Convergence of supercritical fractional flows to the mean curvature flow
aa r X i v : . [ m a t h . A P ] F e b CONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWSTO THE MEAN CURVATURE FLOW
L. DE LUCA, A. KUBIN, AND M. PONSIGLIONE
Abstract.
We consider a core-radius approach to nonlocal perimeters gov-erned by isotropic kernels having critical and supercritical exponents, ex-tending the nowadays classical notion of s -fractional perimeter, defined for0 < s <
1, to the case s ≥ s -fractional perimeters, suitably scaled, Γ-converge to the standard Euclideanperimeter. Under the same scaling, the first variation of such nonlocal perime-ters gives back regularized s -fractional curvatures which, as the core radiusvanishes, converge to the standard mean curvature; as a consequence, we showthat the level set solutions to the corresponding nonlocal geometric flows, suit-ably reparametrized in time, converge to the standard mean curvature flow.Finally, we prove analogous results in the case of anisotropic kernels withapplications to dislocation dynamics. Keywords:
Fractional perimeters; Γ-convergence; Local and nonlocal geo-metric evolutions; Viscosity solutions; Level set formulation; Fractional meancurvature flow; Dislocation dynamics
AMS subject classifications:
Contents
Introduction 21. Supercritical perimeters 52. Proof of Compactness 122.1. Preliminary results 122.2. Proof of Theorem 1.5(i) 153. Proof of the Γ-limit 183.1. Proof of the lower bound 183.2. Proof of the upper bound 223.3. Characterization of sets of finite perimeter 234. Convergence of curvatures and mean curvature flows 244.1. Non-local k sr -curvatures 244.2. The classical mean curvature 304.3. Convergence of k sr -curvature flow to mean curvature flow 325. Stability as r → + and s → + simultaneously 395.1. Γ-convergence and compactness 395.2. Convergence of the k s n r n -curvature flows to the mean curvature flow 456. Anisotropic kernels and applications to dislocation dynamics 476.1. Anisotropic kernels 476.2. Applications to dislocation dynamics 50References 51 Introduction
This paper deals with systems governed by strongly attractive nonlocal poten-tials. Our analysis is geometrical, so that the energy functionals are defined on mea-surable sets rather than on densities, and can be understood as nonlocal perimeters,whose first variation are nonlocal curvatures, driving the corresponding geometricflows.We focus on power law pair potentials acting on measurable sets E ⊂ R d , whosecorresponding nonlocal energy is of the type(0.1) J s ( E ) := Z E Z E − | x − y | d + s d y d x. For − d < s < J s are nonlocal perimeters in the sense of [18]. Such a geometricinterpretation is supported by the fact that, as a consequence of Riesz inequality,balls are minimizers of J s under volume constraints; moreover, the first variationof J s , referred to as nonlocal curvature, is monotone with respect to set inclusion.The latter provides a parabolic maximum principle which yields global existenceand uniqueness of level set solutions to the corresponding geometric evolutions[18, 16].For positive s the kernel in (0.1) is not integrable, and the corresponding energyis infinite. Nevertheless, for 0 < s <
1, changing sign to the interaction and letting E interact with its complementary set instead of itself, gives a finite quantity: thewell-known fractional perimeter [12](0.2) ˜ J s ( E ) := Z E Z E c | x − y | d + s d y d x. In fact, fractional perimeters can been rigorously obtained as limits of renormal-ized Riesz energies by removing the infinite core energy and letting the core radiustend to zero. This has been done in [21], showing that the energies in (0.1) and(0.2) belong to a one parameter family of nonlocal s -perimeters, with − d < s < s = 0 a new perimeter emerges, referred to as0-fractional perimeter.Remarkably, as s → − , s -fractional perimeters, suitably scaled, converge to thestandard perimeter [10, 11, 19, 31, 7, 17], and the corresponding (reparametrizedin time) geometric flows converge to the standard mean curvature flow [26, 16].For s ≥ s = 1 corresponds, at least formally, to the Euclideanperimeter. Notice that for s = 1 the fractional perimeter can be seen, again for-mally, as the square of the (infinite) ˙ H Gagliardo seminorm of the characteristicfunction of E . This fractional energy is particularly relevant in Materials Science,for instance in the theory of dislocations. This is why much effort has been doneto derive the Euclidean perimeter directly as the limit of suitable regularizations ofthe ˙ H seminorm, mainly through phase field approximations [2]. A specific phasefield model for the energy induced by planar dislocations within the Nabarro-Peierlstheory has been introduced in [24] and studied in [22], where the line tension energyinduced by planar dislocations is derived in terms of Γ-convergence. In [20, 4, 13]the authors study dislocation dynamics within the level set formulation ´a la Slepˇcev[32], by considering the geometric flow associated to a suitable regularization of the ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 3 formal first variation of the line tension energy. Numerical schemes have beenimplemented in [3, 15].In this paper we introduce a core-radius approach to renormalize by scaling thegeneralized s -fractional perimeters and curvatures in the critical and supercriticalcases s ≥
1. We show that, as the core-radius tends to zero, the Γ-limit of thenonlocal perimeters is the Euclidean perimeter, the nonlocal curvatures convergeto the standard mean curvature, and the corresponding geometric flows convergeto the mean curvature flow. Moreover, we consider also the anisotropic variantsof such perimeters, with applications to dislocation dynamics. Now we discuss ourresults in more detail.In Section 1 we introduce the core-radius regularized critical and supercriticalperimeters (see (1.4)). In Theorem 1.5 we show that, suitably scaled, they Γ-converge to the Euclidean perimeter. This analysis is very related with, and insome respects generalizes, many results scattered in the literature, mainly for s > s = 1 in [22]. As a byproduct of our Γ-convergence analysis, we providea characterization of finite perimeter sets (Theorem 3.4) in terms of uniformlybounded renormalized supercritical fractional perimeters. Analogous results for0 < s < s = 1 and in [14] for 1 ≤ s < s -parabolic flow and, in turn, on thenotion of s -Laplacian, which is well defined only for 0 < s < s → + and the core-radius vanishes simultaneously(see Theorem 5.1).In Section 6 we generalize our results to the case of possibly anisotropic ker-nels (Subsection 6.1) and we present a relevant application to dislocation dynamics(Subsection 6.2). It is well known that planar dislocation loops formally induce aninfinite elastic energy that can be seen as an anisotropic version of the (squared) ˙ H seminorm of the characteristic function of the slip region enclosed by the dislocationcurve. As mentioned above, renormalization procedures are needed to cut off theinfinite core energy. In [20, 4, 13], the authors consider the geometric evolution ofdislocation loops and face the corresponding renormalization issues: their approachconsists in formally computing the first variation of the infinite energy induced bydislocations, deriving a nonlocal infinite curvature. Then, they regularize such acurvature through convolution kernels, obtaining a finite curvature driving the dy-namics. As the convolution regularization kernel concentrates to a Dirac mass, they L. DE LUCA, A. KUBIN, AND M. PONSIGLIONE recover in the limit a local anisotropic mean curvature flow. The main issue in theiranalysis is that the convolution regularization produces a positive part in the non-local curvature (corresponding to a negative contribution in the normal velocity),concentrated on the scale of the core of the dislocation, giving back an evolutionwhich does not satisfy the inclusion principle. Therefore, solutions exist only forshort time. Moreover, adding strong enough forcing terms, or assuming that thepositive part of the curvature is already concentrated on a point (instead of beingdiffused on the core region), they show that the curvature is in fact monotone withrespect to inclusion of sets; as a consequence, they get a globally defined dynam-ics, converging, as the core-radius vanishes, to the correct anisotropic local meancurvature flow. Here we show that, if one first regularizes the nonlocal perimetersremoving the core energy and then computes the corresponding first variation, thenthe positive part of the curvature is actually concentrated on a point (see Remark6.3), so that the mathematical assumption in [20] is physically correct and fullyjustified through the solid core-radius formalism. Finally, in Subsection 6.2 weshow that the convergence analysis of the geometric flows done in [20] using theapproach [32] can be directly deduced from the analysis developed in Section 4 andSubsection 6.1, providing then a self-contained proof relying on the general theoryof nonlocal evolutions and their stability developed in [18, 16].This paper, together with [21, 16] completes the variational analysis of nonlocalinteractions (0.1) and of the corresponding geometric flows for all positive valuesof d + s . Such functionals account Riesz functionals for − d < s <
0. After suitablerenormalization procedures, the case s = 0 leads to the 0-fractional perimeter,while for 0 < s < s -fractional perimeters. Thispaper focuses on the supercritical case s ≥ Acknoledgments:
The authors are grateful to Annalisa Cesaroni and MatteoNovaga for preliminary discussions at the early stage of the project. The authors aremembers of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loroApplicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Notation:
We work in the space R d where d ≥ { e j } j =1 ,...,d the canonical basis of R d . We denote by R d × p the set of the matrices having d rowsand p columns. The symbol | · | stands for the Lebesgue measure in R d , M( R d ) isthe family of measurable subsets of R d , whereas M f ( R d ) ⊂ M( R d ) is the family ofsubsets if R d having finite measure. We will always assume that every measurableset E coincides with its Lebesgue representative, i.e., with the set of points atwhich E has density equal to one. Moreover, for every p >
0, we denote by H p the p -Hausdorff measure.For every x ∈ R d and for every r >
0, we denote by B ( x, r ) the open ballof radius r centered at x and by B ( x, r ) its closure. Moreover, we set S d − := ∂B (0 , ω d := | B ( x, | and we recallthat H d − ( ∂B ( x, r )) = dω d r d − . Sometimes, we will consider also subsets of R d − .In such a case, we denote by B ′ ( ξ, ρ ) the ball centered at ξ ∈ R d − and having radius ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 5 equal to ρ > ω d − := H d − ( B ′ ( ξ, H d − ( B ′ ( ξ, ρ )) = ω d − ρ d − and H d − ( ∂B ′ ( ξ, ρ )) = ( d − ω d − ρ d − . Furthermore, we set Q := [ − , ) d andfor every ν ∈ S d − we set Q ν := R ν Q , where R ν is a (arbitrarily chosen) rotationsuch that R ν e d = ν .For every set E ∈ M( R d ) we denote by Per( E ) the De Giorgi perimeter of E defined byPer( E ) := sup (cid:26) Z E DivΦ( x ) d x : Φ ∈ C ( R d ; R d ) , k Φ k ∞ ≤ (cid:27) . For every E ∈ M( R d ) , the set ∂ ∗ E identifies the reduced boundary of E and ν E : ∂ ∗ E → R d the outer normal vector field. For all y ∈ R d and for every ν ∈ S d − we set H − ν ( x ) := (cid:8) y ∈ R d : ν · ( y − x ) ≤ (cid:9) , (0.3) H + ν ( x ) := (cid:8) y ∈ R d : ν · ( y − x ) ≥ (cid:9) , (0.4) H ν ( x ) := (cid:8) y ∈ R d : ν · ( y − x ) = 0 (cid:9) . (0.5)For every subset E ⊂ R d the symbol E c denotes its complementary set in R d , i.e., E c := R d \ E .Finally, we denote by C ( ∗ , · · · , ∗ ) a constant that depends on ∗ , · · · , ∗ ; such aconstant may change from line to line.1. Supercritical perimeters
Let s ≥ r >
0, we define the interaction kernel k sr : [0 , + ∞ ) → [0 , + ∞ ) as(1.1) k sr ( t ) := r d + s for 0 ≤ t ≤ r , t d + s for t > r , We note that(1.2) k sr ( lt ) = l − d − s k s rl ( t ) for every r, l, t > . For all r >
0, we define the functional J sr : M( R d ) → [ −∞ ,
0] as J sr ( E ) := Z E Z E − k sr ( | x − y | ) d y d x and for every E ∈ M f ( R d ) we set(1.3) ˜ J sr ( E ) := J sr ( E ) + λ sr | E | , where λ sr := Z R d k sr ( | z | ) d z = ( d + s ) ω d sr s . Notice that for every E ∈ M f ( R d ) J sr ( E ) ≥ − Z E Z R d k sr ( | x − y | ) d y d x = − λ sr | E | , L. DE LUCA, A. KUBIN, AND M. PONSIGLIONE and hence ˜ J sr : M f ( R d ) → [0 , + ∞ ) . Moreover, by the very definition of ˜ J sr in (1.3),for every E ∈ M f ( R d ) we have(1.4) ˜ J sr ( E ) = Z E Z E c k sr ( | x − y | ) d y d x . We first state the following result concerning the pointwise limit of the functionals˜ J sr as r → + . To this purpose, for every s ≥ σ s ( r ) := | log r | if s = 1 d + sd + 1 r − s s − s > . Proposition 1.1.
Let s ≥ and let E ∈ M f ( R d ) be a smooth set. Then, (1.6) lim r → + ˜ J sr ( E ) σ s ( r ) = ω d − Per( E ) , where σ s is defined in (1.5) . In fact, for s > formula (1.6) holds for every set E ∈ M f ( R d ) of finite perimeter. The proof of Proposition 1.1 is postponed and will use, in particular, Proposition1.4 below. For every E ∈ M f ( R d ) we define the functionals F s ( E ) := Z E Z E c ∩ B c ( x, | x − y | d + s d y d x , (1.7) G sr ( E ) := Z E Z E c ∩ B ( x, k sr ( | x − y | ) d y d x , (1.8)and we notice that for every 0 < r < J sr ( E ) = F s ( E ) + G sr ( E ) . Remark 1.2.
It is easy to see that, for every E ∈ M f ( R d ), it holds F s ( E ) ≤ Z E Z B c (0 , | z | d + s d z = dω d s | E | . Let s ≥ r > T sr : R d \ { } → R d as(1.10) T sr ( x ) := − s x | x | d + s if | x | ∈ [ r, + ∞ ) ,xdr d + s − d + sdsr s x | x | d if | x | ∈ (0 , r ) . A direct computation shows that(1.11) Div( T sr ( x )) = k sr ( | x | ) . ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 7
Lemma 1.3.
Let E ∈ M f ( R d ) be a set of finite perimeter. Then, for every
Let E ∈ M f ( R d ) be a set of finite perimeter. Then, for every Proof. First, we notice that, using polar coordinates, for every 0 < r < ν ∈ S d − , it holds Z H − ν (0) ∩ B (0 ,r ) x | x | d · ν d x = − ω d − r , (1.14) Z H − ν (0) ∩ B (0 ,r ) ( x ) · ν d x = − ω d − d + 1 r d +1 , (1.15) Z H − ν (0) ∩ ( B (0 , \ B (0 ,r )) x | x | d + s · ν d x = − ω d − γ s ( r ) , (1.16)where γ s ( r ) := (cid:26) | log r | if s = 1 , r − s − s − if s > . Now we rewrite in a more convenient way the first three addends in the righthandside of (1.12). By (1.14), we get d + sdsr s Z ∂ ∗ E d H d − ( y ) Z E ∩ B ( y,r ) ( y − x ) · ν E ( y ) | x − y | d d x = d + sdsr s Z ∂ ∗ E d H d − ( y ) Z(cid:0) E \ H − νE ( y ) ( y ) (cid:1) ∩ B ( y,r ) ( y − x ) · ν E ( y ) | x − y | d d x − d + sdsr s Z ∂ ∗ E d H d − ( y ) Z(cid:0) H − νE ( y ) ( y ) \ E (cid:1) ∩ B ( y,r ) ( y − x ) · ν E ( y ) | x − y | d d x + d + sdsr s Z ∂ ∗ E d H d − ( y ) Z H − νE ( y ) ( y ) ∩ B ( y,r ) ( y − x ) · ν E ( y ) | x − y | d d x = − d + sdsr s Z ∂ ∗ E d H d − ( y ) Z(cid:0) E △ H − νE ( y ) ( y ) (cid:1) ∩ B ( y,r ) | ( y − x ) · ν E ( y ) || x − y | d d x + ω d − Per( E ) d + sds r − s . (1.17)Analogously, by (1.15), we have − dr d + s Z ∂ ∗ E d H d − ( y ) Z E ∩ B ( y,r ) ( y − x ) · ν E ( y ) d x = 1 dr d + s Z ∂ ∗ E d H d − ( y ) Z(cid:0) E △ H − νE ( y ) ( y ) (cid:1) ∩ B ( y,r ) | ( y − x ) · ν E ( y ) | d x − ω d − Per( E ) 1 d ( d + 1) r − s . (1.18)Furthermore, by using (1.16), we obtain1 s Z ∂ ∗ E d H d − ( y ) Z E ∩ (cid:0) B ( y, \ B ( y,r ) (cid:1) ( y − x ) · ν E ( y ) | x − y | d + s d x = − s Z ∂ ∗ E d H d − ( y ) Z(cid:0) E △ H − νE ( y ) ( y ) (cid:1) ∩ (cid:0) B ( y, \ B ( y,r ) (cid:1) | ( y − x ) · ν E ( y ) || x − y | d + s d x + ω d − Per( E ) 1 s γ s ( r ) . (1.19) ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 9 We notice that 1 s γ s ( r ) + d + sds r − s − d ( d + 1) r − s = σ s ( r ) + α s ;therefore, plugging (1.17), (1.18), (1.19) into (1.12), and using (1.9), we obtain theclaim. (cid:3) We are now in a position to prove Proposition 1.1. Proof of Proposition 1.1. We prove the claim under the assumption that E is smooth.For s > 1, the same proof, with ∂E replaced by ∂ ∗ E , works also for sets E ∈ M f ( R d )having finite perimeter. We will use the decomposition of ˜ J sr in Lemma 1.4. Clearlythe first contribution ω d − Per( E ) (cid:0) σ s ( r ) + α s ) , once scaled by σ s ( r ) converges to ω d − Per( E ) . Now we will prove that all the other contributions, scaled by σ s ( r ),vanish as r → + .2 nd addend: By Remark 1.2, we have thatlim r → + F s ( E ) σ s ( r ) = 0 . rd addend . By the very definition of σ s ( r ) in (1.2) we have that σ s ( r ) r s − isuniformly bounded from below by a positive constant for every 0 < r < , so thatby the change of variable z = x − yr , we have1 σ s ( r ) d + sdsr s Z ∂E d H d − ( y ) Z(cid:0) E △ H − νE ( y ) ( y ) (cid:1) ∩ B ( y,r ) | ( y − x ) · ν E ( y ) || x − y | d d x ≤ C ( d, s ) Z ∂E d H d − ( y ) Z(cid:0) E △ H − νE ( y ) ( y ) (cid:1) ∩ B ( y,r ) r | ( y − x ) · ν E ( y ) || x − y | d d x = C ( d, s ) Z ∂E d H d − ( y ) Z(cid:0) E − yr △ ( H − νE ( y ) ( y ) − yr ) (cid:1) ∩ B (0 , | z · ν E ( y ) || z | d d z , where the last integral vanishes as r → + in virtue of the Lebesgue’s DominatedConvergence Theorem since χ E − yr → χ H − νE ( y ) ( y ) − yr in L .4 th addend . Trivially, we have1 σ s ( r ) 1 dr s Z ∂E d H d − ( y ) Z(cid:0) E △ H − νE ( y ) ( y ) (cid:1) ∩ B ( y,r ) | ( y − x ) · ν E ( y ) | r d d x ≤ σ s ( r ) 1 dr s Z ∂E d H d − ( y ) Z(cid:0) E △ H − νE ( y ) ( y ) (cid:1) ∩ B ( y,r ) | ( y − x ) · ν E ( y ) || x − y | d d x, where the last integral vanishes as shown above. th addend . We first discuss the simpler case s > y ∈ ∂E , using again the change of variable z = x − yr , we have1 σ s ( r ) 1 s Z ∂E d H d − ( y ) Z(cid:0) E △ H − νE ( y ) ( y ) (cid:1) ∩ (cid:0) B ( y, \ B ( y,r ) (cid:1) | ( y − x ) · ν E ( y ) || x − y | d + s d x = r − s σ s ( r ) 1 s Z ∂E d H d − ( y ) Z(cid:0) E − yr △ ( H − νE ( y ) ( y ) − yr ) (cid:1) ∩ (cid:0) B (0 , r ) \ B (0 , (cid:1) | z · ν E ( y ) || z | d + s d z ≤ C ( d, s ) Z ∂E d H d − ( y ) Z(cid:0) E − yr △ ( H − νE ( y ) ( y ) − yr ) (cid:1) \ B (0 , | z | d + s − d z , where the last double integral vanishes as r → + in virtue of the Lebesgue’sDominated Convergence Theorem using that χ E − yr → χ H − νE ( y ) ( y ) − yr in L as r → + and the fact that the function h ( z ) := | z | d + s − is in L ( R d \ B (0 , s > s = 1 since for s = 1the function h ( z ) = | z | d is not in L ( R d \ B (0 , s = 1 and recall that σ ( r ) = | log r | . Since E has smooth boundary, there exists 0 < δ < y ∈ ∂E the sets B − := B ( y − δν E ( y ) , δ ) and B + := B ( y + δν E ( y ) , δ ) satisfy B − ⊂ E \ ∂E , B + ⊂ E c \ ∂E , y ∈ ∂B − ∩ ∂B + . Therefore, we have that(1.20) E △ H − ν E ( y ) ( y ) ⊂ ( H − ν E ( y ) ( y ) \ B − ) ∪ ( H + ν E ( y ) ( y ) \ B + ) , where H ± ν ( y ) are defined in (0.4) and (0.3). Fix y ∈ ∂E and let R y be a rotation of R d such that R y ν E ( y ) = e d . Moreover, we denote by z = ( z ′ , z d ) the points in R d ,so that z ′ = ( z , . . . , z d − ) ∈ R d − . Furthermore, we set R d + := { z ∈ R d : z d ≥ } .By (1.20) we have(1.21) 1 | log r | Z ∂E d H d − ( y ) Z(cid:0) E △ H − νE ( y ) ( y ) (cid:1) ∩ (cid:0) B ( y, \ B ( y,r ) (cid:1) | ( y − x ) · ν E ( y ) || x − y | d +1 d x ≤ | log r | Z ∂E d H d − ( y ) Z(cid:0) H − νE ( y ) ( y ) \ B − (cid:1) ∩ B ( y, | ( y − x ) · ν E ( y ) || x − y | d +1 d x + 1 | log r | Z ∂E d H d − ( y ) Z(cid:0) H + νE ( y ) ( y ) \ B + (cid:1) ∩ B ( y, | ( y − x ) · ν E ( y ) || x − y | d +1 d x = 2 | log r | Per( E ) Z R d + ∩ (cid:0) B (0 , \ B ( δe d ,δ ) (cid:1) z d ( | z ′ | + z d ) d +12 d z d . Therefore, in order to prove that the first double integral in (1.21) vanishes as r → + , it is enough to show that(1.22) Z R d + ∩ (cid:0) B (0 , \ B ( δe d ,δ ) (cid:1) z d ( | z ′ | + z d ) d +12 d z d ≤ C ( d, δ ) , for some finite constant C ( d, δ ) > A δ := { z = ( z ′ , z d ) ∈ R d + \ B ( δe d , δ ) : | z ′ | < δ , z d < δ } , we notice that(1.23) R d + ∩ (cid:0) B (0 , \ B ( δe d , δ ) (cid:1) ⊂ (cid:0) B (0 , \ B (0 , δ ) (cid:1) ∪ A δ . ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 11 Moreover, there exists a constant c δ (take, for instance, c δ = δ ) such that(1.24) A δ ⊂ ˜ A δ := { z = ( z ′ , z d ) ∈ R d + : | z ′ | < δ , z d < c δ | z ′ | } . Therefore, by (1.23) and (1.24), we get Z R d + ∩ (cid:0) B (0 , \ B ( δe d ,δ ) (cid:1) z d ( | z ′ | + z d ) d +12 d z d ≤ Z ˜ A δ z d ( | z ′ | + z d ) d +12 d z d + Z B (0 , \ B (0 ,δ ) z d ( | z ′ | + z d ) d +12 d z d ≤ Z B ′ (0 ,δ ) d z ′ Z δ − √ δ −| z ′ | c δ | z ′ | | z ′ | d +1 d z d + Z B (0 , \ B (0 ,δ ) | z | d d z ≤ c δ δ Z B ′ (0 ,δ ) | z ′ | − d d z ′ + | log δ | =: C ( d, δ ) , i.e., (1.22).6 th addend: We have that1 σ s ( r ) Z E H d − ( E c ∩ ∂B ( x, x ≤ σ s ( r ) dω d | E | → r → + . Thus, the proof of Lemma 1.1 is concluded. (cid:3) We will show that the limit (1.6) is actually a Γ-limit. Theorem 1.5. Let s ≥ and let { r n } n ∈ N ⊂ (0 , + ∞ ) be such that r n → + as n → + ∞ . The following Γ -convergence result holds true. (i) (Compactness) Let U ⊂ R d be an open bounded set and let { E n } n ∈ N ⊂ M( R d ) be such that E n ⊂ U for every n ∈ N and (1.25) ˜ J sr n ( E n ) ≤ M σ s ( r n ) for every n ∈ N , for some constant M independent of n . Then, up to a subsequence, χ E n → χ E strongly in L ( R d ) for some set E ∈ M f ( R d ) with Per( E ) < + ∞ . (ii) (Lower bound) Let E ∈ M f ( R d ) . For every { E n } n ∈ N ⊂ M f ( R d ) with χ E n → χ E strongly in L ( R d ) it holds (1.26) ω d − Per( E ) ≤ lim inf n → + ∞ ˜ J sr n ( E n ) σ s ( r n ) . (iii) (Upper bound) For every E ∈ M f ( R d ) there exists { E n } n ∈ N ⊂ M f ( R d ) suchthat χ E n → χ E strongly in L ( R d ) and ω d − Per( E ) = lim n → + ∞ ˜ J sr n ( E n ) σ s ( r n ) . The proof of Theorem 1.5 will be done in Sections 2 and 3 below.To ease notation, for every r > J sr ( · ) := ˜ J sr ( · ) σ s ( r ) . In view of (1.4), forevery E ∈ M f ( R d ) we have¯ J sr ( E ) = 1 σ s ( r ) Z E Z E c k sr ( | x − y | ) d y d x = 12 σ s ( r ) Z R d Z R d k sr ( | x − y | ) | χ E ( x ) − χ E ( y ) | d y d x . Proof of Compactness This section is devoted to the proof of Theorem 1.5(i). To accomplish this taskwe will need some preliminary results that are collected in Subsection 2.1 below.2.1. Preliminary results. We first recall the following classical result (see also[6, Theorem 3.23]). Theorem 2.1 (Compactness in BV) . Let Ω ⊂ R d be an open set and let { u n } n ∈ N ⊂ BV loc (Ω) with sup n ∈ N (cid:26)Z A | u n ( x ) | d x + | Du n | ( A ) (cid:27) < + ∞ ∀ A ⊂⊂ Ω open . Then, there exist a subsequence { n k } k ∈ N and a function u ∈ BV loc (Ω) such that u n k → u in L (Ω) as k → + ∞ . Now we prove a non-local Poincar´e-Wirtinger type inequality. Lemma 2.2. Let < r < l be such that ω d r d ≤ l d . Let ξ ∈ R d and let u ∈ L ( lQ + ξ ) . Then, for every s ≥ we have (2.1) Z lQ + ξ (cid:12)(cid:12)(cid:12)(cid:12) u ( y ) − | ( lQ + ξ ) \ B ( y, r ) | Z ( lQ + ξ ) \ B ( y,r ) u ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) d y ≤ d d + s l s Z lQ + ξ Z lQ + ξ | u ( y ) − u ( x ) | k sr ( | x − y | ) d y d x . Proof. By translational invariance, it is enough to prove the claim only for ξ = 0.By assumption, for every y ∈ lQ we have | lQ \ B ( y, r ) | ≥ l d − ω d r d ≥ l d . As a consequence, we have Z lQ (cid:12)(cid:12)(cid:12)(cid:12) u ( y ) − | lQ \ B ( y, r ) | Z Q \ B ( y,r ) u ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) d y ≤ Z lQ | lQ \ B ( y, r ) | Z lQ \ B ( y,r ) | u ( y ) − u ( x ) | d x d y = Z lQ | lQ \ B ( y, r ) | Z lQ \ B ( y,r ) | u ( y ) − u ( x ) || y − x | d + s | y − x | d + s d x d y ≤ Z lQ d d + s l d Z lQ \ B ( y,r ) | u ( y ) − u ( x ) || y − x | d + s l d + s d x d y ≤ d d + s l s Z lQ Z lQ | u ( y ) − u ( x ) | k sr ( | y − x | ) d y d x, i.e., (2.1). (cid:3) Lemma 2.3. Let < r < l be such that ω d r d < l d . For every ξ ∈ R d and for every E ∈ M f ( R d ) , it holds (2.2) 1 l d | ( lQ + ξ ) \ E || ( lQ + ξ ) ∩ E |≤ Z lQ + ξ (cid:12)(cid:12)(cid:12)(cid:12) χ E ( x ) − | ( lQ + ξ ) \ B ( x, r ) | Z ( lQ + ξ ) \ B ( x,r ) χ E ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12) d x . ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 13 Proof. We can assume without loss of generality that ξ = 0 . It is enough to prove(2.2) only in the case | lQ ∩ E | ≥ l d ; indeed, once proven the inequality (2.2) insuch a case, if | lQ \ E | ≥ l d , then the set ˜ E = lQ \ E satisfies | lQ ∩ ˜ E | ≥ l d , andhence ˜ E and, in turn, E satisfy (2.2).Let | lQ ∩ E | ≥ l d ; then, for every x ∈ R d we have(2.3) | ( lQ ∩ E ) \ B ( x, r ) | ≥ | lQ ∩ E | − ω d r d ≥ | lQ ∩ E | − l d ≥ | lQ ∩ E | , so that Z lQ (cid:12)(cid:12)(cid:12)(cid:12) χ E ( x ) − | lQ \ B ( x, r ) | Z lQ \ B ( x,r ) χ E ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12) d x = Z lQ ∩ E (cid:12)(cid:12)(cid:12)(cid:12) − | ( lQ \ B ( x, r )) ∩ E || lQ \ B ( x, r ) | (cid:12)(cid:12)(cid:12)(cid:12) d x + Z lQ \ E | ( lQ \ B ( x, r )) ∩ E || lQ \ B ( x, r ) | d x ≥ l d (cid:18)Z lQ ∩ E | ( lQ \ B ( x, r )) \ E | d x + Z lQ \ E | ( lQ \ B ( x, r )) ∩ E | d x (cid:19) = 2 l d Z lQ \ E | ( lQ ∩ E ) \ B ( x, r ) | d x ≥ l d | lQ ∩ E | | lQ \ E | , where in the last inequality we have used formula (2.3). (cid:3) The following result is a localized isoperimetric inequality for the non-localperimeters ˜ J sr . Lemma 2.4. Let s ≥ and let Ω ∈ M f ( R d ) be a bounded set with Lipschitzcontinuous boundary and | Ω | = 1 . For every η ∈ (0 , there exist a constant C ( η, d, s ) > and r > such that for every measurable set A ⊂ Ω with η ≤ | A | ≤ − η , it holds (2.4) Z A Z Ω \ A k sr ( | x − y | ) d y d x ≥ C (Ω , d, s, η ) σ s ( r ) for every r ∈ (0 , r ) . The proof of Lemma 2.4 follows along the lines of [22, Lemma 15], with slightdifferences due to the core radius approach adopted in this paper. Before provingLemma 2.4, we state the following result which is a consequence of [1, Theorem1.4]. Lemma 2.5 ([22]) . Let Ω ∈ M f ( R d ) be a bounded set with Lipschitz continuousboundary and | Ω | = 1 and let φ ∈ C ∞ c ( B (0 , , + ∞ )) be such that R φ d x = 1 and φ > in B (0 , ) . For every δ > we set φ δ ( · ) := δ d φ ( · δ ) . For every η ∈ (0 , there exists a constant C ( φ, η ) > such that for every measurable set A ⊂ Ω with η ≤ | A | ≤ − η and for every δ ∈ (0 , it holds δ Z A Z Ω \ A φ δ ( | x − y | ) d y d x ≥ C (Ω , φ, η ) . The above lemma has been stated and proven in [22, Proposition 14] in the case d = 2 with Ω = ( − , ) but in fact the same proof is not affected neither by thedimension d nor by the specific shape of Ω. We are now in a position to proveLemma 2.4. Proof of Lemma 2.4. Fix η ∈ (0 , r ∈ (0 , 1) and let I ∈ N be such that 2 − I − ≤ r ≤ − I . Notice that(2.5) k sr ( | z | ) ≥ (2 d + s ) min { i,I } if 0 ≤ | z | ≤ − i , with i ∈ N . Let φ and φ δ (for every δ > 0) be as in Lemma 2.5. Now we claim that there existsa constant C ( φ, d, s ) such that(2.6) k sr ( | z | ) ≥ C ( φ, d, s ) I X i =0 (2 s ) i φ − i ( z ) for every z ∈ R d . Before proving the claim we show that (2.6) implies (2.4). Indeed, first notice that | log r | log 2 − ≤ I ≤ | log r | log 2and hence I X i =0 (2 s − ) i = ( I + 1 ≥ | log r | log 2 if s = 1 , (2 I +1 ) s − − s − − ≥ r − s − s − − if s > . so that, recalling the very definition of σ s ( r ) in (1.5), for r small enough we have(2.7) I X i =0 (2 s − ) i ≥ C ( d, s ) σ s ( r ) . Therefore, by applying (2.6) and Lemma 2.5 with δ replaced by 2 − i , we get(2.8) Z A Z Ω \ A k sr ( | x − y | ) d y d x ≥ C (Ω , φ, d, s ) I X i =0 (2 s − ) i i Z A Z Ω \ A φ − i ( x − y ) d y d x ≥ C (Ω , φ, d, s, η ) I X i =0 (2 s − ) i ≥ C ( φ, d, s, η ) σ s ( r ) , where the last inequality follows from (2.7).Now we prove the claim (2.6). Suppose first that 0 ≤ | z | ≤ − I . By applying(2.5) with i = I , we get(2.9) I X i =0 (2 s ) i φ − i ( z ) ≤ sup φ I X i =0 (2 d + s ) i = sup φ I X i =0 d + s ) I − i (2 d + s ) I ≤ sup φ + ∞ X j =0 d + s ) j (2 d + s ) I = 2 d + s d + s − φ (2 d + s ) I ≤ C ( φ, d, s ) k sr ( | z | ) . ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 15 Analogously, if 2 − ¯ ı − < | z | ≤ − ¯ ı for some ¯ ı = 0 , , . . . , I − φ − i ( z ) = 0for every i = ¯ ı + 1 , . . . , I , we have(2.10) I X i =0 (2 s ) i φ − i ( z ) = ¯ ı X i =0 (2 s ) i φ − i ( z ) ≤ sup φ ¯ ı X i =0 (2 d + s ) i ≤ sup φ + ∞ X j =0 d + s ) j (2 d + s ) ¯ ı = 2 d + s d + s − φ (2 d + s ) ¯ ı ≤ C ( φ, d, s ) k sr ( | z | ) , where the last inequality is a consequence of (2.5).Finally, if | z | ≥ φ − i ( z ) = 0 for every i so that(2.11) I X i =0 (2 s ) i φ − i ( z ) = 0 ≤ k sr ( | z | ) . Therefore, by (2.9), (2.10) and (2.11), we deduce (2.6), thus concluding the proofof the lemma . (cid:3) Proof of Theorem 1.5(i). We are now in a position to prove Theorem 1.5(i). Proof. We divide the proof into three steps. Step 1. Let α ∈ (0 , 1) and set l n := r αn for every n ∈ N . Let { Q nh } h ∈ N be adisjoint family of cubes of sidelength l n such that S h ∈ N Q nh = R d . Since | E n | ≤ | U | ,there exists H ( n ) ∈ N , such that, up to permutation of indices,(2.12) | Q nh ∩ E n | ≥ l dn h = 1 , · · · , H ( n ) , | Q nh \ E n | > l dn h ≥ H ( n ) + 1 . For every n ∈ N , we set ˜ E n := H ( n ) [ h =1 Q nh . Let ˜ n ∈ N be such that for all n > ˜ n the pair ( r n , l n ) satisfies the hypothesis ofLemmas 2.2 and 2.3. We claim that there exists a constant C ( d, s ) > | ˜ E n △ E n | ≤ C ( d, s ) l sn σ s ( r n ) M for every n ≥ ˜ n, where M is the constant in (1.25). Indeed, | E n △ ˜ E n | = | ˜ E n \ E n | + | E n \ ˜ E n | = H ( n ) X h =1 | Q nh \ E n | + ∞ X h = H ( n )+1 | E n ∩ Q nh | =2 H ( n ) X h =1 l dn | Q nh \ E n | l dn ∞ X h = H ( n )+1 l dn | E n ∩ Q nh | l dn ≤ + ∞ X h =1 l dn | Q nh \ E n || Q nh ∩ E n |≤ + ∞ X h =1 Z Q nh (cid:12)(cid:12)(cid:12)(cid:12) χ E n ( x ) − | Q nh \ B ( x, r n ) | Z Q nh \ B ( x,r n ) χ E n ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12) d x ≤ + ∞ X h =1 d d + s l sn Z Q nh ∩ E n Z Q nh \ E n k sr n ( | x − y | ) d y d x ≤ C ( d, s ) l sn ˜ J sr n ( E n ) ≤ C ( d, s ) l sn σ s ( r n ) M, where the second inequality follows by formula (2.2), the third inequality is a con-sequence of (2.1), whereas the last one follows directly by (1.25). Step 2. For every n ∈ N let l n and ˜ E n := S H ( n ) h =1 Q nh be as in Step 1. We claimthat there exists a constant C ( α, d, s ) such that for n large enough(2.14) Per( ˜ E n ) ≤ C ( α, d, s ) ¯ J sr n ( E n ) . To ease notation, we omit the dependence on n by setting r := r n , l := l n , E := E n , Q h := Q nh , H := H ( n ), and ˜ E := ˜ E n .We define the family R of rectangles R = ˜ Q ∪ ˆ Q such that ˜ Q and ˆ Q are adjacentcubes (of the type Q h introduced above), ˜ Q ⊂ ˜ E and ˆ Q ⊂ ˜ E c .Notice that(2.15) Per( ˜ E ) ≤ dl d − ♯ R , ¯ J sr ( E ) ≥ d σ s ( r ) X R ∈R Z R ∩ E Z R \ E k sr ( | x − y | ) d y d x . We recall that, by Lemma 2.4, for every rectangle ¯ R given by the union of twoadjacent unitary cubes in R d , there exists ρ > C ( d, s ) := inf (cid:26) σ s ( ρ ) Z F Z ¯ R \ F k sρ ( | x − y | ) d y d x :0 < ρ < ρ , F ∈ M f ( R d ) , F ⊂ ¯ R , ≤ | F | ≤ (cid:27) > . ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 17 Furthermore, by the very definition of σ s ( r ) in (1.5), using that l = r α we have σ s ( r ) l − s = | log( r − α ) | − α if s = 1 d + sd + 1 r (1 − α )(1 − s ) s − s > ( − α σ s ( r − α ) if s = 1 σ s ( r − α ) if s > , so that(2.17) l − s σ s ( r ) ≥ C ( α ) 1 σ s ( r − α ) = C ( α ) 1 σ s ( rl ) . For every set O ∈ M f ( R d ) we set O l := Ol . By (2.15), (1.2), (2.17) and by applying(2.16) with ¯ R = R l for every R ∈ R , for r small enough we obtain¯ J sr ( E ) ≥ C ( d ) σ s ( r ) l d X R ∈R Z R l ∩ E l Z R l \ E l k sr ( | l ( x − y ) | ) d y d x = C ( d ) l − s σ s ( r ) l d − X R ∈R Z R l ∩ E l Z R l \ E l k s rl ( | x − y | ) d y d x ≥ C ( α, d ) l d − X R ∈R σ s ( rl ) Z R l ∩ E l Z R l \ E l k s rl ( | x − y | ) d y d x ≥ C ( α, d ) l d − ♯ R C ( d, s ) ≥ C ( α, d, s )Per( ˜ E ) , i.e., (2.14). Step 3. Here we conclude the proof of the compactness result. We fix α ∈ (1 − s , 1) so that, by (2.13), | E n △ ˜ E n | → n → + ∞ .By assumption and by the very definition of ˜ E n in Step 1, we have that ˜ E n ⊂ U for all n ∈ N . Moreover, by formula (2.14) and by (1.25) for n large enough wehave Per( ˜ E n ) ≤ C ( α, d, s ) ¯ J sr n ( E n ) ≤ C ( α, d, s ) M . It follows that the sequence { χ ˜ E n } n ∈ N satisfies the assumption of Theorem 2.1, andhence there exists a set E ⊂ R d with Per( E ) < + ∞ such that, up to a subsequence, χ ˜ E n → χ E in L ( R d ) as n → + ∞ . Since | E n △ ˜ E n | → n → + ∞ we obtain that χ E n → χ E in L ( U ), i.e., the claim of Theorem 1.5(i). (cid:3) The following result follows by the proof of Theorem 1.5(i). Proposition 2.6. Let s ≥ . Let { r n } n ∈ N ⊂ (0 , + ∞ ) be such that r n → + as n → + ∞ . Let { E n } n ∈ N ⊂ M f ( R d ) be such that χ E n → χ E in L ( R d ) as n → + ∞ ,for some E ∈ M f ( R d ) . If lim sup n → + ∞ ˜ J sr n ( E n ) σ s ( r n ) ≤ M , then E has finite perimeter. Proof. The proof of this corollary is fully analogous to the proof of Theorem 1.5(i),and we adopt the same notation introduced there. Arguing as in the proof of Steps1 and 2 we have that for n large enoughPer( ˜ E n ) ≤ C ( α, d, s ) lim sup n → + ∞ ˜ J sr n ( E n ) σ s ( r n ) ≤ C ( α, d, s ) M , and that if α ∈ (1 − s , 1) , then | ˜ E n △ E n | → n → + ∞ . By assumption, thisimplies that χ ˜ E n → χ E , in L ( R d ) n → + ∞ , and by the lower semicontinuity of the perimeter,Per( E ) ≤ lim inf n → + ∞ Per( ˜ E n ) ≤ C ( α, d, s ) M . (cid:3) Proof of the Γ -limit This section is devoted to the proofs of Theorem 1.5(ii) and (iii), which are thecontent of Subsections 3.1 and 3.2 respectively.3.1. Proof of the lower bound. The proof of Theorem 1.5(ii) closely followsthe strategy used in [22]. We recall that for every ν ∈ S d − , Q ν is a unit squarecentered at the origin with one face orthogonal to ν . Moreover, we recall that H + ν (0) = { x ∈ R d : x · ν ≥ } .The following result is the adaptation to our setting of [22, Lemma 18]. Lemma 3.1. Let s ≥ . For every ε > , there exist r , δ > such that for every ν ∈ S , for every E ∈ M f ( R d ) with (3.1) | ( E △ H − ν (0)) ∩ Q ν | ≤ δ , and for every r < r it holds (3.2) Z Q ν ∩ E Z Q ν ∩ E c k sr ( | x − y | ) d y d x ≥ ω d − (1 − ε ) σ s ( r ) . Proof. Up to a rotation, we can assume that ν = − e d so that Q ν ≡ Q = [ − , ) d and H − ν (0) =: R d + . Let 0 < r < 1. We can assume without loss of generality that E ⊂ Q . Using the change of variable y = x + z we have(3.3) Z Q ∩ E c d x Z Q ∩ E k sr ( | x − y | ) d y = Z Q ∩ E c d x Z { z ∈ R d : x + z ∈ E } k sr ( | − z | ) d z = Z Q ∩ E c d x Z R d k sr ( | z | ) χ E ( x + z ) d z = Z R d k sr ( | z | ) Z R d χ E c ∩ Q ( x ) χ E ( x + z ) d x d z = Z R d k sr ( | z | ) | E c ∩ ( E − z ) ∩ Q | d z = Z R d k sr ( | z | ) m ( z ) d z , where we have set m ( z ) := | E c ∩ ( E − z ) ∩ Q | . ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 19 Let < λ < z ∈ R d be such that | z | ∞ ≤ − λ and z d > 0. Since | ( E − z ) ∩ λQ | = | E ∩ ( λQ + z ) | , by triangular inequality, we get | ( E − z ) ∩ λQ | − | E ∩ λQ | = Z λQ + z χ E d x − Z λQ χ E d x ≥ Z λQ + z χ R d + d x − Z λQ χ R d + d x − Z ( λQ + z ) △ λQ | χ E − χ R d + | d x ≥ λ d − z d − Z U λ,z | χ E − χ R d + | d x , where we have set U λ,z := ( λ + | z | ∞ ) Q \ ( λ − | z | ∞ ) Q and we have used that( λQ + z ) △ λQ ⊂ U λ,z . As a consequence, we deduce that(3.4) m ( z ) = | E c ∩ ( E − z ) ∩ Q | ≥ | E c ∩ ( E − z ) ∩ λQ |≥| E c ∩ λQ | + | ( E − z ) ∩ λQ | − | λQ |≥| E c ∩ λQ | + | E ∩ λQ | + λ d − z d − Z U λ,z | χ E − χ R d + | d x − | λQ | = λ d − z d − Z U λ,z | χ E − χ R d + | d x , where the last equality follows by noticing that | E ∩ λQ | + | E c ∩ λQ | = | λQ | .Let now 0 < δ < to be chosen later on and set A + √ δ := n z ∈ R d : | z | ∞ ≤ √ δ , z d > o . We fix z ∈ A + √ δ and we set J := ⌊ √ δ | z | ∞ ⌋ . We set λ := 1 − √ δ and we cover( λ + 2 J | z | ∞ ) Q \ λ Q with J squared annuli of thickness 2 | z | ∞ , namely we set λ j := λ + 2 j | z | ∞ and U j := λ j Q \ λ j − Q for j = 1 , . . . , J . Moreover, we set˜ λ j := λ + (2 j − | z | ∞ for every j = 1 , . . . , J and we notice that < ˜ λ j < j = 1 , . . . , J . Since z ∈ A + √ δ , we have that | z | ∞ ≤ − ˜ λ J ≤ − ˜ λ j for every j = 1 , . . . , J . Therefore, for every j = 1 , . . . , J we can apply (3.4) with λ = ˜ λ j inorder to get(3.5) m ( z ) ≥ z d ˜ λ d − j − Z U ˜ λj,z | χ E − χ R d + | d x ≥ z d λ d − j − − Z U j | χ E − χ R d + | d x , where we have used also that ˜ λ j − | z | ∞ = λ j − and ˜ λ j + | z | ∞ = λ j so that U ˜ λ j ,z = U j . Summing (3.5) over j = 1 , . . . , J we get Jm ( z ) ≥ z d J X j =1 λ d − j − − Z Q | χ E − χ R d + | d x , which, dividing by J and using discrete Jensen inequality (namely, convextiy),yields(3.6) m ( z ) ≥ z d (cid:16) J J X j =1 λ j − (cid:17) d − − J Z Q | χ E − χ R d + | d x ≥ z d λ d − − | z | ∞ p δ , where in the last inequality we have used (3.1) and the fact that J ≥ √ δ | z | ∞ − 1. Therefore, we have proven that (3.6) holds true whenever z ∈ A + √ δ , whichcombined with (3.3), yields(3.7) Z Q ∩ E c d x Z Q ∩ E k sr ( | x − y | ) d y ≥ λ d − Z A + √ δ z d k sr ( | z | ) d z − p δ Z A + √ δ | z | ∞ k sr ( | z | ) d z . As for the first integral on the right hand side of (3.7), by using polar coordinates z = ρθ with ρ > θ ∈ S d − and using the very definition of σ s ( r ) in (1.5), for δ small enough and for all r < δ we have(3.8) Z A + √ δ z d k sr ( | z | ) d z ≥ Z B (0 ,r ) ∩ R d + z d r d + s d z + Z ( B (0 ,δ ) \ B (0 ,r )) ∩ R d + z d | z | d + s d z = 1 r d + s Z r ρ d d ρ Z S d − ∩ R d + θ d d H d − ( θ )+ Z δ r ρ − s d ρ Z S d − ∩ R d + θ d d H d − ( θ )= ω d − r − s d + 1 + ω d − Z δ r ρ − s d ρ ≥ ω d − σ s ( r ) − ω d − C ( δ , s ) , where C ( δ , s ) := | log δ | if s = 1 δ − s s − s > . Moreover, since | z | ∞ ≤ | z | , it holds(3.9) Z A + √ δ | z | ∞ k sr ( | z | ) d z ≤ Z B (0 , | z | k sr ( | z | ) d z = 1 r d + s Z B (0 ,r ) | z | d z + Z B (0 , \ B (0 ,r ) | z | d + s − d z ≤ C ( d, s ) σ s ( r ) , for some C ( d, s ) > η ( t ) := 1 − (1 − √ t ) d − , and we notice that η ( t ) → t → + . Therefore, by (3.7), (3.8) and (3.9), using that λ d − = 1 − η ( δ ) , wededuce that(3.10) Z Q ∩ E c d x Z Q ∩ E k sr ( | x − y | ) d y ≥ ω d − σ s ( r ) (cid:16) − η ( δ ) − (cid:0) − η ( δ ) (cid:1) C ( δ , s ) σ s ( r ) − p δ C ( d, s ) ω d − (cid:17) , so that, choosing δ > η ( δ ) + 2 p δ C ( d, s ) ω d − ≤ ε ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 21 and r > < r < r ) (cid:0) − η ( δ ) (cid:1) C ( δ , s ) σ s ( r ) ≤ (cid:0) − η ( δ ) (cid:1) C ( δ , s ) σ s ( r ) ≤ ε , by (3.10) we deduce (3.2), thus concluding the proof of the lemma. (cid:3) We are now in a position to prove the Γ-liminf inequality in Theorem 1.5. Proof of Theorem 1.5(ii). We can assume without loss of generality that(3.11) ¯ J sr n ( E n ) = 12 σ s ( r n ) Z R d Z R d k sr n ( | x − y | ) | χ E n ( x ) − χ E n ( y ) | d y d x ≤ C , for some constant C > n . Then, by Corollary 2.6 we have that E has finite perimeter. For every n ∈ N let µ n be the measure on the product space R d × R d defined by µ sn ( A × B ) := 12 σ s ( r n ) Z A Z B k sr n ( | x − y | ) | χ E n ( x ) − χ E n ( y ) | d y d x for every A, B ∈ M( R d ). Then by (3.11), up to a subsequence, µ sn ∗ ⇀ µ s for somemeasure µ s . Now we show that µ s is concentrated on the set D := { ( x, x ) : x ∈ R d } , i.e., that µ s (Ω) = 0 if Ω ∩ D = ∅ . Indeed, let ϕ ∈ C c ( R d × R d ; [0 , + ∞ )) besuch that dist(supp ϕ, D ) = δ for some δ > Z R d × R d ϕ ( x, y ) d µ s ( x, y )= lim n → + ∞ σ s ( r n ) Z R d × R d ϕ ( x, y ) k sr n ( | x − y | ) | χ E n ( x ) − χ E n ( y ) | d y d x ≤ lim n → + ∞ σ s ( r n ) 1 δ d + s Z R d × R d ϕ ( x, y ) d y d x = 0 . Now we define the measure λ s on R d as λ s ( A ) = µ s ( { ( x, x ) : x ∈ A } ) and we claimthat for H d − - a.e. x ∈ ∂ ∗ E it holds(3.12) lim inf l → + λ s ( Q νl ( x )) l d − ≥ lim inf l → + lim inf n → + ∞ µ sn ( Q νl ( x ) × Q νl ( x )) l d − ≥ ω d − , where we have set ν = ν E ( x ) and Q νl ( x ) = x + lQ ν . By (3.12) and Radon-Nikodym Theorem, using the lower semicontinuity of the total variation of measureswith respect to the weak star convergence, we get (1.26).We conclude by proving the claim (3.12). We preliminarily notice that the firstinequality is a consequence of the upper semicontinuity of the total variation ofmeasures on compact sets with respect to the weak star convergence. We pass toprove the second inequality in (3.12). For all x ∈ ∂ ∗ E , we have(3.13) lim l → + Z Q ν | χ E ( x + lx ) − χ H − ν (0) ( x ) | d x = 0 . Fix such a x ∈ ∂ ∗ E . We will adopt a blow-up argument. Consider the sequenceof sets { F n,l } n ∈ N defined by F n,l = x + lE n . By the change of variable x = x + lξ and y = x + lη we have(3.14) 1 l d − µ sn ( Q νl ( x ) × Q νl ( x ))= 12 σ s ( r n ) Z Q ν Z Q ν l d +1 k sr n ( | lξ − lη | ) | χ F n,l ( ξ ) − χ F n,l ( η ) | d ξ d η = l − s σ s ( r n ) Z Q ν Z Q ν k s rnl ( | ξ − η | ) | χ F n,l ( ξ ) − χ F n,l ( η ) | d ξ d η , where in the last equality we have used (1.2). Let 0 < ε < δ , r > l small enough we have(3.15) Z Q ν | χ E ( x + lx ) − χ H − ν (0) ( x ) | d x ≤ δ . Fix such an l ; then, there exists n ( l ) ∈ N such that for n ≥ n ( l ) , it holds(3.16) Z Q ν | χ F n,l ( x ) − χ E ( x + lx ) | d x = 1 l d Z Q νl ( x ) | χ E n ( x ) − χ E ( x ) | d x ≤ δ . By (3.15) and (3.16), using triangular inequality, we obtain | ( F n,l △ H − ν (0)) ∩ Q ν | = Z Q ν | χ F n,l − χ H − ν (0) | d x ≤ δ . Therefore, by applying Lemma 3.1 with k sr = k s rnl and E = F n,l , for n large enough(i.e., in such a way that n ≥ n ( l ) and r n < r l ) we have that(3.17) 12 Z Q ν Z Q ν k s rnl ( | ξ − η | ) | χ F n,l ( ξ ) − χ F n,l ( η ) | d ξ d η ≥ ω d − (1 − ε ) σ s (cid:16) r n l (cid:17) . Now, by the very definition of σ s in (1.5), we have that l − s σ s ( r n ) σ s (cid:16) r n l (cid:17) = ( log l + | log r n || log r n | if s = 11 if s > , so that, in view of (3.14) and (3.17), we deduce that for every 0 < ε < l small enough (depending on ε ), it holdslim inf n → + ∞ l d − µ sn ( Q νl ( x ) × Q νl ( x )) ≥ ω d − (1 − ε ) , whence the second inequality in claim (3.12) follows by the arbitrariness of ε . (cid:3) Proof of the upper bound. The Γ-limsup inequality will be a consequenceof Proposition 1.1 and of standard density results for sets of finite perimeter.We first recall the following fundamental approximation theorem (see, for in-stance, [28, Theorem 13.8]). Theorem 3.2 (Approximation of set with finite perimeter by smooth sets) . A set E ∈ M f ( R d ) has finite perimeter if and only if there exists a sequence { F k } k ∈ N ⊂ M f ( R d ) of open bounded sets with smooth boundary, such that χ F k → χ E (strongly) in L ( R d ) as k → + ∞ , Per( F k ) → Per( E ) as k → + ∞ . (3.18) ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 23 Proof of Theorem 1.5(iii). Let E ∈ M f ( R d ) be a set with finite perimeter. ByTheorem 3.2, there exists a sequence { F k } k ∈ N of open bounded sets with smoothboundary satisfying (3.18) . In view of Proposition 1.1 we have thatlim n → + ∞ ˜ J sr n ( F k ) σ s ( r n ) = ω d − Per( F k ) for every k ∈ N . Therefore, by a standard diagonal argument there exists a sequence { E n } n ∈ N with E n = F k ( n ) for every n ∈ N satisfying the desired properties. (cid:3) Characterization of sets of finite perimeter. As a byproduct of our Γ-convergence analysis, we prove that a set E ∈ M f ( R d ) has finite perimeter if andonly if for all s ≥ r → + ˜ J sr ( E ) σ s ( r ) < + ∞ . We recall the following classical theorem. Theorem 3.3 (Characterization via difference quotients) . Let E ∈ M f ( R d ) . Then E has finite perimeter if and only if there exists C > such that Z R d | χ E ( x + z ) − χ E ( x ) | d x ≤ C | z | for every z ∈ R d . Specifically, it is possible to choose C = Per( E ) . Theorem 3.4. Let E ∈ M f ( R d ) . The following statements hold true. (i) If lim sup r → + ˜ J sr ( E ) σ s ( r ) < + ∞ for some s ≥ , then E is a set of finite perimeter. (ii) If E is a set of finite perimeter then lim sup r → + ˜ J sr ( E ) σ s ( r ) < + ∞ for every s ≥ .More precisely, (3.19) ω d − Per( E ) ≤ lim inf r → + ˜ J sr ( E ) σ s ( r ) ≤ lim sup r → + ˜ J sr ( E ) σ s ( r ) ≤ M ( s, d )Per( E ) , where M ( s, d ) = (cid:26) dω d if s = 1 ω d − if s > . In particular, for s > we have that (3.20) lim r → + ˜ J sr ( E ) σ s ( r ) = ω d − Per( E ) . Remark 3.5. We notice that in the case s = 1 the constant M (1 , d ) = dω d > ω d − ,so that the existence of the limit (3.20) is not proven in such a case. Proof Theorem 3.4: We notice that (i) is an immediate consequence of Proposition2.6 taking E n ≡ E for every n ∈ N . We prove (ii). The Γ-liminf inequality Theorem1.5(ii) implies the first inequality in (3.19). Being the second inequality obvious wepass to the proof of the last one. If s > r → + ˜ J sr ( E ) σ s ( r ) = ω d − Per( E ) . Let now s = 1. Let G r be the functional defined in (1.8); by Theorem 3.3 we obtain G r ( E ) σ ( r ) = 1 | log r | Z E Z E c ∩ B ( x, k r ( | x − y | ) d y d x = 12 | log r | Z R d Z B ( x, | χ E ( x ) − χ E ( y ) | k r ( | x − y | ) d y d x = 12 | log r | Z B (0 , k r ( | h | ) Z R d | χ E ( x + h ) − χ E ( x ) | d x d h ≤ | log r | Per( E ) Z B (0 , | h | k r ( | h | ) d h = dω d E ) (cid:18) d + 1) | log r | (cid:19) . (3.22)Moreover, by Remark 1.2 we have that(3.23) lim r → + F ( E ) σ ( r ) = 0 , where F is the functional defined in formula (1.7).Therefore by formulas (1.9), (3.22), and (3.23) we have(3.24) lim sup r → + ˜ J r ( E ) σ ( r ) = lim sup r → + G r ( E ) σ ( r ) + lim r → + F sr ( E ) σ ( r ) ≤ dω d E ) . thus concluding the proof of (ii). By (3.21) and (3.24) we conclude the proof of(ii). (cid:3) Convergence of curvatures and mean curvature flows In this section we study the behavior of the non-local curvatures corresponding tothe functionals ˜ J sr and of the corresponding geometric flows. Using the approach in[18, 16], it is enough to focus on smooth enough sets. To this purpose, we introducethe class C as the class of the subsets of R d , which are closures of open sets withcompact C boundary. Moreover, we define a notion of convergence in C as follows.Let { E n } n ∈ N ⊂ C we say that E n → E in C as n → + ∞ , for some E ∈ C , if thereexists a sequence of diffeomorphisms { Φ n } n ∈ N converging to the identity in C as n → + ∞ , such that Φ n ( E ) = E n for every n ∈ N . In the following, we will extendthis notion of convergence (in the obvious way) to families of sets { E ρ } ρ ∈ (0 , ⊂ C as the parameter ρ → + .Notice that if E ∈ C , then either E or E c is compact. Therefore, in order todefine the supercritical perimeters and the corresponding curvatures on the whole C ,it is convenient to set ˜ J sr ( E ) := ˜ J sr ( E c ) for every set E ∈ M( R d ) with E c ∈ M f ( R d ).4.1. Non-local k sr -curvatures. Let s ≥ r > E ∈ C . For every x ∈ ∂E we define the k sr -curvature of E at x as(4.1) K sr ( x, E ) := Z R d ( χ E c ( y ) − χ E ( y )) k sr ( | x − y | ) d y. Although this fact may be immediate for the experts, we show that K sr is the firstvariation of the functional ˜ J sr in the sense specified by the following proposition. ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 25 Proposition 4.1 (First variation) . Let s ≥ , r > , and E ∈ C . Let Φ : R × R d → R d be a smooth function, and let { Φ t } t ∈ R be defined by Φ t ( · ) := Φ( t, · ) for every t ∈ R . Assume that { Φ t } t ∈ R is a family of diffeomorphisms with Φ = Id and thatthere exists an open bounded set A ⊂ R d such that (4.2) { x ∈ R d : x = Φ t ( x ) } ⊂ A for all t ∈ R . Setting E t := Φ t ( E ) and Ψ( · ) := ∂∂t Φ t ( · ) (cid:12)(cid:12) t =0 , we have (4.3) dd t ˜ J sr ( E t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z ∂E K sr ( x, E )Ψ( x ) · ν E ( x ) d H d − ( x ) . Proof. By Taylor expansion for every x ∈ R d we have that Φ t ( x ) = x + t Ψ( x )+ o( t ).Therefore the Jacobian J Φ t of Φ t is equal to J Φ t ( x ) := p det( ∇ Φ t ( x ) ∇ Φ t ( x ) ∗ ) = 1 + t Div(Ψ( x )) + o( t ) , where, for every A ∈ R m × k ( m, k ∈ N ), the symbol A ∗ denotes the transpose of thematrix A . By change of variable, it follows that˜ J sr ( E t ) = Z Φ t ( E ) Z Φ t ( E c ) k sr ( | x − y | ) d y d x = Z E Z E c k sr ( | Φ t ( x ) − Φ t ( y ) | ) J Φ t ( x ) J Φ t ( y ) d y d x = Z E Z E c k sr ( | Φ t ( x ) − Φ t ( y ) | ) d y d x + t Z E Z E c k sr ( | Φ t ( x ) − Φ t ( y ) | ) (cid:16) DivΨ( x ) + DivΨ( y ) (cid:17) d y d x + o( t ) Z E Z E c k sr ( | Φ t ( x ) − Φ t ( y ) | ) d y d x . (4.4)Let ( k sr ) ′ : (0 , + ∞ ) → R be the weak derivative of k sr : (0 , + ∞ ) → R , that is equala.e. to ( k sr ) ′ ( h ) := ( < h < r , − ( d + s ) 1 h d + s +1 for h > r . Notice that k sr ∈ W , ( R ). We set K ( t ) := Z E Z E c (cid:16) k sr ( | Φ t ( x ) − Φ t ( y ) | ) − k sr ( | x − y | ) − t ( k sr ) ′ ( | x − y | ) x − y | x − y | · (Ψ( x ) − Ψ( y )) (cid:17) d y d x and we claim that(4.5) lim t → K ( t ) t = 0 . By the fundamental theorem of calculus, we have Z E Z E c (cid:16) k sr ( | Φ t ( x ) − Φ t ( y ) | ) − k sr ( | x − y | ) (cid:17) d y d x = Z t d τ (cid:20) Z E Z E c ( k sr ) ′ ( | Φ τ ( x ) − Φ τ ( y ) | ) Φ τ ( x ) − Φ τ ( y ) | Φ τ ( x ) − Φ τ ( y ) | · (cid:16) ∂ Φ τ ∂τ ( x ) − ∂ Φ τ ∂τ ( y ) (cid:17) d y d x (cid:21) , so that (cid:12)(cid:12)(cid:12)(cid:12) K ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) t Z t (cid:20)Z E Z E c ( k sr ) ′ ( | Φ τ ( x ) − Φ τ ( y ) | ) Φ τ ( x ) − Φ τ ( y ) | Φ τ ( x ) − Φ τ ( y ) | · (cid:16) ∂ Φ τ ∂τ ( x ) − ∂ Φ τ ∂τ ( y ) (cid:17) − ( k sr ) ′ ( | x − y | ) x − y | x − y | · (Ψ( x ) − Ψ( y )) d y d x (cid:21) d τ (cid:12)(cid:12)(cid:12)(cid:12) ≤ | t | Z | t | (cid:12)(cid:12)(cid:12)(cid:12) Z E Z E c ( k sr ) ′ ( | Φ τ ( x ) − Φ τ ( y ) | ) Φ τ ( x ) − Φ τ ( y ) | Φ τ ( x ) − Φ τ ( y ) | · (cid:16) ∂ Φ τ ∂τ ( x ) − ∂ Φ τ ∂τ ( y ) (cid:17) − ( k sr ) ′ ( | x − y | ) x − y | x − y | · (Ψ( x ) − Ψ( y )) d y d x (cid:12)(cid:12)(cid:12)(cid:12) d τ =: 1 | t | Z | t | (cid:12)(cid:12)(cid:12)(cid:12) Z E Z E c (cid:0) f τ ( x, y ) − f ( x, y ) (cid:1) d y d x (cid:12)(cid:12)(cid:12)(cid:12) d τ , where in the last line for every τ ∈ R and for every ( x, y ) ∈ R d × R d we have set f τ ( x, y ) := ( k sr ) ′ ( | Φ τ ( x ) − Φ τ ( y ) | ) Φ τ ( x ) − Φ τ ( y ) | Φ τ ( x ) − Φ τ ( y ) | · (cid:16) ∂ Φ τ ∂τ ( x ) − ∂ Φ τ ∂τ ( y ) (cid:17) . Notice that (4.5) follows if we show that(4.6) Z E Z E c f τ ( x, y ) d y d x → Z E Z E c f ( x, y ) d y d x as τ → . By change of variable, we get(4.7) Z E Z E c f τ ( x, y ) d y d x = Z R d Z R d f ( x, y ) J Φ − τ ( x ) J Φ − τ ( y ) χ E × E c (Φ − τ ( x ) , Φ − τ ( y )) d y d x . Setting C ( J ) := sup τ ∈ (0 , kJ Φ − τ k L ∞ , by (4.2), we have that | f ( x, y ) |J Φ − τ ( x ) J Φ − τ ( y ) χ E × E c (Φ − τ ( x ) , Φ − τ ( y )) ≤ [ C ( J )] | f ( x, y ) | [ χ E ∩ A (Φ − τ ( x )) + χ E ∩ A c ( x )][ χ E c ∩ A (Φ − τ ( y )) + χ E c ∩ A c ( y )] ≤ [ C ( J )] | f ( x, y ) | [ χ A ( x ) + χ E ( x )][ χ A ( y ) + χ E c ( y )] , where the right hand side term is clearly in L ( R d ) . This fact together with (4.7)and J Φ − τ ( x ) J Φ − τ ( y ) χ E × E c (Φ − τ ( x ) , Φ − τ ( y )) → χ E × E c ( x, y ) a.e. as τ → ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 27 yields by Lebesgue Dominated Convergence Theorem, (4.6) and, in turn, (4.5) . By(4.4) and (4.5), and using the divergence theorem, we obtain thatdd t ˜ J sr ( E t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z E Z E c ( k sr ) ′ ( | x − y | ) x − y | x − y | · (Ψ( x ) − Ψ( y )) d y d x + Z E Z E c k sr ( | x − y | )(DivΨ( x ) + DivΨ( y )) d y d x = Z E Z E c ( k sr ) ′ ( | x − y | ) x − y | x − y | · (Ψ( x ) − Ψ( y )) d y d x + Z E c (cid:20) − Z E ( k sr ) ′ ( | x − y | ) Ψ( x ) · x − y | x − y | d x + Z ∂E k sr ( | x − y | ) Ψ( x ) · ν E ( x ) d H d − ( x ) (cid:21) d y + Z E (cid:20)Z E c ( k sr ) ′ ( | x − y | ) Ψ( y ) · x − y | x − y | d y − Z ∂E k sr ( | x − y | ) Ψ( y ) · ν E ( y ) d H d − ( y ) (cid:21) d x = Z ∂E Ψ( x ) · ν E ( x ) Z R d ( χ E c ( y ) − χ E ( y )) k sr ( | x − y | ) d y d H d − ( x )= Z ∂E K sr ( x, E )Ψ( x ) · ν E ( x ) d H d − ( x ) , whence (4.3) follows. (cid:3) In Proposition 4.2 we prove some qualitative properties of the curvatures K sr defined in (4.1), which imply in particular that K sr are non-local curvatures in thesense of [18, 16]. Proposition 4.2. For every s ≥ , < r < the functionals K sr defined in (4.1) satisfy the following properties: (M) Monotonicity: If E, F ∈ C with E ⊆ F , and if x ∈ ∂F ∩ ∂E , then K sr ( x, F ) ≤ K sr ( x, E ) ; (T) Translational invariance: for any E ∈ C , x ∈ ∂E , y ∈ R d , K sr ( x, E ) = K sr ( x + y, E + y ) ; (S) Symmetry: For all E ∈ C and for every x ∈ ∂E it holds K sr ( x, E ) = −K sr ( x, R d \ ◦ E ) , where ◦ E denotes the interior of E . (B) Lower bound on the curvature of the balls: (4.8) K sr ( x, B (0 , ρ )) ≥ for all x ∈ ∂B (0 , ρ ) , ρ > Uniform continuity: There exists a modulus of continuity ω r such that thefollowing holds. For every E ∈ C , x ∈ ∂E , and for every diffeomorphism Φ : R d → R d of class C , with Φ = Id in R d \ B (0 , , we have |K sr ( x, E ) − K sr (Φ( x ) , Φ( E )) | ≤ ω r ( k Φ − Id k C ) . Proof. We prove separately the properties above. Property (M): Let E, F ∈ C such that E ⊆ F , then − χ F ≤ − χ E and χ F c ≤ χ E c .Therefore for all x ∈ ∂E ∩ ∂F , we have K sr ( x, F ) = Z R d ( χ F c ( y ) − χ F ( y )) k sr ( | x − y | ) d y ≤ Z R d ( χ E c ( y ) − χ E ( y )) k sr ( | x − y | ) d y = K sr ( x, E ) . Property (T): Let E ∈ C , x ∈ ∂E and y ∈ R d . By the change of variable ζ = η − y ,we obtain K sr ( x + y, E + y ) = Z R d ( χ E c + y ( η ) − χ E + y ( η )) k sr ( | x + y − η | ) d η = Z R d ( χ E c ( ζ ) − χ E ( ζ )) k sr ( | x − ζ | ) d ζ = K sr ( x, E ) . Property (S): Let E ∈ C and x ∈ ∂E , then we have K sr ( x, E ) = Z R d ( χ E c ( y ) − χ E ( y )) k sr ( | y − x | ) d y = − Z R d ( χ E ( y ) − χ E c ( y )) k sr ( | x − y | ) d y = −K sr ( x, R d \ ◦ E ) . Property (B): Let ρ > x ∈ ∂B (0 , ρ ) . Since B (2¯ x, ρ ) ⊂ B c (0 , ρ ) = R d \ B (0 , ρ ) ,we get K sr (¯ x, B (0 , ρ )) = Z R d ( χ B c (0 ,ρ ) ( y ) − χ B (0 ,ρ ) ( y )) k sr ( | ¯ x − y | ) d y ≥ Z R d ( χ B (2¯ x,ρ ) ( y ) − χ B (0 ,ρ ) ( y )) k sr ( | ¯ x − y | ) d y = 0 , (4.9)where in the last equality we have used the change of variable z = 2¯ x − y and theradial symmetry of k sr to deduce that Z R d χ B (2¯ x,ρ ) ( y ) k sr ( | ¯ x − y | ) d y = Z R d χ B (0 ,ρ ) ( z ) k sr ( | ¯ x − z | ) d z . Hence, by formula (4.9), (4.8) follows. Property (UC): Let Φ : R d → R d be a diffeomorphism of class C , with Φ( y ) = y for all | y − x | ≥ E := Φ( E ) . Let moreover θ k sr : [0 , + ∞ ) → R be thefunction defined by θ k sr ( η ) := R B (0 ,η ) k sr ( | z | ) d z . Fix ε > η ε > θ k sr ( η ε ) , θ k sr (2 η ε ) ≤ ε . Notice that (cid:12)(cid:12)(cid:12)(cid:12) Z B ( x,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | x − y | ) d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ θ k sr ( η ε ) , (4.11) (cid:12)(cid:12)(cid:12)(cid:12) Z B (Φ( x ) ,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ θ k sr ( η ε ) , (4.12) (cid:12)(cid:12)(cid:12)(cid:12) Z B (Φ( x ) , η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ θ k sr (2 η ε ) . (4.13) ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 29 By (4.11), (4.12), and (4.10), using triangular inequality, we have |K sr ( x, E ) − K sr (Φ( x ) , Φ( E )) |≤ ε (cid:12)(cid:12)(cid:12)(cid:12) Z B c ( x,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | x − y | ) d y − Z B c (Φ( x ) ,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (cid:12)(cid:12)(cid:12)(cid:12) Z B c ( x,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | x − y | ) d y − Z Φ( B c ( x,η ε )) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Z Φ( B c ( x,η ε )) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y − Z B c (Φ( x ) ,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y (cid:12)(cid:12)(cid:12)(cid:12) . (4.14)By the change of variable z = Φ( y ) and using that Φ( y ) = y if | y − x | ≥ (cid:12)(cid:12)(cid:12)(cid:12) Z B c ( x,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | x − y | ) d y − Z Φ( B c ( x,η ε )) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z B c ( x,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | x − y | ) d y − Z B c ( x,η ε ) ( χ E c ( z ) − χ E ( z )) k sr ( | Φ( x ) − Φ( z ) | ) J Φ( z ) d z (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z B c ( x,η ε ) (cid:12)(cid:12)(cid:12) k sr ( | x − y | ) − k sr ( | Φ( x ) − Φ( y ) | ) J Φ( y ) (cid:12)(cid:12)(cid:12) d y = Z B c ( x, (cid:12)(cid:12)(cid:12) k sr ( | x − y | ) − k sr ( | Φ( x ) − y | ) (cid:12)(cid:12)(cid:12) d y (4.15) + Z B ( x, \ B ( x,η ε ) (cid:12)(cid:12)(cid:12) k sr ( | x − y | ) − k sr ( | Φ( x ) − Φ( y ) | ) J Φ( y ) (cid:12)(cid:12)(cid:12) d y . (4.16)Now, assuming that k Φ − Id k C is small enough, by using Lagrange Theorem onecan show that(4.17) Z B c ( x, (cid:12)(cid:12)(cid:12) k sr ( | x − y | ) − k sr ( | Φ( x ) − y | ) (cid:12)(cid:12)(cid:12) d y ≤ ω ( k Φ − Id k C ) Z B c ( x, | x − y | d + s +1 d y ≤ ε , for some modulus of continuity ω . Analogously, for k Φ − Id k C small enough, onecan easily check that(4.18) Z B ( x, \ B ( x,η ε ) (cid:12)(cid:12)(cid:12) k sr ( | x − y | ) − k sr ( | Φ( x ) − Φ( y ) | ) J Φ( y ) (cid:12)(cid:12)(cid:12) d y ≤ Z B ( x, \ B ( x,η ε ) (cid:12)(cid:12)(cid:12) k sr ( | x − y | ) − k sr ( | Φ( x ) − Φ( y ) | ) (cid:12)(cid:12)(cid:12) d y + Z B ( x, \ B ( x,η ε ) k sr ( | Φ( x ) − Φ( y ) | ) (cid:12)(cid:12) − J Φ( y ) (cid:12)(cid:12) d y ≤ ε . Therefore, by (4.15), (4.16), (4.17), (4.18) we deduce that(4.19) (cid:12)(cid:12)(cid:12)(cid:12) Z B c ( x,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | x − y | ) d y − Z Φ( B c ( x,η ε )) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . In the end, we observe that, for k Φ − Id k C small enough, it holdsΦ( B c ( x, η ε )) △ B c (Φ( x ) , η ε ) ⊂ B (Φ( x ) , η ε ) , which, in view of (4.13), yields (cid:12)(cid:12)(cid:12)(cid:12) Z Φ( B c ( x,η ε )) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y − Z B c (Φ( x ) ,η ε ) ( χ E c ( y ) − χ E ( y )) k sr ( | Φ( x ) − y | ) d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ θ k sr (2 η ε ) ≤ ε . (4.20)Plugging (4.19) and (4.20) into (4.14) , we conclude the proof of property (UC) andof the whole proposition. (cid:3) The classical mean curvature. For every E ∈ C , and for every x ∈ ∂E ,we denote by K ( x, E ) the (scalar) mean curvature of ∂E at x , i.e., the sum of theprincipal curvatures of ∂E at x . It is well-known that K is nothing but the firstvariation of the perimeter. Let E ∈ C , x ∈ ∂E and assume that ν E ( x ) = e d ; thenin a neighborhood of x = ( x ′ , x d ) ∈ ∂E we have that ∂E is the graph of a C -function f : B ′ ( x ′ , r ) → R , for some r > f ( x ′ ) = 0 so that B ( x, r ) ∩ E = { ( x ′ , x d ) ∈ B ( x, r ) : x d ≤ f ( x ′ ) } . In this case the mean curvature of ∂E at x isgiven by(4.21) K ( x, E ) =Div (cid:18) − D f p | D f | (cid:19) ( x ′ ) = − d − X i =1 ∂ ∂ x i f ( x ′ )= − ω d − Z S d − θ ∗ D f ( x ′ ) θ d H d − ( θ ) , where θ ∗ is the row vector obtained by transposing the (column) vector θ andD f ( x ′ ) denotes the Hessian matrix of f evaluated at x ′ . Proposition 4.3. The standard mean curvature K satisfies the following proper-ties: (M) Monotonicity: If E, F ∈ C with E ⊆ F , and if x ∈ ∂F ∩ ∂E , then K ( x, F ) ≤ K ( x, E ) ; ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 31 (T) Translational invariance: For every E ∈ C , x ∈ ∂E , y ∈ R d , it holds: K ( x, E ) = K ( x + y, E + y ) ; (B) Lower bound on the curvature of the balls: K ( x, B (0 , ρ )) ≥ for all x ∈ ∂B (0 , ρ ) , ρ > Symmetry: For every E ∈ C and for every x ∈ ∂E it holds K ( x, E ) = −K ( x, R d \ ◦ E ) . (UC’) Uniform continuity: Given R > , there exists a modulus of continuity ω R such that the following holds. For every E ∈ C , x ∈ ∂E , such that E hasboth an internal and external ball condition of radius R at x , and for everydiffeomorphism Φ : R d → R d of class C , with Φ = Id in R d \ B (0 , , wehave (4.22) |K ( x, E ) − K (Φ( x ) , Φ( E )) | ≤ ω R ( k Φ − Id k C ) . Proof. We prove only the property (UC’) , since the check of the remaining prop-erties is straightforward. Let R > E ∈ C be such that E satisfies both aninternal and external ball condition of radius R at a point x ∈ ∂E . In order to getthe claim, we can always assume without loss of generality that k Φ − Id k C ≤ E ∩ B ( x, r ) = { z ∈ B ( x, r ) : g ( z ) < } , for some r > g ∈ C ( B ( x, r )). Moreover, in suitable coordinates wehave that x = 0, D g (0) = e d and, for all i = j , with i, j = 1 , · · · , d , ∂ g∂z i ∂z j (0) = 0.Then, it is well known that(4.23) K (0 , E ) =Div τ (cid:16) D g | D g | (cid:17) (0) = d − X i =1 ∂ g∂z i (0) , where Div τ denotes the tangential divergence operator. Since the mean curvatureis invariant by translations and rotations, up to small perturbations of Φ in C wemay assume, without loss of generality, that Φ(0) = 0 and that the normal to Φ( E )at Φ(0) = 0 is still e d . SinceΦ( E ) ∩ B (0 , ˜ r ) = { y ∈ B (0 , ˜ r ) : g (Φ − ( y )) < } for some ˜ r > 0, setting h := g ◦ Φ − , we have(4.24) K (0 , Φ( E )) = 1 | D h (0) | d − X j =1 ∂ h∂y j (0) − | D h (0) | d − X j =1 ∂h∂y j (0) ∂ | D h | ∂y j (0) . Therefore, using thatD h (0) =D g (0) DΦ − (0) = e d DΦ − (0) ,∂ h∂y j ∂y k (0) = d − X i =1 ∂ g∂z i (0) ∂ Φ − i ∂y j (0) ∂ Φ − i ∂y k (0) + ∂ Φ − d ∂y j ∂y k (0) , we have(4.25) 1 | D h (0) | (cid:12)(cid:12)(cid:12)(cid:12) d − X j =1 ∂ h∂y j (0) − d − X i =1 ∂ g∂z i (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | D h (0) | (cid:12)(cid:12)(cid:12)(cid:12) d − X j =1 ∂ Φ − d ∂y j (0) + d − X i =1 ∂ g∂z i (0) (cid:16) d − X j =1 (cid:16) ∂ Φ − i ∂y j (0) (cid:17) − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + | D h (0) − || D h (0) | (cid:12)(cid:12)(cid:12)(cid:12) d − X i =1 ∂ g∂z i (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C h k D Φ − k C + k D g k C k Id − (DΦ − ) k C + k D g k C k Id − DΦ − k C i ≤ C (cid:16) R (cid:17) k Id − Φ k C . Similar computations (that are left to the reader) show that(4.26) 1 | D h (0) | (cid:12)(cid:12)(cid:12)(cid:12) d − X j =1 ∂h∂y j (0) ∂ | D h | ∂y j (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) R (cid:17) k Id − Φ k C . Therefore, (4.22) follows from (4.23), (4.24), (4.25) and (4.26) . (cid:3) Convergence of k sr -curvature flow to mean curvature flow. We nowprove that the viscosity solutions to the k sr -curvature flow converge to the classicalmean curvature flow as r → + . To this end, we will adopt notation and we willuse results in [16].We preliminarily notice that since the curvatures K sr defined in (4.1) satisfyproperty (UC) in Proposition 4.2, they also satisfy property (UC’) in Proposition4.3 (with K replaced by K sr ). As a consequence K and K sr (for every 0 < r < s ≥ 1) satisfy the following continuity property:(C) Continuity: If { E n } n ∈ N ⊂ C , E ∈ C and E n → E in C , then the corre-sponding curvatures of E n at x converge to the curvature of E at x forevery x ∈ ∂E n ∩ ∂E .Such a property, together with properties (M) and (T) (see Propositions 4.3 and4.2), implies that the functionals K and K sr (for every s ≥ r ∈ (0 , K and K sr satisfy also properties(B) and (UC’) (referred to as (C’) in [16]), they both satisfy the assumptionsof [16, Theorem 2.9] that guarantee existence and uniqueness of suitably definedviscosity solutions of the corresponding geometric flows. We refer to [16, Definition2.3] for the precise definition of viscosity solution in this setting, while for thereader’s convenience we state the existence and uniqueness result specialized to thegeometric evolutions considered in this paper. Proposition 4.4. Let s ≥ and r > . Let u ∈ C ( R d ) be a uniformly continuousfunction with u = C in R d \ B (0 , R ) for some C , R ∈ R with R > . Then,there exists a unique viscosity solution - in the sense of [16, Definition 2.3] - u sr : R d × [0 , + ∞ ) → R to the Cauchy problem (4.27) ( ∂ t u ( x, t ) + | D u ( x, t ) |K sr ( x, { y : u ( y, t ) ≥ u ( x, t ) } ) = 0 u ( x, 0) = u ( x ) . ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 33 Moreover, the same result holds true if K sr is replaced by (any multiple of ) K . We will show that, as r → + , the scaled k sr -curvatures σ s ( r ) K sr converge to ω d − K on regular sets. In view of [16, Theorem 3.2] , such a result will be crucialin order to prove the convergence of the corresponding geometric flows. We firstprove the following result by adopting the same strategy used in [26, Proposition2]. Lemma 4.5. Let M, N ∈ R ( d − × ( d − and let { M r } r> , { N r } r> ⊂ R ( d − × ( d − be such that M r → M, N r → N as r → + . Then, for every δ > it holds lim r → + (cid:18) σ s ( r ) (cid:16) Z F r,δ k sr ( | y | ) d y − Z F r,δ k sr ( | y | ) d y (cid:17)(cid:19) = Z S d − θ ∗ ( N − M ) θ d H d − ( θ ) , (4.28) where F r,δ := { y = ( y ′ , y d ) ∈ B (0 , δ ) : ( y ′ ) ∗ M r y ′ ≤ y d ≤ ( y ′ ) ∗ N r y ′ }F r,δ := { y = ( y ′ , y d ) ∈ B (0 , δ ) : ( y ′ ) ∗ N r y ′ ≤ y d ≤ ( y ′ ) ∗ M r y ′ } . Proof. For every α > G α := { y = ( y ′ , y d ) ∈ R d − × R : y ′ = ρθ, ≤ ρ ≤ α, θ ∈ S d − ,ρ θ ∗ M r θ ≤ y d ≤ ρ θ ∗ N r θ } Therefore, for r small enough, F r,δ = G δ ∩ B (0 , δ ) , F r,δ ∩ B (0 , r ) = G r ∩ B (0 , r ) , F r,δ \ B (0 , r ) = (cid:0) ( G δ \ G r ) ∩ ( B (0 , δ ) (cid:1) ∪ (cid:0) G r \ B (0 , r ) (cid:1) . It follows that Z F r,δ k sr ( | y | ) d y = Z G δ k sr ( | y | ) d y − Z G δ \ B (0 ,δ ) k sr ( | y | ) d y = Z G r k sr ( | y | ) d y + Z G δ \G r k sr ( | y | ) d y − Z G δ \ B (0 ,δ ) | y | d + s d y = Z G r ∩ B (0 ,r ) k sr ( | y | ) d y + Z G r \ B (0 ,r ) k sr ( | y | ) d y + Z G δ \G r | y | d + s d y − Z G δ \ B (0 ,δ ) | y | d + s d y = Z G r ∩ B (0 ,r ) r d + s d y + Z G r \ B (0 ,r ) | y | d + s d y + Z G δ \G r | y | d + s d y − Z G δ \ B (0 ,δ ) | y | d + s d y = Z G r r d + s d y + Z G δ \G r | y | d + s d y − Z G δ \ B (0 ,δ ) | y | d + s d y − Z G r \ B (0 ,r ) r d + s d y + Z G r \ B (0 ,r ) | y | d + s d y . (4.29) We set A r := { θ ∈ S d − : θ ∗ ( M r − N r ) θ ≤ } and we notice that Z G r r d + s d y + Z G δ \G r | y | d + s d y = 1 r d + s Z A r d H d − ( θ ) Z r d ρ ρ d − Z ρ θ ∗ N r θρ θ ∗ M r θ d y d + Z A r d H d − ( θ ) Z δr d ρ ρ d − Z ρ θ ∗ N r θρ θ ∗ M r θ ρ + y d ) d + s d y d = 1 d + 1 r d +1 r d + s Z A r θ ∗ ( N r − M r ) θ d H d − ( θ )+ Z A r d H d − ( θ ) Z δr d ρ ρ d − Z θ ∗ N r θθ ∗ M r θ ρ ( ρ + ρ t ) d + s d t = r − s d + 1 Z A r θ ∗ ( N r − M r ) θ d H d − ( θ )+ Z A r d H d − ( θ ) Z δr d ρ ρ s Z θ ∗ N r θθ ∗ M r θ ρ t ) d + s d t , (4.30)where in the last but one equality we have used the change of variable y d = ρ t .Moreover, trivially we have(4.31) Z G δ \ B (0 ,δ ) | y | d + s d y ≤ C ( δ ) , for some C ( δ ) > r small enough, G r \ B (0 , r ) ⊂ (cid:0) B ′ (0 , r ) \ B ′ (0 , r − cr ) (cid:1) × [ − cr , cr ]for some constant c > r ; as a consequence, |G r \ B (0 , r ) | ≤ Cr d +2 , whence we deduce that(4.32) Z G r \ B (0 ,r ) r d + s d y ≤ Cr − s , Z G r \ B (0 ,r ) | y | d + s d y ≤ Z G r \ B (0 ,r ) r d + s d y ≤ Cr − s . Therefore, by (4.29), (4.30), (4.31) and (4.32), we obtain(4.33) 1 σ s ( r ) Z F r,δ k sr ( | y | ) d y = r − s ( d + 1) σ s ( r ) Z A r θ ∗ ( N r − M r ) θ d H d − ( θ )+ 1 σ s ( r ) Z A r d H d − ( θ ) Z δr d ρ ρ s Z θ ∗ N r θθ ∗ M r θ ρ t ) d + s d t + f ( r ) , ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 35 where f ( r ) → r → + .Now we set A r := { θ ∈ S d − : θ ∗ ( N r − M r ) θ ≤ } ;by arguing as in the proof of (4.33) we obtain(4.34) 1 σ s ( r ) Z F r,δ k sr ( | y | ) d y = r − s ( d + 1) σ s ( r ) Z A r θ ∗ ( M r − N r ) θ d H d − ( θ )+ 1 σ s ( r ) Z A r d H d − ( θ ) Z δr d ρ ρ s Z θ ∗ M r θθ ∗ N r θ ρ t ) d + s d t + f ( r ) , where f ( r ) → r → + . Therefore by formulas (4.33) and (4.34), using that A r ∪ A r = S d − , we get (cid:18) σ s ( r ) (cid:16) Z F r,δ k sr ( | y | ) d y − Z F r,δ k sr ( | y | ) d y (cid:17)(cid:19) = (cid:20) r − s ( d + 1) σ s ( r ) (cid:21) Z S d − θ ∗ ( N r − M r ) θ d H d − ( θ )+ 1 σ s ( r ) Z S d − d H d − ( θ ) Z δr d ρ ρ s Z θ ∗ N r θθ ∗ M r θ ρ t ) d + s d t + f ( r ) − f ( r ) . (4.35)Since r − s ( d + 1) σ s ( r ) = (cid:26) d +1) | log r | if s = 1 s − d + s if s > , and recalling that M r and N r converge to M and N , respectively, we getlim r → + r − s ( d + 1) σ s ( r ) Z S d − θ ∗ ( N r − M r ) θ d H d − ( θ )= s = 1 ,s − d + s Z S d − θ ∗ ( N − M ) θ d H d − ( θ ) if s > s ≥ r ) and the very definition of σ s ( r ) in (1.5),we have lim r → + σ s ( r ) Z δr ρ s ρ t ) d + s d ρ = d + 1 d + s , which, by the Dominate Convergence Theorem, yieldslim r → + σ s ( r ) Z S d − d H d − ( θ ) Z δr d ρ ρ s Z θ ∗ N r θθ ∗ M r θ ρ t ) d + s d t = lim r → + Z S d − d H d − ( θ ) Z θ ∗ N r θθ ∗ M r θ d t σ s ( r ) Z δr ρ s ρ t ) d + s d ρ = d + 1 d + s Z S d − θ ∗ ( N − M ) θ d H d − ( θ ) . (4.37)By formulas (4.35), (4.36) and (4.37) we obtain (4.28). (cid:3) Theorem 4.6. Let s ≥ . Let { E r } r> ⊂ C be such that E r → E in C as r → + ,for some E ∈ C . Then, for every x ∈ ∂E ∩ ∂E r for every r > , it holds lim r → + K sr ( x, E r ) σ s ( r ) = ω d − K ( x, E ) . Proof. Let x ∈ ∂E ∩ ∂E r for all r > 0. By Proposition 4.2 we have that thecurvatures K sr satisfy properties (S) and (T); moreover, it is easy to check that K sr are invariant by rotations. Therefore, we can assume without loss of generalitythat E and { E r } r> are compact, and that x = 0, ν E (0) = ν E r (0) = e d for all r > 0. Since E r → E in C as r → + we have that there exist δ > φ, φ r : B ′ (0 , δ ) → R such that φ r → φ in C as r → + , φ (0) = φ r (0) = 0 ,D φ (0) = D φ r (0) = 0 and ∂E ∩ B (0 , δ ) = { ( y ′ , φ ( y ′ )) : y ′ ∈ B ′ (0 , δ ) } ∩ B (0 , δ ) ,∂E r ∩ B (0 , δ ) = { ( y ′ , φ r ( y ′ )) : y ′ ∈ B ′ (0 , δ ) } ∩ B (0 , δ ) ,E ∩ B (0 , δ ) = { ( y ′ , y d ) : y ′ ∈ B ′ (0 , δ ) , y d ≤ φ ( y ′ ) } ∩ B (0 , δ ) ,E r ∩ B (0 , δ ) = { ( y ′ , y d ) : y ′ ∈ B ′ (0 , δ ) , y d ≤ φ r ( y ′ ) } ∩ B (0 , δ ) . Let η > δ small enough we have(4.38) (cid:12)(cid:12)(cid:12) φ r ( y ′ ) − 12 ( y ′ ) ∗ D φ r (0) y ′ (cid:12)(cid:12)(cid:12) ≤ η | y ′ | for every 0 < r < , y ′ ∈ B ′ (0 , δ ) . We define the following sets A ( r ) := { y = ( y ′ , y d ) ∈ B (0 , δ ) : − φ r ( y ′ ) ≤ y d ≤ φ r ( y ′ ) } ,B ( r ) := { y = ( y ′ , y d ) ∈ B (0 , δ ) : φ r ( y ′ ) ≤ y d ≤ − φ r ( y ′ ) } ,C ( r ) := (cid:0) E cr \ B ( r ) (cid:1) ∩ B (0 , δ )= { y = ( y ′ , y d ) ∈ B (0 , δ ) : y d ≥ max { φ r ( y ′ ) , − φ r ( y ′ ) }} ,D ( r ) := (cid:0) E r \ A ( r ) (cid:1) ∩ B (0 , δ )= { y = ( y ′ , y d ) ∈ B (0 , δ ) : y d ≤ min { φ r ( y ′ ) , − φ r ( y ′ ) }} , where the equalities above are understood in the sense of measurable sets, i.e., upto negligible sets. We notice that E r ∩ B (0 , δ ) = A ( r ) ∪ D ( r ) , E cr ∩ B (0 , δ ) = B ( r ) ∪ C ( r ) , Z C ( r ) k sr ( | y | ) d y = Z D ( r ) k sr ( | y | ) d y , ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 37 whence we deduce that K sr (0 , E r ) = Z B (0 ,δ ) ( χ E cr ( y ) − χ E r ( y )) k sr ( | y | ) d y + Z B c (0 ,δ ) ( χ E cr ( y ) − χ E r ( y )) k sr ( | y | ) d y = Z R d ( χ B ( r ) ( y ) − χ A ( r ) ( y )) k sr ( | y | ) d y + Z R d ( χ C ( r ) ( y ) − χ D ( r ) ( y )) k sr ( | y | ) d y + Z B c (0 ,δ ) ( χ E cr ( y ) − χ E r ( y )) k sr ( | y | ) d y = Z R d ( χ B ( r ) ( y ) − χ A ( r ) ( y )) k sr ( | y | ) d y + Z B c (0 ,δ ) ( χ E cr ( y ) − χ E r ( y )) k sr ( | y | ) d y . (4.39)Trivially, (cid:12)(cid:12)(cid:12) Z B c (0 ,δ ) ( χ E cr ( y ) − χ E r ( y )) k sr ( | y | ) d y (cid:12)(cid:12)(cid:12) ≤ dω d δ − s s . In order to study the limitlim r → + σ s ( r ) Z R d ( χ B ( r ) ( y ) − χ A ( r ) ( y )) k sr ( | y | ) d y , we define the following sets A − ( r ) := n y = ( y ′ , y d ) ∈ B (0 , δ ) : − 12 ( y ′ ) ∗ D φ r (0) y ′ + η | y ′ | ≤ y d ≤ 12 ( y ′ ) ∗ D φ r (0) y ′ − η | y ′ | o ,A + ( r ) := n y = ( y ′ , y d ) ∈ B (0 , δ ) : − 12 ( y ′ ) ∗ D φ r (0) y ′ − η | y ′ | ≤ y d ≤ 12 ( y ′ ) ∗ D φ r (0) y ′ + η | y ′ | o ,B − ( r ) := n y = ( y ′ , y d ) ∈ B (0 , δ ) :12 ( y ′ ) ∗ D φ r (0) y ′ + η | y ′ | ≤ y d ≤ − 12 ( y ′ ) ∗ D φ r (0) y ′ − η | y ′ | o ,B + ( r ) := n y = ( y ′ , y d ) ∈ B (0 , δ ) :12 ( y ′ ) ∗ D φ r (0) y ′ − η | y ′ | ≤ y d ≤ − 12 ( y ′ ) ∗ D φ r (0) y ′ + η | y ′ | o . By (4.38) we have that A − ( r ) ⊂ A ( r ) ⊂ A + ( r ) , B − ( r ) ⊂ B ( r ) ⊂ B + ( r ) , and hence Z R d ( χ B − ( r ) ( y ) − χ A + ( r ) ( y )) k sr ( | y | ) d y ≤ Z R d ( χ B ( r ) ( y ) − χ A ( r ) ( y )) k sr ( | y | ) d y ≤ Z R d ( χ B + ( r ) ( y ) − χ A − ( r ) ( y )) k sr ( | y | ) d y. (4.40)Then, by applying Lemma 4.5 and using (4.40), we obtain − Z S d − θ ∗ (D φ (0) + 2 η Id) θ d H d − ( θ ) ≤ lim inf r → + σ s ( r ) Z R d ( χ B ( r ) ( y ) − χ A ( r ) ( y )) k sr ( | y | ) d y ≤ lim sup r → + σ s ( r ) Z R d ( χ B ( r ) ( y ) − χ A ( r ) ( y )) k sr ( | y | ) d y ≤ − Z S d − θ ∗ (D φ (0) − η Id) θ d H d − ( θ ) . (4.41)Therefore, by (4.39) and (4.41), we get − Z S d − θ ∗ (D φ (0) + 2 η Id) θ d H d − ( θ ) ≤ lim inf r → + σ s ( r ) K sr (0 , E r ) ≤ lim sup r → + σ s ( r ) K sr (0 , E r ) ≤ − Z S d − θ ∗ (D φ (0) − η Id) θ d H d − ( θ ) , which, sending η to 0 and using (4.21) implies the claim. (cid:3) We are now in a position to state the main result of this section. Theorem 4.7. Let s ≥ be fixed. Let u ∈ C ( R d ) be a uniformly continuousfunction with u = C in R d \ B (0 , R ) for some C , R ∈ R with R > . Forevery r > , let u sr : R d × [0 , + ∞ ) → R be the viscosity solution to the Cauchyproblem (4.27) . Then, setting v sr ( x, t ) := u sr ( x, tσ s ( r ) ) for all x ∈ R d , t ≥ , wehave that, for every T > , v sr uniformly converge to u as r → + in R d × [0 , T ] ,where u : R d × [0 , + ∞ ) → R is the viscosity solution to the classical mean curvatureflow (4.42) ( ∂ t u ( x, t ) + | D u ( x, t ) | ω d − K ( x, { y : u ( y, t ) ≥ u ( x, t ) } ) = 0 u ( x, 0) = u ( x ) . Proof. We preliminarily notice that, by an easy scaling argument, the functions v sr are viscosity solution to ( ∂ t v ( x, t ) + | D v ( x, t ) | σ s ( r ) K sr ( x, { y : v ( y, t ) ≥ v ( x, t ) } ) = 0 v ( x, 0) = u ( x ) . By Theorem 4.6 we have that , as r → + the scaled k sr -curvatures σ s ( r ) K sr convergeto ω d − K on regular sets. Moreover, by Propositions 4.2 and 4.3, K sr (for every r ∈ (0 , K satisfy properties (M), (T), (S), (UC’). Furthermore, for every ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 39 ρ > x ∈ ∂B (0 , ρ ), by Proposition 4.2, we have that K sr ( x, B (0 , ρ ) ≥ r ∈ (0 , K sr ( x, B (0 , ρ )) < + ∞ . This triviallyimplies the following property:(UB) There exists K > r ∈ (0 , K sr ( x, B (0 , ρ )) ≥ − Kρ for all ρ > x ∈ ∂B (0 , ρ ) and sup r ∈ (0 , K sr ( x, B (0 , ρ )) < + ∞ for all ρ > x ∈ ∂B (0 , ρ ) .Properties (M), (T), (S), (UC’) (referred to as (C’) in [16]) and (UB), together withthe convergence of the curvatures on regular sets, are exactly the assumptions of[16, Theorem 3.2], which, in our case, establishes the convergence of v sr to u locallyuniformly in R d × [0 , T ] for every T > (cid:3) Stability as r → + and s → + simultaneously In this section we study Γ-convergence and compactness properties for the s -fractional perimeters ˜ J sr defined in (1.4) when r → + and s → ¯ s (with ¯ s ≥ s = 1 , the case¯ s > { r n } n ∈ N ⊂ (0 , 1) and { s n } n ∈ N ⊂ (1 , + ∞ ) be such that r n → + and s n → + as n → + ∞ . Recalling the definitions of σ s ( r ) in (1.5) and α s in (1.13), we set(5.1) β ( r n , s n ) := σ s n ( r n ) + α s n = d + s n d + 1 r − s n n − s n − d + 1and we notice that(5.2) lim n → + ∞ β ( r n , s n ) ≥ lim n → + ∞ r − s n n − s n − n → + ∞ Z r n ρ − s n d ρ ≥ lim n → + ∞ Z r n ρ − d ρ = lim n → + ∞ | log r n | = + ∞ . -convergence and compactness. In Theorem 5.1 below we study the Γ-convergence of the functionals β ( r n ,s n ) ˜ J s n r n as n → + ∞ . Theorem 5.1. Let { r n } n ∈ N ⊂ (0 , and { s n } n ∈ N ⊂ (1 , + ∞ ) be such that r n → + and s n → + as n → + ∞ . The following Γ -convergence result holds true. (i) (Compactness) Let U ⊂ R d be an open bounded set and let { E n } n ∈ N ⊂ M( R d ) be such that E n ⊂ U for every n ∈ N and ˜ J s n r n ( E n ) ≤ M β ( r n , s n ) for every n ∈ N ,for some constant M independent of n . Then up to a subsequence, χ E n → χ E strongly in L ( R d ) for some set E ∈ M f ( R d ) with Per( E ) < + ∞ . (ii) (Lower bound) Let E ∈ M f ( R d ) . For every { E n } n ∈ N ⊂ M f ( R d ) with χ E n → χ E strongly in L ( R d ) it holds ω d − Per( E ) ≤ lim inf n → + ∞ ˜ J s n r n ( E n ) β ( r n , s n ) . (iii) (Upper bound) For every E ∈ M f ( R d ) there exists { E n } n ∈ N ⊂ M f ( R d ) suchthat χ E n → χ E strongly in L ( R d ) and ω d − Per( E ) = lim sup n → + ∞ ˜ J s n r n ( E n ) β ( r n , s n ) . Proof of compactness. We start by proving the compactness property Theo-rem 5.1(i). To this purpose, we first prove the following lemma which correspondsto Lemma 2.4 when both r and s vary. Lemma 5.2. Let { r n } n ∈ N ⊂ (0 , and { s n } n ∈ N ⊂ (1 , + ∞ ) be such that r n → + and s n → + as n → + ∞ . Let Ω ∈ M f ( R d ) be a bounded set with Lipschitzcontinuous boundary and | Ω | = 1 . For every η ∈ (0 , there exist a constant C (Ω , d, S, η ) > and ¯ n ∈ N such that for every measurable set A ⊂ Ω with η ≤| A | ≤ − η it holds Z A Z Ω \ A k s n r n ( | x − y | ) d y d x ≥ C (Ω , d, S, η ) β ( r n , s n ) for every n ≥ ¯ n , where S := sup n ∈ N s n .Proof. The proof is fully analogous to the one of Lemma 2.4; we sketch only themain differences. For every n ∈ N , let I n ∈ N be such that 2 − I n − ≤ r n ≤ − I n .Let φ and φ δ (for every δ > 0) be as in Lemma 2.5. By arguing verbatim as in theproof of (2.6) (see (2.9), (2.10), and (2.11)), for every n ∈ N and for every z ∈ R d we have(5.3) k s n r n ( | z | ) ≥ d + s n − d + s n φ I n X i =0 (2 s n ) i φ − i ( z ) ≥ d +1 d +1 − φ I n X i =0 (2 s n ) i φ − i ( z ) . Moreover, since | log r n | log 2 − ≤ I n ≤ | log r n | log 2 , setting m ( S ) := inf s ∈ (1 ,S ] s − s − − ,we get I n X i =0 (2 s n − ) i = (cid:0) s n − (cid:1) I n +1 − s n − − ≥ r − s n n − s n − − r − s n n − s n − s n − s n − − ≥ m ( S )2 d + 1 d + s n (cid:16) d + s n d + 1 r − s n n − s n − (cid:17) ≥ m ( S )2 d + 1 d + S (cid:16) d + s n d + 1 r − s n n − s n − d + 1 r − s n n − s n − (cid:17) ≥ m ( S )2 d + 1 d + S β ( r n , s n ) , (5.4)where in the last inequality we have used that, in view of (5.1), r − snn − s n − ≥ (cid:3) With Lemma 5.2 in hand, we can prove Theorem 5.1(i), whose proof closelyfollows the one of Theorem 1.5(i). We sketch only the main differences. Proof of Theorem 5.1(i). We preliminarily notice that, up to a subsequence, thefollowing limit exists(5.5) lim n → + ∞ ( s n − | log r n | =: λ ; ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 41 clearly, λ ∈ [0 , + ∞ ] . We first prove the claim under the assumption λ = 0 . Let α ∈ (0 , 1) and for every n ∈ N we set l n := r αn ( s n − 1) ; therefore, since λ ∈ (0 , + ∞ ] ,lim n → + ∞ r n l n = lim n → + ∞ r − αn s n − . By adopting the same notation as in Subsection 2.2 we set˜ E n := H ( n ) [ h =1 Q nh , where { Q nh } h ∈ N is a family of pairwise disjoint cubes of sidelength l n which coversthe whole R d and satisfies (2.12).By arguing verbatim as in the proof of (2.13) one can show that there exists n ′ ∈ N such that(5.6) | ˜ E n △ E n | ≤ l s n n β ( r n , s n ) M for every n ≥ n ′ . We observe that(5.7) lim n → + ∞ l s n n β ( r n , s n ) = lim n → + ∞ r αs n n ( s n − s n (cid:16) d + s n d + 1 r − s n n − s n − d + 1 (cid:17) = lim n → + ∞ r − s n + αs n n d + s n d + 1 ( s n − s n − = 0 . Now, setting S := sup n ∈ N s n , we claim that there exists a constant C ( α, d, S ) suchthat for n large enough(5.8) Per( ˜ E n ) ≤ C ( α, d, S ) ˜ J s n r n ( E n ) β ( r n , s n ) . In order to prove (5.8), we argue as in Step 2 in Subsection 2.2. We define thefamily R of rectangles R = ˜ Q nh ∪ ˆ Q nh such that ˜ Q nh and ˆ Q nh are adjacent, ˜ Q nh ⊂ ˜ E n and ˆ Q nh ⊂ ˜ E cn .Notice that(5.9) Per( ˜ E n ) ≤ dl d − n ♯ R , ˜ J s n r n ( E n ) β ( r n , s n ) ≥ d β ( r n , s n ) X R ∈R Z R ∩ E n Z R \ E n k s n r n ( | x − y | ) d y d x . Moreover, by Lemma 5.2, for every rectangle ¯ R given by the union of two adjacentunitary cubes in R d , there exists ¯ n ∈ N such that(5.10) C ( d, λ ) := inf (cid:26) β ( ρ n , s n ) Z F Z ¯ R \ F k s n ρ n ( | x − y | ) d y d x : n ≥ ¯ n, F ∈ M f ( R d ) , F ⊂ ¯ R , ≤ | F | ≤ (cid:27) > . Furthermore, by the very definition of β ( r n , s n ) in (5.1), we have β ( r n , s n ) l − s n n β (cid:0) r n l n , s n (cid:1) =1 + ( d + 1)( l − s n n − d + s n ) (cid:0) r − s n n − l − s n n (cid:1) + ( s n − l − s n n , whence, using that l n = r αn ( s n − 1) and (5.5), we deduce(5.11) lim n → + ∞ β ( r n l n , s n ) β ( r n ,s n ) l − snn = ( e λα − e λ − e λα if λ = + ∞ λ = + ∞ . For every set O ∈ M f ( R d ) we set O l n := Ol n . By (5.9), (1.2), (5.11) and by applying(5.10) with ¯ R = R l n for every R ∈ R , for n large enough we obtain˜ J s n r n ( E n ) β ( r n , s n ) ≥ C ( d ) β ( r n , s n ) l dn X R ∈R Z R ln ∩ E ln Z R ln \ E ln k s n r n ( | l n ( x − y ) | ) d y d x = C ( d ) l − s n n β ( r n , s n ) l d − n X R ∈R Z R ln ∩ E ln Z R ln \ E ln k s nrnln ( | x − y | ) d y d x ≥ C ( α, d, λ ) l d − n X R ∈R β ( r n l n , s n ) Z R ln ∩ E ln Z R ln \ E ln k s nrnln ( | x − y | ) d y d x ≥ C ( α, d, λ ) l d − n ♯ R C ( d, λ ) ≥ C ( α, d, λ )Per( ˜ E n ) , i.e., (5.8). Therefore, using (5.6), (5.7) and (5.8), by arguing as in Step 3 of theproof of Theorem 1.5(i), we get the claim whenever (5.5) is satisfied.Finally, if lim n → + ∞ ( s n − | log r n | = 0 , taking l n = r αn (with α ∈ (0 , n → + ∞ l s n n β ( r n , s n ) = 0 , lim n → + ∞ β ( r n , s n ) l − s n n β (cid:0) r n l n , s n (cid:1) = 11 − α , which used in the above proof, in place of (5.7) and (5.11), respectively, imply theclaim also in this case. (cid:3) The following result follows by arguing as in the proof of Proposition 2.6, usingnow the estimates in the proof of Theorem 5.1(i) instead of the ones in the proofof Theorem 1.5 (i). Proposition 5.3. Let { E n } n ∈ N ⊂ M f ( R d ) be such that χ E n → χ E in L ( R d ) as n → + ∞ , for some E ∈ M f ( R d ) . If lim sup n → + ∞ ˜ J s n r n ( E n ) β ( r n , s n ) < + ∞ , then E is a set of finite perimeter. Proof of the lower bound. In order to prove the Γ-liminf inequality Theorem5.1(ii), we first need the following result, which is the analogous to Lemma 3.1under our assumptions on { s n } n ∈ N and { r n } n ∈ N . ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 43 Lemma 5.4. Let { r n } n ∈ N ⊂ (0 , and { s n } n ∈ N ⊂ (1 , + ∞ ) be such that r n → + and s n → + as n → + ∞ . For every ε > there exist δ > and ¯ n ∈ N such thatfor every ν ∈ S d − , for every E ∈ M f ( R d ) with | ( E △ H − ν (0)) ∩ Q ν | < δ and for every n ≥ ¯ n it holds Z Q ν ∩ E Z Q ν ∩ E c k s n r n ( | x − y | ) d y d x ≥ ω d − (1 − ε ) β ( r n , s n ) . Proof. By arguing as in the proof of Lemma 3.1 (see (3.10)) one can prove that(5.12) Z Q ν ∩ E Z Q ν ∩ E c k s n r n ( | x − y | ) d y d x ≥ ω d − β ( r n , s n ) (cid:16) − η ( δ ) − − η ( δ ) β ( r n , s n ) δ − s n − s n − − C ( d ) p δ (cid:17) , where η ( t ) → t → < δ < η ( δ ) + 2 C ( d ) p δ ≤ ε . Furthermore, sincelim n → + ∞ δ − s n − s n − | log δ | and lim n → + ∞ β ( r n , s n ) = + ∞ , we have that there exists ¯ n ∈ N such that(5.14) 1 − η ( δ ) β ( r n , s n ) δ − s n − s n − ≤ ε n ≥ ¯ n . Therefore, by (5.12), (5.13) and (5.14), we get the claim. (cid:3) Proof of Theorem 5.1(ii). The proof closely follows the one of Theorem 1.5(ii). Wecan assume without loss of generality that(5.15) 12 β ( r n , s n ) Z R d Z R d k s n r n ( | x − y | ) | χ E n ( x ) − χ E n ( y ) | d y d x ≤ C , for some constant C > n . Then, by Corollary 5.3 we have that E has finite perimeter. For every n ∈ N let µ n be the measure on the product space R d × R d defined by µ n ( A × B ) := 12 β ( r n , s n ) Z A Z B k s n r n ( | x − y | ) | χ E n ( x ) − χ E n ( y ) | d y d x for every A, B ∈ M( R d ). By arguing as in the proof of Theorem 1.5(ii) we havethat, up to a subsequence, µ n → µ as n → + ∞ for some measure µ concentratedon the set D := { ( x, x ) : x ∈ R d } . Therefore, by using the same Radon-Nykodymargument exploited in the proof of Theorem 1.5(ii), it is enough to show that for H d − - a.e. x ∈ ∂ ∗ E (5.16) lim inf l → + µ ( Q νl ( x ) × Q νl ( x ) l d − ≥ lim inf l → + lim inf n → + ∞ µ n ( Q νl ( x ) × Q νl ( x )) l d − ≥ ω d − , where we have set ν := ν E ( x ) and Q lν ( x ) := x + lQ ν . In order to prove (5.16)we adopt the same strategy used to prove (3.12). More precisely, setting F n,l = x + lE n , in place of (3.14) we have(5.17) 1 l d − µ n ( Q νl ( x ) × Q νl ( x ))= l − s n β ( r n , s n ) Z Q ν Z Q ν k s nrnl ( | ξ − η | ) | χ F n,l ( ξ ) − χ F n,l ( η ) | d ξ d η , and, in place of (3.17),(5.18) 12 Z Q ν Z Q ν k s nrnl ( | ξ − η | ) | χ F n,l ( ξ ) − χ F n,l ( η ) | d ξ d η ≥ ω d − (1 − ε ) β (cid:16) r n l , s n (cid:17) , which is a consequence of Lemma 5.4. Therefore, sincelim n → + ∞ l − s n β ( r n , s n ) β (cid:16) r n l , s n (cid:17) = 1 , by (5.17) and (5.18), we getlim inf l → + µ ( Q νl ( x ) × Q νl ( x )) l d − ≥ (1 − ε ) ω d − , whence (5.16) follows by the arbitrariness of ε . (cid:3) Proof of the upper bound. In order to prove the Γ-limsup inequality, we needthe following result which is the analogous of Proposition 1.1 when both r and s vary. Proposition 5.5. Let E ∈ M f ( R d ) be a smooth set. Then lim n → + ∞ ˜ J s n r n ( E ) β ( r n , s n ) = ω d − Per( E ) . Proof. By Lemma 1.4 and by formula (1.9) we have˜ J s n r n ( E ) β ( r n , s n ) = ω d − Per( E ) + 1 β ( r n , s n ) F s n ( E ) − β ( r n , s n ) Z ∂E d H d − ( y ) Z ( E △ H − νE ( y ) ( y )) ∩ B ( y,r n ) r ns n | ( y − x ) · ν E ( y ) || x − y | d (cid:18) d + s n ds n (cid:19) d x + 1 β ( r n , s n ) Z ∂E d H d − ( y ) Z ( E △ H − νE ( y ) ( y )) ∩ B ( y,r n ) r ns n | ( y − x ) · ν E ( y ) | r nd d d x − β ( r n , s n ) 1 s n Z ∂E d H d − ( y ) Z ( E △ H − νE ( y ) ( y )) ∩ ( B ( y, \ B ( y,r n )) | ( y − x ) · ν E ( y ) || x − y | d + s n d x − β ( r n , s n ) 1 s n Z E H d − ( E c ∩ ∂B ( x, x , where H − ν ( y ) is the set defined in (0.3). Therefore, by arguing verbatim as in theproof of Proposition 1.1 and using (5.2), we deduce the claim. (cid:3) With Proposition 5.5 in hand, the proof of Theorem 5.1(iii) is fully analogous tothe one of Theorem 1.5(iii) and is omitted. ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 45 Convergence of the k s n r n -curvature flows to the mean curvature flow. Here we study the convergence of the curvatures K s n r n defined in (4.1) to the classicalmean curvature K in (4.21) when r n → + and s n → + simultaneously. As inSubsection 4.3 we use such a result to deduce the convergence of the correspondinggeometric flows.First we prove the following lemma which is the analogous of Lemma 4.5 in thecase treated in this section. Lemma 5.6. Let { s n } n ∈ N ⊂ (1 , + ∞ ) and { r n } n ∈ N ⊂ (0 , be such that r n → + and s n → + as n → + ∞ . Let M, N ∈ R ( d − × ( d − and let { M n } n ∈ N , { N n } n ∈ N ⊂ R ( d − × ( d − be such that M n → M, N n → N as n → + ∞ . Then, for every δ > ,it holds lim n → + ∞ (cid:18) β ( r n , s n ) (cid:16) Z F n,δ k s n r n ( | y | ) d y − Z F n,δ k s n r n ( | y | ) d y (cid:17)(cid:19) = Z S d − θ ∗ ( N − M ) θ d H d − ( θ ) , (5.19) where F n,δ := { y = ( y ′ , y d ) ∈ B (0 , δ ) : ( y ′ ) ∗ M n y ′ ≤ y d ≤ ( y ′ ) ∗ N n y ′ }F n,δ := { y = ( y ′ , y d ) ∈ B (0 , δ ) : ( y ′ ) ∗ N n y ′ ≤ y d ≤ ( y ′ ) ∗ M n y ′ } . Proof. By arguing verbatim as in the proof of (4.35) one can show that1 β ( r n , s n ) Z F n,δ k s n r n ( | y | ) d y − β ( r n , s n ) Z F n,δ k s n r n ( | y | ) d y = r n − s n ( d + 1) β ( r n , s n ) Z S d − θ ∗ ( N n − M n ) θ d H d − ( θ )+ 1 β ( r n , s n ) Z S d − d H d − ( θ ) Z δr n d ρ ρ s n Z θ ∗ N n θθ ∗ M n θ ρ t ) d + sn d t , + η n , (5.20)where η n → n → + ∞ . It is easy to see thatlim n → + ∞ r n − s n β ( r n , s n ) = 0 , whence we get(5.21) lim n → + ∞ r − s n n ( d + 1) β ( r n , s n ) Z S d − θ ∗ ( N n − M n ) θ d H d − ( θ ) = 0 . Now we claim that for every t ∈ R (5.22) lim n → + ∞ β ( r n , s n ) Z δr n ρ s n ρ t ) d + sn d ρ = 1 , which in view of (5.20) and (5.21) and of the Dominate Convergence Theorem,implies (5.19). In order to prove (5.22), we first notice that1 β ( r n , s n ) Z δr n ρ s n ρ t ) d + sn = 1 β ( r n , s n ) Z δr n ρ s n d ρ − β ( r n , s n ) Z δr n ρ s n (cid:18) − ρ t ) d + sn (cid:19) d ρ = 1 β ( r n , s n ) r − s n n − δ − s n s n − − β ( r n , s n ) Z δr n ρ s n (cid:18) − ρ t ) d + sn (cid:19) d ρ , so that, by the very definition of β in (5.1), it is enough to show that(5.23) lim sup n → + ∞ β ( r n , s n ) Z δr n ρ s n (cid:18) − ρ t ) d + sn (cid:19) d ρ = 0 . As for the proof of (5.23) we argue as follows. First we notice that, setting S :=sup n ∈ N s n , (1 + ρ t ) d + sn ≤ C ( d, S, t ) ρ , so that, for n large enough,1 β ( r n , s n ) Z δr n ρ s n (cid:18) − ρ t ) d + sn (cid:19) d ρ = 1 β ( r n , s n ) Z δr n ρ s n (1 + ρ t ) d + sn − ρ t ) d + sn d ρ ≤ β ( r n , s n ) Z δr n C ( d, S, t ) ρ − s n d ρ → n → + ∞ , thus concluding the proof of (5.23) and of the whole lemma. (cid:3) By using Lemma 5.6 in place of Lemma 4.5 in the proof of Theorem 4.6, one canprove the following result. Theorem 5.7. Let { r n } n ∈ N ⊂ (0 , and s n ⊂ (1 , + ∞ ) be such that r n → + and s n → + as n → + ∞ . Let { E n } n ∈ N ⊂ C such that E n → E in C as n → + ∞ , forsome E ∈ C . Then for every x ∈ ∂E ∩ ∂E n for every n ∈ N , lim n → + ∞ K s n r n ( x, E n ) β ( r n , s n ) = ω d − K ( x, E ) . Finally, by using Theorem 5.7 in place of Theorem 4.6 in the proof of Theorem4.7, one can prove the following result which provides the convergence of the k s n r n -nonlocal curvature flows when r n → + and s n → + . Theorem 5.8. Let { r n } n ∈ N ⊂ (0 , and s n ⊂ (1 , + ∞ ) be such that r n → + and s n → + as n → + ∞ . Let u ∈ C ( R d ) be a uniformly continuous functionwith u = C in R d \ B (0 , R ) for some C , R ∈ R with R > . For every n ∈ N , let u s n r n : R d × [0 , + ∞ ) → R be the viscosity solution to the Cauchy problem (4.27) (with r and s replaced by r n and s n , respectively). Then, setting v s n r n ( x, t ) := u s n r n ( x, t β ( r n ,s n ) ) for all x ∈ R d , t ≥ , we have that, for every T > , v s n r n uniformlyconverge to u as n → + ∞ in R d × [0 , T ] , where u : R d × [0 , + ∞ ) → R is the viscositysolution to the classical mean curvature flow (4.42) . ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 47 Anisotropic kernels and applications to dislocation dynamics In this section we study the asymptotic behavior of supercritical nonlocal perime-ters and the corresponding geometric flows in the case of anisotropic kernels. More-over, we present an application to the dynamics of dislocation curves in two dimen-sions.6.1. Anisotropic kernels. Let g ∈ C ( S d − ; (0 , + ∞ )) be such that g ( ξ ) = g ( − ξ )for every ξ ∈ S d − . For every s ≥ r > k g,sr : R d \ { } → (0 , + ∞ ) as k g,sr ( x ) := g (cid:0) x | x | (cid:1) k sr ( | x | ) , where k sr is defined in(1.1) . Here we study the asymptotic behavior, as r → + of the functionals ˜ J g,sr :M f ( R d ) → [0 , + ∞ ) defined by(6.1) ˜ J g,sr ( E ) := Z E Z E c k g,sr ( y − x ) d y d x . In Proposition 6.1 below we will show that the functionals ˜ J g,sr scaled by σ s ( r )converge as r → + to the anisotropic perimeter Per g defined on finite perimetersets as(6.2) Per g ( E ) := Z ∂ ∗ E ϕ g ( ν E ( x )) d H d − ( x ) , where the density ϕ g is given by(6.3) ϕ g ( ν ) := Z { ξ ∈ S d − : ξ · ν ≥ } g ( ξ ) ξ · ν d H d − ( ξ ) , for every ν ∈ S d − . Proposition 6.1. For every s ≥ and for every set E ∈ M f ( R d ) of finite perimeterit holds lim r → + ˜ J g,sr ( E ) σ s ( r ) = Per g ( E ) . Proof. First we claim the following anisotropic version of formula (1.12): Z E Z E c ∩ B ( x, k g,sr ( y − x ) d y d x = d + sdsr s Z ∂ ∗ E d H d − ( y ) Z E ∩ B ( y,r ) g (cid:16) y − x | x − y | (cid:17) ( y − x ) | x − y | d · ν E ( y ) d x − dr d + s Z ∂ ∗ E d H d − ( y ) Z E ∩ B ( y,r ) g (cid:16) y − x | x − y | (cid:17) ( y − x ) · ν E ( y ) d x + 1 s Z ∂ ∗ E d H d − ( y ) Z E ∩ ( B ( y, \ B ( y,r )) g (cid:16) y − x | x − y | (cid:17) ( y − x ) · ν E ( y ) | x − y | d + s d x − s Z E d x Z E c ∩ ∂B ( x, g (cid:16) y − x | x − y | (cid:17) d H d − ( y ) . (6.4)If g ∈ C ( S d − ), the proof of (6.4) is identical to the proof of (1.12), noticing that ∇ g (cid:0) x | x | (cid:1) · T sr ( x ) = 0 for every x ∈ R d \ { } , with T sr defined in (1.10). The case g ∈ C ( S d − ) follows by standard density arguments. (cid:3) In Theorem 6.2 below we will see that the functionals ˜ J g,sr actually Γ-converge,as r → + , to Per g . Theorem 6.2. Let s ≥ and let { r n } n ∈ N ⊂ (0 , + ∞ ) be such that r n → as n → + ∞ . The following Γ -convergence result holds true. (i) (Compactness) Let U ⊂ R d be an open bounded set and let { E n } n ∈ N ⊂ M( R d ) be such that E n ⊂ U for every n ∈ N and ˜ J g,sr n ( E n ) ≤ M σ s ( r n ) for every n ∈ N , for some constant M independent of n . Then, up to a subsequence, χ E n → χ E strongly in L ( R d ) for some set E ∈ M f ( R d ) with Per( E ) < + ∞ . (ii) (Lower bound) Let E ∈ M f ( R d ) . For every { E n } n ∈ N ⊂ M f ( R d ) with χ E n → χ E strongly in L ( R d ) it holds Per g ( E ) ≤ lim inf n → + ∞ ˜ J g,sr n ( E n ) σ s ( r n ) . (iii) (Upper bound) For every E ∈ M f ( R d ) there exists { E n } n ∈ N ⊂ M f ( R d ) suchthat χ E n → χ E strongly in L ( R d ) and Per g ( E ) = lim n → + ∞ ˜ J g,sr n ( E n ) σ s ( r n ) . Proof. The proof of the compactness property (i) follows by Theorem 1.5(i), oncenoticed that there exist two positive constants c < c such that c ≤ g ( θ ) ≤ c forevery θ ∈ S d − . As for the proof of the Γ-liminf inequality in (ii) one can argueverbatim as in the proof of Theorem 1.5(ii), using the following inequality(6.5) Z Q ν Z Q ν g (cid:16) x − y | x − y | (cid:17) k sr ( | x − y | ) | χ E ( x ) − χ E ( y ) | d y d x ≥ (1 − ε ) σ s ( r ) ϕ g ( ν ) , in place of (3.2). The proof of (6.5) under the assumptions of Lemma 3.1 is identicalto the proof of Lemma 3.1 (see (3.8)).Finally, the Γ-limsup inequality (iii) follows as in the isotropic case Theorem1.5(iii) using Proposition 6.1 in place of Proposition 1.1. (cid:3) We introduce the notion of k g,sr curvature and we study the convergence as r → + of the corresponding geometric flows.Let s ≥ r > E ∈ C . For every x ∈ ∂E we define the k g,sr -curvature of E at x as(6.6) K g,sr ( x, E ) := Z R d ( χ E c ( y ) − χ E ( y )) k g,sr ( x − y ) d y. Remark 6.3. We notice that for every E ∈ C and for every x ∈ ∂E it holds(6.7) K g,sr ( x, E ) = Z R d k g,sr ( x − y ) d y − Z E k g,sr ( x − y ) d y = Z R d k g,sr ( z ) d z − k g,sr ∗ χ E ( x )= (cid:16) − k g,sr + Z R d k g,sr ( z ) d z δ (cid:17) ∗ χ E ( x ) , where ∗ denotes the convolution operator and δ is the Dirac delta centered at 0 . By(6.7) we have that K g,sr is exactly the type of curvatures considered in [20, formula(1.4)]. We remark that the positive part of the curvature K g,sr is concentrated on apoint. This is why, as already observed in [20], the curvature K g,sr , although having ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 49 a positive contribution, still satisfies the desired monotonicity property with respectto set inclusion (see the proof of (M) in Proposition 4.2).We first show that K g,sr is the first variation of ˜ J g,sr in the sense specified by thefollowing proposition, which is the anisotropic analogous of Proposition 4.1. Proposition 6.4. Let s ≥ , r > , and E ∈ C . Let Φ : R × R d → R d be as inProposition 4.1 . Setting E t := Φ t ( E ) and Ψ( · ) := ∂∂t Φ t ( · ) (cid:12)(cid:12) t =0 , we have dd t ˜ J g,sr ( E t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z ∂E K g,sr ( x, E )Ψ( x ) · ν E ( x ) d H d − ( x ) . Proof. If g ∈ C , then the proof is fully analogous to the proof of Proposition 4.1 .The case when g ∈ C follows by a density argument, using that if { g n } n ∈ N ⊂ C ( S d − ; (0 , + ∞ )) uniformly converges to g , E n → E in C and x n → x , then K g n ,sr ( x n , E n ) converge to K g,sr ( x, E ) . Such a continuity property can be proved asin Proposition 4.2 (UC). (cid:3) By arguing as in the proof of Proposition 4.2 one can show that the curva-tures K g,sr satisfy properties (M), (T), (S), (B), (UC). Now we introduce the (local)anisotropic curvatures K g, defined as follows. Let E ∈ C and x ∈ ∂E ; in aneighborhood of x , ∂E is the graph of function f ∈ C ( H ν E ( x ) ( x )) (see (0.5) forthe definition of H ν ( x )) with D f ( x ) = 0 (here and below D f and D f are com-puted with respect to a given system of orthogonal coordinates on H ν E ( x ) ( x )). Theanisotropic mean curvature of ∂E at x is given by(6.8) K g, ( x, E ) = − Z H νE ( x ) ( x ) ∩ S d − g ( ξ ) ξ ∗ D f ( x ) ξ d H d − ( ξ ) . One can check that K g, is the first variation of Per g in the sense specified byProposition 4.1 (we refer to [8] for the first variation formula of generic anisotropicperimeters, while we leave to the reader the computations for the specific anisotropicdensity ϕ g considered here, defined in (6.3)). Notice that if g ≡ K g, = ω d − K where K is the classical mean curvature defined in (4.21). Moreover, onecan check that K g, satisfies properties (M), (T), (S), (B), (UC’) in Proposition 4.3.In Proposition 6.5 below we show that the curvatures K g,sr converge, as r → + ,to the anisotropic curvature K g, . Theorem 6.5. Let s ≥ . Let { E r } r> ⊂ C be such that E r → E in C as r → + ,for some E ∈ C . Then, for every x ∈ ∂E ∩ ∂E r for all r > , it holds (6.9) lim r → + K g,sr ( x, E r ) σ s ( r ) = K g, ( x, E ) . Proof. The proof of (6.9) is fully analogous to the one of Theorem 4.6 and inparticular it is based on a suitable anisotropic variant of Lemma 4.5. In fact,Lemma 4.5 can be extended also to the anisotropic case with (4.28) replaced bylim r → + (cid:18) σ s ( r ) (cid:16) Z F r,δ g (cid:16) y | y | (cid:17) k sr ( | y | ) d y − Z F r,δ g (cid:16) y | y | (cid:17) k sr ( | y | ) d y (cid:17)(cid:19) = Z S d − g ( θ, θ ∗ ( N − M ) θ d H d − ( θ ) . (6.10) If ν E ( x ) = e d , one can argue verbatim as in the proof of Theorem 4.6, clearly using(6.10) in place of (4.28). The same proof with only minor notational changes canbe adapted also to the case ν E ( x ) = e d . (cid:3) We are now in a position to state our result on the convergence of the geometricflows of K g,sr as r → + , whose proof is omitted, being fully analogous to the oneof Theorem 4.7. Theorem 6.6. Let s ≥ be fixed. Let u ∈ C ( R d ) be a uniformly continuousfunction with u = C in R d \ B (0 , R ) for some C , R ∈ R with R > . Forevery r > , let u sr : R d × [0 , + ∞ ) → R be the viscosity solution to the Cauchyproblem ( ∂ t u ( x, t ) + | D u ( x, t ) |K g,sr ( x, { y : u ( y, t ) ≥ u ( x, t ) } ) = 0 u ( x, 0) = u ( x ) . Then, setting v sr ( x, t ) := u sr ( x, tσ s ( r ) ) for all x ∈ R d , t ≥ , we have that, forevery T > , v sr uniformly converge to u as r → + in R d × [0 , T ] , where u : R d × [0 , + ∞ ) → R is the viscosity solution to the anisotropic mean curvature flow ( ∂ t u ( x, t ) + | D u ( x, t ) |K g, ( x, { y : u ( y, t ) ≥ u ( x, t ) } ) = 0 u ( x, 0) = u ( x ) . Applications to dislocation dynamics. Here we apply the results in Sub-section 6.1 to the motion of curved dislocations in the plane. To this purpose, webriefly recall and describe, in an informal way, some notions about the isotropic lin-earized elastic energy induced by planar dislocations; such notions are well knownto experts and we refer to classic books such as [23] for an exhaustive monographyon this subject.Let E be a bounded region of the plane R = R ∩{ z ∈ R : z = 0 } , representinga plastic slip region with Burgers vector b = e = (1 , , J ( E ) := µ π Z E Z E c | x − y | (cid:16) ν − ν x + 1 − ν − ν x (cid:17) d y d x , where µ > ν ∈ ( − , ) are the shear modulus and the Poisson’s ratio, respec-tively. Formula (6.11) can be deduced by [23, formula (4-44)], by integrating byparts. Clearly, the energy J in (6.11) is always infinite whenever E is non-empty.It is well understood that such an infinite energy should be suitably truncatedthrough ad hoc core regularizations, specific of the microscopic details of the un-derlying crystal. The specific choice of the core regularization, giving back thephysically relevant (finite) elastic energy induced by the dislocation is, for our pur-poses, irrelevant. Here we adopt the energy-renormalization procedure introducedin (6.1). First we set(6.12) g ( ξ ) := µ π (cid:16) ν − ν ξ + 1 − ν − ν ξ (cid:17) , for every ξ ∈ S , and we notice that the energy in (6.11) can be (formally) rewritten as J ( E ) = Z E Z E c g (cid:16) x − y | x − y | (cid:17) | x − y | d y d x = Z E Z E c k g ( x − y ) d y d x , ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 51 where k g is defined by k g ( z ) := g ( z | z | ) | z | . The core-regularization of J is given bythe functional ˜ J g, r defined by (6.1), where the parameter r > E , with Burgers vector equal to e , governed by a self-energy releasemechanism. We consider a geometric evolution, that can be formally understoodas the gradient flow of the self-energy ˜ J g, r with respect to an L structure on the(graphs locally describing the) evolving dislocation curve. If the energy were thestandard perimeter, this evolution would be nothing but the standard mean curva-ture flow. Notice that the energy considered here is nonlocal; moreover, althoughit is derived from isotropic linearized elasticity, it has in fact an anisotropic depen-dence (induced by the direction of the given Burgers vector) on the normal to thecurve. Another possible source of anisotropy is the so-called mobility, dependingon the microscopic details of the underlying crystalline lattice; here, for simplicity,we assume that such a mobility is in fact isotropic, equal to one. The dynamicsdiscussed above corresponds to the geometric evolution where the normal velocityof the evolving dislocation curve at any point x is given by −K g, r , defined in (6.6).In order to study the asymptotic behavior, as r → + , of the dynamics describedabove we use the results developed in Subsection 6.1. First, we notice that that thefunction g defined in (6.12) is continuous (actually, it is smooth) and even, so thatit satisfies the assumptions required in Subsection 6.1. Moreover, recalling (6.3)and (6.8), for the choice of g in (6.12), an easy computation shows that ϕ g ( ν ) = µ π (cid:16) ν − ν (1 + ν ) + 1 − ν − ν (1 + ν ) (cid:17) , for every ν ∈ S , K g, ( x, E ) = µ π (cid:16) ν − ν ( ν E ( x )) + 1 − ν − ν ( ν E ( x )) (cid:17) , for every E ∈ C , x ∈ ∂E . Therefore, by Theorem 6.6, the unique (in the level set sense) dislocation dy-namics described above, converges, as r → + , to a degenerate evolution where thedislocation disappear instantaneously. 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Nonlinear Anal. (2003), 79–115. ONVERGENCE OF SUPERCRITICAL FRACTIONAL FLOWS TO MCF 53 (Lucia De Luca) Istituto per le Applicazioni del Calcolo “Mauro Picone” IAC-CNR.Via dei Taurini 19, I-00185 Roma, Italy Email address , L. De Luca: [email protected] (Andrea Kubin) Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Univer-sit`a di Roma. P.le Aldo Moro 5, I-00185 Roma, Italy Email address , A. Kubin: [email protected] (Marcello Ponsiglione) Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Uni-versit`a di Roma. P.le Aldo Moro 5, I-00185 Roma, Italy Email address , M. Ponsiglione:, M. Ponsiglione: