A Comparison Between the Nonlocal and the Classical Worlds: Minimal Surfaces, Phase Transitions, and Geometric ﬂows

A comparison between the nonlocal and the classical worlds: minimal surfaces, phase transitions, and geometric ﬂows

Serena Dipierro ∗ March 31, 2020

The nonlocal world presents an abundance of sur- and the σ-perimeter of a set E in Rn as the prises and wonders to discover. These special proper- σ-interaction between E and its complement Ec, ties of the nonlocal world are usually the consequence namely n c of long-range interactions, which, especially in pres- Perσ(E, R ) := Iσ(E,E ). (2) ence of geometric structures and nonlinear phenomena, end up producing a variety of novel patterns. We To deal with local minimizers, given a domain Ω ⊂ n will briefly discuss some of these features, focusing on R (say, with sufficiently smooth boundary), it is also the case of (non)local minimal surfaces, (non)local convenient to introduce the notion of σ-perimeter of phase coexistence models, and (non)local geometric a set E in Ω. To this end, one can consider the in- flows. teraction in (2) as composed by four different terms (by considering the intersections of E and Ec with Ω and Ωc), namely we can rewrite (2) as

1 Nonlocal minimal surfaces n c Perσ(E, R ) = Iσ(E ∩ Ω,E ∩ Ω) + I (E ∩ Ω,Ec ∩ Ωc) + I (E ∩ Ωc,Ec ∩ Ω) (3) In [CRS10], a new notion of nonlocal perimeter has σ σ c c c been introduced, and the study of the correspond- + Iσ(E ∩ Ω ,E ∩ Ω ). ing minimizers has started (related energy functionals had previously arisen in [Vis91] in the con- Among the four terms in the right hand side of (3), text of phase systems and fractals). The simple, the first three terms take into account interactions in but deep idea, grounding this new definition con- which at least one contribution comes from Ω (specif- c sists in considering pointwise interactions between ically, all the contributions to Iσ(E ∩Ω,E ∩Ω) come c c disjoint sets, modulated by a kernel. The proto- from Ω, while the contributions to Iσ(E ∩Ω,E ∩Ω ) c c type of these interactions considers kernels which and Iσ(E ∩ Ω ,E ∩ Ω) come from the interactions of c have translational, rotational, and dilation invari- points in Ω with points in Ω ). The last term in (3) is structurally different, since it only takes into account arXiv:2003.13234v1 [math.AP] 30 Mar 2020 ance. Concretely, given σ ∈ (0, 1), one considers the n contributions coming from outside Ω. It is therefore σ-interaction between two disjoint sets F and G in R as defined by natural to define the σ-perimeter of a set E in Ω as the collection of all the contributions in (3) that take ZZ dx dy into account points in Ω, thus defining Iσ(F,G) := n+σ , (1) F ×G |x − y| c Perσ(E, Ω) := Iσ(E ∩ Ω,E ∩ Ω) c c c c (4) ∗University of Western Australia, Department of Mathe- + Iσ(E ∩ Ω,E ∩ Ω ) + Iσ(E ∩ Ω ,E ∩ Ω), matics and Statistics, 35 Stirling Highway, WA 6009, Crawley, Australia. Email address: [email protected] see Figure 1

1 integral fashion weighted by the interaction kernel, in a precise form given by (5). This interpretation is indeed closely related with the classical mean curvature, in which the size of E is measured only “in the vicinity of the boundary”, since (up to normalizing constants) Z 1 1 HE(x) = lim ΞE(y) dy, for x ∈ ∂E. ρ&0 n+1 ρ Bρ(x)

The settings in (2) and (5) also reveal the “fractional notion” of these nonlocal minimal surfaces. As a mat- Figure 1: Long-range interactions leading to the fractional ter of fact, if one considers the fractional Gagliardo perimeter of the set E in Ω. seminorm defined, for every s ∈ (0, 1), by s This notion of perimeter recovers the classi- ZZ |f(x) − f(y)|2 s n [f]H (R ) := n+2s dx dy, cal one as σ % 1 in various senses, see n× n |x − y| e.g. [BBM01,Dáv02,ADPM11,CV11,Lom18], and in- R R deed the analogy between the minimizers of (4) with it follows that respect to prescribed sets outside Ω (named “nonlo- 2 cal minimal surfaces” in [CRS10]) and the minimiz- [ΞE] σ/2 n = 8 Perσ(E). H (R ) ers of the classical perimeter (the “classical minimal surfaces”) has been widely exploited in [CRS10] as Moreover, defining the fractional Laplacian as well in a series of subsequent articles, such as [CV13]. Z f(x) − f(y) Notwithstanding the important similarities between s (−∆) f(x) := n+2s dy, n |x − y| the nonlocal and the classical settings, many striking R differences arise, as we will also describe in this note. and, for a smooth set E, identifying a.e. the func- Minimizers, and more generally critical points, of tion ΞE with the σ-perimeter satisfy (with a suitable, weak or vis-  −1 if x is in the interior of E, cosity, meaning, and in a principal value sense) the  c equation ΞeE(y) := 1 if x is in the interior of E ,  Z 0 if x ∈ ∂E, σ ΞE(y) HE(x) := n+σ dy = 0, n |x − y| R (5) one sees that, for x ∈ ∂E, for x ∈ (∂E) ∩ Ω, Z σ ΞeE(y) being HE(x) = n+σ dy n |x − y| ( R −1 if x ∈ E, Z Ξ (y) − Ξ (x) (6) ΞE(y) := c eE eE 1 if x ∈ E , = n+σ dy n |x − y| R see [CRS10, CL]. The quantity Hσ can be seen as a σ/2 E = −(−∆) ΞeE(x). “nonlocal mean curvature”, and indeed it approaches 1 in various senses the classical mean curvature HE In this sense, one can relate “geometric” equations, (though important differences arise, see [AV14]). In- such as (5), with “linear” equations driven by the terestingly, the nonlocal mean curvature measures fractional Laplacian (as in (6)), in which however the the size of E with respect to its complement, in an nonlinear feature of the problem is encoded by the

2 fact that the equation takes place on the boundary 2.1 Interior regularity for classical of a set (once again, however, sharp differences arise minimal surfaces between nonlocal minimal surfaces and solutions of linear equations, as we will discuss in the rest of this Classical minimal surfaces are smooth up to dimen- note). sion 7. This has been established in [DG65] when n = 3, in [Alm66] when n = 4 and in [Sim68] when n 6 7. From the considerations above, we see how the Also, it was conjectured in [Sim68] that this regu- study of nonlocal minimal surfaces is therefore re- larity result was optimal in dimension 7, suggesting lated to the one of hypersurfaces with vanishing as a possible counterexample in dimension 8 the cone nonlocal mean curvature, and a direction of re- 4 4 search lies in finding correspondences between criti- {(x, y) ∈ R × R s.t. |x| < |y|}. cal points of nonlocal and classical perimeters. With respect to this point, we recall that double heli- This conjecture was indeed positively assessed coids possess both vanishing classical mean curva- in [BDGG69], thus showing that classical minimal ture and vanishing nonlocal mean curvature, as no- surfaces can develop singularities in dimension 8 and ticed in [CDDP16]. Also, nonlocal catenoids have higher. been constructed in [DdPW18] by bifurcation meth- Besides the smoothness, the analyticity of minimal ods from the classical case. Differently from the lo- surfaces in dimension up to 7 has been established cal situation, the nonlocal catenoids exhibit linear in [DG57, Mos60, DG61]. See also [Tru71, Rei64] for growth at infinity, rather than a logarithmic one, see more general results. also [CV]. Comprehensive discussions about the regularity The cases of hypersurfaces with prescribed non- of classical minimal surfaces can be found e.g. local mean curvature and of minimizers of nonlo- in [Sim83, Sim93, DHT10, DL16, CF17]. cal perimeter functionals under periodic conditions have also been considered in the literature, see 2.2 Interior regularity for nonlocal e.g. [DdPDV16,CFW18a,CFW18b,CFSMW18]. For minimal surfaces completeness, we also mention that we are restrict- ing here to the notion of nonlocal minimal surfaces Differently from the classical case, the regularity the- of fractional type, as introduced in [CRS10]. Other ory for nonlocal minimal surfaces constitutes a chal- very interesting, and structurally different, notions of lenging problem which is still mostly open. Till now, nonlocal minimal surfaces have also been considered a complete result holds only dimension 2, since the in the literature, see [MRT19b,CSV19,MRT19a] and smoothness of nonlocal minimal surfaces has been es- the references therein. tablished in [SV13]. In higher dimensions, the results in [CV13] give that nonlocal minimal surfaces are smooth up to 2 Interior regularity for mini- dimension 7 as long as σ is sufficiently close to 1: mal surfaces namely, for every n ∈ N ∩ [1, 7] there exists σ(n) ∈ [0, 1) such that all σ-nonlocal minimal surfaces in di- A first step to understand the geometric properties mension n are smooth provided that σ ∈ (σ(n), 1). of (both classical and nonlocal) minimal surfaces is The optimal value of σ(n) is unknown (except to detect their regularity properties and the possible when n = 1, 2, in which case σ(n) = 0). singularities. While the regularity theory of classical This result strongly relies on the fact that the minimal surfaces is a rather well established topic, nonlocal perimeter approaches the classical perime- many basic regularity problems in the nonlocal mini- ter when σ % 1, and therefore one can expect that mal surfaces setting are open, and they require brand when σ is close to 1 nonlocal minimal surfaces inherit new ideas to be attacked. the regularity of classical minimal surfaces.

3 On the other hand, we remark that it is not possi- Interestingly, the notion of stability in the nonlocal ble to obtain information on the regularity of nonlo- regime provides stronger information with respect to cal minimal surfaces from the asymptotics of the frac- the classical counterpart: for instance, if E is stable tional perimeter as σ & 0. Indeed, it has been proved for the nonlocal perimeter in B2R, then in [DFPV13] that the fractional perimeter converges, roughly speaking, to the Lebesgue measure of the set n−1 when σ & 0, and so in this case minimizers can be Per(E,BR) 6 CR , (7) as wild as they wish. As a matter of fact, the smoothness obtained where C > 0 is a constant depending only on n and σ. in [SV13] and [CV13] is of C1,α-type, the improve- ment to C∞ has been obtained in [BFV14]. We stress that the left hand side in (7) involves Indeed, nonlocal minimal surfaces enjoy an “im- the classical perimeter (not the nonlocal perimeter), provement of regularity” from locally Lipschitz hence an estimate of this type is quite informative to C∞ (more precisely, from locally Lipschitz to C1,α also for minimizers (not only for stable sets). In a for any α < σ, thanks to [FV17], and from C1,α for sense, up to now, the perimeter estimate in (7) is some α > σ/2 to C∞, thanks to [BFV14]). the only regularity results known for nonlocal minimal surfaces (and, more generally, for nonlocal stable It is an open problem to establish whether smooth surfaces) in any dimension (differently from [SV13] nonlocal minimal surfaces are actually analytic. It and [CV13], this estimate does not imply the smooth- is also open to determine whether or not singular ness of the surface, but only a bound on the perime- nonlocal minimal surfaces exist (a preliminary analy- ter). sis performed in [DdPW18] for symmetric cones may lead to the conjecture that nonlocal minimal surfaces are smooth up to dimension 6, but completely new Moreover, the right hand side of (7) is uniform phenomena may arise in dimension 7). with respect to the external data of E. In partic- Quantitative versions of the regularity results ular, wild data are shown to have a possible impact for nonlocal minimal surfaces have been obtained on the perimeter of E near the boundary, but not in in [CSV19]. the interior (for instance, nonlocal minimal surfaces (and, more general, nonlocal stable surfaces) in B As a matter of fact, in [CSV19] also more general 2 have a uniformly bounded perimeter in B ). This is a interaction kernels have been considered, and regu- 1 remarkable property, heavily relying on the nonlocal larity results have been obtained for more general structure of the problem, which has no counterpart critical points than minimizers. In particular, one in the classical case. As an example, one can con- can consider stable surfaces with vanishing nonlocal sider a family of parallel hyperplanes, which is a local mean curvature as critical points with “nonnegative minimizer for the perimeter: since each hyperplane second variations” of the functional. More precisely, produces a certain perimeter contribution in B , no one says that E is “stable” for the nonlocal perime- 1 uniform bound can be attained in this situation. In ter in Ω if Per (E, Ω) < +∞ and for every vector σ this sense, a bound as in (7) prevents arbitrary fam- field X = X(x, t) ∈ C2( n × (−1, 1); n), which is R R ilies of possibly perturbed hyperplanes to be stable compactly supported in Ω and whose integral flow t surfaces for the nonlocal perimeter. is denoted by ΦX , and for every ε > 0, there exists t0 > 0 such that t 2 Perσ ΦX (E) ∪ E − Perσ(E) + εt > 0 Concerning the regularity of stable sets for the non- t 2 local perimeter, we also mention that half-spaces are and Perσ ΦX (E) ∩ E − Perσ(E) + εt > 0. the only stable cones in R3, if the fractional parame- for all t ∈ (−t0, t0). ter σ is sufficiently close to 1, as proved in [CCS].

4 3 The Dirichlet problem for order to obtain a solution of (8), it is crucial that minimal surfaces the Lipschitz seminorm of the minimizer in XM is in fact strictly smaller than M (as long as M is chosen The counterpart of the theory of minimal surfaces conveniently large), so to obtain an interior mini- consists in finding solutions with graphical structure mum of the area functional, and thus find (8) as the for given boundary or exterior data. In the classical Euler-Lagrange equation of this minimization proce- setting, this corresponds to studying graphs with van- dure (the Lipschitz bound permitting the use of uni- ishing mean curvature inside a given domain with a formly elliptic regularity theory for PDEs, leading to prescribed Dirichlet datum along the boundary of the the desired smoothness of the solution inside the do- domain. Its nolocal counterpart consists in studying main). graphs with vanishing nonlocal mean curvature inside Finding sufficient and necessary conditions for this a given domain with a prescribed datum outside this procedure to work, and, in general, for obtaining solu- domain. We discuss now similarities and differences tions of (8) has been a classical topic of investigation. between these two problems. The main lines of this research took into account a “bounded slope condition” on the domain and the 3.1 The Dirichlet problem for classical datum that allows one to exploit affine functions as minimal surfaces barriers. In this, the convexity of Ω played an important role for the explicit construction of linear bar- A classical problem in geometric analysis is to seek riers. As a matter of fact, the first existence results hypersurfaces with vanishing mean curvature and for problem (8) dealt with the planar case n = 2 and prescribed boundary data. Namely, given a smooth convex domains Ω, see [Ber10, Haa27, Rad30]. domain Ω ⊂ Rn and a boundary datum ϕ ∈ C(∂Ω), Existence results in higher dimensions for convex the problem is to find u ∈ C2(Ω) ∩ C(Ω) that solves domains have been established in [Gil63, Sta63]. the Dirichlet problem The optimal conditions for existence results were  discovered in [JS68] and rely on the notion of “mean div √ ∇u = 0 in Ω, convexity” of the domain: namely, if Ω has C2 bound- 1+|∇u|2 (8) ary, problem (8) is solvable for every continuous  u = ϕ on ∂Ω. boundary datum ϕ if and only if the mean curva- A classical approach to this problem (often referred ture of Ω is nonnegative (when n = 2, the notion of to with the name of “Hilbert-Haar existence theory” mean convexity boils down to the usual convexity). see Chapter 3.1 of [MM84] and the references therein) For general results of this flavor, see Theorem 16.11 consists in fixing M > 0 and using the Ascoli Theo- in [GT01] and the references therein. rem to minimize the area functional among functions in 3.2 The Dirichlet problem for nonlo-

0,1 cal minimal surfaces XM := u ∈ C (Ω) with u = ϕ on ∂Ω Given a smooth domain Ω ⊂ Rn, one considers the and [u]C0,1(Ω) 6 M , cylinder Ω? := Ω × R and looks for local minimizers where, as customary, we denote the Lipschitz semi- of the nonlocal perimeter among all the sets with pre- norm of u by scribed datum outside Ω? (more precisely, one seeks |u(x) − u(y)| local minimizers of the nonlocal perimeter in Ω,e for ? [u]C0,1(Ω) := sup . every smooth and bounded Ωe ⊂ Ω , see [Lom18] for x,y∈Ω |x − y| x6=y all the details of this construction). To use this direct minimization approach, one needs In [DSV16], it is shown that this problem is solv- of course to check that XM 6= ∅. Furthermore, in able in the class of graphs. More precisely, if E is a

5 Figure 2: An example of stickiness (the red is the prescribed Figure 3: Another example of stickiness (the red is the pre- set, the green is the s-minimizer). scribed set, the green is the s-minimizer).

? local minimizer of the σ-perimeter in Ω such that its Interestingly, one of the main ingredients in the ? datum outside Ω has a graphical structure, namely proof of the genericity of the boundary discontinuity ? n in [DSVb] consists in an “enhanced boundary regular- E \ Ω = {xn+1 < u0(x), x ∈ R \ Ω}, (9) ity”: namely, nonlocal minimal surfaces in the plane n for some continuous function u0 : R → R, then E with a graphical structure that are continuous at the ? has a graphical structure inside Ω as well, that is 1, 1+σ boundary are automatically C 2 at the boundary ? E ∩ Ω = {xn+1 < u(x), x ∈ Ω}. (10) (that is, boundary continuity implies boundary dif- ferentiability). for some continuous function u : Rn → R. A byproduct of this construction, taking into Remarkably, in general this problem may lose con- account also the boundary properties obtained tinuity at the boundary of Ω?. That is, in the setting in [CDSS16], is also that nonlocal minimal surfaces of (9) and (10), it may happen that with a graphical structure in the plane exhibit a “but- lim u(x) 6= lim u0(x). (11) terfly effect” for their boundary derivatives: namely, x→∂Ω x→∂Ω x∈Ω x∈Ωc while a nonlocal minimal graph that is continuous at The first example of this quite surprising phe- the boundary has also a finite derivative there, an ar- nomenon was given in [DSV17], and this is indeed bitrarily small perturbation of the exterior graph will part of a general and remarkable structure of non- produce not only a jump at the boundary but also an local minimal surfaces that we named “stickiness”: infinite boundary derivative (that is, the boundary namely, nonlocal minimal surfaces have the tendency derivative switches from a finite, possibly zero, value to stick at the boundary of the domain (even when to infinity only due to an arbitrarily small perturba- the domain is convex, in sharp contrast with the pat- tion of the datum). We refer to [DDFV] for a survey tern exhibited by classical minimal surfaces). See e.g. about this type of phenomena. Figures 2 and 3 for some qualitative examples. It is worth recalling that the boundary discontinu- Recently, in [DSVb], it was discovered that the ity of nonlocal minimal surfaces is not only a special boundary discontinuity pointed out in (11) is indeed feature of the nonlocal world, when compared to the a “generic” phenomenon at least in the plane: namely case of classical minimal surfaces, but also a special if a given graphical external datum produces a con- feature of the nonlinear equation of nonlocal mean tinuous minimizer, then an arbitrarily small pertur- curvature type, when compared to the case of linear bation of it will produce a minimizer exhibiting the fractional equations: indeed, as detailed in [ROS14], boundary discontinuity. solutions of linear equations of the type (−∆)su = f

6 in a domain Ω with an external datum are typi- 4 Growth and Bernˇste˘ınprop- cally Höldercontinuous up to the boundary of Ω, erties of minimal graphs and no better than this (instead, nonlocal minimal surfaces are typically discontinuous at the boundary, A classical question, dating back to [Ber27] is whether but when they are continuous they are also differen- or not minimizers of a given geometric problem which tiable). have a global graphical structure are necessarily hyperplanes. If this feature holds, we say that the prob- Though this discrepancy between nonlocal mini- lem enjoys the “Bernˇste˘ınproperty”. mal surfaces and solutions of linear equations seems This question is also related with the growth of a bit surprising at a first glance, a possible explana- these minimal graphs at infinity. We now discuss tion for it lies in the “accuracy” of the approximation and compare the classical and the nonlocal worlds that a linear equation can offer to an equation of ge- with respect to these features. ometric type, such as the one related to the nonlocal mean curvature. Namely, linear equations end up be- 4.1 Growth and Bernˇste˘ınproperties ing a good approximation of the geometric equation of classical minimal graphs with respect to the normal of the surface itself: hence, when the surface is “almost horizontal” the graphical The Bernˇste˘ınproblem for classical minimal surfaces structure of the geometric equation is well shadowed asks whether or not solutions of by its linear counterpart, but when the surface is “al- ! most vertical” the linear counterpart of the geometric ∇u n div = 0 in R equation should take into account not the graph of p1 + |∇u|2 the original surface but its “inverse” (i.e., the graph describing the surface in the horizontal, rather than are necessarily affine functions. vertical, direction). When n = 2, a positive answer to this question was provided in [Ber27]. The development of the theory We also refer to [DSVa] for a first analysis of the related to the classical Bernˇste˘ınproblem is closely stickiness phenomenon in dimension 3. linked to the regularity theory of classical minimal surfaces, since, as proved in [DG65], the falsity of the Bernˇste˘ınproperty in Rn would imply the existence Many examples of nonlocal minimal surfaces ex- of a singular minimal surface in Rn. hibiting the stickiness phenomenon are studied in detail in [BLV19], also with respect to a suitable param- In light of this connection, the Bernˇste˘ın prop- eter measuring the weighted measure of the datum at erty for classical minimal surfaces was established infinity. in [DG65] for n = 3, in [Alm66] for n = 4, and in [Sim68] up to n = 7 (compare with the references in Section 2.1). A counterexample when n = 8 In addition, very accurate and fascinating numeri- was then provided in [BDGG69], by constructing an cal simulations on the stickiness phenomenon of non- eight-dimensional graph in R9 which has vanishing local minimal surfaces have been recently performed mean curvature without being affine. in [BLN19]. As a counterpart of this kind of problems, a precise growth analysis of classical minimal surfaces We also remark that nonlocal minimal surfaces with graphical structure has been obtained in [Fin63] with a graphical structure enjoy special regularity when n = 2 and in [BDGM69] for all dimensions (see properties: for instance they are smooth inside their also [Tru72, BG72, Sim76, Kor86, DSJ11] for related domain of minimization, as established in [CC19]. results).

7 More precisely, if u is a solution of for a suitable constant C > 0, depending only on n and σ. ! ∇u n On the one hand, comparing (12) and (13), we div = 0 in BR ⊂ R , p1 + |∇u|2 observe that the nonlocal world offers us a wealth of surprises: first of all, the estimate in (13) is of then polynomial type, rather than of exponential type, as it was the bound in (12). Roughly speaking, this    sup u − u(0) is due to the fact that two different and sufficiently B2R sup |∇u| 6 exp C 1 +  , (12) close“ends” of a nonlocal minimal surface repel each B R R other (compare the discussion about the nonlocal catenoid on page 3). Moreover, the right hand side for some C > 0 depending only on n. Interestingly, of (13) presents the oscillation of the solution in the this exponential type of gradient estimate is optimal, same ball B , while (12) needed to extend the right as demonstrated in [Fin63]. R hand side to a larger ball such as B2R. On the other hand, the nonlocal world maintains 4.2 Growth and Bernˇste˘ınproperties its own special difficulties: for instance, differently of nonlocal minimal graphs from the classical case, it is an open problem to determine whether the estimate in (13) is optimal. The Bernˇste˘ınproperties of nonlocal minimal graphs have been investigated in [FV17, FV19]. In partic- See Table 1 for a sketchy description of some simi- ular, it is proved in these works that the falsity of larities and differences between classical and nonlocal the nonlocal Bernˇste˘ınproperty in Rn would imply minimal surfaces. the existence of a singular nonlocal minimal surface in Rn, thus providing a perfect counterpart of [DG65] to the nonlocal setting. 5 Phase coexistence models Combining this with the regularity results for non- A physical situation in which minimal interfaces nat- local minimal surfaces in [SV13] and [CV13], one ob- urally arise occurs in the mathematical description of tains that the Bernˇste˘ınproperty holds true for solu- phase coexistence (or phase transition). In the classi- tions u : Rn → R of the σ-mean curvature equation cal setting, the ansatz is that the short range particle in (5) provided that either n ∈ {1, 2}, or n 6 7 and σ interaction produces a surface tension that leads, on is sufficiently close to 1. a large scale, to interfaces of minimal area. A modern It is an open problem to prove or disprove the variation of this model takes into account long-range Bernˇste˘ınproperty for nonlocal minimal graphs u : particle interactions, and we describe here in which n sense this model is related to the theory of (non)local R → R in the general case when n > 2 and σ ∈ (0, 1) (the analysis on symmetric cones performed minimal surfaces. in [DdPW18] might suggest that new phenomena could arise in dimension 7). 5.1 Classical phase coexistence mod- Concerning the growth of nonlocal minimal sur- els faces, it has been established in [CC19] that if u : One of the most popular phase coexistence models is n B2R ⊂ R → R is a solution of the σ-mean curvature the one producing the so-called Allen-Cahn equation equation in B2R, then −∆u = u − u3. (14)  sup u − inf un+1+σ BR sup |∇u| C 1 + BR , (13) Equation (14) has a variational structure, corre- 6  R  BR sponding to critical points of an energy functional

8 classical minimal surfaces nonlocal minimal surfaces interior regularity up to n = 7 (optimal) known when n = 2, and up to n = 7 if σ is large enough (optimality unknown, no singular example available) boundary regularity yes for convex (or mean boundary discontinuity convex) domains (but enhanced diﬀer- entiable regularity if continuous) Bernˇste˘ın property up to n = 7 (optimal) known when n = 2, and up for u : Rn → R to n = 7 if σ is large enough (optimality unknown, no counterexample available) gradient growth at exponential (optimal) polynomial (optimality un- inﬁnity known)

Table 1: Classical versus nonlocal minimal surfaces

n of the type • for any uε : R → [−1, 1] converging to u in L1 ( n), we have that Z loc R F(u; Ω) = F(u) := |∇u(x)|2 + W (u(x)) dx, (15) Ω lim inf Fε(uε) F0(u), (18) ε&0 > where W is a “double-well” potential attaining its n n minima at the “pure phases” −1 and +1. • given u : R → [−1, 1], there exists uε : R → 1 n A classical link between the functional in (15) [−1, 1]converging to u in Lloc(R ), such that and the perimeter can be expressed in terms of Γ- lim sup Fε(uε) 6 F0(u), (19) convergence, as stated in [MM77]. More precisely, ε&0 given a smooth domain Ω ⊂ Rn and ε > 0, one can consider the rescaled functional where, for a suitable c > 0, Z 2 1  Fε(u; Ω) = Fε(u) := ε |∇u(x)| + W (u(x)) dx.  if u = ΞE Ω ε c Per(E, Ω) (16)  for some set E, F0(u; Ω) = F0(u) := Interestingly, the functional Fε in (16) inherits the   minimization properties of the functional F in (15) +∞ otherwise. after a dilation: for instance, if u is a minimizer for F (20) in every ball, then This type of “functional convergence” has also a “ge- x ometric counterpart”, given by the locally uniform uε(x) := u (17) convergence of levels sets, as established in [CC95]. ε More precisely, if u is a minimizer for F in ev- is a minimizer for Fε in every ball. Moreover, it holds ery ball and uε is as in (17), then uε converges, 1 n that: up to subsequences, in Lloc(R ) to some u = ΞE,

9 with E minimizing the perimeter, and the level sets where W is a “double-well” potential attaining its of uε approach ∂E locally uniformly, that is, given minima at the “pure phases” −1 and +1, and QΩ is any ϑ ∈ (0, 1), any R > 0 and any δ > 0 there ex- the cross-type domain given by ists ε0 ∈ (0, 1) such that, if ε ∈ (0, ε0), we have that (Ω × Ω) ∪ (Ω × Ωc) ∪ (Ωc × Ω). [ |uε| < 1 − θ ∩ BR ⊆ Bδ(p). (21) The rationale behind this choice of QΩ is to collect p∈∂E all the couples of points in which at least one of the The level set convergence in (21) relies on suitable two lies in Ω, so to comprise all the point interactions “energy and density estimates” stating that the en- that involve Ω (a similar choice was performed in the ergy behaves as an “interface” and, in a neighborhood geometric problem (4)). of the interface, both the phases have positive densi- The Γ-convergence theory for the functional in (25) ties. More precisely, if R > 1 and u is a minimizer has been established in [SV12]. For this, one needs of F in BR+1, then to introduce a family of rescaled functional with different scaling properties depending on the fractional n−1 FBR (u) 6 CR , (22) parameter s. More precisely, it is convenient to deﬁne for some C > 0, and, for every ϑ1, ϑ2 ∈ (0, 1), Z |u(x) − u(y)|2 if |u(0)| 6 ϑ1 then Eε(u; Ω) = Eε(u) := aε n+2s dx dy QΩ |x − y| n {u > ϑ2} ∩ BR > cR Z n (23) + bε W (u(x)) dx, and {u 6 −ϑ2} ∩ BR > cR . Ω (26)

5.2 Long-range phase coexistence where models  ε2s−1 if s ∈ (1/2, 1),  In the recent literature a number of models have been −1 aε := | log ε| if s = 1/2, introduced in order to study phase coexistence driven  by long-range particle interactions, see e.g. [AB98, 1 if s ∈ (0, 1/2), GP06]. and We describe here a simple model of this kind, com-  paring the results obtained in this situation with the ε−1 if s ∈ (1/2, 1),  classical ones presented in Section 5.1. −2s −1 −1 bε := ε aε = ε | log ε| if s = 1/2, We focus on a nonlocal version of the Allen-Cahn ε−2s if s ∈ (0, 1/2). equation (14) given by The coeﬃcients aε and bε are tuned with respect (−∆)su = u − u3, (24) to the energy of a one-dimensional layer joining −1 and +1 and, in this way, the functional Eε in (26) sat- with s ∈ (0, 1). Equation (24) has a variational struc- isﬁes some interesting Γ-convergence properties. Re- ture, corresponding to critical points of an energy markably, these properties are very sensitive to the functional of the type fractional parameter s: indeed, when s < 1/2 the Γ-convergence theory obtains the σ-perimeter intro- Z |u(x) − u(y)|2 E(u; Ω) = E(u) := dx dy duced in (4), with σ := 2s, but when s > 1/2, the |x − y|n+2s nonlocal character of the problem is lost in the limit QΩ (25) Z and the Γ-convergence theory boils down to the clas- + W (u(x)) dx, sical one in (18), (19) and (20). Ω

10 More precisely, one can deﬁne We refer to Table 2 for a schematic representation of similarities and diﬀerences of classical and nonlo-  if s ∈ [1/2, 1) cal phase transitions in terms of Γ-limits and density  c Per(E, Ω) and u = Ξ estimates.  E   for some set E, In addition, long- phase transition models and non-   local minimal surfaces have close connections with  spin models and discrete systems, see [ABCP96, E0(u; Ω) = E0(u) := if s ∈ (0, 1/2)  GL97, GLM00, ABC06, dlLV10,BP13, AG16] and the c Per2s(E, Ω) and u = ΞE  references therein. In particular, in [CDV17] the  for some set E,  reciprocal approximation of ground states for long-   range Ising models and nonlocal minimal surfaces has  +∞ otherwise. been established and the relation between these two problems has been investigated in detail.

In this sense, the functional E0 is the “natural replacement” of the one in (20) to deal with the Γ- 6 Geometric flows convergence of nonlocal phase transitions, in the sense that the statements in (18), (19) and (20) hold Another classical topic in geometric analysis consists true in this case with Fε replaced by Eε and F0 re- in the study of evolution equations describing the mo- placed by E0. tion of set boundaries. In this framework, at any time t, one prescribes the speed V of a hypersur- For additional discussions of nonlocal Γ- face ∂E in the direction of the outer normal to E . convergence results, see [DMV]. t t In this section, in light of the setting in (5), we The convergence of level sets in (21) has also a will describe analogies and differences between the perfect counterpart for the minimizers of E, as es- classical and the nonlocal mean curvature flows. tablished in [SV14]. These results also rely on suitable density estimates which guarantee that (23) also 6.1 Mean curvature flow holds true for minimizers of E. One of the widest problem in geometric analysis fo- However, in this case, an energy estimate as in (22) cuses on the case in which the normal velocity V of exhibits a significant difference, depending on the the flow is taken to be the classical mean curvature. fractional exponent s. Indeed, in this framework, we This flow decreases the classical perimeter and has a have that if R > 2 and u is a minimizer of E in BR+1, number of remarkable properties. A convenient de- then scription of the mean curvature flow can be given in terms of viscosity solutions, using a level set ap-  CRn−1 if s ∈ (1/2, 1), proach, see [ES91]. See also [CGG91, AAG95] for  related viscosity methods. If the initial hypersurface EBR (u) 6 CR log R if s = 1/2, (27) CRn−2s if s ∈ (0, 1/2), satisfies a Lipschitz condition, then the mean curvature flow possesses a short time existence theory, for some C > 0. see [EH91]. In general, the flow may develop singular- In spite of its quantitative difference with (22), the ities, and the understanding of the different possible estimate in (27) is sufficient to determine that the singularities is a very rich and interesting field of in- interface contribution is “negligible” with respect to vestigation in itself, see e.g. [Bra78, Eck04, Man11]. the Lebesgue measure of large balls. Moreover, such an estimate is in agreement with the energy of one- The planar case n = 2 presents some special fea- dimensional layers. ture. In this case, the flow becomes “curve shorten-

11 phase transitions s ∈ phase transitions s = phase transitions s ∈ (0, 1/2) 1/2 (1/2, 1] Γ-limit nonlocal perimeter classical perimeter classical perimeter energy estimates Rn−2s R log R Rn−1

Table 2: Classical versus nonlocal phase transitions

ing” in the sense that its special characteristic is to erally, the results in [Hui98] can be seen as a quan- decrease the length of a given closed curve as fast titative refinement of those in [GH86, Gra87]. See as possible. In this setting, it was proved in [Gra87] also [Hui84] for related results. that the curve shortening flow makes every smooth In any case, for the mean curvature flow, the for- closed embedded curves in the plane shrink smoothly mation of neckpinch singularity for the evolving hy- to round points (i.e., the curve shrinks becoming persurface can only occur (and it occurs) in dimen- “closer and closer to a circle”). More precisely: on sion n > 3, see [Hui90, GKS18]. the one hand, as shown in [GH86], if the initial curve is convex, then the evolving curve remains convex 6.2 Nonlocal mean curvature flow and becomes asymptotically circular as it shrinks; on the other hand, as shown in [Gra87], if the curve is A natural variant of the flow discussed in Section 6.1 only embedded, then no singularity develops before consists in taking as normal velocity the nonlocal the curve becomes convex, then shrinking to a round mean curvature of the evolving set, as given in (5). point. The main feature of this evolution problem is that it These results can be also interpreted in light of a decreases the nonlocal perimeter defined in (4). “distance comparison property”. Namely, given two This evolution problem emerged in [Imb09, CS10]. points p and q on the evolving surface, one can con- A detailed viscosity solution theory for the nonlocal sider the “extrinsic” distance d = d(p, q), given by mean curvature flow has been presented in [CMP13], the Euclidean distance of p and q, and the arc-length see also [CMP15] for more general results. n “intrinsic” distance ` = `(p, q), that measures the dis- In the viscosity setting, given an initial set E ⊂ R , tance of p and q along the curve. For a closed curve the evolution of E is described by level sets of a func- of length L, one can also define tion uE = uE(x, t) which solves a parabolic problem driven by the nonlocal mean curvature. More pre- L π` λ := sin . cisely, the evolution of E is trapped between an outer π L and an inner flows defined by Then, as proved in [Hui98], we have that the mini- + n E (t) := {x ∈ R s.t. uE(x, t) > 0} mal ratio ρ := min d is nondecreasing under the curve λ and E−(t) := {x ∈ n s.t. u (x, t) > 0}. shortening flow (and, in fact, strictly increasing un- R E less the curve is a circle). In this spirit, the ratio ρ The ideal case would be the one in which these two plays the role of an improving isoperimetric quantity flows coincide “up to a smooth manifold” which de- which measures the deviation of the evolving curve scribes the evolving surface, i.e. the case in which from a round circle. In particular, if a curve devel- oped a “neckpinch”, it would present two points for + − ΣE(t) := E (t) \ E (t) which d = 0 and ` > 0, hence ρ = 0: as a conse- (28) = {x ∈ n s.t. u (x, t) = 0} quence, the monotonicity proved in [Hui98] immedi- R E ately implies that planar curves do not develop neck- is a nice hypersurface, but, in general, such a property pinches under the curve shortening flow. More gen- is not warranted by the notion of viscosity solutions.

12 A short-time existence theory of smooth solutions nonlocal mean curvature flow of the cross E in (29) of the nonlocal mean curvature flow has been re- does not develop any fattening and, more precisely, cently established in [JL19] under the assumption the evolution of E would be E itself (in a sense, we that the initial hypersurface is of class C1,1. As far switch from a fattening situation, produced by sin- as we know, it is still an open problem to determine gular kernels with slow decay, to a “pinning effect” whether the nonlocal mean curvature flow possesses produced by smooth and compactly supported ker- a short-time existence theory of smooth solutions for nels). Lipschitz initial data (this would provide a complete counterpart of the classical results in [EH91]). Furthermore, there are cases in which the fattening phenomena are different according to the type of lo- It is certainly interesting to detect suitable infor- cal or nonlocal mean curvature flow that we consider mation that are preserved by the nonlocal mean cur- (see [ES91]). For instance, if the initial set is made vature flow. For instance: the fact that the nonlocal by two tangent balls in the plane such as mean curvature flow preserves the positivity of the nonlocal mean curvature itself has been established E := B (−1, 0) ∪ B (1, 0), (30) in [SV19], and the preservation of convexity has been 1 1 proved in [CNR17]. then, for the nonlocal mean curvature flow, using the An important difference between the classical and notation in (28), we have that ΣE(t) has empty in- the nonlocal mean curvature flows consists in the terior for all t > 0. This situation is different with “fattening phenomena” of viscosity solutions, in respect to the classical mean curvature flow, which which the set in (28) may develop a nonempty inte- immediately develops fattening (see [BP94]). rior (thus failing to be a nice hypersurface), as investigated in [CDNV19]. For instance, one can consider The nonlocal mean curvature flow (and its many the case in which the initial set is the “cross in the variants) are a great source of intriguing questions plane” given by and mathematical adventures. Among the interesting questions remained open, we mention that it is 2 E = {(x, y) ∈ R s.t. |x| > |y|}. (29) not known whether the nonlocal mean curvature flow possesses a nonlocal version of the monotonicity for- In this situation, using the notation in (28), one can mula in [Hui98], or what a natural replacement of it prove that Σ (t) is nontrivial and immediately de- E could be. This question is also related to a deeper un- velops a positive measure: more precisely, we have derstanding of the singularities of the nonlocal mean that ΣE(t) contains the ball B 1 , for some c > 0. ct 1+s curvature flow. On the one hand, this result is in agreement with the classical case, since also the classical mean curvature flow develops this kind of situations when start- Acknowledgments ing from the set in (29) (see [ES91]). On the other hand, the fattening phenomena for The author is member of INdAM/GNAMPA and the nonlocal situation offers a number of surprises AustMS. Her research is supported by the Australian with respect to the classical case (see [CDNV19]). Research Council Discovery Project DP170104880 First of all, the quantitative properties of the inter- “N.E.W. Nonlocal Equations at Work” and by the action kernels play a decisive role in the development DECRA Project DE180100957 “PDEs, free bound- of the fat portions of ΣE(t) and in general on the evo- aries and applications”. lution of the set E. For instance, if the interaction It is a pleasure to thank Enrico Valdinoci for 1 kernel |x−y|n+σ in (1) is replaced by a different kernel his comments on a preliminary version of this that is smooth and compactly supported, then the manuscript.

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