Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

evolution is an analogue of the classical (isotropic) mean curvature flow, which corresponds to the case ϕ(x) = ψ(x) = |x| and it is studied as model of crystal growth, see [5,6,13,14]. Existence and uniqueness of the level set flow associated to (1.1) have been obtained for general mobilities ψ and purely crystalline norms ϕ in [13,14], in the viscosity setting, whereas the case of general norms ϕ with convex mobilities ψ has been treated in [5,6], in the distributional setting.
In this paper we consider the evolution of subgraphs of entire Lipschitz functions, in the case in which either ϕ is regular (see (2.2)) or ψ is a norm. In particular, we will assume that there exists a Lipschitz continuous function u 0 : R N → R such that the initial set E 0 coincides with {(x, z) | z u 0 (x)}. By monotonicity of the flow and invariance with respect to translations, the evolution E t is defined for all times and is the subgraph of a Lipschitz function, that is, E t = {(x, z) | z u(x, t)}. In Section 2 we describe the main properties of this flow.
Since the evolution is defined for all times, we are interested in the analysis of the long time patterns of the evolution. We recall that evolution of entire Lipschitz graphs in the isotropic setting has been considered in [9], see also [8], whereas the case of fractional mean curvature flow has been considered recently by two of the authors in [4].
As in the isotropic case, the long time attractors of the flow starting from entire Lipschitz graphs are self-similar expanding solutions, defined as follows: Definition 1.1. An expanding homothetic solution is a solution to (1.1) such that E t = λ(t)E 1 where λ(1) = 1 and λ ′ (t) 0 for t > 1. This is equivalent to assume that E 1 is a solution to (1.2) c(p · ν(p)) = −ψ(ν(p))H ϕ (p, E 1 ) for some c 0.
In Section 3 we characterize the graphical expanding solutions as evolutions issuing from cones, that is, from Lipschitz graphs of positive 1-homogeneous functions, see Theorem 3.1 and Proposition 3.3. Moreover, in Theorem 3.4 we show that Lipschitz graphical evolutions to (1.1) asymptotically approach self-similar expanding solutions, in an appropriate rescaled setting, provided the initial graph is a sublinear perturbations of a cone. More precisely, we introduce the following time rescaling: (1.3) τ (t) := log(2t + 1) 2 so that the evolution (1.1) of the rescaled flow is governed by the geometric law (1.4) ∂ τp ·ν(p) = −p ·ν(p) − ψ(ν(p))H ϕ (p,Ẽ τ ) and we show thatẼ τ converge locally in Hausdorff sense as τ → +∞ to a solution to (1.2), with c = 1.
In the rest of the paper we analyze the long time behavior of the flow (1.1) without rescaling in the case of regular anisotropies, see assumption (2.2) below. In order to rule out possible oscillations in time, it is necessary to assume some decay condition of the initial data at infinity, and in particular we will consider initial graphs which are asymptotically flat or asymptotically approaching mean convex cones.
In Section 4 we consider self-similar expanding solutions starting from Lipschitz mean convex cones. In Theorem 4.3, by constructing appropriate barriers, we show that such solutions are stable with respect to perturbations vanishing at infinity.
In Section 5 we prove that if the initial surface is asymptotically approaching a hyperplane, then the evolution asymptotically flattens out in Hausdorff sense. This result is obtained by comparison with large Wulff shapes, and by constructing appropriate 1-dimensional periodic barriers to the evolution. This approach also provides a different proof for the same result in the isotropic setting, which was obtained by integral estimates on the flow, see [8,10].

Preliminary definitions and results
We recall some definitions for anisotropies and related geometric flows (see for instance [3]).
Definition 2.1. Let ϕ : R N +1 → [0, +∞) be a positively 1-homogeneous convex map, such that ϕ(p) > 0 for all p = 0. We associate to the surface tension the anisotropy ϕ 0 : R N +1 → [0+∞) defined as ϕ 0 (q) := sup ϕ(p) 1 p·q, which is again convex and positively 1-homogeneous. The anisotropic mean curvature of a set E at a point p ∈ ∂E is defined as when ϕ is regular, where ν(p) is the exterior normal vector to ∂E at p, and div τ is the tangential divergence, whereas in the general case it is defined using the subdifferential, H ϕ (p, E) ∈ div τ (∂ϕ(ν(p))).
Definition 2.2 (W ϕ 0 -condition). We define the Wulff shape as the convex compact set We say that C ⊆ R N +1 satisfies the interior (resp. exterior) RW ϕ 0 -condition at x ∈ ∂C if there exists y x ∈ R N +1 such that RW ϕ 0 + y x ⊆ C and x ∈ ∂(RW ϕ 0 + y x ) (resp. there exists y x such that RW ϕ 0 + y x ⊆ R N +1 \ C and x ∈ ∂(RW ϕ 0 + y x )).
Remark 2.5. When ϕ is sufficiently regular, that is, (2.2) holds, the solution to (2.4) is intended in the sense of viscosity solutions, see [2,12], whereas in the general case in which ϕ is not smooth and ψ is a norm, the solution is intended as the level set distributional solution defined in [5]. We recall also that the level set distributional solution is the locally uniform limit of viscosity solutions to (2.4), when we approximate the anisotropy and the mobility with regular ones, see [6].
We recall the following result about well posedness of the flow (1.1).
provide a solution (in the appropriate sense) to (1.1). Moreover, if U 0 , V 0 are two uniformly continuous functions such that U 0 V 0 , then U (p, t) V (p, t) for all t > 0 and p ∈ R N +1 .
Finally, if U 0 is Lipschitz continuous with Lipschitz constant C, then where the constant C ′ depends on C. In particular, if there exists a direction ω ∈ R N +1 such that U 0 (p + λω) > U 0 (p) for every λ > 0 and every p ∈ R N +1 , then U (p + λω, t) > U (p, t) for every t > 0, λ > 0, p ∈ R N +1 .
Proof. For the existence and uniqueness of solutions to (2.4), and the comparison principle, in the case that (2.2) holds we refer to [2], whereas for the general case in which (2.3) holds we refer to [5]. The last two properties are a consequence of the comparison principle and the fact that the differential operator is invariant by translations. Indeed, if U 0 is Lipschitz continuous, and C is the Lipschitz constant of U 0 , then for every fixed h ∈ R N +1 , then U h (p, t) := U (p + h, t) ± C|h| is a solution to (2.4) with initial datum U 0 (p + h) ± C|h|. Since U 0 (p + h) − C|h| U 0 (p) U 0 (p+h)+C|h|, by comparison there holds that U (p+h, t)−C|h| U (p, t) U (p+h, t)+C|h|, which implies that A similar argument shows that, if U 0 (p + λω) > U 0 (p) for every λ > 0 then U (p + λω, t) > U (p, t).
Finally, observe that if the initial datum is a cone, that is, V 0 (p) = C|p − p 0 |, for some p 0 ∈ R N +1 , then by uniqueness, using the positive 1-homogeneity of the Euclidean norm and the scaling properties of the operator, we get that the solution to (2.7) satisfies V (p − p 0 , t) = 1 r V (r(p − p 0 ), r 2 t) for every r > 0 and t 0, p ∈ R N +1 . This implies in particular that Therefore, to prove the Hölder continuity of U , we proceed as follows. We fix p 0 and observe that U 0 (p) C|p − p 0 | + U 0 (p 0 ). Hence, by comparison and using (2.5), we get By translation invariance of the operator, we conclude that for every s > 0, The other inequality is obtained analogously, taking as initial datum −C|p − p 0 |.
Remark 2.7. It is easy to check that the rescaled Wulff shape RW ϕ 0 , for R > 0, satisfies is the level set solution to (1.1), with initial datum W ϕ 0 as defined in Theorem 2.6.
In this paper we consider the case in which the initial datum E 0 is the subgraph of an entire Lipschitz function. Up to a rotation of coordinates, we may assume that A direct application of Theorem 2.6 gives the following result on the evolution of Lipschitz graphs.
Corollary 2.8. Assume that E 0 satisfies (2.6). Then the level set flow satisfies for some K > 0 depending only on the Lipschitz constant ∇u 0 ∞ of u 0 . When ϕ is regular, that is, (2.2) holds, then u is the viscosity solution to When ϕ is not regular and (2.3) holds, then the solution is intended in the distributional sense as in [5], and coincides with the locally uniform limit of viscosity solutions to (2.7) when ϕ, ψ are approximated by regular functions, see [6].
If u 0 is only Lipschitz continuous we approximate u 0 with initial data in C 2,α (R N ), and then conclude by stability of solutions to (2.7) with respect to local uniform convergence.
C and let (2.2) hold. Then the viscosity solution u of (2.7) is uniformly of class C 1,α (R N ), for any fixed t > 0 and for all α ∈ (0, 1), with C 1,α norm bounded independently of t > 0.
Proof. First of all let u(·, t) be the viscosity solution to (2.7). By Corollary 2.8, it is Lips- are respectively a supersolution and a subsolution to (2.7), so that by comparison principle, see Theorem 2.6, we get Moreover for every t τ > 0, the functions u(x, t) ± sup x |u(x, τ ) − u 0 (x)| are respectively a supersolution and a subsolution to (2.7) with initial datum u(x, τ ), whence for all x ∈ R N and t > 0.

Self-similar expanding solutions and convergence of the rescaled flow
We discuss the properties of solutions to (1.1) starting from Lipschitz cones, that is, subgraphs of Lipschitz continuous and positively 1-homogeneous functionsū: Then we consider the long time behavior of solutions starting from sublinear perturbations of Lipschitz cones.
Theorem 3.1. Letū : R N → R be as in (3.1) and letĒ 0 be the subgraph ofū. Then, for every t > 0, the evolutionĒ t of (1.1) with initial datumĒ 0 satisfies for p ∈ ∂Ē t that is, the flow starting fromĒ 0 is an expanding homothetic solution to (1.1). WritingĒ t as subgraph of a functionū(·, t) we have that for all T > 0 Proof. By (3.1) we get that for every r > 0, there holds rĒ 0 =Ē 0 , and H ϕ (rp, rĒ 0 ) = r −1 H ϕ (p,Ē 0 ) for all r > 0 and all p ∈ ∂Ē 0 . Therefore, by uniqueness of solutions and by the rescaling properties of the operator, we get thatĒ t = rĒ t/r 2 for all t, r > 0. The previous rescaling identity givesū Letting r := t − 1 2 for t > 0, we get This implies that, if p ∈ ∂Ē 1 then pt Substituting in (1.1) we get thatĒ 1 solves (1.2) with c −1 = 2. The same argument holds substituting t = 1 with another time t > 0. We observe that, by rescaling property (3.3), and by the fact thatū(·, t) is Lipschitz continuous with the same Lipschitz constant asū, for every T > 0 and t > 0 we have Sending t → +∞, we get the result. We now show that, ifĒ 0 satisfies the exterior and interior W ϕ 0 −condition, the expanding solution is asymptotic at infinity to the initial cone, by comparison with the shrinking Wulff shapes constructed in Remark 2.7.
Note that by positive 1-homogeneity of the functionū, for every R > 0, there exists K > 0 such thatĒ 0 satisfies the exterior and interior RW ϕ 0 −condition at every (x,ū(x)) ∈ ∂Ē 0 , with |x| K. Therefore we get where y x , y x are introduced in Definition 2.2. By comparison (see Remark 2.7 and Theorem 2.6) we have that where R(t) := R 2 − 2N ψt is defined in Remark 2.7. From the previous inclusions, we deduce that, for all |x| K and t < R 2 2N ψ , where C is a constant which depends on ϕ and on the Lipschitz constant ofū. This implies the conclusion, sending R → +∞.
A similar argument can be used to prove the same result whenū ∈ C 1 (R N \ {0}). For it we define v λ (x) :=ū(λe + x) −ū(λe) where λ > 0 and e ∈ R N with |e| = 1. For every compact set K, since ∇ū is continuous in R N \ {0}, we get that as λ → +∞ uniformly in x and e. Hence, letting f e (x) := ∇ū(e) · x we conclude that uniformly in e, with |e| = 1. This implies that there are functions α(r), ε(r) > 0, r > 0, with α(r) → ∞ and ε(r) → 0 as r → ∞ so that for large |x|, there exist y x , y x ∈ R N +1 such that . So, the thesis follows from the same argument as above.
Remark 3.2. If we assume that ∇ 2 ϕ, ψ belong to C 0,β (R N +1 \ {0}) for some β ∈ (0, 1), then there exists C > 0 such that Indeed due to (3.4) it is sufficient to check that there exists C > 0 such that sup x∈R N |div(∇ x ϕ(−∇ū(x, 1), 1))| C, and this is a consequence of Proposition 2.9. Moreover, if (3.5) is satisfied, then |ū t (x, t)| C ′ t − 1 2 , for some C ′ > 0 depending on C and on the Lipschitz norm of u 0 . Integrating in t we get hence, for every T > 0 we conclude that On the other hand, every homothetically expanding solution E t to (1.1) which is the subgraph of a Lipschitz continuous function has a backward in time extension which starts from a subgraph of a suitable Lipschitz and 1-homogeneous function. Proof. Since E 1 solves (1.2), we have that Therefore the solution u to (2.7) in R N × (1, +∞) with initial datum u(x, 1) = u 1 (x) is given by By definition λ(t) is well defined for every t > t 0 = 1 − 1 2c , and so u can be extended to a continuous function in R N × (t 0 , +∞), which is Lipschitz continuous in x, with the same Lipschitz constant as u 1 . We compute now the limit as t → t + 0 , that is the limit as λ(t) → 0 + in (3.6). For r ∈ (0, 1), we define v r (x) := ru 1 x r .
Note that (v r ) r are equi-Lipschitz and locally bounded so that, up to subsequences, there exists the locally uniform limit of v r as r → 0 + by Arzelà-Ascoli Theorem.
In order to conclude, we need to show that such limit is unique, i.e., it does not depend on the subsequence. If this is true, it is easy to check thatū(x) := lim r→0 + ru 1 x r satisfies (3.1). To prove the whole convergence, we observe that by (3.6) v r (x) = u(x, λ −1 (r)) ∀r > 0.
Then u n is the solution to (2.7) in R N ×(0, +∞) with initial datum u n (x, 0) = u(x, λ −1 (r n )) = v rn (x). Recalling that v rn (x) →v(x) locally uniformly, and that v rn andv are equi-Lipschitz functions, by Corollary 2.8, we get that, up to subsequences, u n (x, t) →v(x, t) locally uniformly in (x, t), for some functionv(x, t). By stability of solutions to (2.7) with respect to uniform convergence, we get thatv(x, t) is the solution to (2.7) in R N ×(0, +∞) with initial datumv(x, 0) =v(x). Now observe that by definition u n (x, t) = u(x, t + λ −1 (r n )) → u(x, t + t 0 ) for every t > 0, thereforev(x, t) = u(x, t + t 0 ) for every t > 0. This implies that the limitv is independent of the subsequence r n .
We now provide the locally uniform convergence to self-similar expanding solutions, in the rescaled setting (1.3), if the initial Lipschitz graph is a sublinear perturbation of a cone. A similar result has been obtained in [9] for the isotropic case, and in [4] for the fractional mean curvature flow. Note that, if the flow E t is the subgraph of the solution u(x, t) to (2.7), then the rescaled flowẼ τ defined in (1.3) is the subgraph of the rescaled function Theorem 3.4. Let u 0 be a Lipschitz continuous function, such that there existū which satisfies (3.1), and constants K > 0, δ ∈ (0, 1) for which there holds Letũ(y, τ ) be the rescaled function as defined in (3.7), where u is the solution to (2.7) with initial data u 0 and letū(x, t) be the solution to (2.7) with initial datumū. Then lim τ →+∞ũ (y, τ ) =ū y, 1 2 locally uniformly.
Proof. We argue as in the proof of [4, Theorem 5.1]. Let χ : (0, +∞) → (0, +∞) be a smooth function such that χ(k) ≡ 0 if k < 1 and χ(k) ≡ 1 if k > 2. Define for r > 1 Then our assumption implies that By the comparison principle we deduce that where u, u r are the solutions to (2.7) respectively with initial data u 0 and u r 0 . Then passing to the rescaled functionsũ andũ r as defined in (3.7), we obtain On the other hand,ū r δ |x|. Letū ±r be the solution to (2.7) with initial datum respectivelyū(x)± 2K r δ |x|. By the comparison principle (3.9)ū −r (x, t) u r (x, t) ū +r (x, t) ∀x ∈ R N , t > 0.
Recalling thatū(x) ± 2K r δ |x| are Lipschitz functions with Lipschitz constant bounded by Dū ∞ + 2K, by Corollary 2.8 we get that there exists B depending only on Dū ∞ and K such thatū −r y, Therefore by (3.8) we conclude that u −r y, for all y ∈ R N , τ > 0 and r > 1. Notice thatū ±r y, 1 2 →ū y, 1 2 as r → +∞, locally uniformly in y by stability of solutions with respect to local uniform convergence, sinceū(x) ± 2K r δ |x| →ū(x) locally uniformly. Therefore, taking r = e τ in the previous inequality and letting τ → +∞, we obtain the local uniform convergence ofũ. Remark 3.5. If u 0 −ū ∈ L ∞ (R N ), the convergence result in Theorem 3.4 can be strengthened to uniform convergence. In the isotropic case, the uniform convergence has been obtained in [9] under the assumptions of Theorem 3.4, by using maximum principle and integral estimates.

Stability of self-similar solutions asymptotic to mean convex cones
In this section we assume that the anisotropy is regular, that is, (2.2) holds, and we address the issue of the stability with respect to perturbations vanishing at infinity in the case of mean convex cones. The same problem has been considered in the isotropic setting in [8].
Note that the assumption in (4.1) implies that the epigraph ofū, that is, the set {(x, z) | z ū(x)} is a mean convex set.
We first show that mean convexity is preserved and that the homothetic solutionū(x, t) always lies aboveū.
Proof. Condition (4.1) implies thatū(x) is a stationary subsolution to (2.7), so by comparison u(x, t) ū(x). Again by comparison and the semigroup property we get that for all s > 0, there holdsū(x, t + s) ū(x, t).
Choosing r sufficiently small, we get that div(∇ x ϕ(−∇b r (x), 1)) < 0 in the viscosity sense for all x. Now, let us fix r 0 , and b r 0 the corresponding function such that the previous condition is satisfied. In particular b r 0 is a stationary subsolution to (2.7). We claim that there exists λ > 0 such that is the solution to (2.7) with initial datumū. Note that the function w(x, t) is a subsolution to (2.7), since it is the supremum between two subsolutions. Therefore to check the claim, it is sufficient to show that u(x, 0) = u 0 (x) w(x, 0) = sup ū(x) − m, λb r 0 x λ − δ . First of all, by assumption we know that u 0 (x) ū(x) − m for all x. On the other hand, observe that if λ > max(R(δ), m/r 0 ), by definition of χ and the positive 1-homogeneity ofū, there holds that for |x| R(δ) < λ, On the other hand if |x| R(δ), by assumption and our construction of b r , u 0 (x) ū(x)− δ λb r 0 x λ − δ. Therefore, by comparison, (4.2) holds for every λ > max(R(δ), m/r 0 ). We fix λ 0 which satisfies this condition. Now, observe that by Lipschitz continuity and Lemma 4.
, for all |x| 2λ 0 and for all t t δ and then, in turn, by (4.2), we get that u(x, t) ū(x) for all t t δ and |x| 2λ 0 . On the other hand, if |x| > 2λ 0 there holds by definition that λ 0 b r 0 x λ 0 − δ =ū(x) − δ and then again by (4.2), u(x, t) ū(x) − δ for all t, for all |x| > 2λ 0 . So, we get the conclusion.
Proof. First of all we observe that by Proposition 4.2 and the comparison principle, there holds that for all δ > 0 there exists t δ such that for all t > t δ , We fix now ε > 0 and R ε > 0 such that u 0 (x) ū(x) + ε for all |x| > R ε . Therefore, by Lemma 4.1, we get that for all t > 0, Observe now that by (3.3) and Lipschitz continuitȳ Sinceū(0, 1) > 0 by Lemma 4.1, there exists t ε > 0 sufficiently large such that Therefore, by (4.4), (4.5), and by comparison we get that for every t > 0 By (4.6) and (4.3) we get that for ε > 0, δ > 0 We conclude sending t → +∞, and recalling Theorem 3.1. In case that ∇ 2 ϕ, ψ are more regular, we conclude recalling Remark 3.2.

Stability of hyperplanes
In this section we show that hyperplanes are stable with respect to the flow (1.1), if the anisotropy is regular, that is, (2.2) holds. In particular we show that if the initial datum is flat at infinity, then the solution stabilizes to the hyperplane at which the initial datum is asymptotic. The same result has been obtained in the isotropic case by integral estimates and comparison with large balls in [8,10], and by using the heat kernel, in dimension 2, in [18]. Here we provide a different proof based on construction of suitable periodic barriers.
We start with a preliminary lemma on 1-dimensional periodic barriers.
, and let f 0 : R → R be a function of class C 1,1 (R) which is Zperiodic and odd. Let u(x, t) be the solution to (2.7) with initial datum u 0 (x) := f 0 (x · e 1 ). Then lim t→+∞ u(x, t) = 0 uniformly in C(R N ).
Proof. By uniqueness we have that where f : R×[0, +∞) → R satisfies f (r, 0) = f 0 (r), f (r, t) = −f (−r, t) and f (r+z, t) = f (r, t) for every z ∈ Z. Notice also that Differentiating in time the anisotropic perimeter of the graph of u, for all t > 0 we get that depends only on the initial function f 0 . It follows that there exists a sequence of times t n → +∞ such that f t (·, t n ) → 0 in L 2 ([0, 1]) as n → +∞. Recalling Proposition 2.10, up to extracting a further subsequence we can also assume that u(·, t n ) →ū(x) uniformly in C 1 (R N ) as n → +∞, whereū(x) =f (x · e 1 ) withf Z-periodic.
Recalling that we are assuming (2.2), we get that by the strong maximum principle, and the periodicity ofū, we conclude thatū ≡ 0.
Theorem 5.2. Let E 0 ⊆ R N +1 such that ∂E 0 is a Lipschitz surface and assume that there exists a half-space H for which Proof. We may assume without loss of generality that H = {(x, z) ∈ R n × R | z 0}. Moreover, by the assumption that lim R→+∞ d(E 0 \ B(0, R), H \ B(0, R)) = 0, there exist two Lipschitz functions u 0 , v 0 : R N → R such that lim |x|→+∞ u 0 (x) = 0 = lim |x|→+∞ v 0 (x) and t), v(x, t) are the solutions to (2.7) with initial datum u 0 , v 0 .
We claim that lim t→+∞ u(x, t) = 0 = lim t→+∞ v(x, t) uniformly. If the claim is true, then the result follows.
It is sufficient to prove the claim only for u (since for v is completely analogous), moreover, we restrict to the case in which u 0 0 (or equivalenty u 0 0). Indeed the general case is easily obtained by using as barriers the solutions with initial data u + 0 = max(u 0 , 0) and u − 0 = min(u 0 , 0). So, we prove the claim only for the case u 0 0. Note that by comparison, since the constants are stationary solutions, 0 u(x, t) max u 0 for all x ∈ R N , t > 0.
First of all we prove that inf