Nonlocal diffusion of smooth sets

We consider normal velocity of smooth sets evolving by the $s-$fractional diffusion. We prove that for small time, the normal velocity of such sets is nearly proportional to the mean curvature of the boundary of the initial set for $s\in [\frac{1}{2}, 1)$ while, for $s\in (0, \frac{1}{2})$, it is nearly proportional to the fractional mean curvature of the initial set. Our results show that the motion by (fractional) mean curvature flow can be approximated by fractional heat diffusion and by a diffusion by means of harmonic extension of smooth sets.


Introduction
For N ≥ 2, we let Ω 0 be a bounded open set of R N with boundary Γ 0 . Consider the heat equation with initial data the indicator function of the set Ω 0 : (1.1) for some time t 1 > 0. In 1992, Bence-Merriman-Osher [7] provided a computational algorithm for tracking the evolution in time of the set Ω 0 whose boundary Γ 0 moves with normal velocity proportional to its classical mean curvature. At time t 1 > 0, they considered Bence-Merriman-Osher [7] applied iteratively this procedure to generate a sequence of sets (Ω j ) j≥0 and conjectured in [7] that their boundaries Γ j evolved by mean curvature flow. Later Evans [20] provided a rigorous proof for the Bence-Merriman-Osher algorithm by means of the level-set approach to mean curvature flow developed by Osher-Sethian [42], Evans-Spruck [21][22][23][24] and Chen-Giga-Goto [13]. For related works in this direction, we refer the reader to [5,31,37,38,40,43,47] and references therein.
Recently Caffarelli and Souganadis considered in [12] nonlocal diffusion of open sets E ⊂ R N given by where We consider the fractional heat kernel K s with Fourier transform given by K(ξ, t) = e −t|ξ| 2s . It satisfies It follows that the unique bounded solution to (1.2) is given by Keywords: Motion by fractional mean curvature flow; Fractional heat equation; Fractional mean curvature; Harmonic extension. 1 By solving a finite number of times (1.2) for a small fixed time step σ s (h), the authors in [12] find a discrete family of sets for a suitable scaling function σ s to be defined below. It is proved in [12] that as nh → t, ∂E h nh converges, in a suitable sense, to Γ t . Here, the family of hypersurface {Γ t } t>0 , with Γ 0 = ∂E, evolves under generalized mean curvature flow for s ∈ [ 1 2 , 1) and under generalized fractional mean curvature flow for s ∈ (0, 1 2 ). We refer the reader to [12,20,36] for the notion generalized (nonlocal) mean curvature flow which considers the level sets of viscosity solutions to quasilinear parabolic integro-differential equations.
In the present paper, we are interested in the normal velocity of the sets E t := x ∈ R N : K s (·, σ s (t)) ⋆ τ E (x) > 0 (1.4) as they depart from a sufficiently smooth initial set E 0 := E. We consider here and in the following for some radially symmetric function P s ∈ C 1 (R N ).
(1.5) We make the following assumptions: for some constants C N,s , C N,s > 0. In the Section 1.1 below, we provide examples of valuable kernels K s satisfying the above properties. Now, as we shall see below (Lemma 2.1), for t > 0 small, ∇ x u(x, σ s (t)) = 0 for all x ∈ B(y, σ s (t) 1 2s ) and y ∈ ∂E. Hence ∂E t is a C 1 hypersurface, for small t > 0. For t > 0 and y ∈ ∂E, we let v = v(t, y) be such that y + vtν(y) ∈ ∂E t ∩ B(y, σ s (t) 1 2s ), (1.8) where ν(y) is the unit exterior normal of E at y. In the spirit of the work of Evans [20] on diffusion of smooth sets, we provide in this paper an expansion of v(t, y) as t → 0. It turns out that v(0, y) is proportional to the fractional mean curvature of ∂E at y for s ∈ (0, 1/2) and v(0, y) is proportional to the classical mean curvature of ∂E at y for s ∈ [1/2, 1). We notice that it is not a priori clear from (1.8), that v remains finite as t → 0. This is where the (unique) appropriate choice of σ s (t) enters during our estimates. Here and in the following, we define σ s (t) = t 2s 1+2s for s ∈ (0, 1/2), t s for s ∈ (1/2, 1) (1.9) and for s = 1/2, σ s (t) is the unique positive solution to Before stating our main result, we recall that for s ∈ (0, 1 2 ) and ∂E is of class C 1,β for some β > 2s, the fractional mean curvature of ∂E is defined for x ∈ ∂E as On the other hand, if ∂E is of class C 2 then the normalized mean curvature of ∂E is given, for x ∈ ∂E, by see also (2.4) and [25]. Having fixed the above definitions, we now state our main result.
Theorem 1.1. We let s ∈ (0, 1) and E ⊂ R N , N ≥ 2. We assume, for s ∈ (0, 1/2), that ∂E is of class C 1,β for some β > 2s and that ∂E is of class C 3 , for s ∈ [1/2, 1). Then, as t → 0, the expansion of v(t, y), defined in (1.8), is given, locally uniformly in y ∈ ∂E, by v(t, y) = where H s and H are respectively the fractional and the classical mean curvatures of ∂E and the positive constants a N,s , b N,1/2 and c N,s are given by and P s (y) := K s (y, 1).
Some remarks are in order. The assumption of E being of class C 3 in Theorem 1.1 is motivated by the result of Evans in [20], where in the case s = 1 and K 1 , the heat kernel, he obtained v = (N − 1)H(0) + O(t 1 2 ). We notice that from our argument below, we cannot improve the error term o t (1) in the case s ∈ (0, 1/2) even if E is of class C ∞ . This is due to the definition of the fractional mean curvature H s as a principal value integral. We finally remark, in the particular case, that .
1.1. Some applications of Theorem 1.1. We next put emphasis on two valuable examples where Theorem 1.1 applies. 1) Fractional heat diffusion of smooth sets. We recall, see e.g. [8,46], that the fractional heat kernel K s satisfies (1.5), (1.6) and (1.7) with (1.11) We recall that K s is known explicitly only in the case s = 1/2, where K 1/2 (y, t) In this case Theorem 1.1 provides an approximation of the (fractional) mean curvature motion by fractional heat diffusion of smooths sets, thereby extending, in the fractional setting, Evan's result in [20] on heat diffusion of smooth sets. 2) Diffusion of smooth sets by Harmonic extension. We consider the Poisson kernel on the half space R N +1 12) where p N,s := Thanks to the result of Caffarelli and Silvestre in [11], the function It is clear that K s (x, t) := K s (x, t where σ s (t) is given by (1.9) and (1.10). Therefore this Harmonic extension yields an approximation of (fractional) mean curvature motion of smooth sets. We conclude Section 1 by noting that the notion of nonlocal curvature appeared for the first time in [12]. Later on, the study of geometric problems involving fractional mean curvature has attracted a lot of interest, see [1,10,10], the survey paper [25] and the references therein. While the mean curvature flow is well studied, see e.g. [2,4,19,29,30,35], its fractional counterpart appeared only recently in the literature, see e.g. [14][15][16][17][18]36,45].

Acknowledgments
This work is supported by the Alexander von Humboldt foundation and the German Academic Exchange Service (DAAD). Part of this work was done while the authors were visiting the International Center for Theoretical Physics (ICTP) in December 2019 within the Simons associateship program.

Preliminary results and notations
Unless otherwise stated, we assume for the following that E is an open set of class C 1,β , with 0 ∈ ∂E and the unit normal of ∂E at 0 coincides with e N . We denote by Q r = B N −1 r × (−r, r) the cylinder of R N centred at the origin with B N −1 r the ball of R N −1 centred at the origin with radius r > 0. Decreasing r, if necessary, we may assume that In the following, for f, g : We denote by o t (1) any function that tends to zero when t → 0. If in addition, ∂E is of class C 3 , then for and the normalized mean curvature of ∂E at 0 is given by Recall that the unit exterior normal ν(y ′ ) := ν(y ′ , γ(y ′ )) of E and the volume element dσ(y ′ ) on ∂E ∩ Q r are given by We finally note, in view of (1.5) and (1.6), that we have for some positive constant C = C(N, s). We start with the following result.
Then there exist t 0 , C > 0, only depending on N, s, β and E, such that for all t ∈ (0, t 0 ) and z ∈ B As a consequence, for all t ∈ (0, t 0 ), the set is of class C 1 .
By a change of variable, (1.5) and (1.6), we have Integrating by parts, we have By a change of variable, (1.5), (1.6) and the fact that provided r. Next, using (2.6) and recalling that z ∈ B t 1 2s , we then have From this and (2.11), we deduce that Combining this with (2.9) and (2.10), we get Therefore (2.7) follows. Finally (2.8) follows from the inverse function theorem and the fact that w is of class In the sequel, we will need the following lemmas to estimate some error terms.
Lemma 2.2. For s ∈ (0, 1), we let E ⊂ R N be a set of class C 1,β , for some β > 2s, as in Section 2. For r > 0, we set where C is a positive constant depending only on N , β, s and E. Proof Then, by (2.6), where C is a positive constant depending on N , β and s and which may change from a line to another. Next, using (2.6), (1.7) and the dominate convergence theorem, we obtain   as t → 0.
3. Proof of Theorem 1.1 in the case s ∈ (0, 1 2 ) In this section, we start by the following preliminary result.
where B r is the ball of R N centered at the origin and of radius r > 0. By integration by parts, we have Then by a change of variable and (2.5), we have ∂E∩Br K s (y ′ , y N − vtθ, σ s (t))ν N (y)dσ(y) = By the Fundamental Theorem of calculus, we can write In the following, we let ε(y ′ ) := γ(y ′ ) − vtθ.
as t → 0, where H(0) is the mean curvature of ∂E at 0.
The proof is then ended.