Antonin Chambolle

Daniele De Gennaro

Massimiliano Morini

MINIMIZING MOVEMENTS FOR ANISOTROPIC AND INHOMOGENEOUS MEAN CURVATURE FLOWS

In this paper we address anisotropic and inhomogeneous mean curvature flows with forcing and mobility, and show that the minimizing movements scheme converges to level set/viscosity solutions and to distributional solutions à la Luckhaus-Sturzenhecker to such flows, the latter result holding in low dimension and conditionally to the convergence of the energies. By doing so we generalize recent works concerning the evolution by mean curvature by removing the hypothesis of translation invariance, which in the classical theory allows one to simplify many arguments.

Introduction

In this paper we deal with the anisotropic, inhomogeneous mean curvature flow with forcing and mobility. By inhomogeneous we mean that the flow is driven by surface tensions depending on the position in addition to the orientation of the surface. The evolution of sets t → E t ⊆ R N considered is (formally) governed by the law [START_REF] Allen | A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing[END_REF] V (x, t) = ψ(x, ν Et (x)) -H ϕ Et (x) + f (x, t) , x ∈ ∂E t , t ∈ (0, T ), where V (x, t) is the (outer) normal velocity of the boundary ∂E t at x, ϕ(x, p) is a given anisotropy representing the surface tension, H ϕ is the anisotropic mean curvature of ∂E t associated to ϕ, ψ(x, p) is an anisotropy evaluated at the outer unit normal ν Et (x) to ∂E t which represents a velocity modifier (also called the mobility term), and f is the forcing term. We will be mainly concerned with smooth anisotropies (and the regularity assumptions will be made precise later on): in this case, the curvature H ϕ is the first variation of the anisotropic and inhomogeneous perimeter associated to the anisotropy ϕ (in short, ϕ-perimeter) defined as [START_REF] Almeida | Mean curvature flow with obstacles[END_REF] P ϕ (E) := ˆ∂ * E ϕ(x, ν E (x)) dH N -1 (x)

for any set E of finite perimeter (where ∂ * E denotes the reduced boundary of E) and, if E is sufficiently smooth, it takes the form

H ϕ E (x) = div(∇ p ϕ(x, ν E (x)))
, where with ∇ p we denote the gradient made with respect to the second variable. Note that evolution (1) can be red as the motion of sets in R N , when the latter is endowed with the Finsler metric induced by the anisotropy (see Remark 4.13). Equation ( 1) is relevant in Material Sciences, Crystal Growth, Image Segmentation, Geometry Processing and other fields see e.g. [START_REF] Allen | A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing[END_REF][START_REF] Desbrun | Implicit fairing of irregular meshes using diffusion and curvature flow[END_REF][START_REF] Gurtin | Toward a nonequilibrium thermodynamics of two-phase materials[END_REF][START_REF] Sethian | Level set methods and fast marching methods[END_REF][START_REF] Taubin | A signal processing approach to fair surface design[END_REF].

The mathematical literature for inhomogeneous mean curvature flows is not as extensive as in the homogeneous case, mainly due to the difficulties arising from the lack of translational invariance. Indeed, assuming that the evolution is invariant under translations allows to simplify many arguments used in the classical proofs of, for example, comparison results and estimates on the speed of evolution. In the homogeneous case the well-posedness theory is nowadays well established and quite satisfactory, both in the local and nonlocal case, and even in the much more challenging crystalline case (that is, when the anisotropy ϕ is piecewise affine) see [START_REF] Almeida | Mean curvature flow with obstacles[END_REF][START_REF] Almgren | Curvature-driven flows: a variational approach[END_REF][START_REF] Bellettini | Anisotropic motion by mean curvature in the context of Finsler geometry[END_REF][START_REF] Chambolle | Existence and uniqueness for anisotropic and crystalline mean curvature flows[END_REF][START_REF] Chambolle | A nonlocal mean curvature flow and its semiimplicit time-discrete approximation[END_REF], 1 [START_REF] Chambolle | Nonlocal curvature flows[END_REF][START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF][START_REF] Giga | Motion by crystalline-like mean curvature: a survey[END_REF][START_REF] Laux | Convergence of the thresholding scheme for multi-phase meancurvature flow[END_REF][START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF][START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF] to cite a few. Concerning the inhomogeneous mean curvature flow, we cite [START_REF] Huisken | Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature[END_REF][START_REF] Huisken | Geometric evolution equations for hypersurfaces[END_REF] where the short time existence of smooth solutions on manifolds is shown, and [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF][START_REF] Ilmanen | The level-set flow on a manifold[END_REF], where the viscosity level set approach (introduced for the homogeneous evolution in [START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF][START_REF] Evans | Motion of level sets by mean curvature. I[END_REF]) is extended, respectively, to the equation [START_REF] Allen | A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing[END_REF] and to the Riemannian setting.

In the present work we implement the minimizing movement approach à la Almgren-Taylor-Wang (in short, ATW scheme) [START_REF] Almgren | Curvature-driven flows: a variational approach[END_REF] to prove existence via approximation of a level set solution to the generalized anisotropic and inhomogeneous motion [START_REF] Allen | A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing[END_REF]. To carry on this scheme (which has only been sketched in [START_REF] Bellettini | Anisotropic motion by mean curvature in the context of Finsler geometry[END_REF], but lacks a formal proof) we gain insights from [START_REF] Chambolle | Nonlocal curvature flows[END_REF]. We also show that, under the additional hypothesis of convergence of the energies (4) and low dimension [START_REF] Chambolle | Nonlocal curvature flows[END_REF](which are nowadays classical for this approach), the same approximate solutions provide in the limit a suitable notion of "BV-solutions", also termed distributional solutions, see [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF][START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF].

There are many more concepts of weak solution for the mean curvature flow. In particular, we cite the diffuse-interface approximation provided by the Allen-Cahn equation [START_REF] Evans | Phase transitions and generalized motion by mean curvature[END_REF][START_REF] Ilmanen | Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature[END_REF][START_REF] Hensel | A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness[END_REF][START_REF] Laux | Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow[END_REF] and the threshold dynamic scheme [START_REF] Merriman | Motion of multiple junctions: A level set approach[END_REF][START_REF] Esedoglu | Threshold dynamics for networks with arbitrary surface tensions[END_REF] (see also the relative entropy methods of [START_REF] Laux | Convergence of the thresholding scheme for multi-phase meancurvature flow[END_REF]). Other recent results concern the weak-strong uniqueness problem, which consists in proving that weak solutions coincide with the smooth ones as long as the latter exist. After classical works concerning viscosity solutions, a new definition of "BV-solution" (whose existence is proved via the Allen-Cahn approximation scheme) allows the authors in [START_REF] Hensel | A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness[END_REF][START_REF] Laux | Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow[END_REF] to prove weak-strong uniqueness for isotropic and anisotropic mean curvature flows. This result is based upon the so-called optimal dissipation inequality satisfied by their weak solution. In general, it is very difficult to say if the ATW scheme could satisfy such a property, mainly because of the "degeneracy" of the dissipation term in the incremental problem defined via the distance function. Even if all these results concern the translationally invariant case, a study of some of these properties in the inhomogeneous setting seems very interesting and challenging.

Other remarks on possible research directions are the following. To begin with, the new arguments which are used to compensate the lack of translation invariance are based on the locality of the anisotropic curvature H ϕ associated with a smooth anisotropy ϕ. This implies that the proofs are not straightforwardly adaptable to the so-called "variational curvatures" considered in [START_REF] Chambolle | Nonlocal curvature flows[END_REF], which are non-local in nature. On the other hand, since the crystalline curvatures are highly nonlocal and degenerate operators (see e.g. [START_REF] Chambolle | Existence and uniqueness for anisotropic and crystalline mean curvature flows[END_REF][START_REF] Novaga | Anisotropic and crystalline mean curvature flow of meanconvex sets[END_REF]), they do not fall in the theory constructed in the present work. In principle, it would be possible to follow the same perturbative study conducted in [START_REF] Chambolle | Existence and uniqueness for anisotropic and crystalline mean curvature flows[END_REF] in order to prove at least existence for an inhomogeneous and crystalline mean curvature flow. However, a satisfactory characterization of the limiting motion equation bearing a comparison principle is lacking so far.

This work can be seen as a first step towards constructing a general theory of motions driven by non-translationally invariant and possibly nonlocal curvatures, in the spirit of [START_REF] Chambolle | Nonlocal curvature flows[END_REF].

1.1. Main results. Now briefly recall the minimizing movements procedure in order to state the main results of the paper. Given an initial bounded set E 0 and a parameter h > 0, we define the discrete flow E (h) t := T h,t-h E (h) t-h for any t ≥ h and E (h) t = E 0 for t ∈ [0, h), where the functional T h,t is defined for t ≥ 0 as follows: for any bounded set E we set T h,t E (or, sometimes, T - h,t E) as the minimal solution to the problem [START_REF] Almgren | Curvature-driven flows: a variational approach[END_REF] min

P ϕ (F ) + ˆF sd ψ E (x) h + [ t h ]h+h [ t h ]h f (x, s) ds dH N -1 (x) : F is measurable ,
where sd ψ E (x) is the signed geodesic distance between x and E induced by the anisotropy ψ (see [START_REF] Novaga | Anisotropic and crystalline mean curvature flow of meanconvex sets[END_REF] for the precise definition) and [s] = max{n ≤ s, n ∈ N ∪ {0}} denotes the integer part of a non-negative real number s ∈ [0, +∞). We will then define T + h,t E as the maximal solution to the problem above. Any L 1 -limit point as h → 0 of the family {E (h) t } t≥0 will be called a flat flow. In the whole paper we will assume that ϕ ∈ E (see Definition 2.2) and ψ is an anisotropy as in Definition 2.1,

∀t ∈ [0, +∞) it holds f (•, t) ∈ C 0 (R N ), ∥f ∥ L ∞ (R N ×[0,+∞)) < ∞. (H0)
With more effort one could weaken the hypothesis and require ´t 0 f (•, s) ds to be continuous (see [START_REF] Chambolle | Implicit time discretization of the mean curvature flow with a discontinuous forcing term[END_REF]). For the sake of simplicity we will require the global-in-time boundedness. We prove existence and H ölder regularity for flat flows.

Theorem 1.1 (Existence of flat flows). Let E 0 be a bounded set of finite perimeter and ϕ, ψ, f satisfy (H0). Fix T > 0. For any h > 0, let {E (h) t } t∈[0,T ) be a discrete flow with initial datum E 0 . Then, there exists a family of sets of finite perimeter {E t } t∈[0,T ) and a subsequence h k ↘ 0 such that

E (h) t → E t in L 1 ,
for a.e. t ∈ [0, T ). Such flow satisfies the following regularity property: there exists a constant c, depending on T , such that for every 0 ≤ s ≤ t < T ,

|E s △E t | ≤ c|t -s| 1/2 , P ϕ (E t ) ≤ P ϕ (E 0 ) + c.
Subsequently, we will show that flat flow s are distributional solutions, as defined in [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF]. We will require additional hypothesis: firstly, low dimension [START_REF] Chambolle | Nonlocal curvature flows[END_REF] (linked to the complete regularity of the ϕ-perimeter minimizer, compare [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF][START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF]), moreover

∃ c ψ > 0 s.t. |ψ(x, v) -ψ(y, v)| ≤ c ψ |x -y|, ∀x, y ∈ R N , v ∈ S N -1 , (H1) f ∈ C 0 (R N × [0, ∞)]). (H2)
Theorem 1.2 (Existence of distributional solutions). Assume (H0), (H1), (H2) and [START_REF] Chambolle | Nonlocal curvature flows[END_REF]. For any T > 0, if

(4) lim k→∞ ˆT 0 P ϕ (E (h k ) t ) = ˆT 0 P ϕ (E t ), then {E t } t∈[0,T ] is a distributional solution (1)
with initial datum E 0 in the following sense:

(1) for a.e. t ∈ [0, T ) he set E t has weak ϕ-curvature H ϕ Et (see [START_REF] Desbrun | Implicit fairing of irregular meshes using diffusion and curvature flow[END_REF] for details) satisfying

ˆT 0 ˆ∂ * Et |H ϕ Et | 2 < ∞; (2) there exist v : R N × (0, T ) → R with ´T 0 ´∂ * Et v 2 dH N -1 dt < ∞ and v(•, t) ∂Et ∈ L 2 (∂E t ) for a.e. t ∈ [0, T ), such that - ˆT 0 ˆ∂ * Et vη dH N -1 dt = ˆT 0 ˆ∂ * Et H ϕ Et -f η dH N -1 dt (5) ˆT 0 ˆEt ∂ t η dx dt + ˆE0 η(•, 0) dx = - ˆT 0 ˆ∂ * Et ψ(•, ν Et )vη dH N -1 dt, ( 6
)
for every η ∈ C 1 c (R N × [0, T ))
. The definitions 1), 2) extend to our case the definition of BV -solutions of [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF] and the distributional solutions of [START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF]. We recall that hypothesis (4) ensures that the evolving sets avoid the so-called "fattening" phenomenon. It is known that this hypothesis is satisfied in the case of evolution of convex or mean-convex sets, see e.g. [START_REF] Novaga | Anisotropic and crystalline mean curvature flow of meanconvex sets[END_REF][START_REF] Philippis | Implicit time discretization for the mean curvature flow of mean convex sets[END_REF][START_REF] Fuchs | Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets[END_REF], but in general is not known under which general hypothesis it is valid. We also remark that the proof of the theorem above provides a detailed proof of [10, Theorem 3.2], which had only been sketched. Moreover, we bypass the use of a Bernstein-type result (which is usually employed) by a double blow-up technique.

In the second part of the work we will focus on the level set approach. Briefly, given an initial compact set E 0 , we set u 0 such that {u 0 ≥ 0} = E 0 and we look for a solution u in the viscosity sense (in a sense made precise in Definition 4.5) to ( 7)

∂ t u + ψ(x, -∇u) (div∇ p ϕ(x, ∇u(x)) -f (x, t)) = 0 u(•, t) = u 0 .
Classical remarks ensure that any level set {u ≥ s} is evolving following the mean curvature flow (1). To prove existence for [START_REF] Barles | A new approach to front propagation problems: theory and applications[END_REF] we use an approximating procedure. For h > 0 and t ∈ (0, +∞) we set iteratively

u ± h (•, t) = u 0 for t ∈ [0, h) and for t ≥ h u + h (x, t) := sup s ∈ R : x ∈ T + h,t-h {u + h (•, t -h) ≥ s} u - h (x, t) := sup s ∈ R : x ∈ T - h,t-h {u - h (•, t -h) > s} ,
where the operator T ± h,t has been previously introduced. We remark that these are maps piecewise constant in time, since T ± h,t = T ± h,[t/h]h , which are only upper and lower semicontinuous in space respectively. Then, we will pass to the limit h → 0 on the families {u ± h } h to find functions u + , u - which are viscosity sub -and supersolution respectively of equation [START_REF] Barles | A new approach to front propagation problems: theory and applications[END_REF]. Passing to the limit as h → 0 in our case is not straightforward. The main issue is that we do not have an uniform estimate on the modulus of continuity of the functions u h (compare [START_REF] Chambolle | Nonlocal curvature flows[END_REF]) and thus we can not pass to the (locally) uniform limit of the sequence. (More precisely, our best estimate contained in Lemma 4.8 decays too fast as h → 0 to provide any useful information). Nonetheless, motivated by [START_REF] Barles | Front propagation and phase field theory[END_REF][START_REF] Barles | An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications[END_REF][START_REF] Barles | A new approach to front propagation problems: theory and applications[END_REF] we can define the half-relaxed limits

u + (x, t) := sup (x h ,t h )→(x,t) lim sup h→0 u + h (x h , t h ) u -(x, t) := inf (x h ,t h )→(x,t) lim inf h→0 u - h (x h , t h ), (8)
and prove that the functions defined above are sub -and supersolutions, respectively, to [START_REF] Barles | A new approach to front propagation problems: theory and applications[END_REF]. The main difficulty in this regard is that we need to work with just semicontinuous functions in space, as in the translationally invariant setting one can easily prove the uniform equicontinuity of the approximating sequence. We prove the following.

Theorem 1.3. Assume (H0), (H1) and f ∈ C 0 (R n × [0, +∞)). The function u + (respectively u -) defined in (8) is a viscosity subsolution (respectively a viscosity supersolution) of [START_REF] Barles | A new approach to front propagation problems: theory and applications[END_REF].

Thanks to the results of [START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF] we then prove that, under the additional hypothesis

∇ x ∇ p ϕ(•, p) and ∇ 2 p ϕ(•, p) are Lipschitz, uniformly for p ∈ S N -1 ∇ 2 p ϕ 2 (x, p) is uniformly elliptic in p, uniformly in x ψ(•, p) Lipschitz continuous, uniformly in p f (•, t) Lipschitz continuous, uniformly in t, (H3)
the following uniqueness result holds.

Theorem 1.4. Assume (H0) and (H3). If u 0 is a continuous function which is spatially constant outside a compact set, equation [START_REF] Barles | A new approach to front propagation problems: theory and applications[END_REF] with initial condition u 0 admits a unique continuous viscosity solution u given by [START_REF] Bellettini | Anisotropic motion by mean curvature in the context of Finsler geometry[END_REF]. In particular, u + = u -= u is the unique continuous viscosity solution to [START_REF] Barles | A new approach to front propagation problems: theory and applications[END_REF] and u ± h → u as h → 0, locally uniformly.

The previous result yields a proof of consistency between the level set approach and the minimizing movements one to study the evolution [START_REF] Allen | A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing[END_REF]. We recall that it has been established for the classical mean curvature flow in [START_REF] Chambolle | An algorithm for mean curvature motion[END_REF], in the anisotropic but homogeneous case in [START_REF] Eto | An area-minimizing scheme for anisotropic mean-curvature flow[END_REF] and in a very general nonlocal setting in [START_REF] Chambolle | Nonlocal curvature flows[END_REF].

Preliminaries

We start introducing some notations. We consider 0 ∈ N. We will use both B r (x) and B(x, r) to denote the Euclidean ball in R N centered in x and of radius r; with B N -1 r (x) we denote the Euclidean ball in R N -1 centered in x and of radius r; with S N -1 we denote the sphere ∂B 1 (0) ⊆ R N ; with Sym N the symmetric real matrices of size N × N . In the following, we will always speak about measurable sets and refer to a set as the union of all the points of density 1 of that set i.e. E = E (1) . If not otherwise stated, we implicitly assume that the function spaces considered are defined on R N , e.g L ∞ = L ∞ (R N ); the space C 0 denotes the space of continuous functions. Moreover, we often drop the measure with respect to which we are integrating, if clear from the context. Definition 2.1. We define anisotropy (sometimes defined as an elliptic integrand ) a function ψ with the following properties: ψ(x, p) : R N × R N → [0, +∞) is a continuous function, which is convex and positively 1-homogeneous in the second variable, such that 1

c ψ |p| ≤ ψ(x, p) ≤ c ψ |p|
for any point x ∈ R N and vector p ∈ R N .

We remark that, as standard, we define a real function f positively 1-homogeneous if for any λ ≥ 0, it holds f (λx) = λf (x). In particular, the anisotropies that we will consider are not symmetric. In the following, we will always denote the gradient of an anisotropy with respect to the first (respectively second) variable as ∇ x ψ (respectively ∇ p ψ). We then recall the definition of some well-known quantities (see [START_REF] Bellettini | Anisotropic motion by mean curvature in the context of Finsler geometry[END_REF]). Define the polar function of an anisotropy ψ, denoted with ψ • , as

(9) ψ • (•, ξ) := sup p∈R N {ξ • p : ψ(•, p) ≤ 1} .
Using the definition it is easy to see that for all p, ξ ∈ R N it holds

ψ(•, p)ψ • (•, ξ) ≥ p • ξ, -ψ(•, -p)ψ • (•, ξ) ≤ p • ξ.
Furthermore, one can prove that (see [START_REF] Bellettini | Anisotropic motion by mean curvature in the context of Finsler geometry[END_REF]) for p ̸ = 0

ψ • (∇ p ψ) = 1, ψ(∇ p ψ • ) = 1, (ψ • ) • = ψ.
We define for any x, y ∈ R N the geodesic distance induced by ψ, or ψ-distance in short, as

dist ψ (x, y) := inf ˆ1 0 ψ • (γ(t), γ(t)) dt : γ ∈ W 1,1 ([0, 1]; R N ), γ(0) = x, γ(1) = y .
We remark that this function is not symmetric in general. We define the signed distance function

from a closed set E ⊆ R N as (10) sd ψ E (x) := inf y∈E dist ψ (y, x) -inf y / ∈E dist ψ (x, y),
so that sd ψ E ≥ 0 on E c and sd ψ E ≤ 0 in E. We remark that the bounds stated in Definition 2.1 imply

(11) 1 c ψ dist ≤ dist ψ ≤ c ψ dist,
where here and in the following we will denote with dist, sd the Euclidean distance and signed distance function respectively. We define the ψ-balls as the balls associated to the ψ-distance, that is B ψ ρ (x) := {y ∈ R N : dist ψ (y, x) < ρ}, which in general are not convex nor symmetric. Definition 2.2. We say that an anisotropy ϕ is a regular elliptic integrand, and write ϕ ∈ E , if there exists two constants λ ≥ 1, l ≥ 0 such that if ϕ(x, •) S N -1 ∈ C 2,1 (S N -1 ) and for every x, y, e ∈ R N , ν, ν ′ ∈ S N -1 one has:

1 λ ≤ ϕ(x, ν) ≤ λ, |ϕ(x, ν) -ϕ(y, ν)| + |∇ p ϕ(x, ν) -∇ p ϕ(y, ν)| ≤ l|x -y| |∇ p ϕ(x, ν)| + ∥∇ 2 p ϕ(x, ν)∥+ ∥∇ 2 p ϕ(x, ν) -∇ 2 p ϕ(x, ν ′ )∥ |ν -ν ′ | ≤ λ e • ∇ 2 p ϕ(x, ν)[e] ≥ |e -(e • ν)ν| 2 λ .
Given any set of finite perimeter E, one can define the ϕ-perimeter P ϕ as follows

P ϕ (E) := ˆ∂ * E ϕ(x, ν E (x)) dH N -1 (x),
where ∂ * E is the reduced boundary of E and ν E is the measure-theoretic outer normal, see [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF] for further references on sets of finite perimeter. The ϕ-perimeter of a set of finite perimeter E in an open set A is defined as

P ϕ (E; A) := ˆ∂ * E∩A ϕ(x, ν E (x)) dH N -1 (x).
We remark that, by definition of regular elliptic integrand, for any set E of finite perimeter it holds 1 λ P (E) ≤ P ϕ (E) ≤ λP (E).

Some additional remarks on this definition can be found in [START_REF] Philippis | Regularity of free boundaries in anisotropic capillarity problems and the validity of Young's law[END_REF]. We just recall the submodularity property of the ϕ-perimeter, which can be proved for instance by using the formulae for the reduced boundary and measure-theoretic normal of union and intersection of sets of finite perimeter (see [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF]).

Proposition 2.3 (Submodularity property). For any two sets E, F ⊆ R N of finite perimeter, one has

(12) P ϕ (E ∪ F ) + P ϕ (E ∩ F ) ≤ P ϕ (E) + P ϕ (F ).
Moreover, by homogeneity, [START_REF] Cagnetti | A second order minimality condition for the Mumford-Shah functional[END_REF] and recalling that for any set E of finite perimeter it holds

Dχ E = -ν E dH N -1
∂ * E we have the following equivalent definitions

P ϕ (E) = sup ˆRN -Dχ E • ξ : ξ ∈ C 1 c (R N ; R N ), ϕ • (•, ξ) ≤ 1 (13) = sup ˆE div ξ dH N -1 : ξ ∈ C 1 c (R N ; R N ), ϕ • (•, ξ) ≤ 1 .
Concerning the regularity property of the ϕ-perimeter minimizers, we refer to [START_REF] Schoen | Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II[END_REF]. We just recall the following results. Given two anisotropies ϕ, ψ ∈ E , we define the "distance" between them as dist E (ϕ, ψ) := sup{|ϕ(x, p) -ψ(x, p)|

+ |∇ p ϕ(x, p) -ψ(x, p)| + |∇ 2 p ϕ(x, p) -∇ 2 p ψ(x, p)| : x ∈ R N , p ∈ S N -1 },
where | • | denotes the Euclidian norm. Given ϕ ∈ E , we recall that E is a 0-minimizer for the ϕ-perimeter if for any x ∈ R N , r > 0

P ϕ (E; B r (x)) ≤ P ϕ (F ; B r (x))
for every F ⊂ R N such that F △E ⊂⊂ B r . Then, some regularity properties of minimizers of ϕ-perimeter can be found in the theorems of part II.7 and II.8 in [START_REF] Schoen | Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II[END_REF], which are recalled below.

Theorem 2.4. Assume ϕ ∈ E . Then, for any 0-minimizer E of the ϕ-perimeter, the reduced boundary ∂ * E of the set E is of class C 1,1/2 and the singular set Σ := ∂E \ ∂ * E satisfies

H N -3 (Σ) = 0.
Theorem 2.5. Let m > 0, α ∈ (0, 1). Then, there exists ε = ε(m, α) > 0 with the following property:

let ϕ = ϕ(p) ∈ E , ϕ ∈ C 3,α (R N \ {0}) with ∥ϕ| S N -1 ∥ C 3,α ≤ m and dist E (ϕ, | • |) ≤ ε.
Then, for any 0-minimizer E of the ϕ-perimeter, the reduced boundary ∂ * E of the set E is of class C 1,1/2 and the singular set

Σ := ∂E \ ∂ * E satisfies H N -7 (Σ) = 0.
We sum up these hypotheses that yield the complete regularity of minimizers of parametric elliptic integrands: either ϕ ∈ E and N ≤ 3, or N ≤ 7 and the hypotheses of Theorem 2.5 are satisfied. ( 14)

2.1. The first variation of the ϕ-perimeter. In this section we compute the first variation of the ϕ-perimeter and define some additional operators associated to it.

Assume E is of class C 2 . Let X be a smooth and compactly supported vector field and assume Ψ(x, t) =: Ψ t (x) is the associated flow. To simplify the notation, we write ν(x, t) = ∇ x sd Ψ(E,t) (x).

By classical formulae (see e.g. [START_REF] Cagnetti | A second order minimality condition for the Mumford-Shah functional[END_REF]) we can compute the following. For the sake of brevity, we avoid writing the evaluation ϕ = ϕ(x, ν E (x)), if not otherwise specified, and assume that all the integrals are made with respect to the Hausdorff (N -1)-dimensional measure H N -1 .

d dt t=0 P ϕ (E t ) = d dt t=0 ˆ∂E ϕ(Ψ t (x), ν(Ψ t (x), t))JΨ t = ˆ∂E ∇ x ϕ • X + ∇ p ϕ • (-∇ τ (X • ν) + Dν[X]) + ϕ div τ X (15) = ˆ∂E ∇ x ϕ • X + ∇ p ϕ • (-∇ τ (X • ν) + Dν[X]) + div τ (ϕX) -∇ϕ • X + (∇ϕ • ν)(X • ν) = ˆ∂E ∇ x ϕ • X + ∇ p ϕ • (-∇ τ (X • ν) + Dν[X]) -∇ x ϕ • X -Dν[∇ p ϕ] • X + div τ (ϕX) + (∇ϕ • ν)(X • ν) = ˆ∂E -∇ p ϕ • ∇ τ (X • ν) + (∇ x ϕ • ν)(X • ν) + (Dν[∇ p ϕ] • ν) (X • ν) + div τ (ϕX) = ˆ∂E div τ (∇ p ϕ(X • ν)) -∇ p ϕ • ∇ τ (X • ν) + (X • ν)(∇ x ϕ • ν) = ˆ∂E (div τ ∇ p ϕ)(X • ν) + ∇ p ϕ • ∇ τ (X • ν) -∇ p ϕ • ∇ τ (X • ν) + (∇ x ϕ • ν)(X • ν) = ˆ∂E (X • ν) (div τ ∇ p ϕ + ∇ x ϕ • ν) = ˆ∂E (X • ν) div∇ p ϕ
where the last equality follows from the definition of div τ and the fact that ϕ is 1-homogeneous with respect to the p variable, since

div∇ p ϕ = div τ ∇ p ϕ + i ν i (∂ xi ∇ p ϕ) [ν] = div τ ∇ p ϕ + i ν i ∇ p (∂ xi ϕ) • ν + ν • ∇ 2 p ϕDν [ν] = div τ ∇ p ϕ + ∇ x ϕ • ν.
Therefore, we define the first variation of a C 2 -regular set E, induced by the vector field X, as ( 16)

δP ϕ (E)[X • ν] := ˆ∂E (X(x) • ν(x)) div∇ p ϕ(x, ν(x)) dH N -1 (x)
and the ϕ-curvature of the set E as

(17) H ϕ E (x) := div∇ p ϕ(x, ν(x)
). If we now consider equation [START_REF] Chambolle | Implicit time discretization of the mean curvature flow with a discontinuous forcing term[END_REF], we develop the tangential gradient to find

∇ p ϕ • (-∇ τ (X • ν) + Dν[X]) = ∇ p ϕ • (-∇ τ X[ν] -Dν[X] + Dν[X]) = 0.
This shows that for any set E of class C 2 it holds

δP ϕ (E)[X • ν] := ˆ∂E (∇ x ϕ • X + ϕ div τ X) dH N -1 ,
where we dropped the evaluation of ϕ at (x, ν E (x)). We remark that the expression on the right hand side makes sense even if the set E is just of finite perimeter. Defining the ϕ-divergence operator div ϕ as div ϕ X := ∇ x ϕ • X + ϕ div τ X, [START_REF] Philippis | Regularity of free boundaries in anisotropic capillarity problems and the validity of Young's law[END_REF] we are led to define the distributional ϕ-curvature of a set E of finite perimeter as an operator H ϕ E ∈ L 1 (∂E) (if it exists) such that the following representation formula holds

(19) ˆ∂E div ϕ X dH N -1 = ˆ∂E H ϕ E ν E • X dH N -1 , ∀X ∈ C ∞ c (R N ; R N ).
The previous computations allow to say that the distributional ϕ-curvature can be expressed as [START_REF] Philippis | Implicit time discretization for the mean curvature flow of mean convex sets[END_REF] if the set is of class C 2 . Finally, since ϕ is a regular elliptic integrand, one can prove the following monotonicity result.

Lemma 2.6. Let E, F be two C 2 sets of finite ϕ-perimeter with E ⊆ F , and assume that x ∈ ∂F ∩ ∂E: then H ϕ F (x) ≤ H ϕ E (x). Proof. Since the anisotropy is smooth, we can expand the curvature formula [START_REF] Philippis | Implicit time discretization for the mean curvature flow of mean convex sets[END_REF] as ( 20)

H ϕ = tr ∇ x ∇ p ϕ(x, ν) + ∇ 2 p ϕ(x, ν)Dν
and compare H ϕ E with H ϕ F . We consider separately the two terms appearing in [START_REF] Esedoglu | Threshold dynamics for networks with arbitrary surface tensions[END_REF]. The first one depends on ν just by the value it has at the point x. Therefore, since ν E (x) = ν F (x) we have the equality. The second one falls in the classical framework of smooth anisotropies that do not depend on the space variable. Since Dν F ≤ Dν E (as matrices) one concludes the proof. □

The minimizing movements approach

In this section we follow the work of [START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF] (see also [START_REF] Almgren | Curvature-driven flows: a variational approach[END_REF][START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF]) to prove the existence for the mean curvature flow via the minimizing movements approach. We recall that in the whole paper we will assume the hypothesis (H0).

3.1. The discrete scheme. In this subsection we will define the discrete scheme approximating the weak solution of the mean curvature flow, and we shall study some of its properties.

We define the following iterative scheme. Given h > 0, f ∈ L ∞ (R N × [0, ∞)) and t ≥ h, and given a bounded set of finite perimeter F , we minimize the energy functional

(21) F F h,t (E) = P ϕ (E) + 1 h ˆE sd ψ F (x) dx -ˆE F h (x, t) dx
in the class of all measurable sets E ⊆ R N , and where we have set

F h (x, t) := t+h t f (x, s) ds.
Equivalently, we could define the energy functional as

F F h,t (E) = P ϕ (E) + 1 h ˆE△F |sd ψ F | -ˆE F h (x, t) dx,
which agrees with [START_REF] Eto | An area-minimizing scheme for anisotropic mean-curvature flow[END_REF] up to a constant. Then, we denote

T h,t F = E ∈ argmin F F h,t .
We will refer to this minimizing procedure as the incremental problem. It is well-known (compare [START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF] and [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF]Proposition 17.8]) that a minimimum of (21) of class C 2 satisfies the Euler-Lagrange equation ( 22)

ˆ∂E H ϕ E X • ν E dH N -1 = - ˆ∂E 1 h sd ψ F (x) -F h (x, t) X(x) • ν E (x) dH N -1 (x) for all X ∈ C ∞ c (R N ; R N ).
We can then define the discrete flow, which can be seen as a discretein-time approximation of the mean curvature flow starting from the initial set E 0 . We define iteratively the discrete flow by setting E (h) t = E 0 for t ∈ [0, h) and ( 23)

E (h) t = T h,t-h E (h) t-h = T h,([ t h ]-1)h E (h) t-h , t ∈ [h, +∞),
where [•] denotes the integer part of a real number. This section is devoted to recall and prove some estimates on the discrete flow. The first one is a well-known existence result.

Lemma. For any measurable function g : R N → R such that min{g, 0} ∈ L 1 loc , the problem min P (E) + ˆE g : E is of finite perimeter admits a solution.

Consider now F as a bounded set of finite perimeter. Then, the function g = sd ψ F /h -F h is coercive, thus min{g, 0} ∈ L 1 . Therefore, by the previous result and by classical arguments see [START_REF] Chambolle | Nonlocal curvature flows[END_REF]Proposition 6.1] for a proof, one can prove the following result. Lemma 3.1. For any given set F of finite perimeter, the problem (21) admits a solution E, which satisfies the discrete dissipation inequality

P ϕ (E) + 1 h ˆE△F |sd ψ F | ≤ P ϕ (F ) + ˆE\F F h (x, t) dx - ˆF \E F h (x, t) dx.
Moreover, the problem (21) admits a minimal and a maximal solution.

We define T + h,t F (respectively T - h,t F ) as the maximal (respectively minimal) solution to (21) having as initial datum F . In the following, whenever no confusion is possible, we shall write T h,t instead of T - h,t . A comparison result holds. We will consider just bounded sets as datum for the problem ( 21), but the same result holds in general for unbounded sets (see also Section 4.1 for the case of unbounded sets with bounded boundary). The proof of this result is classical (see e.g. [START_REF] Chambolle | Nonlocal curvature flows[END_REF]) and it is based on the submodularity of the perimeter [START_REF] Chambolle | Existence and uniqueness for anisotropic and crystalline mean curvature flows[END_REF]. We will omit it. Lemma 3.2 (Weak comparison principle). Assume that F 1 , F 2 are bounded sets with F 1 ⊂⊂F 2 and consider g 1 , g 2 ∈ L ∞ with g 1 ≥ g 2 . Then, for any two solutions E i , i = 1, 2 of the problems

min P ϕ (E) + ˆE sd ψ Fi h + g i : E is of finite perimeter ,
we have

E 1 ⊆ E 2 . If, instead, F 1 ⊆ F 2 ,
then we have that the minimal (respectively maximal) solution to [START_REF] Eto | An area-minimizing scheme for anisotropic mean-curvature flow[END_REF] for i = 1 is contained in the minimal (respectively maximal) solution to (21) for i = 2.

We now prove the volume-density estimates for minimizers of problem [START_REF] Eto | An area-minimizing scheme for anisotropic mean-curvature flow[END_REF]. This result is based on the minimality properties of almost-minimizers for perimeters induced by regular elliptic integrands (see [START_REF] Philippis | Regularity of free boundaries in anisotropic capillarity problems and the validity of Young's law[END_REF]Remark 1.9] for further results). These estimates have the disadvantage that the smallness condition on the radius depends on the parameter h. Subsequently, we will recall a finer result in the spirit of [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF], where we can drop this dependence by making some restrictions on the balls considered. Lemma 3.3. Let g ∈ L ∞ and assume E minimizes the functional

F (F ) = P ϕ (F ) + ˆF g among all measurable subsets of R N . Then the density estimate σρ N ≤ |B ρ (x) ∩ E| ≤ (1 -σ)ρ N σρ N -1 ≤ P ϕ (E; B ρ (x)) ≤ (1 -σ)ρ N -1 (24) holds for all x ∈ ∂ * E, 0 < ρ < (2λ∥g∥ ∞ ) -1 := ρ 0 , for a suitable σ = σ(N, c ψ , λ).
Proof. By minimality,

P ϕ (E) ≤ P ϕ (F ) + ∥g∥ ∞ |E△F | ∀F ⊆ R N ,
thus [18, Lemma 2.8] implies the thesis. □ Remark 3.4. We remark that the previous result allows us to choose the minimal solution to [START_REF] Eto | An area-minimizing scheme for anisotropic mean-curvature flow[END_REF] to be an open set, and the maximal one to be a closed set. This follows from the fact that the density estimates imply that the boundary of any minimizer has zero measure.

We now recall [START_REF] Chambolle | Existence and uniqueness for anisotropic and crystalline mean curvature flows[END_REF]Lemma 3.7], which is an anisotropic version of [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF]Remark 1.4]. It provides volume-density estimates for minimizers of [START_REF] Eto | An area-minimizing scheme for anisotropic mean-curvature flow[END_REF] starting from E, uniform in ψ and h, holding in the exterior of E. We remark that, even if in the reference the anisotropy ϕ considered did not depend on x, all the arguments hold with minor modifications also in our case. We recall the proof of this result, as similar techniques will be used later on. Lemma 3.5. Let E be a bounded, closed set, h > 0 , and g ∈ L ∞ (R N ). Let E ′ be a minimizer of

P ϕ (F ) + ˆF sd ψ E h + g.
Then, there exists σ > 0, depending on λ, and r 0 ∈ (0, 1), depending only on N, λ, G := ∥g∥ L ∞ (F ) , with the following property: if

x is such that |E ′ ∩ B s (x)| > 0 for all s > 0 and B r (x) ∩ E = ∅ with r ≤ r 0 , then (25)
|E ′ ∩ B r (x)| ≥ σr N . Analogously, if x is such that |B s (x) \ E ′ | > 0 for all s > 0 and B r (x) ⊆ E with r ≤ r 0 , then |B r (x) \ E ′ | ≥ σr N .
Proof. For all s ∈ (0, r), set E ′ (s) := E ′ \ B s (x). Note that, for a.e. s we have

P ϕ (E ′ (s)) = P ϕ (E ′ ) -P ϕ (E ′ ∩ B s (x)) + ˆE′ ∩∂Bs(x) (ϕ(x, ν(x)) + ϕ(x, -ν(x))) dH N -1 (x),
where ν denotes the outer normal vector of the set

E ′ ∩ ∂B s (x). Since E ′ ∩ B s (x) ⊂ E c and sd ψ E ≥ 0 in E c , one has ´E′ ∩Bs(x)
sd ψ E ≥ 0, and therefore the minimality of E ′ implies

P ϕ (E ′ ∩ B s (x)) + ˆE′ ∩Bs(x) g ≤ ˆE′ ∩∂Bs(x) (ϕ(x, ν(x)) + ϕ(x, -ν(x))) dH N -1 (x).
By the bound on the ϕ-perimeter and using the classical isoperimetric inequality (whose constant is denoted

C N ) we obtain 2λH N -1 (E ′ ∩ ∂B s (x)) ≥ 1 λ P (E ′ ∩ B s (x)) + ˆE′ ∩Bs(x) g ≥ 1 λ C N |E ′ ∩ B s (x)| N -1 N -∥g∥ ∞ |E ′ ∩ B s (x)| ≥ C N 2λ |E ′ ∩ B s (x)| N -1 N , provided |E ′ ∩ B s (x)| 1/N ≤ C N /(2λ∥g∥ ∞ )
, which is true if r 0 is small enough. Since the rhs is positive for every s, we conclude

(26) d ds |E ′ ∩ B s (x)| 1 N ≥ C N 4λ 2 N for a.e. s ∈ (0, r).
The thesis follows by integrating the above differential inequality. The other case is analogous. □ Remark 3.6. Requiring that the anisotropy ψ is bounded uniformly from above and below ensures that the results of the previous Lemmas 3.3 and 3.5 can be read in terms of the ψ-balls. For example, for any r ≥ 0 and x ∈ R N , equation ( 25) could be read as

|E ′ ∩ B ψ r (x)| ≥ σc -N ψ r N , provided x is such that |E ′ ∩ B ψ s (x)|
> 0 for all s > 0 and B ψ r (x) ∩ E = ∅, and holds for all r ≤ r 0 /c ψ . Here, σ is as in Lemma 3.5 and depends only on λ. Analogous statements holds for Lemma 3.9.

We now provide some estimates on the evolution of balls under the discrete flow. We start by a simple remark concerning the boundedness of the evolving sets.

Remark 3.7. A simple estimate on the energies implies that the minimizers of ( 21) are bounded whenever F is bounded. Indeed, assume F ⊆ B R and consider

B ρ (x) ∩ (E \ B R ) ̸ = ∅: testing the minimality of E against F we easily deduce R 2h |B ρ (x) ∩ E| ≤ ˆE∩Bρ(x) sd ψ F h ≤ P ϕ (F ) + ∥F h (•, t)∥ ∞ |E△F | ≤ P ϕ (F ) + ∥f ∥ ∞ (|F | + |E|).
Employing the density estimates of Lemma 3.5 and sending R → ∞, we get a contradiction, as the isoperimetric inequality implies that |E| is bounded since F F h,t (F ) < ∞. We now want to prove finer estimates on the speed of evolution of balls. These estimates are classically a crucial step in order to prove existence of the flow. In the case under study, the main difficulties come from the inhomogeneity of the functionals considered, as in the homogeneous case convexity arguments easily yield the boundedness result, for example. We will use a "variational" approach in the spirit of [START_REF] Chambolle | Nonlocal curvature flows[END_REF] (but see also [START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF]Lemma 3.8] for a different proof relying more on the smoothness of the evolving set). Lemma 3.8. For every R 0 > 0 there exist h 0 (R 0 ) > 0 and C(R 0 , ϕ, ψ, f ) > 0 with the following property: For all R ≥ R 0 , h ∈ (0, h 0 ), t > 0 and x ∈ R N one has

(27) T h,t (B R (x)) ⊃ B R-Ch (x).
Proof. We divide the proof into three steps. In the following, the constants σ, r 0 are those of Lemma 3.5. We will assume x = 0 for simplicity. We fix R ≥ R 0 and denote E := T h,t B R .

Step 1. We prove that, given a ∈ (0, σ), ε ∈ (0, 1), we can ensure

|B R(1-ε) \ E| < a R N (1 -ε) N for h small enough. Indeed, assume by contradiction |B R(1-ε) \ E| ≥ a R N (1 -ε) N . Testing the minimality of E against B R , we obtain ˆ(B R(1-ε) \E)∪(E\B R ) |sd ψ B R | h ≤ 1 h ˆBR △E |sd ψ B R | ≤ P ϕ (B R )- ˆBR \E F h + ˆE\B R F h ,
and estimating

|sd ψ B R | ≥ Rε/c ψ on B R(1-ε) \ E, we get Rε hc ψ |B R(1-ε) \ E| ≤ P ϕ (B R ) + ∥f ∥ ∞ ω N R N + |B R(1+ε) \ B R | + ˆE\B R(1+ε) F h - |sd ψ B R | h .
Taking h ≤ ε/(c ψ ∥f ∥ ∞ ), the last term on the rhs is negative, thus

Rε hc ψ |B R(1-ε) \ E| ≤ P ϕ (B R ) + ∥f ∥ ∞ R N (ω N + 2 N +1 ε).
We employ the hypothesis to obtain

a hc ψ ε(1 -ε) N R N +1 ≤ c ψ N ω N R N -1 + cR N , a contradiction for h ≤ ca ε (1 -ε) N min{1, R 2 }, where c is a constant depending on N, ϕ, ψ, ∥f ∥ ∞ .
Step 2. Using Step 1, we prove that B R/2 ⊂ E for h small. Assume that R ≤ r 0 : by following the second part of the proof of Lemma 3.5 we obtain equation [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF], which reads d ds

|B s \ E| 1/N ≥ C N 4λ 2 N = σ 1/N for a.e s ∈ (0, R).
Applying the previous step with ε = 1/4, a = σ/3 N , it holds |B 3R/4 \ E| ≤ σR N /4 N for all h ≤ c(N, ϕ, ψ, f )R. Therefore, one deduces the existence of a positive extinction radius

(28) R * = 3R 4 - |B 3R/4 \ E| 1/N σ 1/N ≥ R 2 such that |B R * \ E| = 0,
which proves the claim. Clearly, taking h ≤ cR 0 the smallness assumption on h is uniform for R ≥ R 0 .

If R ≥ r 0 one simply uses a covering argument. For any x ∈ B R-r0 , applying the previous result to the ball B r0 (x) and using the comparison principle of Lemma 3.2, we conclude that ∀h ≤ c r 0 it holds x∈B R-r 0 B r0/2 (x)⊂⊂E.

Step 3. We conclude the proof. By the previous two steps and Remark 3.7, taking h small enough, we see that ρ := sup{r > 0 : |B r \ E| = 0} ∈ (R/2, +∞). We can assume ρ ≤ R, otherwise the result of the lemma is trivial. Consider the vector field

∇ p ϕ x, x |x| ∈ C 1 (R N , R N ).
Then, recalling (13), we get P ϕ (G) ≥ -´RN Dχ G • ∇ p ϕ(x, x/|x|) for all G set of finite perimeter and

P ϕ ((1 + ε)B ρ ) = ˆRN Dχ (1+ε)Bρ • -∇ p ϕ x, x |x| . Setting W ε = (1 + ε)B ρ \ E, by submodularity on (1 + ε)B ρ , E and exploiting the minimality of E, we obtain ˆRN ∇ p ϕ x, x |x| • Dχ Wε = ˆRN ∇ p ϕ x, x |x| • Dχ (1+ε)Bρ -Dχ (1+ε)Bρ∩E ≤ P ϕ ((1 + ε)B ρ ∩ E) -P ϕ ((1 + ε)B ρ ) ≤ P ϕ (E) -P ϕ ((1 + ε)B ρ ∪ E) ≤ 1 h ˆW ε sd ψ B R - ˆWε F h (x, t) dx.
We conclude, using the divergence theorem ,

ˆW ε -div∇ p ϕ x, x |x| ≤ 1 h ˆW ε sd ψ B R + ∥f ∥ ∞ |W ε |.
Dividing by |W ε | and sending ε → 0 we obtain

∂Bρ∩E -div∇ p ϕ x, x |x| dH N -1 ≤ 1 c ψ ρ -R h + ∥f ∥ ∞ .
Exploiting the regularity assumptions on ϕ, we remark that

|div∇ p ϕ| = |tr ∇ x ∇ p ϕ + ∇ 2 p ϕ∇(x/|x|) | ≤ C 1 + 1 |x| .
Thus, we obtain

-C 1 + 1 ρ ≤ ρ -R h , which implies that ρ ∈ (0, ρ 1 ) ∪ (ρ 2 , R) for ρ 1,2 = R -Ch ∓ (R -Ch) 2 -4Ch /2, as long as h ≤ R 2 0 /(4C). Since the choice ρ ≤ ρ 1 < R/2
is not admissible, we conclude the proof by estimating

ρ 2 = R -Ch + R -Ch 2 1 - 4Ch (R -Ch) 2 -1 ≥ R -Ch - Ch R -Ch ,
from which the thesis follows. □

The proof of the previous result can be employed to prove an estimate from above of the evolution speed of the flow, as the following result shows. Since the proof follows the same lines and is easier in this case, we only sketch it. Lemma 3.9. Fix T > 0 and R 0 > 0. Then, there exist positive constants C = C(ϕ, ψ, f, R 0 ) and

h 0 = h 0 (R 0 ) such that, for every R ≥ R 0 and h ≤ h 0 , if E 0 ⊆ B R , then E (h) t ⊆ B R+CT for all t ∈ (0, T ).
Proof. Choose h small as in the previous result and set

ρ = inf{r > 0 : |E \ B r | = 0} ∈ (R/2, +∞).
We can assume ρ ≥ R, otherwise the result is trivial. Defining W ε = E \ (1 -ε)B ρ and reasoning as before we obtain

ˆRN ∇ p ϕ x, x |x| • Dχ W ε = ˆRN ∇ p ϕ x, x |x| • Dχ (1-ε)Bρ∪E -Dχ (1-ε)Bρ ≥ -P ϕ ((1 -ε)B ρ ∪ E) + P ϕ ((1 -ε)B ρ ) ≥ -P ϕ (E) + P ϕ ((1 -ε)B ρ ∩ E) ≥ 1 h ˆW ε sd ψ B R - ˆWε F h (x, t) dx.
As in the previous proof, we arrive at

ρ -R h ≤ C 1 + 1 ρ , which implies that ρ ≤ ρ 2 = R + Ch + (R + Ch) 2 + 4Ch /2 ≤ R + Ch, up to changing C. □ 3.2.
Existence of flat flows. In the following, we will prove that the discrete flow (defined in ( 23)) defines a discrete-in-time approximation of a weak solution to the mean curvature flow, which is usually known as a "flat" flow (because the approximating surfaces ∂ * E (h) t converge in the "flat" distance of Whitney to the limit ∂ * E t , see [START_REF] Almgren | Curvature-driven flows: a variational approach[END_REF]).

We start by proving uniform bounds on the distance between two consecutive sets of the discrete flow and on the symmetric difference between them. We introduce the time-discrete normal velocity: for all t ≥ 0 and x ∈ R N , we set

v h (x, t) := 1 h sd ψ E (h) t-h (x) for t ∈ [h, +∞) 0 for t ∈ [0, h).
The following result provides a bound on the L ∞ -norm of the discrete velocity. Since the proof is essentially the same of [38, Lemma 2.1], we will omit it. The only difference is that we use the upper and lower bounds of ( 11) to work with Euclidean balls.

Lemma 3.10. There exists a positive constant c ∞ depending only on N, ψ with the following property. Let E 0 be a bounded set of finite perimeter and let {E (h) t } t∈(0,T ) be a discrete flow starting from E 0 . Then, sup

E (h) t △E (h) t-h |v h (•, t)| ≤ c ∞ h -1/2
for all h sufficiently small.

The following result can be found in [START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF]Proposition 3.4] (see also [START_REF] Fusco | Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane[END_REF]Lemma 2.2]): it provides an estimate on the volume of the symmetric difference of two consecutive sets of the discrete flow. The proof is analogous to the one in the reference. Lemma 3.11. There exists a constant C such that for every t ≥ h the discrete flow E (h) t satisfies for all h sufficiently small (29)

|E (h) t+h △E (h) t | ≤ C lP ϕ (E (h) t ) + 1 l ˆE(h) t △E (h) t+h |sd ψ E (h) t | ∀l ≤ c √ h,
where c is a positive constant depending on N, ψ.

We are now able to prove an uniform bound on the perimeter of the evolving sets. The proof follows [START_REF] Fusco | Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane[END_REF]Proposition 2.3]. Lemma 3.12. For any initial bounded set E 0 of finite ϕ-perimeter and h small enough, the discrete flow

{E (h) t } satisfies P ϕ (E (h) t ) ≤ C T ∀t ∈ (0, T ), for a suitable constant C T = C T (T, E 0 , f, ϕ, ψ).

Proof. By testing the minimality of E

(h) t against E (h) t-h we obtain ∀t ∈ [h, T ) (30) P ϕ (E (h) t ) + 1 h ˆE(h) t △E (h) t-h |sd ψ E (h) t-h | ≤ P ϕ (E (h) t-h ) + ∥f ∥ ∞ |E (h) t △E (h) t-h |.
Combining this estimate with (29) for l = 2Ch∥f ∥ ∞ ≪ √ h, where C is the constant appearing in equation [START_REF] Hensel | A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness[END_REF], we obtain for h sufficiently small [START_REF] Huisken | Geometric evolution equations for hypersurfaces[END_REF] P ϕ (E

(h) t ) + 1 2h ˆE(h) t △E (h) t-h |sd ψ E (h) t-h | ≤ 1 + 2C 2 h∥f ∥ 2 ∞ P ϕ (E (h) t-h )
Iterating the previous estimate, we find

P ϕ (E (h) t ) ≤ (1 + 2C 2 ∥f ∥ ∞ h) [ t h ]-1 P ϕ (E (h) h ).
In order to estimate P ϕ (E (h) h ) we start by observing that Remark 3.7, for h = h(E 0 ) small enough, implies

E (h) h ⊆ B 2r
, where E 0 ⊆ B r . Therefore, by [START_REF] Huisken | Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature[END_REF] for t = h we obtain P ϕ (E

(h) h ) ≤ P ϕ (E 0 ) + c and we conclude P ϕ (E (h) t ) ≤ C T (P ϕ (E 0 ) + 1). □
We then present a sketch of the proof of the local Hölder continuity in time of the discrete flow, uniformly in h, which can be deduced as in [START_REF] Fusco | Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane[END_REF]Proposition 2.3]. We highlight the main differences. Proposition 3. [START_REF] Chambolle | A nonlocal mean curvature flow and its semiimplicit time-discrete approximation[END_REF]. Let E 0 be an initial bounded set of finite ϕ-perimeter and T > 0. Then, for h small enough, for a discrete flow {E

(h) t } starting from E 0 it holds |E (h) t △E (h) s | ≤ C T |t -s| 1/2 ∀h ≤ t ≤ s < T, for a suitable constant C T = C T (T, E 0 , f, ϕ, ψ).
Proof. Following the previous proof, employing again (31) we find

P ϕ (E (h) 2h ) + 1 2 ˆE(h) 2h △E (h) h |v h (•, 2h)| + 1 2 ˆE(h) h △E (h) 0 |v h (•, h)| ≤ (1 + ch)P ϕ (E (h) h ) + 1 2 ˆE(h) h △E (h) 0 |v h (•, h)| ≤ (1 + ch) P ϕ (E (h) h ) + ˆE(h) h △E (h) 0 |v h |(•, h) ≤ (1 + ch) 2 P ϕ (E 0 ).
Iterating, we conclude as before (32)

[T /h] k=1 ˆE(h) kh △E (h) (k-1)h |v h (•, kh)| ≤ C T (P ϕ (E 0 ) + 1).
Therefore, combining the previous results and applying [START_REF] Hensel | A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness[END_REF] with l = h ≪ √ h, we obtain

(33) ˆT h |E (h) t △E (h) t-h | ≤ c [T /h] k=1 hP ϕ (E (h) kh ) + ˆE(h) kh △E (h) (k-1)h |v h (•, kh)| ≤ C T (P ϕ (E 0 ) + 1) .
The proof then follows the one of [25, Proposition 2.3], from equation (2.5) onward. □

We finally prove the main result of this section, the existence of flat flows.

Proof of Theorem 1.1. The proof is classical and we only sketch it. By the uniform equicontinuity of the approximating sequence of Proposition 3.13 and compactness of sets of finite perimeter (by Lemma 3.9 and 3.12) we can use the Ascoli-Arzelà theorem to prove that the sequence (E

(h k ) t
) k∈N converges in L 1 to sets E t for all times t ≥ 0 and that the family {E t } t≥0 satisfies the 1/2-Hölder continuity property, locally uniformly in time. The other property is then easily deduced. □ 3.3. Existence of distributional solutions. From Theorem 1.1 we deduce the existence of a subsequence (h k ) k≥0 such that (34)

Dχ E (h k ) t * ⇀ Dχ Et ∀t ≥ 0.
We will also assume (4), remarking that it implies [START_REF] Ishii | Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor[END_REF] lim

k→∞ P ϕ (E (h k ) t ) = P ϕ (E t ) for a.e. t ∈ [0, +∞).
Our aim is to derive ( 5) and ( 6) from the Euler-Lagrange equation ( 22) and passing to the limit h → 0. To achieve this, we will prove that the discrete velocity is a good approximation (up to multiplicative factors) of the discrete evolution speed of the sets. Notice that ( 5) is a weak formulation of (1), while [START_REF] Barles | Front propagation and phase field theory[END_REF] establishes the link between v and the velocity of the boundaries of E t . Indeed, law (1) can be interpreted as looking for a family {E t } t≥0 of sets, whose normal vector ν Et and ϕ-curvature H ϕ Et are well-defined objects and a function v

: [0, ∞) × R N → R such that for every t ∈ [0, +∞) and x ∈ ∂E t (36) v(x, t) = -H ϕ Et (x) + f (x, t) V (x, t) = ψ(x, ν Et (x))v(x, t),
where V represents the normal velocity of evolution, obtained as the limit as h → 0 (in a suitable sense) of the ratio

χ Et -χ E t-h h .
In this whole section we will assume that hypothesis ( 14) holds. In particular, the sets defining the discrete flow are smooth hypersurfaces in R N . Moreover, we require hypotheses (H1) to hold.

We start by estimating in time the L 2 -norm of the discrete velocity. The proof is the same as the one presented in [41, Lemma 3.6], up to using the density estimates on the ϕ-perimeter of Lemma 3.3 and considering the ψ-balls instead of the Euclidean one. Proposition 3.14. Let {E (h) t } t≥0 be a discrete flow starting from an initial bounded set E 0 of finite ϕ-perimeter. Then, for any T > 0 and for h small enough, it holds

ˆT 0 ˆ∂E (h) t v 2 h dH N -1 dt ≤ C T ,
for a suitable constant C T = C T (T, E 0 , ϕ, ψ, f ).

Recalling now the Euler-Lagrange equation [START_REF] Evans | Phase transitions and generalized motion by mean curvature[END_REF] and Lemma 3.12 we conclude (37)

ˆT 0 ˆ∂E (h) t H ϕ E (h) t 2 = ˆT 0 ˆ∂E (h) t (v h -F h ) 2 ≤ C T ,
We now prove an estimate on the error between the discrete velocity ψ(•, ν Et )v h (•, t) and the discrete time derivative of χ h . The proof of this result is based on a double blow-up argument, and the smoothness of sets (locally) minimizing the ϕ-perimeter is essential. We will split the proof in various lemmas: the first one concerns the composition of blow-ups, and is a well-known result to the experts. We present a simple proof since we could not find a reference.

Lemma 3.15 (Composition of blow-ups). Consider

0 < β < β ′ < 1. Assume that (A h ) h∈[0,1]
is a family of measurable sets such that the following blow-ups converge as h → 0

A h -x h h β → A 1 in L 1 loc h -(β ′ -β) A 1 → A 2 in L 1 loc ,
where

x h ∈ ∂A h for all h ∈ [0, 1]. Then, if the composition of the blow-ups h -β ′ (A h -x h ) converges in L 1 loc , the limit coincides with A 2 .
Proof. We can assume wlog x h = 0. Denote with

A 3 = L 1 loc -lim h→0 h -β ′ A h .
We fix a ball B M and ε > 0. There exists h * such that ∀h ≤ h * it holds

|((h -β ′ A h )△A 3 ) ∩ B M | ≤ ε, |((h -β ′ +β A 1 )△A 2 ) ∩ B M | ≤ ε.
We fix h and wlog assume M h β ′ -β ≤ 1. Taking h < h suitably small (depending on h, ε), we can ensure β) . We can then estimate

|(( h-β A h )△A 1 ) ∩ B 1 | ≤ εh N (β ′ -β) . Since h-β h -(β ′ -β) > h -β ′ , there exists h < h such that h-β ′ = h-β h -(β ′ -
|(A 3 △A 2 ) ∩ B M | ≤ |(A 3 △ h-β ′ A h ) ∩ B M | + |((h -β ′ +β )A 1 △( h-β ′ A h )) ∩ B M | + |((h -β ′ +β A 1 )△A 2 ) ∩ B M | ≤ 2ε + h -N (β ′ -β) |(A 1 △( h-β A h )) ∩ B M h β ′ -β | ≤ 2ε + h -N (β ′ -β) |(A 1 △( h-β A h )) ∩ B 1 | ≤ 3ε.

□

We now compute some estimates on the normal vector on the boundary of the evolving sets, following the proof of [41, Lemma 4.2] (see also [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF]Proposition 2.2]). We fix c ∞ as the constant appearing in Lemma 3.10.

In the sequel, we will denote by ω(h) a modulus of continuity, that is a continuous increasing function ω : [0, 1] → R with ω(0) = 0, which can possibly change from statement to statement and line to line to absorb constants independent of h. Lemma 3.16. Assume (H0) and (H1). For given constants 1/2 < β ′ < α < 1 and T > 2, there exists a modulus of continuity ω with the following property. Consider t ∈ [2h, T ] and

x h ∈ ∂E (h) t such that (38) |v h (t, y)| ≤ h α-1 ∀y ∈ B c∞ √ h (x h ) ∩ (E (h) t △E (h) t-h ). Then, there exists ν ∈ S N -1 such that |ν E (h) t (•) -ν| ≤ ω(h) in B h β ′ (x h ) ∩ ∂E (h) t |ν E (h) t-h (•) -ν| ≤ ω(h) in B h β ′ (x h ) ∩ ∂E (h) t-h . ( 39
)
Proof. We fix 1 2 < β < β ′ < α and 0 < R < h s , s = t, t -h, we find (40)

P ϕ (E (h) s , B Rh β (x h )) ≤ P ϕ (G, B Rh β (x h )) + 1 h ˆG△E (h) s |sd ψ E (h) s-h | + ˆG△E (h) s |F h |,
for any set G of finite perimeter such that G△E (h) s ⊂⊂B Rh β (x h ). Using Lemma 3.10, the 1-Lipschitz regularity of sd ψ and (38), we deduce

|v h (s, y)| ≤ c ψ Rh β-1 + c ∞ h -1/2 ≤ (1 + c ∞ )h -1/2 for any y ∈ B Rh β (x h ) ∩ (E (h)
s △F ). Plugging this inequality in [START_REF] Merriman | Motion of multiple junctions: A level set approach[END_REF], we find (41)

P ϕ (E (h) s , B Rh β (x h )) ≤ P ϕ (G, B Rh β (x h )) + 1 + c √ h |F △E (h) s | + ∥f ∥ ∞ |G△E (h) s |.
We then introduce the blown-up sets for s = t, t -h, defined as

E (h),β s := h -β E (h) s -x h .
Rescaling equation [START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF], we easily find that

E (h),β s is a (Λ h , r h )-minimizer of the ϕ(x h +h β •, •)-perimeter, with Λ h = (1 + c)h β-1/2 , r h = h 1/2-β .
Moreover, scaling the density estimates [START_REF] Fuchs | Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets[END_REF] we have a uniform bound on the perimeters of the sets E (h),β s in each ball B R . By compactness, there exist two sets

E β 1 , E β 2 such that E (h),β t → E β 1 , E (h),β t-h → E β 2 in L 1 loc .
Then, by scaling and ( 38) we find

|sd ψ E (h),β t-h (•)| ≤ c ∞ h α-β on B h 1/2-β (0) ∩ (E (h),β t △E (h),β t-h ),
thus we easily conclude that

E β := E β 1 = E β 2 .
By Lemma 3.9 we can assume that x h → x 0 as h → 0, up to subsequences. Moreover, by closeness of Λ h -minimizers under L 1 loc -convergence (see e.g. [18, Theorem 2.9]), one can see that E β is a 0-minimizer for the ϕ(x 0 , •)-perimeter. Thus, by complete regularity, it is a smooth C 2 set. We can then employ the classic blow-up theorem to deduce that, for a fixed β ′ ∈ (β, α), the blow-up h -(β ′ -β) E β converges to a half-space H = {x • ν ≤ 0} as h → 0. Finally, the blow-ups

E (h),β ′ s := E (h)
s -x h h β ′ admit a converging subsequence by compactness of sets of finite perimeter and by rescaling equation [START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF]. Thus, the previous Lemma 3. [START_REF] Chambolle | Implicit time discretization of the mean curvature flow with a discontinuous forcing term[END_REF] implies

E (h),β ′ s → H in L 1 loc
as h → 0. To conclude, the ε-regularity Theorem for Λ-minimizers (see e.g. [18, Theorem 3.1]) ensures that E (h),β ′ s are uniformly C 1, 1 2 sets in B 1 (0) for s = t, t -h as h → 0. □

We recall here an approximation result proved in [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF] (see also [START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF] for a more detailed proof). We remark that the proof of this result is purely geometric and does not rely on the variational problem satisfied by the sets

E (h) t , E (h) t-h .
Corollary (Corollary 4.3 in [START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF]). Under the hypotheses of Lemma 3.16, fix 0 < β < α and let C h β be the open cylinder defined as

C h β (x h , ν) := x ∈ R N : |(x -x h ) • ν| < h β 2 , (x -x h ) -((x -x h ) • ν) ν < h β 2 .
Then, it holds

ˆCh β /2 (x h ,ν) (χ E (h) t -χ E (h) t-h ) dx - ˆ∂E (h) t ∩C h β /2 (x h ,ν) sd E (h) t-h dH N -1 ≤ ω(h) ˆCh β /2 (x h ,ν) |χ E (h) t -χ E (h) t-h |.
Carefully inspecting the proof, one indeed proves that there exists a geometric constant C such that for any y ∈

B N -1 h β /2 (x h ) (42) |sd E (h) t-h (y, f (h) t (y)) 1 + |∇f (h) t (y)| 2 -f (h) t (y) -f (h) t-h (y) | ≤ ω(h)|f (h) t (y) -f (h) t-h (y)|,
where we set

∂E (h) s ∩ C = {(y, f (h) s (y)) ∈ R N -1 × R, |y| ≤ h β /2}, for s = t, t -h.
We briefly recall some classical results. Consider an anisotropy ψ, independent of the position. It is well-known that, for any closed set G ⊆ R N , setting sd ψ G as the distance from G induced by ψ • , then the gradient of sd ψ G exists almost everywhere and satisfies the eikonal equation (for a proof see for instance [10, Remark 2.2]) [START_REF] Sethian | Level set methods and fast marching methods[END_REF] ψ(∇sd ψ G ) = 1 almost everywhere. Moreover, in this particular case, in the definition of dist ψ we can consider just straight lines as follows from a simple application of Jensen's inequality: for any curve γ as in the definition of dist ψ , we have

ˆ1 0 ψ • ( γ(t)) dt ≥ ψ • ˆ1 0 γ = ψ • (y -x).
Proposition 3.17 (Estimate on almost flat sets). Under the hypotheses of Lemma 3.16 and with the same notation, fix β ∈ (0, α) and let C h β be the open cylinder defined as

C h β (x h , ν) := x ∈ R N : |(x -x h ) • ν| < h β 2 , (x -x h ) -((x -x h ) • ν) ν < h β 2 .
Then, it holds

ˆCh β /2 (x h ,ν) (χ E (h) t -χ E (h) t-h ) dx - ˆ∂E (h) t ∩C h β /2 (x h ,ν) ψ(x, ν E (h) t ) sd ψ E (h) t-h dH N -1 ≤ ω(h) ˆCh β /2 (x h ,ν) |χ E (h) t -χ E (h) t-h |.
Proof. We recall that the modulus of continuity ω may change from line to line to absorb constants independent of h.

From the previous Lemma 3. [START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF] we know that, for h suitably small, both ∂E

t-h in C h β /2 (x h , ν) can be written as graphs of functions of class C 1, 1 2 . Up to a change of coordinates, we can assume wlog that x h = 0, ν = e N . For simplicity, we set C = C h β /2 (0, e N ). We thus find

∂E (h) s ∩ C = {(y, f (h) s (y)) ∈ R N -1 × R, |y| ≤ h β /2}
for s = t, t -h, where f

(h) s : B N -1 h β /2 → R are C 1, 1 2 functions with ∥∇f (h) s ∥ L ∞ (B h β /2 ) ≤ ω(h).
We want to prove the following slightly stronger pointwise inequality: namely, that for any point x = (y, f

(h) t (y)) ∈ ∂E (h) t ∩ C, it holds (44) sd ψ E (h) t-h (x) ψ(x, ν E (h) t (x)) 1 + |∇f (h) t (y)| -f (h) t (y) -f (h) t-h (y) ≤ ω(h)|f (h) t (y) -f (h) t-h (y)|.
Integrating the previous inequality over C yields the thesis. Clearly, it is enough to prove [START_REF] Taubin | A signal processing approach to fair surface design[END_REF] at each point

x such that |sd ψ E (h) t-h (x)| > 0. We thus fix x = (y, f (h) t (y)) ∈ ∂E (h) t ∩ C and denote by x ′ := (y, f (h)
t-h (y)). We remark that these points depend on h, but we drop the subscript to ease notation. It can be assumed without loss of generality that x / ∈ E (h) t-h , as the other case is analogous.

Step 1 We now prove that, with the notation previously introduced, it holds

(45) |sd ′ E (h) t-h (x) -sd ψ E (h) t-h (x)| ≤ ω(h)|f (h) t (y) -f (h) t-h (y)|,
where sd ′ denotes the signed distance function induced by the anisotropy ψ(x ′ , •). Let γ be a smooth curve, with γ(0) = x, γ(1) ∈ ∂E . Firstly, we remark that one could assume

(46) γ([0, 1]) ⊆ B(x, 2c 2 ψ |f (h) t (y) -f (h) t-h (y)|)
Indeed, if it were not the case, the lower bounds contained in ( 11) and ( 42) allow us to estimate

ˆ1 0 ψ • (γ, γ) dt ≥ 1 c ψ ˆ1 0 | γ| dt ≥ 2c ψ |f (h) t (y) -f (h) t-h (y)| ≥ 2c ψ sd E (h) t-h (x) ≥ 2 sd ψ E (h) t-h (x), (47)
a contradiction for h small. We can reason analogously for sd ′ E (h) t-h

. In particular, we can consider just curves having length

´1 0 | γ| ≤ c|f (h) t (y) -f (h)
t-h (y)|. Therefore, we obtain (by homogeneity)

sd ψ E (h) t-h (x) ≤ ˆ1 0 ψ • (γ, γ) dt ≤ ˆ1 0 ψ • (x ′ , γ) dt + sup ν∈S N -1 , t∈[0,1] |ψ(γ(t), ν) -ψ(x ′ , ν)| ˆ1 0 | γ| ≤ ˆ1 0 ψ • (x ′ , γ) dt + c ω(h)|f (h) t (y) -f (h) t-h (h)|,
and, taking the inf γ , we obtain sd ψ

E (h) t-h (x) ≤ sd ′ E (h) t-h (x) + ω(h)|f (h) t (y) -f (h) t-h (y)|.
The converse inequality can be proved analogously, yielding (45).

Therefore, in what follows we will consider always the anisotropy frozen in x ′ , and use sd ′ instead of sd ψ . Finally, let p ∈ ∂E (h) t-h a minimizer for the definition of sd ′ E (h) t-h (x). In the following, with Π v H z, Π H z we denote respectively the projection on the hyperplane H of z along the direction v and the orthogonal projection of z on H.

Step 2. In this step we assume that E (h) t-h ∩ C coincides with the halfspace H = p + {z • ν ≤ 0} intersected with the same cylinder and prove claim [START_REF] Taubin | A signal processing approach to fair surface design[END_REF]. To this aim, we start noticing that by translation we may assume p = 0 and that sd ′ H (z+ξ) = sd ′ H (z) for all z ∈ R N and for all ξ orthogonal to ν. Hence, in fact,

(48) sd ′ H (z) = sd ′ H ((z • ν)ν) = (z • ν)sd ′ H (ν) . Therefore, sd ′
H is differentiable everywhere, with ∇sd ′ H = sd ′ H (ν)ν. Recalling the eikonal equation [START_REF] Sethian | Level set methods and fast marching methods[END_REF], it must hold sd ′ H (ν) = 1/ψ(x ′ , ν) and in turn, from (48), and choosing z = x, we have (49

) sd ′ H (x)ψ(x ′ , ν) = x • ν = sd H (x). We remark that sd ′ H (x) = sd ′ E (h) t-h
(x) by [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF], thus we conclude (44) by combining (49) with [START_REF] Schoen | Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II[END_REF].

Step 3. We now conclude in the general case. With the notation introduced at the end of Step 1, set ν = ν E (h) t-h (p), and consider the half-space H = p + {z • ν ≤ 0} and w := x ′ -Π H (x ′ ) as depicted in Figure 1. We shall prove that

|w| ≤ ω(h)|f (h) t (y) -f (h) t-h (y)|.
To see this, we start by remarking that (39) implies

|e N -e N (e N • ν E (h) t-h )| ≤ ω(h) in ∂E (h) t-h ∩ C, implying e N • ν E (h) t-h
≥ 1 -ω(h), and thus, for any versor v tangent to ∂E

(h) t-h ∩ C one has |v • e N | ≤ ω(h). Therefore, we have (x ′ -p) • e N ≤ ω(h)|x ′ -p| and also x ′ -p |x ′ -p| • ν = x ′ -p |x ′ -p| • (e N (ν • e N ) + ν -e N (ν • e N )) ≤ ω(h) + |ν -e N (ν • e N )| = ω(h) + 1 -|ν • e N | 2 1/2 ≤ 3 ω(h),
by choosing h small. Up to defining √ ω as ω, using the previous estimate and the bounds (46) we see that

(50) |w| = |x ′ -p| x ′ -p |x ′ -p| • ν ≤ ω(h)|x ′ -p| ≤ ω(h)|f (h) t (y) -f (h) t-h (y)|. H x x ′ E (h) t-h E (h) t p Π ν H x w Π e N H x γ ν Figure 1.
The situation in the proof of the lemma.

We now remark that sd ′

E (h) t-h (x) = sd ′ H (x)
(by convexity of the anisotropy ψ(x ′ , •)) and so, applying the previous step to H and using also (45), we get

sd ψ E (h) t-h (x) ψ(x, ν E (h) t (x)) 1 + |∇f (h) t (y)| -|x -Π e N H x| ≤ ω(h)|x -Π e N H x|.
We conclude ( 44) by estimating

|x -Π e N H x| -|x -x ′ | ≤ |x ′ -Π e N H x| = |w|/|ν • e N | ≤ ω(h) 1 -ω(h) |f (h) t (y) -f (h) t-h (y)|,
where we used (50). We conclude the proof by a simple change of coordinates and using ( 44) to find

ˆ∂E (h) t ∩C ψ(x, ν E (h) t (x)) sd ψ E (h) t-h (x) dH N -1 - ˆBh β /2 f (h) t (y) -f (h) t-h (y) dy = ˆBh β /2 ψ((y, f (h) t (y)), ν E (h) t (y, f (h) t (y))) sd ψ E (h) t-h (y, f (h) t (y)) 1 + |∇f (h) t (y)| 2 -(f (h) t (y) -f (h) t-h (y)) dy ≤ ω(h) ˆBh β /2 |f (h) t -f (h) t-h | dy.
□ Finally, we are able to prove that the error generated by approximating the discrete velocity with v h goes to zero as h → 0. We follow the lines of [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF]Proposition 2.2]. Proposition 3.18 (Error estimate). Under the hypothesis of Lemma 3.16, the error in the discrete curvature equation vanishes in the limit h → 0, namely

(51) lim h→0 1 h ˆT 0 ˆE(h) t η dx - ˆE(h) t-h η dx dt - ˆT 0 ˆ∂E (h) t ψ(x, ν E (h) t )v h η dH N -1 (x) dt = 0 for all η ∈ C 1 c (R N × [0, T )). Proof. We fix t ∈ [2h, ∞) and α ∈ ( 1 2 , N +2 2N +2 ). For any point x h ∈ ∂E (h) t
we define the open set A x h defined as follows:

(i) if (38) holds, we set A x h = C h β /2 (x h , ν), with the notations of Corollary 3.17;

(ii) otherwise we set

A x h = B(x h , c ∞ √ h)
, where c ∞ is the constant of Lemma 3.10.

By Lemma 3.10, the family {A x h :

x h ∈ ∂E (h) t } is a covering of E (h) t △E (h) t-h
. By a simple application of Besicovitch's theorem (see e.g. [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF]), we find a finite collection of points

I ⊆ ∂E (h) t such that {A x h } x h ∈I is a covering of E (h) t △E (h)
t-h with the finite intersection property. We proceed to estimate (51) on each A x h belonging to this family.

Estimate in case (i)

We use Proposition 3.17 to deduce ˆAx h (χ

E (h) t -χ E (h) t-h )η dx - ˆ∂E (h) t ∩Ax h ψ(x, ν E (h) t )sd ψ E (h) t-h η dH N -1 ≤ |η(x h , t)| ˆAx h (χ E (h) t -χ E (h) t-h ) - ˆ∂E (h) t ∩Ax h ψ(x, ν E (h) t )sd ψ E (h) t-h dH N -1 + ˆAx h (χ E (h) t -χ E (h) t-h )(η -η(x h , t)) - ˆ∂E (h) t ∩Ax h (η -η(x h , t)) ψ(x, ν E (h) t ) sd ψ E (h) t-h dH N -1 ≤ C(ω(h)∥η∥ ∞ + h β ∥∇η∥ ∞ ) ˆAx h |χ E (h) t -χ E (h) t-h | dH N -1 + ch β ∥∇η∥ ∞ P (E (h) t , A x h ). ( 52
)
Estimate in case (ii) By assumption ∃y ∈ B c∞

√ h (x h ) ∩ (E (h) t △E (h) t-h ) such that |v h (t, y)| > h α-1 . We can assume wlog y ∈ E (h) t . We then have B(y, h α /(2c ψ )) ⊆ R N \ E (h) t-h and sd ψ E (h) t-h > h α /(2c 2
ψ ) on B(y, h α /(2c ψ )). Since h α << h 1/2 , we can use the density estimates of Lemma 3.3 to deduce

ch (N +1)α-1 ≤ ˆB(y,h α /(2c ψ ))∩(E (h) t △E (h) t-h ) |v h | dx.
Analogously, recalling also Lemma 3.10, we deduce

ˆB(x h ,c∞ √ h)∩∂E (h) t |ψ(x, ν E (h) t-h ) sd ψ E (h) t-h | dH N -1 (x) ≤ ch N 2 .
Combining the two previous equations and

B(y, h α /(2c ψ )) ⊆ B(y, c √ h), we infer ˆAx h |χ E (h) t -χ E (h) t-h | + ˆAx h ∩∂E (h) t |ψ(x, ν E (h) t-h ) sd ψ E (h) t-h | dH N -1 ≤ ch N 2 -(N +1)α+1 ˆAx h ∩(E (h) t △E (h) t-h ) |ψ(x, ν E (h) t-h )v h |. ( 53
)
Summing over x h ∈ I both (52) and (53), and using the local finiteness of the covering, we get

ˆ(χ E (h) t -χ E (h) t-h )η dx - ˆ∂E (h) t ψ(x, ν E (h) t )sd ψ E (h) t-h η dH N -1 ≤ x h ∈I ˆAx h (χ E (h) t -χ E (h) t-h )η dx - ˆ∂E (h) t ∩Ax h ψ(x, ν E (h) t )sd ψ E (h) t-h η dH N -1 ≤ c ω(h)∥η∥ ∞ + h β ∥∇η∥ ∞ + h N 2 -(n+1)α+1 ∥η∥ ∞ • • P (E (h) t ) + |E (h) t △E (h) t-h | + ˆE(h) t △E (h) t-h |v h |
where the last constant c depends on N, ψ. We then use Lemma 3.12, ( 32) and ( 33) to conclude

ˆT 2h 1 h ˆE(h) t η dx - ˆE(h) t-h η dx - ˆT h ˆ∂E (h) t ψ(x, ν E (h) t )v h η dH N -1 ≤ c ω(h)∥η∥ ∞ + h β ∥∇η∥ ∞ + h N 2 -(n+1)α+1 ∥η∥ ∞ ,
where c = c(E 0 , f, T, ψ) and T is chosen such that spt η⊂⊂R N × [0, T ]. The conclusion follows using the definition of α and taking the limit h → 0.

□

The proof of our main theorem of this section is now a consequence of the previous results. In particular, hypothesis [START_REF] Ilmanen | The level-set flow on a manifold[END_REF] and [START_REF] Ishii | Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor[END_REF] imply that the discrete flow converges to the flat flow in the sense of varifolds and this allows to prove (5), while ( 6) is a consequence of Proposition 3.18. In order to prove the convergence of the approximations in time of the forcing term, we need to require additionally that (H2) holds.

Proof of Theorem 1.2. Firstly, combining [START_REF] Hutchinson | Second fundamental form for varifolds and the existence of surfaces minimising curvature[END_REF]Theorem 4.4.2] with the bounds contained in [START_REF] Laux | Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow[END_REF] and in Proposition 3.14, we conclude the existence of functions v, H ϕ , f :

R N × [0, ∞) → R satisfying ˆT 0 ˆ∂Et |v| 2 + |H ϕ | 2 + | f | 2 dH N -1 dt ≤ C T
and the following properties

lim k ˆT 0 ˆ∂E (h k ) t v h k η dH N -1 dt = ˆT 0 ˆ∂Et ηv dH N -1 dt lim k ˆT 0 ˆ∂E (h k ) t F h k (x, t)η dH N -1 dt = ˆT 0 ˆ∂Et η f dH N -1 dt lim k ˆT 0 ˆ∂E (h k ) t H ϕ E (h k ) t η dH N -1 dt = ˆT 0 ˆ∂Et ηH ϕ dH N -1 dt, (54) for any η ∈ C 0 c (R N × [0, T ))
. We now employ an approximation procedure to prove that H ϕ (•, t) is the ϕ-mean curvature of E t for a.e. t ∈ [0, ∞), following the lines of [START_REF] Luckhaus | Implicit time discretization for the mean curvature flow equation[END_REF][START_REF] Mugnai | Global solutions to the volume-preserving meancurvature flow[END_REF]. Fixed t ∈ [0, +∞) and ε > 0, set ν ε a continuous function such that ´∂Et (ν Et -ν ε ) 2 dH N -1 < ε. Then, by [START_REF] Ilmanen | The level-set flow on a manifold[END_REF] one could prove that lim k→∞ ´∂E (h k )

t (ν E (h k ) t -ν ε ) 2 dH N -1 < ε.
Considering test functions in (54) of the form η(x, t) = a(t)g(x), one has for a.e. t ∈ [0, +∞)

lim k ˆ∂E (h k ) t H ϕ E (h k ) t g dH N -1 = ˆ∂Et H ϕ g dH N -1 .
Thus, for a.e. t ∈ [0, +∞) and for any

X ∈ C 0 c (R N ; R N ) it holds lim k ˆ∂E (h k ) t H ϕ E (h k ) t ν E (h k ) t • X dH N -1 = ˆ∂Et H ϕ ν Et • X dH N -1
by approximating the normal vectors of E (h k ) t with ν ε . Furthermore, by the convergence (34) and the hypothesis [START_REF] Ishii | Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor[END_REF] we can use the Reshetnyak's continuity theorem (see e.g. [4, Theorem 2.39]), ensuring

ˆ∂E (h k ) t L(x, ν E (h k ) t ) dH N -1 → ˆEt L(x, ν Et ) dH N -1 as k → ∞, for any L ∈ C 0 c (R N × R N ). We choose L(x, ν) = div ϕ X for some X ∈ C 1 c (R N ; R N ) to obtain ˆ∂Et div ϕ X dH N -1 = lim k ˆ∂E (h k ) t div ϕ X dH N -1 = lim k ˆ∂E (h k ) t X • ν E (h k ) t H ϕ E (h k ) t dH N -1 = ˆ∂Et X • ν Et H ϕ dH N -1 ,
which shows that H ϕ (•, t) is the ϕ-mean curvature of the set E t for a.e. t ∈ [0, +∞). Moreover, we remark that F h k (x, t) → f (x, t) for every (x, t), thus for any test function η ∈ C 0 c (R N × [0, +∞)) and t ∈ [0, +∞) we have

ˆ∂E (h) t F h k (x, t)η(x, t) dH N -1 x - ˆ∂E f η dH N -1 x ≤ ˆ∂E (h) t F h k η - ˆ∂Et F h k η + ˆ∂Et |F h k -f |η ≤ ∥f ∥ ∞ ∥η∥ ∞ P (E (h) t ) -P (E t ) + ˆ∂Et |F h k -f |η → 0
applying the dominated convergence theorem and recalling Lemma 3.9. Thus, f = f . We then prove (5) by passing to the limit in the Euler-Lagrange equation [START_REF] Evans | Phase transitions and generalized motion by mean curvature[END_REF].

To prove [START_REF] Barles | Front propagation and phase field theory[END_REF] we employ Proposition 3.18: for every η ∈ C 0 c (R N ×[0, T )), by a change of variables we have that

ˆT h ˆE(h) t η dx - ˆE(h) t-h η dx dt = ˆT h ˆE(h) t (η(x, t) -η(x, t -h)) dx dt -h ˆE0 η dx
where we have used that

E (h) t = E 0 for t ∈ [0, h). Therefore, a simple convergence argument yields lim h→0 1 h ˆT h ˆE(h) t η dx - ˆE(h) t-h η dx dt = - ˆT h ∂ t η(x, t) dx dt - ˆE0 η.
Combining the previous estimate with Proposition 3.18 and passing to the limit, we obtain (6). □

Viscosity solutions

In this section we will prove the existence of another weak notion of solution for the mean curvature flow starting from a compact set. We will follow the so-called level set approach based on the theory of viscosity solution. We recall that in the first part we work with the standing assumptions of the paper (H0). Additionally, we require (H1).

4.1. The discrete scheme for unbounded sets. In this short subsection we will define the discrete evolution scheme for unbounded sets having compact boundary. The idea would be to define this evolution simply as the complement of the evolution of the complementary set, but since the anisotropies we are considering are not symmetric, we need additional care.

We recall that, given an anisotropy ϕ, we define φ(x, ν) := ϕ(x, -ν). This anisotropy has all the properties of the original one, concerning regularity and bounds. We start remarking the following simple fact. One can see that dist ψ (x, y) = dist ψ (y, x), since for any curve γ ∈

W 1,1 ([0, 1]; R N ), γ(0) = x, γ(1) = y, a simple change of variable yields ˆ1 0 ψ • (γ(t), γ(t)) dt = ˆ1 0 ψ • γ(1 -t), - d dt (γ(1 -t)) dt = ˆ1 0 (ψ • )(η(t), η(t)) dt, for η(t) = γ(1 -t), once one sees that (ψ • )(•, ν) = sup ψ(•,ξ)≤1 ξ • (-ν) = sup ψ(•,-ξ)≤1 (-ξ) • ν = ( ψ) • (•, ν).
Therefore, by definition of signed distance we have (55) sd ψ E (x) = -sd ψ E c (x). For every compact set F and h > 0, t ≥ 0, we will denote by T ± h,t F the maximal and the minimal solution to problem [START_REF] Eto | An area-minimizing scheme for anisotropic mean-curvature flow[END_REF], according to Lemma 3.1 with P ϕ and sd ψ , respectively, replaced by P φ and sd ψ . Finally, for every set E with compact boundary we define

(56) T ± h,t E := T ∓ h,t E c c .
As in the case for compact sets, we set T h,t E := T - h,t E. Given an open, unbounded set E 0 having compact boundary, we can then define the discrete flow {E (h) t } t≥0 as follows:

E (h) t := E 0 for t ∈ [0, h) and E (h) t = T h,t E (h)
t-h , ∀t ∈ [h, +∞). One easily checks that analogous results to Lemmas 3.2, 3.9 and 3.8 hold also for this problem. We state the corresponding results.

h > 0, t ≥ 0. Then, T h,t F 1 ⊆ T h,t F 2 .
Lemma 4.2. For any T > 0 there exists a constant C T (ϕ, ψ, f, T ) such that for every R > 0 the following holds.

If the initial open set E ⊃ B c R , then E (h) t ⊃ B c C T R for all t ∈ [0, T ]. Lemma 4.3. For every R 0 > 0 there exist h 0 (R 0 ) > 0 and C(R 0 , ϕ, ψ, f ) > 0 with the following property: For all R ≥ R 0 , h ∈ (0, h 0 ), t > 0 and x ∈ R N one has T h,t ((B R (x)) c ) ⊆ (B R-Ch (x)) c .
We now state a comparison principle between bounded and unbounded sets, following the line of [14, Lemma 6.10].

Lemma 4.4. Let E 1 be a compact set and let E 2 be an open, unbounded set, with compact boundary, and such that

E 1 ⊆ E 2 . Then, for every h ∈ (0, 1), t ≥ 0 it holds T ± h,t E 1 ⊆ T ± h,t E 2 . Proof. We fix h ∈ (0, 1), t ∈ [0, T ] for T > 0. Set R > 0 such that E 1 , E c
2 ⊆ B R and note that by Lemmas 3.2 and 3.9 (applied to P φ instead of P ϕ ) we get (57)

T + h,t E 2 c ⊆ T - h,t E c 2 ⊆ T - h,t B R ⊆ B C T R , for some C T (ϕ, ψ, f, T ). Since T - h,t E c 2 is the minimal solution of min P φ(E) + 1 h ˆE sd ψ E c 2 (x) dx -ˆE F h (x, t) dx ,
considering the change of variables Ẽ = E c and using (55), we easily conclude that

T + h,t E 2 = T - h,t E c 2 c
is the maximal solution of

min P ϕ ( Ẽ) + 1 h ˆBC T R sd ψ E2 - 1 h ˆẼ c sd ψ E2 - ˆẼ c F h (x, t) dx - 1 h ˆBC T R sd ψ E2 .
we then note that

ˆBC T R sd ψ E2 = ˆẼ sd ψ E2 χ B C T R + ˆẼ c sd ψ E2 ,
for every Ẽ such that Ẽc ⊆ B C T R . By (57), we conclude that T + h,t E 2 is the maximal solution of (58

) min P ϕ ( Ẽ) + 1 h ˆẼ sd ψ E2 χ B C T R - ˆẼ c F h (x, t) dx : Ẽc ⊆ B C T R .
Analogously, one proves that T - h,t E 2 is the minimal solution of (58). Finally, we remark that sd

ψ Es χ B C T R ≤ sd ψ E1 and that T ± h,t E 1 ∪ T ± h,t E 2 , T ± h,t E 1 ∩ T ± h,t E 2 are
both admissible competitors for (58), one argues exactly as in the proof of Lemma 3.2 to conclude

T ± h,t E 1 ⊆ T ± h,t E 2 . □ 4.2.
The level set approach. We recall that in this section we assume (H0), (H1). Consider a function u : R N × [0, +∞) → R whose spatial superlevel sets {u(•, t) ≥ s} evolve according to the mean curvature equation

V (x, t) = -ψ(x, ν {u(•,t)≥s} ) H ϕ {u(•,t)≥s} (x) -f (x, t) for x ∈ ∂{u(•, t) ≥ s}.
The function u then satisfies (recalling that -∇u/|∇u| is the outer normal vector to the superlevel set {u(•, t) ≥ u(x, t)}) the equation

∂ t u = |∇u|V (x) = -ψ(x, -∇u) H ϕ {u(•,t)≥u(x,t)} (x) -f (x, t) = -ψ(x, -∇u) (div∇ p ϕ(x, -∇u) -f (x, t)) = -ψ(x, -∇u) i ∂ xi ∂ p ϕ(x, -∇u) -∇ 2 p ϕ(x, -∇u) : ∇ 2 u -f (x, t) : = -ψ(x, -∇u) H(x, ∇u, ∇ 2 u) -f (x, t) ,
where we defined the Hamiltonian H :

R N × R N \ {0} × Sym N → R as (59) H(x, p, X) := i ∂ xi ∂ pi ϕ(x, -p) -∇ 2 p ϕ(x, -p) : X.
We therefore focus on solving the parabolic Cauchy problem (60)

∂ t u + ψ(x, -∇u) H(x, ∇u, ∇ 2 u) -f (x, t) = 0 u(•, t) = u 0 .
The appropriate setting for this type of geometric evolution equations is the one of viscosity solutions, in the framework of [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF][START_REF] Ishii | Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor[END_REF] (see also [START_REF] Chambolle | Nonlocal curvature flows[END_REF]). We will focus on the evolution of sets with compact boundary on compact time intervals of the form [0, T ]. We now define the notion of admissible test function. In the following, with a small abuse of language, we will say that a function u : R n ×[0, T ] → R is constant outside a compact set if there exists a compact set K ⊂ R N such that u(•, t) is constant in R N \ K for every t ∈ [0, T ] (with the constant possibly depending on t).

Definition 4.5. Let ẑ = (x, t) ∈ R N × (0, T ) and let A ⊆ (0, T ) be any open interval containing t. We will say that η ∈ C 0 (R N × Ā) is admissible at the point ẑ if it is of class C 2 in a neighborhood of ẑ, if it is constant out of a compact set, and, in case ∇η(ẑ) = 0, the following holds: for all (x, t) ∈ R N × A, and there exist numbers a, b > 0 such that

|η(x, t) -η(ẑ) -η t (ẑ)(t -t)| ≤ a|x -x| 3 + b|t -t| 2 .
We then recall one of the equivalent definitions of viscosity solutions.

Definition 4.6. An upper semicontinuous function u :

R N × [0, T ] → R (in short, u ∈ usc(R N × [0, T ]))
, constant outside a compact set, is a viscosity subsolution of the Cauchy problem (60) if u(•, 0) ≤ u 0 and for all z := (x, t) ∈ R N × (0, T ) and all C ∞ -test functions η such that η is admissible at z and u -η has a maximum at z (in the domain of definition of η) the following holds:

i) If ∇η(z) = 0, then it holds (61) η t (z) ≤ 0 ii) If ∇η(z) ̸ = 0, then (62)
∂ t η(z) + ψ(z, -∇η(z)) H(z, ∇η(z), ∇ 2 η(z)) -f (z, t) ≤ 0. A lower semicontinuous function u : R N × [0, T ] → R (in short, u ∈ lsc(R N × [0, T ]))
, constant outside a compact set, is a viscosity supersolution of the Cauchy problem (60) if u(•, 0) ≥ u 0 and for all z := (x, t) ∈ R N × [0, T ] and all C ∞ -test functions η such that η is admissible at z and u -η has a minimum at z (in the domain of definition of η) the following holds:

i) If ∇η(z) = 0, then η t (z) ≥ 0; ii) If ∇η ̸ = 0 then ∂ t η(z) + ψ(z, -∇η(z)) H(z, ∇η(z), ∇ 2 η(z)) -f (z, t) ≤ 0.
Finally, a function u is a viscosity solution for the Cauchy problem (60) if it is both a subsolution and a supersolution of (60).

Remark. By classical arguments, one could assume that the maximum of u -η is strict in the definition of subsolution above (an analogous remark holds for supersolutions).

Remark. We remark that, if -u is a subsolution to (60) with initial datum -u 0 , then u is a supersolution for (60) for the initial datum u 0 and where ϕ, ψ are replaced by φ, ψ respectively, as defined in Section 4.1.

We will first prove existence for viscosity solutions of (60) via an approximation-in-time technique, and then prove uniqueness of solutions to (60) to link the approximate solution to the mean curvature flow equation. We would like to proceed with the classical construction of e.g. [START_REF] Chambolle | An algorithm for mean curvature motion[END_REF][START_REF] Chambolle | Nonlocal curvature flows[END_REF][START_REF] Eto | An area-minimizing scheme for anisotropic mean-curvature flow[END_REF], but in our case the lack of continuity of the evolving functions forces us to be particularly careful with the procedure.

We use the shorthand notation of lsc for lower semicontinuous and usc for upper semicontinuous. Given a bounded, usc function v which is constant outside a compact set, we define the transformation

(63) T + h,t v(x) = sup s : x ∈ T + h,t {v ≥ s} .
Firstly, we see that T + h,t v(x) ∈ R, as v is bounded. Moreover, it turns out that the function T + h,t v is usc, bounded and constant outside a compact set. Indeed, definition (63) is equivalent to

T + h,t v(x) = inf s : x / ∈ T + h,t {v ≥ s} = inf s∈R s + 1 (T + h,t {v≥s}) c (x) ,
where 1 A (x) is the indicatrix function of a set A, being 0 on the set and +∞ outside. By definition, 1 A is an usc function for any open set A. Thus, recalling Remark 3.4, in the equation above we are taking the infimum of a family of usc functions, which is then a usc function. The other two properties follows from the previous study of the discrete evolution. Analogously, given a bounded lsc function g, we define

(64) T - h,t g(x) = sup s : x ∈ T - h,t {g > s} = sup s∈R s -1 T - h,t {g>s}
, which is now a bounded lsc function (as sup of lsc functions), constant outside a compact set.

We are now ready to give the definition of the discrete-in-time approximations of sub -and super solution to (60). Given an initial compact set E 0 , set u 0 as a (uniformly) continuous function, spatially constant outside a compact set, such that {u 0 ≥ 0} = E 0 . We remark that for every s ∈ R, the superlevel set {u 0 ≥ s} is either compact or it is unbounded with compact boundary. Then, for h > 0 we introduce the following family of maps as u ± h (•, t) = u 0 for t ∈ [0, h) and (65)

u ± h (•, t) := T ± h,t-h u ± h (•, t -h)
{u + h (•, t) > λ} ⊆ T + h,t-h {u + h (•, t -h) ≥ λ} ⊆ {u + h (•, t) ≥ λ} (67) {u - h (•, t) > λ} ⊆ T - h,t-h {u - h (•, t -h) > λ} ⊆ {u - h (•, t) ≥ λ}. Proof. Fix x ∈ R N , t ∈ [0, h).
For any given σ < u - h (x, h) we have that there exists a sequence

(s n ) ↗ σ so that x ∈ T - h,t-h {u 0 > s n } ⊆ T + h,t-h {u 0 ≥ s n }. Thus, u + h (x, t) ≥ σ.
We then conclude by induction. Then, (67) follows easily by the definition (65). □

We then prove that the half-relaxed limits (in the spirit of [START_REF] Barles | Front propagation and phase field theory[END_REF], see also the references therein) of the families of functions u ± h u + (x, t) := sup

(x h ,t h )→(x,t) lim sup h→0 u + h (x h , t h ) u -(x, t) := inf (x h ,t h )→(x,t) lim inf h→0 u - h (x h , t h ), (68)
are (respectively) sub -and supersolutions in the viscosity sense of (60), see Theorem 1.3 (note that, by definition, u + is usc, while u -is lsc). The proof of this result is the subject of the following section and we recall that the hypothesis required are (H0), (H1) and

f ∈ C 0 (R N × [0, ∞)) only.
Once the existence of sub -and super-solutions to the equation is settled, we need to properly define the notion of level-set solution to the mean curvature flow. To do so, we first prove uniqueness for (60) via a comparison principle and under additional hypothesis. Then, we show that the evolution of the zero superlevel set of the solution does not depend on the choice of the initial function u 0 . We start with a comparison result between u + , u -and u 0 at the initial time: it will ensure that the classical hypothesis for the comparison principle are satisfied. We first prove an estimate for the speed of decay of the level sets of the evolving functions. While it will only be needed in the following section, in the proof of the forthcoming Lemma 4.9 we will use similar techniques, so we preferred to state it here. Lemma 4.8. Let u + (x, t) be the function defined in (68), let σ ∈ R. Assume that, for a suitable x 0 and R > 0, it holds B(x 0 , R) ⊆ {u + (•, t 0 ) ≥ σ} . Then, there exists C = C(R, ϕ, ψ, f ) such that B(x 0 , R -C(t -t 0 )) ⊆ {u + (•, t) ≥ σ} for every t ≤ t 0 + R/(2C). An analogous statement holds for u -by considering its open sublevel sets.

Proof. We focus on the case {u + (•, t 0 ) ≥ σ} bounded, the other case being analogous. By assumption, for any R 0 < R, if h is small enough, we have B(x 0 , R 0 ) ⊆ {u + h (•, t 0 ) ≥ σ}. Set C = C(R 0 /2, ϕ, ψ, f ) as the constant of Lemma 3.8. Let R n be defined recursively following law [START_REF] Giga | Motion by crystalline-like mean curvature: a survey[END_REF], that is R n+1 = R n -Ch, as long as R n ≥ R 0 /2. By simple iteration we find that R n = R 0 -nCh, as long as R n ≥ R 0 /2, which can be ensured enforcing h n ≤ R 0 /(2C). Therefore, for any t ≥ t 0 such that t -t 0 ≤ R 0 /(2C), we set n = [(t -t 0 )/h] and send h → 0 to deduce (recalling also Lemma 3.2)

{u + (•, t) ≥ σ} ⊃ B(x 0 , R 0 -C(t -t 0 )).
Since the choice of R 0 is arbitrary, we conclude. □

We are now ready to prove a comparison result for the functions u ± and a continuity estimate at the initial time t = 0. Lemma 4.9. For any (x, t) ∈ R N × [0, +∞) it holds

u -(x, t) ≤ u + (x, t).
Moreover u -(•, 0) = u + (•, 0) = u 0 , so that there exists a modulus of continuity ω such that ∀x, y ∈ R N u + (x, 0) -u -(y, 0) ≤ ω(|x -y|).

Proof. The proof of the first inequality essentially follows from (66) and the definition of u ± . To prove the equality at the initial time t = 0, we start by remarking that u + (•, 0) ≥ u 0 as can be seen taking sequences of the form (x h , 0) in (68). Then, consider ω as a continuous, strictly increasing modulus of continuity for u 0 . We can also see that ∀ε > 0 {u 0 ≤ u 0 (x) + ε} ⊇ B(x, ω -1 (ε)) by uniform continuity. Thus, reasoning iteratively as in Lemma 4.8 and using (67), we obtain that there exists h 0 (ε) such that ∀h ≤ h 0 it holds

{u + h (•, t) ≤ u + h (x, 0)+ε} ⊇ T + h,t-h {u 0 > u 0 (x) + ε} c = T - h,t-h {u 0 ≤ u 0 (x)+ε} ⊇ B(x, ω -1 (ε/2)),
as long as t ≤ (ω -1 (ε) -ω -1 (ε/2))/(2C) =: t ε , and where we recalled that u ± h (•, 0) = u 0 . Now, fix σ > 0, x ∈ R N such that u(x, 0) > σ and a sequence (

x h k , t h k ) → (x, 0) such that lim k + h k (x h k , t h k ) > σ. Then, for k large enough (x h k , t h k ) ∈ B(x, ω -1 (ε/2)) × [0, t ε ) and so we conclude σ < lim k u + h (x h k , t h k ) ≤ u 0 (x, 0) + ε.
Letting ε → 0 we conclude u(•, 0) + ≤ u 0 . The proof for u -is essentially the same. The last claim follows from the previous one, recalling that ω is a modulus of uniform continuity for u 0 . □

In order to prove a comparison principle for (60), we will need to assume (H3). Under these additional hypotheses, we are able to prove uniqueness for the parabolic Cauchy problem (60). The proof of this result follows from [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF]Theorem 4.2]: we will just show in detail that the assumption of the aforementioned theorem hold in our case, following [8, Proposition 6.1] and [26, pag. 463].

Proof of Theorem 1.4. The proof of this result essentially follows from [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF]Theorem 4.2], combined with the existence result of Theorem 1.3. Referring to the notation of [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF], we firstly remark that in our case Ω = R N , thus the parabolic boundary of U = Ω × [0, T ] is simply ∂ p U = R N × {0}. Therefore, the initial conditions (A1) -(A3) are all verified by Lemma 4.9. We then define the continuous Hamiltonian F :

[0, T ] × R N × (R N \ {0}) × M N ×N → R as follows (69) F (t, x, p, X) := ψ(x, -p) - i ∂ xi ∂ pi ϕ(x, -p) + ∇ 2 p ϕ(x, -p) : X + f (x, t) ,
and focus on the conditions (F 1), (F 3) -(F 5), (F 6 ′ ), (F 7), (F 9), (F 10) that F must satisfy. The assumptions (F 1), (F 3)-(F 5), (F 9) are easily checked. (F 6 ′ ) follows from the Lipschitz regularity of ϕ and ψ, as ∀t ∈

[0, T ], x ∈ R N , |p| ≥ ρ, |q| + |X| ≤ R one has |F (t, x, p, X) -F (t, x, q, X)| ≤ c ψ |p -q| - i ∂ xi ∂ pi ϕ(x, -p) + ∇ 2 p ϕ(x, -p) : X + ψ(x, -q) - i (∂ xi ∂ pi ϕ(x, -p) -∂ xi ∂ pi ϕ(x, -q)) + ∇ 2 p ϕ(x, -p) -∇ 2 p ϕ(x, -q) : X ≤ c R |p -q| 1 + 1 |p| + c R |p -q| ≤ c R,ρ |p -q|.
For (F 7), we remark that the first term in the parenthesis in (69) is 0-homogeneous in p, while the second one is (-1)-homogeneous in p but 1-homogeneous in X. Lastly, we sketch how to prove (F 10). Since it concerns the X-terms, we focus simply on

∇ 2 p ϕ(x, -p) : X = tr ∇ 2 p ϕ(x-, p) X T .
Multiplying by ϕ(x, -p), we rewrite ϕ(x, -p)tr ∇ 2 p ϕ(x-, p) X T = tr(A(x, -p)X T ), where A = B -(∇ p ϕ ⊗ ∇ p ϕ) , with B being the uniformly elliptic operator 1 2 ∇ 2 p ϕ 2 . We can then factorize B = L LT , with L being a nondegenerate, lower triangular matrix. Then, following the proof of [8, Proposition 6.1] and [26, pg. 463], we obtain (F 10). □

Once uniqueness is settled, one can finally define the notion of level set solution to the mean curvature flow as follows.

Definition 4.10. Let E 0 be a compact initial set. Define a uniformly continuous, bounded function u 0 : R N → R such that {u 0 ≥ 0} = E 0 . Then, let u : R N × [0, +∞) → R be the unique continuous viscosity solution to (60) given by Theorem 1.4. Then, the family E t := {u + (•, t) ≥ 0} t≥0 will be called the level set solution to the mean curvature flow. This definition is well posed since the Hamiltonian defined in (59) satisfies the so-called geometricity condition. Namely, one can easily check that for any λ ̸ = 0, p ∈ R N \ 0, q ∈ R N and any symmetric N × N matrix X one has

H(x, λp, λX + p ⊗ q + q ⊗ p) = λ |λ| H(x, p, X).
Thus, one can prove by classical arguments (see e.g. [14, Remark 3.9]) the following result.

Lemma 4.11. Let u 0 , ũ0 two initial data for (60) such that {u 0 ≥ 0} = {ũ 0 ≥ 0}. Then, denoting by u, ũ the corresponding solutions to (60), one has

{u(•, t) ≥ 0} = {ũ(•, t) ≥ 0} for all t ∈ [0, T ],
and the same identity holds for the open superlevel sets.

4.3.

Proof of Theorem 1.3. In this section we will prove that the limiting functions u ± are respectively a viscosity sub -and supersolutions to (60). We remark that we work assuming (H0), (H1) and that f ∈ C 0 (R N × [0, +∞)). We will be following the structure of the proof of [START_REF] Chambolle | Nonlocal curvature flows[END_REF]Theorem 6.16], but taking into account the weaker definition of u + holding in our case. We will be using the O, o notations with respect to h → 0 and focus on proving that u + is a subsolution. The proof for u -is analogous.

Proof of Theorem 1.3. Consider u + as defined in (8): we need to prove that it is a subsolution.

In the following, we will denote u := u + and u h := u + h . Let η(x, t) be an admissible test function in z := (x, t)∈ R N × (0, T ) and assume that (x, t) is a strict maximum point for u -η. Assume furthermore that u -η = 0 in such point. We need to show that either (61) or (62) holds at z. Case 1. Let us first assume that ∇η(z) ̸ = 0. By classical arguments, we can assume that z is a strict maximum point and that η is smooth. By the definition of u, there exists a sequence zk := (x h k , th k ) → z such that lim k u h k (z k ) = u(z). We remark that we can substitute the functions u h k for t > 0 with their usc envelope in time, without changing the value of u. Indeed, the usc envelope of u h k is the function at all discrete times lh k is given by max{u

h k (•, (l -1)h k ), u h k (•, lh k )}
and coincides with u h k elsewhere. Since now u h k is usc in time and space, by standard arguments (compare e.g. [5, Lemma 6.1]), there exists a radius ρ > 0 such that all functions u h k -η achieve a local maximum in B ρ (z) at points z k = (x k , t k ). Then, passing to a further subsequence we can ensure that z k → w ∈ B ρ (z), and we use the definition of u to obtain

(u -η)(w) ≥ lim sup k (u h k -η)(z k ) ≥ lim sup k (u h k -η)(z k ) = (u -η)(z).
Therefore, w = z by maximality. Thus we can assume that each function u h k -η achieves a local maximum in B ρ (z) at a point z h k =: (x k , t k ) and that u h k (z h k ) → u(z) as k → ∞. Finally, we can assume also that ∇η(x k , t k ) ̸ = 0 for k large enough.

Step 1. We start defining an appropriate set which is then used as a competitor for the minimality of the level sets of the functions u h . From the previous computations, one has in particular that (70)

u h (x, t) ≤ η(x, t) + c k where c k := u h k (x k , t k ) -η(x k , t k ), with equality if (x, t) = (x k , t k ). Let σ > 0 and set η σ h k (x) := η(x, t k ) + c k + σ 2 |x -x k | 2 .
Then, for all

x ∈ R N , u h k (x, t k ) ≤ η σ h k (x) with equality if and only if x = x k . We set l k = u h k (x k , t k ) = η σ h k (x k ).
We fix ε > 0, to be chosen later, and write

E ε,k := {u h k (•, t k -h k ) ≥ l k -ε}. We define 1 (71) W ε := T + h,t k -h k E ε,k \ η σ h k (•) > l k + ε . We immediately see that W ε → {x k } in the Kuratowski sense as ε → 0 since by (67) (72) {u h k (•, t k ) > l k -ε} \ η σ h k (•) > l k + ε ⊆ W ε ⊆ {u h k (•, t k ) ≥ l k -ε} \ η σ h k (•) > l k + ε , see
also (78) below. Then, we check that |W ε | > 0 for all ε small enough. By the continuity of η σ and |∇η(z)| ̸ = 0, for any ε there exist a radius r ε such that

W ε ⊇ B(x k , r ε ) ∩ T + h,t k -h k E ε,k . Furthermore, for any ε > 0, using (67) again yields x k ∈ T + h k ,t k -h k {u h k (•, t k -h k ) ≥ l k -ε},
and the latter set coincides with the closure of its points of density 1 by Lemma 3.3. Thus, x k satisfies lower density estimates and so we conclude that |W ε | > 0. Now, assume E ε,k is bounded. By minimality we have

P ϕ (T + h,t k -h k E ε,k ) + 1 h k ˆT + h,t k -h k E ε,k sd ψ E ε,k (x) dx + ˆWε F h k (x, t k -h k ) dx ≤ P ϕ T + h,t k -h k E ε,k ∩ {η σ h k > l k + ε} + 1 h k ˆ T + h,t k -h k E ε,k ∩{η σ h k >l k } sd ψ E ε,k . (73)
Adding to both sides the term P ϕ {η σ h k > l k + ε} ∪ T + h,t k -h k E ε,k and using the submodularity (12), we obtain

P ϕ ({η σ h k > l k + ε} ∪ W ε ) -P ϕ ({η σ h k > l k + ε}) + 1 h k ˆWε sd ψ E ε,k (x) dx + ˆWε F h k (x, t k -h k ) dx ≤ 0. By (70), {u h k (•, t k -h k ) ≥ l k -ε} ⊆ {η(•, t k -h k ) ≥ l k -c k -ε}, therefore it holds P ϕ ({η σ h k > l k + ε} ∪ W ε ) -P ϕ ({η σ h k > l k + ε}) + 1 h k ˆWε sd ψ {η(•,t k -h k )≥l k -c k -ε} (x) dx + ˆWε F h k (x, t k -h k ) dx ≤ 0. ( 74
)
If instead E ε,k is an unbounded set with compact boundary, we replace inequality (73) by

P ϕ (T h,t k -h k E ε,k ) + 1 h k ˆ T + h,t k -h k E ε,k ∩B R sd ψ E ε,k (x) dx + ˆWε F h k (x, t k -h k ) dx ≤ P ϕ ( T + h,t k -h k E ε,k ∩ {η σ h k > l k + ε}) + 1 h k ˆ T + h,t k -h k E ε,k ∩{η σ h k >l k +ε}∩B R sd ψ E ε,k , 1
We need to define the sets Wε in this way (compare the different definition in [START_REF] Chambolle | Nonlocal curvature flows[END_REF]) since firstly, we can not rule out that the inclusions in (72) are strict, and secondly it is not clear if otherwise |Wε| > 0.

for R > 0 sufficiently large, see (58). Then, one can argue as before to obtain (74).

Step 2. We estimate the first two terms in (74). The quantity P ϕ ({η σ h k > l k + ε} ∪ W ε ) -P ϕ ({η σ h k > l k + ε}) can be estimated as done in Lemma 3.8. Indeed, we consider the vector field v = ∇ p ϕ(x, ∇η σ h k ) in ( 13) and we use the divergence theorem to get

P ϕ ({η σ h k > l k + ε} ∪ W ε ) -P ϕ ({η σ h k ≥ l k + ε}) ≥ ˆ∂({η σ h k >l k +ε}∪Wε) v • ν - ˆ∂{η σ h k >l k +ε} v • ν = |W ε | Wε div v, (75)
where ν denotes the unit outer vector to the set we are integrating on. We then remark that ffl Wε div v → H .

For any z ∈ W ε , we have

(77) η(z, t k ) + c k + σ 2 |z -x k | 2 ≤ l k + ε.
Since, in turn, η(z, t k ) + c k > l k -ε it follows that σ|z -x k | 2 < 4ε and thus, for ε small enough, In order to prove this result, we start remarking that for k → ∞ and choosing ε ≪ h k , one has sd ψ {η(•,t k -h k )≥l k -c k -ε} (z) → 0 (as z → x k for ε → 0 and x k ∈ {η(•, t k ) ≥ l k -c k }). In particular, recalling the bounds [START_REF] Chambolle | An algorithm for mean curvature motion[END_REF] Then, we consider a geodesic curve for the definition of sd ψ {η(•,t k -h k )≥l k -c k -ε} (z): if this distance is positive, we choose γ : [0, 1] → R N with γ(0) = z, γ(1) = y, with y as before, otherwise we take γ such that γ(0) = y, γ(1) = z. In the following, we will assume sd ψ {η(•,t k -h k )≥l k -c k -ε} (z) > 0, the other case being analogous. Recalling [START_REF] Cagnetti | A second order minimality condition for the Mumford-Shah functional[END_REF]

(x k )} (x k ) -F h k (x k , t k -h k ) ≤ 0.
Letting simultaneously σ → 0 and k → ∞, recalling the continuity properties of H ϕ , we deduce (62). Indeed the sets {η σ h k > η σ h k (x k )} are converging in C 2 to the set {η > η(x)}, x k → x and thus H ϕ

{η σ h k >η σ h k (x k )} (x k ) → H ϕ {η>η(x)} (x)
, and we conclude the proof of this step. Case 2. Now we consider the case ∇η(x, t) = 0 and we show that ∂ t η(x, t) ≤ 0. The proof follows the line of the one in [START_REF] Chambolle | Nonlocal curvature flows[END_REF], we just highlight the differences.

Since ∇η(z) = 0, there exist a, b > 0 such that |η(x, t) -η(z) -∂ t η(z)(t -t)| ≤ a|x -x| 3 + b|t -t| 2 , thus, we can define η(x, t) = ∂ t η(z)(t -t) + 2a|x -x| 3 + 2b|t -t| 2 ηk (x, t) = η(x, t) + 1 k( t -t) .

We remark that u -η achieves a strict maximum in z and the local maxima of u -ηk in R N × [0, t] are in points (x k , t k ) → z as k → ∞, with t n ≤ t. From now on, the only difference from [START_REF] Chambolle | Nonlocal curvature flows[END_REF] is in the case x k = x for an (unrelabeled) subsequence. We assume x k = x ∀k > 0 and define b k = t -t k > 0 and the radii Passing to the limit k → ∞, we conclude that ∂ t η(z) ≤ 0. □

We conclude with two remarks concerning some possible generalizations of the results presented.

Remark 4.12. The results presented in this work can be immediately extended to unbounded initial open sets E 0 , whose boundary is compact. Indeed, defining the discrete flow as

E (h) t = E 0 if t ∈ [0, h), otherwise by induction E (h) t = T - h,t E (h)
t-h , where the operator T - h,n is the one defined in (56), this evolution is uniquely characterized by the one of the complement. Thus, all the results presented in this paper can be extended to this particular unbounded case. Remark 4.13. Following the lines of [START_REF] Bellettini | Anisotropic motion by mean curvature in the context of Finsler geometry[END_REF] (in the spirit of [START_REF] Almgren | Curvature-driven flows: a variational approach[END_REF]) one can see that the results of this paper may be extended to prove existence of flat flows and level set solutions to the mean curvature flow on R N endowed with the geometric structure induced by a Finsler metric ϕ • . For example, the perimeter functional in this setting is defined as follows. Given a set E of finite perimeter, its (intrinsic) perimeter is

P ϕ • (E) = ˆ∂ * E ϕ(x, ν E (x)) dH N -1 ϕ • (x),
where the Hausdorff measure

H N -1 ϕ •
is the one induced by the metric ϕ • . In particular, one can compute dH N -1 ϕ • (x) = ω N |B ϕ • (x)| -1 dH N -1 (x) (see [START_REF] Bellettini | Anisotropic motion by mean curvature in the context of Finsler geometry[END_REF]), thus this approach is equivalent to consider in our framework a slightly different (but still regular) anisotropy, namely ϕ * (x, ν) := ω N |B ϕ • (x)| -1 ϕ(x, ν). In particular, this approach leads to considering the evolution of hypersurfaces E t moving according to the evolution law V ϕ • (x, t) = -H Et (x) + f (x, t) x ∈ ∂E t , t ∈ (0, T ) where now V ϕ • represents the speed of evolution along the anisotropic normal outer vector n ϕ • (x) = ∇ p ϕ(x, ν E (x)) and H is the "intrinsic" mean curvature, thus the first variation of the perimeter P ϕ • . Recalling that n ϕ • (x) • ν E (x) = ϕ(x, ν E (x)), we see that the hypersurfaces are evolving with a normal (in the Euclidean sense) velocity given by the law V (x, t) = ϕ(x, ν Et (x)) -H ϕ * Et (x) + f (x, t) .

After this transformation, we can apply the results previously proved.

1 2

 1 -β /c ψ . Testing the minimality of E (h)

  to be used in the definition of the geodesic distance sd ψ E (h) t-h

Lemma 4 . 1 .

 41 Let F 1 ⊆ F 2 be open, unbounded sets with compact boundary and fix

  (78) W ε ⊆ B c √ ε (x k ). By a Taylor expansion, for every z ∈ W ε we have(79) η(z, t k -h k ) = η(z, t k ) -h k ∂ t η(z, t k ) + h 2 k ˆ1 0 (1 -s)∂ 2 tt η(z, t k -sh k ) ds.Then, we consider y, y e ∈ {η(•,t k -h k )(y) = l k -c k -ε}being respectively, a point of minimal ψ-distance and Euclidean distance from z. Claim: We claim that it holds (80) |z -y| = O(h k ).

  one has |z -y e | ≤ c 2 ψ |z -y| ≤ c 3 ψ |sd ψ {η(•,t k -h k )≥l k -c k -ε} (z)| → 0 as k → ∞. By (77) we deduce in particular η(z, t k ) + c k < l k + ε, that is, (81) 0 ≤ η(z, t k ) -η(y, t k -h k ) ≤ 2ε,and the same inequality substituting y e to y. Thus, one hasη(z, t k ) -η(y e , t k -h k ) = ∇η(y, t k -h k ) • (z -y e ) -h k ∂ t η(y, t k -h k ) + O(|z -y e | 2 + h 2 k ) which we combine with ∇η(y, t k -h k ) • (z -y e ) = ±|∇η(y, t k -h k )| |z -y e | (see[START_REF] Chambolle | Nonlocal curvature flows[END_REF] for details) and (81) to get |z -y e | |∇η(y, t k -h k )| ≤ 2ε + O(h k ) + O(|z -y e | 2 ).Recalling that |∇η(y,t k -h k )| ≥ c > 0 for h k small enough, we divide by |∇η(y, t k -h k )| to conclude |z -y e | = O(h k ) as ε ≪ h k .Finally, employing again[START_REF] Chambolle | An algorithm for mean curvature motion[END_REF], we prove the claimed (80).

r k := 2

 2 Cb k ,where C is the constant of Lemma 4.8. Taking k large enough, by Lemma 4.8 the balls B(•, r k ) have an extinction time greater than 2( t -t k ). We then haveB(x, r k ) ⊆ {η k (•, t k ) ≤ ηk (x, t k ) + 2ar 3 k } ⊆ {u(•, t k ) ≤ u(x, t k ) + 2ar 3 k },by maximality of u -ηk at z k . Since the balls B(•, r k ) are not vanishing, we concludex ∈ {u(•, t) ≤ u(x, t k ) + 2ar 3 k }. Finally, we use again the maximality of u -η at z and the choice of r k to obtainη(x, t k ) -η(z) t k -t = η(x, t k ) -η(z) -b k ≤ u(x, t k ) -u(x, t) -b k ≤ -2ar 3 k -b k = c b k .

  for t ≥ h. We easily see that the maps above are functions (as implied by the comparison principle contained in Lemmas 3.2, 4.1 and 4.4) piecewise constant in time (asT ± h,t = T ± h,[t/h]h). Moreover, by the previous remarks, we have that u + h (•, t) is an usc function, while u - h (•, t) is a lsc function, for every t ∈ [0, +∞). Some further properties of the approximating scheme are listed below. Lemma 4.7. For any h > 0, t ≥ 0 we have the following. It holds

	(66)	u -h (•, t) ≤ u + h (•, t).
	Furthermore, given any λ ∈ R and t ≥ h it holds

  >l k } (x k ) and ffl Wε F h k (x, t k -h k ) dx → F h k (x k , t k -h k ) as ε → 0 by continuity.Step 3. We bound the distance term in (74) by showing that(76) 1 h k sd ψ {η(•,t k -h k )=l k -c k -ε} (z) ≥ ∂ t η(z, t k ) -O(h k ) ψ(y, -∇η(y, t k -h k )) + O(h k )

	ϕ
	{η σ h k

  , we haveη(z, t k -h k ) = η(y, t k -h k ) + ˆ1 0 ∇η(γ, t k -h k ) • γ dt ≥ η(y, t k -h k ) -ˆ1 0 ψ(γ, -∇η(γ, t k -h k ))ψ • (γ, γ) dt ≥ η(y, t k -h k ) -ψ(y, -∇η(y, t k -h k )) sd ψ {η(•,t k -h k )=l k -c k -ε} (z) -∇η(γ, t k -h k )) -ψ(y, -∇η(y, t k -h k ))) ψ • (γ, γ) dt ≥ η(y, t k -h k ) -ψ(y, -∇η(y, t k -h k )) + c|z -y| sd ψ {η(•,t k -h k )=l k -c k -ε} (z), where in the last line we reasoned as in (47) to obtain the bound sup t |γ(t)-y| ≤ c|z -y|. Recalling (80) one has(82) η(z, t k -h k ) ≥ η(y, t k -h k ) -ψ(y, -∇η(y, t k -h k )) sd ψ {η(•,t k -h k )=l k -c k -ε} (z) + o(h k ). Combining (79) with (82) and using (81), we deducesd ψ {η(•,t k -h k )=l k -c k -ε} (z) ψ(y, -∇η(y, t k -h k )) + o(h k ) ≥ -2ε + h k ∂ t η(z, t k ) -h 2 -s)∂ 2 tt η(z, t k -sh k ) ds.Note that, in view of (77) and[START_REF] Chambolle | An algorithm for mean curvature motion[END_REF],|η(z, t k ) -η(y, t k )| ≤ cε + ch k = O(h k ), provided ε ≪ h k andsmall enough. We then conlude (76) by combining the previous inequality with (78),(80) as1 h k sd ψ {η(•,t k -h k )=l k -c k -ε} (z) ≥ ∂ t η(z, t k ) -2ε h k -O(h k ) -O h k (1) ψ(y, -∇η(y, t k -h k )) = ∂ t η(x k , t k ) + O( √ ε) -2ε h k -O(h k ) -O h k (1) ψ(x k , -∇η(x k , t k -h k )) + O( √ ε) + O(h k ). We conclude the proof by employing (74), (75) and (76), dividing by |W ε | and sending ε → 0 to obtain∂ t η(x k , t k ) -O h k (1) ψ(x k , -∇η(x k , t k )) + O(h k ) + H ϕ

	-(1 Step 4. {η σ ˆ1 0 (ψ(γ, k ˆ1 0 h k ≥η σ h k

Acknowledgements. The authors want to thank the referees for the careful reading of the manuscript and their comments, which helped improve the paper. D. De Gennaro has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 94532 .