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

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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 phenom- ena, 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 func- tionals 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 2 (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 jx − yj 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 curva- ture, 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 2 @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 2 (0; 1), by s This notion of perimeter recovers the classi- ZZ jf(x) − f(y)j2 s n [f]H (R ) := n+2s dx dy; cal one as σ % 1 in various senses, see n× n jx − yj e.g. [BBM01,D´av02,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 jx − yj 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- 8 −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 2 @E; σ ΞE(y) HE(x) := n+σ dy = 0; n jx − yj R (5) one sees that, for x 2 @E, for x 2 (@E) \ Ω; Z σ ΞeE(y) being HE(x) = n+σ dy n jx − yj ( R −1 if x 2 E; Z Ξ (y) − Ξ (x) (6) ΞE(y) := c eE eE 1 if x 2 E ; = n+σ dy n jx − yj 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- f(x; y) 2 R × R s.t. jxj < jyjg: 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 2 N \ [1; 7] there exists σ(n) 2 [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 2 (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.
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