∈ (∂E) ∩ B δ (x) and δ is sufficiently small, then ν(y) = ν(x) + O(δ). Moreover, for any ∈ (0, δ], the (n − 2)-dimensional contribution of ∂E in ∂B (x) coincides, up to higher orders in δ, with the one of the (n − 2)-dimensional sphere, that is ω n−1 n−2 , see Figure 14.
Now we show that, if n 3 and E is a bounded set with smooth boundary, Once again, more general (and subtle) statements hold true, see [MS02,DFPV13] for details.
To prove (A.3), we denote by Γ(x) := 1 (n − 2) ω n |x| n−2 the fundamental solution 7 of the Laplace operator when n 3, that is −∆Γ(x) = δ 0 (x), 7 It is interesting to understand how the fundamental solution of the Laplacian also occurs when n = 2. In this case, we observe that if c E := ∂E ν(y) dH n−1 (y), then of course where δ 0 is the Dirac's Delta centered at the origin. Then, from (1.13), lim s 0 s Per s (E, R n ) = 1 2 (n − 2) ∂E ∂E ν(x) · ν(y) |x − y| n−2 dH n−1 (y) dH n−1 (x) We remark that formula (A.3) is actually a particular case of a more general phenomenon, described in [DFPV13]. For instance, if the following limit exists Notice indeed that (A.3) is a particular case of (A.4), since when E is bounded, then a(E) = 0. Equation (A.4) has also a suggestive interpretation, since it says that, in a sense, as s 0, the fractional perimeter is a convex interpolation of measure contributions inside the reference set Ω: namely it weights the measures of two contributions of E and the complement of E inside Ω by a convex parameter a(E) ∈ [0, 1] which in turn takes into account the behavior of E at infinity.
Appendix B. A sketchy discussion on the asymptotics of the s-mean curvature As s 1/2, the s-mean curvature recovers the classical mean curvature (see [AV14] for details). A very natural question raised to us by Jun-Cheng Wei dealt with the asymptotics as s 0 of the s-mean curvature. Notice that, by (A.3), we know that 2s times the s-perimeter approaches ω n times Hence, we write 1 |x − y| 2s = exp (−2s log |x − y|) = 1 − 2s log |x − y| + O(s 2 ), thus 1 2s ν(x) · ν(y) log |x − y| dH n−1 (x) dH n−1 (y) + O(s) and one can use the same fundamental solution trick as in the case n 3. the volume. Since the variation of the volume along normal deformations is 1, if one is allowed to "exchange the limits" (i.e. to identify the limit of the variation with the variation of the limit), then she or he may guess that 2s times the s-mean curvature should approach ω n . This is indeed the case, and higher orders can be computed as well, according to the following observation: if E has smooth boundary, p ∈ ∂E and E ⊆ B R (p) for some R > 0, then as s 0. To prove this, we first observe that, up to a translation, we can take p = 0. Moreover, since E lies inside B R , In addition, since ∂E is smooth, we have that (possibly after a rotation) there exists δ o ∈ 0, min{1, R} such that, for any δ ∈ (0, δ o ], E ∩ B δ contains {x n −M |x | 2 } and is contained in {x n M |x | 2 }. Here, M > 0 only depends on the curvatures of E and we are using the notation x = (x , x n ) ∈ R n−1 × R (notice also that since we took p = 0, the ball B δ is actually centered at p). Therefore, we have that In this way, a cancellation gives that As a consequence, for any σ ∈ [0, s], if s ∈ (0, 1/4), Therefore, we use this inequality with σ := 0 and σ := s and the Dominated Convergence Theorem, to find that Hence, since we can now take δ arbitrarily small, we conclude that In view of this, and recalling (1.14) and (B.2), we find that This proves (B.1).
A rigorous statement for this claim will be given in the forthcoming Lemma C.1: for the moment, we remark that the expression above is related with the Jacobi field along surfaces of vanishing nonlocal mean curvature. We refer to [?delpino] for full details about this type of formulas. See in particular formula (1.6) there, which gives the details of this formula, Lemma A.2 there, which shows that, as s 1/2, the first integral approaches the Laplace-Beltrami operator and Lemma A.4 there, which shows that the latter integral produces, as s 1/2, the norm squared of the second fundamental form, in agreement with the classical case.
Here, for simplicity, we reduce to the case in which E is a graph and we consider a small normal deformation of its boundary, plus an additional small translation, and we write the resulting manifold as an appropriate normal deformation. The details go as follows: Lemma C.1. Let Σ ⊂ R n be a graph of class C 2 , and let E be the corresponding epigraph. Let ν = (ν 1 , . . . , ν n ) be the exterior normal of Σ = ∂E.
An interesting consequence of Lemma C.1 is that graphs with vanishing nonlocal mean curvature cannot have horizontal normals, as given by the following result: Theorem C.2. Let E ⊂ R n . Suppose that ∂E is globally of class C 2 and that H s E (x) = 0 for any x ∈ ∂E.
To prove Theorem C.2, we first compare deformations and translations of a graph. Namely, we show that a normal deformation of size εν n of a graph with normal ν = (ν 1 , . . . , ν n ) coincides with a vertical translation of the graph itself, up to order of ε 2 . The precise result goes as follows: Lemma C.3. Let Σ ⊂ R n be a graph of class C 2 globally, and let E be the corresponding epigraph. Let ν = (ν 1 , . . . , ν n ) be the exterior normal of Σ = ∂E.
Given ε > 0, let Then, if ε is sufficiently small, Σ ε is a graph, for some epigraph E ε , and there exists a C 2 -diffeomorphism Ψ of R n that is Cε 2 -close to the identity in C 2 (R n ), for some C > 0, such that Ψ(E ε ) = E + εe n .
As a consequence, by the Global Inverse Function Theorem (see e.g. Corollary 4.3 in [Pal59]), we have that y is a global diffeomorphism of class C 2 of R n−1 , we write x = x (y ) the inverse diffeomorphism and we have that Σ ε = y , γ x (y ) + ε 1 + ∇γ x (y ) 2 , y ∈ R n−1 .
From Lemma C.3 here and Theorem 1.1 in [Coz15], we obtain: