Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global $\frac{1}{n+1}$-H\"older continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the H\"older exponent to $\frac12$ in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.


Introduction
Many processes in material sciences such as phase transformation, crystal growth, grain growth, stress-driven rearrangement instabilities, etc. can be modelled as geometric interface motions, in which surface tensions act as a principal driving force (see e.g., [49,15,53,40] and references therein). A typical example of such a motion is anisotropic mean curvature flow: given a norm φ on R n (called anisotropy), the equation for the anisotropic mean curvature flow of hypersurfaces parametrized as Γ t reads as β(ν)V = −div Γt [∇φ(ν)] on Γ t , (1.1) where V denotes the normal velocity of Γ t in the direction of the unit outer normal ν of Γ t and β is the mobility, a positive kinetic coefficient [30]. Anisotropic mean curvature flow is called crystalline provided the boundary of the Wulff shape W φ := {φ ≤ 1} lies on finitely many hyperplanes; in this quite interesting case, equation (1.1) must be properly interpreted, due to the nondifferentiability of φ ; see for instance [1,29,52,28,12,13,31,20,32,17,18]. Equation (1.1) (sometimes referred to as the two-phase evolution) can be generalized to the case of networks in the plane, and more generally to the case of partitions of space (sometimes called the multiphase case): here the evolving sets are intrisically nonsmooth, since the presence of triple junctions (in the plane), or multiple lines, quadruple points etc. (in space) during the flow is unavoidable. It must be stressed that evolutions of partitions received recently a lot of attention from the mathematical community [51,25,26,23,45,39,38,50] both as a natural generalization of the case of two phases, and because they model a variety of physical phenomena, such as grain growth and evolution of multicrystals [40,14].
The presence of singularities at finite time is a common feature of mean curvature flow type motions, both in the two-phase case [33,34,35,36,44], and in the multiphase case (see for instance [45]). This phenomenon justifies to introduce and study some notion of weak solution, defined globally in time. This has been done in several different ways: just to quote a few, the Brakke varifold-solution [15], the viscosity solution (see [27] and references therein), the Ilmanen elliptic regularization [37], the level-set theoretic subsolution and the minimal barrier solution (see [6] and references therein), the Almgren-Taylor-Wang [2] and Luckhaus-Sturzenhecker [42] solutions, next included by De Giorgi into his notion of minimizing movement and generalized minimizing movement (GMM) [21,22]; see also [24,47]. Some of those solutions (e.g., the Brakke solution [15,54], the GMM solution [11], the elliptic regularization [50]) can be adapted to treat the multiphase case at least in the Euclidean case, especially those that do not rely heavily on the comparison principle. Also, the existence of a distributional solution of mean curvature evolution of partitions on the torus using the time thresholding method introduced in [46] has been proved in [41]; see also [39].
The aim of the present paper is to prove the existence of a GMM for anisotropic mean curvature flow of partitions with no restrictions on the space dimension, in the presence of a set of mobilities and forcing terms, and to point out some qualitative properties of this weak evolution, which are obtained via a comparison argument with a GMM of each single phase considered separately.
Let us recall the definition of GMM for partitions from [22] (see Definition 2.6 for the notion of bounded partition). Definition 1.1 (Generalized minimizing movement for partitions). Let P b (N + 1) be the set of all bounded (N + 1) -partitions of R n (Definition 2.6) endowed with the L 1 (R n ) -convergence, and let F : where φ j and ψ j are norms on R n , called anisotropies and mobilities, respectively, H i ∈ L 1 loc (R n ), i = 1, . . . , N, and H N +1 ∈ L 1 (R n ) are driving forces, P φ j (A j ) is the φ j -anisotropic perimeter, A = (A 1 , . . . , A N +1 ), B = (B 1 , . . . , B N +1 ) and d ψ j (·, E) is the ψ j -distance function from E ⊆ R n . We say that a map M : [0, +∞) → P b (N + 1) is a GMM associated to F starting from G  Our first result (see Theorems 4.1 and 4.2 for the precise statements) extends the existence results of [11] to the case with anisotropies, mobilities and external forces. We also improve the 1 n+1 -Hölder regularity in time of GMM proven in [11] to 1/2 -Hölder continuity in the two-phase case, without any restriction on the anisotropies.  To prove Theorem 1.2 we establish uniform density estimates for minimizers of F using the method of cutting out and filling in with balls, an argument of [42]. At this level the presence of mobilities does not create any new substantial problem. While in the two-phase case we do not need any assumption on the anisotropies, in the multiphase case assumption (3.1) is needed to get the lower volume density estimate for minimizers which is important in the proof of time-continuity of GMM.
In case of partitions with the same anisotropies and the same mobilities and without forcing, the Hölder exponent of GMM can be improved to 1/2 (see Theorem 5.1). Denoting by F 2 the restriction of F to two-phase case without forcing (see (5.2)), this can be done using the comparison property (Theorem 5.2) between the minimizers of F and the minimizers of F 2 . This comparison result also enables us to get a weak comparison flow of corresponding multiphase and two-phase flows (Theorem 5.5): . . , N }, and C N +1 be any convex sets such that C i ⊂ G i for any i ∈ {1, . . . , N + 1} and let L i ∈ M M (F 2 , C i ) be the unique minimizing movement starting from C i . Then for any M ∈ GM M (F, G),

2)
and Note that the comparison principle (1.3) implies that any bounded partition will disappear in the long run; moreover, (1.2) allows to estimate the extinction time of the i -th bounded phase (Corollary 5.6).
Finally, let us mention that a natural problem remains open, namely the consistency of GMM with the classical solution, provided the latter exists, at least on a short time interval. Such a result has been proven by Almgren-Taylor-Wang in [2] in the two-phase case without mobility; the proof is based on various stability properties of the flow, and using comparison arguments. It has also been proven by Almgren-Taylor [1] in the two-phase crystalline case. However, consistency is not known in the case of networks in the plane (and a fortiori for partitions in space), even in the Euclidean case without mobilities and forcing.
The paper is organized as follows. In Section 2 we introduce the notation, some results from the theory of sets of finite perimeter, and the definition of a partition. In Section 3 we prove the density estimates for almost minimizers. The existence of generalized minimizing movements (Theorem 1.2) is established in Section 4. In Section 5 we improve the Hölder regularity of GMM (Theorem 1.3) and provide some weak comparison principles.

Notation and preliminaries
In this section we introduce the notation and collect some important properties of sets of locally finite perimeter. The standard references for BV -functions and sets of finite perimeter are [4,43].
We use N 0 to denote the set of all nonnegative integers. The symbol B r (x) stands for the open ball in R n centered at x ∈ R n of radius r > 0. The characteristic function of a Lebesgue measurable set F is denoted by χ F and its Lebesgue measure by |F |; we set also ω n := |B 1 (0)|. We denote by E c the complement of E in R n .
Given a norm ψ in R n and a nonempty set E ⊆ R n , d ψ (·, E) stands for the ψ -distance from E, i.e., When ψ is Euclidean for simplicity we drop the dependence on ψ. We also write diam ψ E := sup{ψ(x − y) : x, y ∈ E} to denote the ψ -diameter of E.
By O(R n ) (resp. O b (R n ) ) we denote the collection of all open (resp. open and bounded) subsets of R n . The set of L 1 loc (R n ) -functions having locally bounded total variation in R n is denoted by BV loc (R n ) and the elements of are called locally finite perimeter sets. Given a E ∈ BV loc (R n , {0, 1}) we denote by a) P (E, Ω) := Ω |Dχ E | the perimeter of E in Ω ∈ O(R n ); b) ∂E the measure-theoretic boundary of E : c) ∂ * E the reduced boundary of E; d) ν E the outer generalized unit normal to ∂ * E.
For simplicity, we set P (E) := P (E, R n ) provided E ∈ BV (R n ; {0, 1}). Further, given a Lebesgue measurable set E ⊆ R n and α ∈ [0, 1] we define Unless otherwise stated, we always suppose that any locally finite perimeter set E we consider coincides with E (1) (so that by [43,Eq. 15.3] ∂E coincides with the topological boundary). We recall that ∂ * E = ∂E and Dχ E = ν E dH n−1 ∂ * E, where H n−1 is the (n − 1) -dimensional Hausdorff measure in R n and is the symbol of restriction.
The following generalizes the notion of the perimeter. Given When Ω = R n , we write P φ (E) := P φ (E, R n ), and when φ is Euclidean, we write P in place of P φ .
The collection of all N -partitions of R n is denoted by P(N ). Our assumptions The elements of P(N ) are denoted by calligraphic letters A, B, C, . . . and the components of A ∈ P(N ) by the corresponding roman letters (A 1 , . . . , A N ).
Let φ 1 , . . . , φ N be norms in R n and set Φ := {φ 1 , . . . , φ N }. The functional is called the anisotropic perimeter, or Φ -perimeter of the partition A in Ω. For simplicity, we write Per Φ (A) := Per Φ (A, R n ). For shortness, we also set Per Φ = Per when all φ i are Euclidean. Since N is finite, there exist 0 i.e., on a generalized hypersurface Σ ij := Ω ∩ ∂ * A i ∩ ∂ * A j dividing the phase i from the phase j the perimeter contributes where ν Σ ij is the generalized unit normal to Σ ij pointing for instance from A i to A j . We set where ∆ is the symmetric difference of sets, i.e., We say that the sequence . From (2.6) and [11, Theorem 3.2] we get Then there exist a partition A ∈ P(N ) and a subsequence {A (l k ) } converging to A in L 1 loc (R n ) as k → +∞.
Note that A∆B ⊂⊂ R n for every A, B ∈ P b (N + 1), and therefore, In view of (2.6) and [11, Theorem 3.10] we have the following compactness result.
Then there exist A ∈ P b (N + 1) and a subsequence {A (k l ) } converging to A in A j ⊆ Ω. 6

Density estimates for almost minimizers
In this section we prove density estimates for almost minimizers (see Theorem 3.2). In the two-phase case without mobility, density estimates have been proven in [2,42] (see also the proof of Theorem 4.2 for the case with mobility and forcing) and in the isotropic N -phase case is proven in [11]. The proof of Theorem 3.2 is similar to [11,Theorem 3.6], however, some (technical) difficulties arise when two anisotropies differ too much and this is why we need assumption (3.5) for proving the lower-density estimates.
or also an almost-minimizer for short) if Theorem 3.2 (Density estimates for almost minimizers). Assume that the entries of Φ satisfy (2.5). Let A ∈ P(N ) be a (Λ 1 , Λ 2 , r 0 , α 1 , α 2 ) -minimizer and i ∈ {1, . . . , N }. Then either A i = ∅ or for any x ∈ ∂A i and r ∈ (0, r 0 ] where where and Proof. Without loss of generality, we assume i = 1 and We start by proving (3.

2) and (3.3). Let us show
we have A∆B ⊂⊂ B s for every s ∈ (r, r 0 ) and thus, by almost minimality, the definition (2.7) of |A∆B| and the essential disjointness of A j , and for any j = 2, . . . , N, (3.13) By (2.5), the essential disjointness of A j and (3.10) we have thus, (3.13) and (3.10) imply

By (2.5), (2.4) and the essential disjointness of
1 ∩ ∂B r ) to both sides of (3.9) and using (3.14) By the choice of r 0 in (3.4) we have, for l = 1, 2, and whence, repeating for instance the arguments of the proof of [11,Eq. 3.19], we obtain |A From (3.9) and the definition of r 0 for all r ∈ (0, r 0 ] we get Now we prove (3.6) and (3.7). Note that assumption (3.5) implies r 0 , γ N > 0. Let us show Since B (j) ∆A ⊂⊂ B s for every s ∈ (r, r 0 ), by the almost minimality of A (recall that r 0 ≤ r 0 ) and the equality |A∆B (j) | = 2|A 1 ∩ B r | one has (3.18) Using the equality the analogue of (3.12) with j = 1 and also (3.17) in (3.18) we establish Summing these inequalities in j ∈ I 1 and using (2.5) we get where |I 1 | is the number of elements of I 1 . By the definition of I 1 , and by (2.5) to both sides of (3.16) and get nα l −n+1 β l for l = 1, 2 and therefore and thus, by (3.20) and the isoperimetric inequality, Now integrating we get γ n N ω n r n ≤ |A 1 ∩ B r | and (3.6) follows. Finally, since from (3.2), (3.6) and the relative isoperimetric inequality we deduce (3.7).
The following volume-distance comparison appeared in a similar form also in [2,11,42] and will be used in the proof of the existence of GMM.

21)
whenever x ∈ ∂A. Then for any ℓ > 0 and B ∈ BV (R n ; {0, 1}) one has Proof. We follow [11,Proposition 4.5] with minor modifications and we give the details for the convenience of the reader. Define By the Chebyshev inequality, Let us estimate |E|. By a covering argument, one can find a finite family of disjoint balls Analogously, if ℓ < r 0 , then |E| ≤ 5 n ω n θ P (A j ) ℓ. Now (3.22) follows from the inequality |B∆A| ≤ |E| + |F | and estimates for |A| and |B|.

Existence of GMM for bounded partitions
Given a norm ψ in R n and E, F ⊆ R n set Given a family Ψ := {ψ 1 , . . . , ψ N +1 } of norms ψ i in R n , and A, B ∈ P b (N + 1), we set where N + 1 ≥ 2. In the literature Ψ is called the set of mobilities. Since N is finite, there exist 0 < c Ψ ≤ C Ψ < +∞ such that Observe that for every B ∈ P b (N + 1) the map σ Ψ (·, B) is L 1 (R n ) -lower semicontinuous in P b (N + 1).
The main result of this section is the following, which generalizes [11, Theorems 4.9 and 5.1] to the anisotropic case with mobilities; recall that κ N +1 is defined in (3.1).
Proof. We give only few details of the proof since it can be done following the arguments of the proofs of [11, Theorems 4.9 and 5.1].
Step 1: Existence of minimizers. Given A ∈ P b (N + 1) and λ ≥ 1, the problem has a solution. Moreover, every minimizer A(λ) = (A 1 (λ), . . . , A N +1 (λ)) satisfies the bound We omit the proof since it is proven along the same lines as [11,Theorem 4.2] using the anisotropic Comparison Theorem with convex sets 1 and the inequality d ψ (·, E 0 ) > 0 in any F ⊂ R n \ E 0 .

By (4.12)
Thus, from (4.14) we get  4.1. Two-phase case. When N = 1, repeating the arguments of [42] in our more general setting, we can improve the Hölder exponent of GMM to 1/2 without any restriction on the anisotropies. and In addition, if |G 1 \ G 1 | = 0, then (4.16) holds for any t, t ′ ≥ 0 with |t − t ′ | < 1.
The proof runs along the same lines of Theorem 4.1 with however an improved bound for the radii in the proof of the density estimates, see (4.30) below. We need to make a detailed proof since this will be used in the proof of Theorem 5.1.

Proof. Letting
and Note that, except for the presence of A 1 Hdx, F 2 is of the form of the Almgren-Taylor-Wang functional.
We divide the proof into five steps.
Step 1: Existence of minimizers. Let E 0 ∈ BV (R n ; {0, 1}) be such that E 0 ⊂ D. Since φ is a norm, d E 0 ψ ≥ 0 in R n \ E 0 and H ≥ 0 in R n \ D, as in the Euclidean two-phase case (see e.g. [3]) we can use the Comparison Theorem with the convex set D to establish the existence of a minimizer of F 2 (·, E 0 , λ) and also that every minimizer E λ satisfies E λ ⊆ D.
Step 2: Unconstrained density estimates for minimizers. Let E λ minimize F 2 (·, E 0 , λ) and x 0 ∈ E λ ∆E 0 be such that d(x 0 , ∂E 0 ) ≥ r 1 for some r 1 > 0 satisfying Notice that there are no restrictions on r 1 > 0; in addition x 0 needs not be on ∂E λ . Let us show that for any r ∈ (0, r 1 ) ; for any r ∈ (0, r 1 ) .
We prove only (a), since the proof of (b) is similar. For shortness we write B r := B r (x 0 ). Fix any r ∈ (0, r 1 ) such that By the minimality of E λ we have for any s > r. The choice of x 0 and the definition of d E 0 ψ imply d E 0 ψ ≥ 0 in B r 1 and hence using (4.21), (3.12) (applied with φ j = φ and A j = E λ ) and the inclusion (E λ \ B r )∆E λ ⊂ B r , from (4.22) we get (4.23) The definition of φ, (2.5), the isoperimetric inequality and the Hölder inequality yield Recall by Step 1 that E λ ⊆ D. Thus from the inequality and (4.20), and therefore, from (4.24) we deduce for any r ∈ (0, r 1 ).
The next step is valid in the two-phase case. We miss the proof of a similar statement in the multiphase case because we are not able to prove the analogue of Step 2 2 .
We essentially follow the arguments of [42,48]. Let 2 In the multiphase case we miss the analogue of (4.23), that was obtained neglecting the term (4.22). For instance, in the planar 4 -phase, at a triple junction involving φ1, φ2, φ3 and surrounded by the fourth phase having φ4 as surface tension, it is conceivable that, if φ1, φ2, φ3 are quite large compared to φ4 , then around the triple point, the fourth phase appears after one minimization step.

Assume by contradiction that there exists
, where for shortness we drop the dependence of r on n, λ, ǫ, Φ and Ψ . Setting B r := B r (x 0 ) without loss of generality we also suppose that (4.21) holds. First we assume x 0 ∈ E λ \ E 0 . Then the minimality of E λ implies F 2 (E λ , E 0 , λ) ≤ F 2 (E λ \ B r , E 0 , λ) so that, similarly to (4.23), By the Hölder inequality, the inclusion E λ ⊂ D and (4.20), therefore, by (2.5) and the isoperimetric inequality, This and (4.25) imply By (4.1), the choice of x 0 and the definition of r one has d E 0 ψ ≥ c Ψ d(·, ∂E 0 ) ≥ 2c Ψ r in B r . Thus, from (4.27) and (2.5) we get This, the inequality H n−1 (E λ ∩ ∂B r ) ≤ nω n r n−1 and Step 2 (a) imply Therefore, by the definition of C 1 and r, and repeat the similar arguments above.
Before passing to the next step let us define Step 4: Uniform density estimates for minimizers. Given λ > C 5 and a minimizer E λ of F 2 (·, E 0 , λ), following arguments of [42,48] let us show that and for any x ∈ ∂E λ and r ∈ (0, For any r as in (4.30) and y ∈ B r (x) one has Let us prove the lower volume density estimates. For shortness set B r := B r (x). Let r ∈ (0, C 3 λ −1/2 ) be such that (4.21) holds. As in the proof of Step 2, from the inequality By (4.31), the choice of r and the equality Furthermore, using λ > C 5 , as in (4.26) Therefore, from (4.32) it follows that Adding E λ ∩∂Br φ(ν Br )dH n−1 to both sides of (4.33), and using (2.5) and the isoperimetric inequality we get Integrating this over r we get the lower volume density estimate in (4.28).
For what concerns the upper perimeter density estimate in (4.29) we observe that from (4.33) and (2.5) it follows that 2c Φ P (E λ , B r ) ≤ (2C Φ + c Φ )nω n r n−1 for a.e. r ∈ (0, C 3 λ −1/2 ). Since r → P (E λ , B r ) is non-decreasing and leftcontinuous, this inequality holds for all r. Finally the lower perimeter density estimate follows from (4.28) and the relative isoperimetric inequality for the ball.

Improved time Hölder regularity
In this section we show that when φ i = φ and ψ i = ψ for any i = 1, . . . , N + 1, the time Hölder continuity exponent of GMM for partitions can be improved to 1/2 . The result follows from the generalization of [42] in the previous section (Theorem 4.2) combined with with a comparison (Theorem 5.2 below) between a multiphase flow and a two-phase flow starting from just one of the phases and its complement. Arguments from our main continuity result (in Theorem 4.1) are needed to reconnect both flows in the limit.
where C 6 is given in (4.38). In addition, if Recall that, by Theorem 4.1, for any G ∈ P b (N + 1), GM M (F, G) is non-empty, each M ∈ GM M (F, G) is locally 1/(n + 1) -Hölder continuous and (G) for any t ≥ 0 .
Besides F we need to consider also the functional F 2 defined (up to constants) in (4.17) with H = 0, i.e., We start with a comparison result: this is the key point of the proof of Theorem 5.1 since it allows to compare the evolution of a single phase with the multiphase case.
Inserting (5.6) with G = A and E = F i in (5.5) and using (2.4) we get By assumption 2g ′ i − g i + g j > 0 a.e., and hence F i ⊆ A i up to a negligible set. The case i = N + 1 is similar.
Proof. Let λ h → +∞ be such that Proof. Recall that anisotropic mean curvature flow with a mobility starting from a bounded convex set C is uniquely defined [8], coincides with the GMM starting from C and extincts at a finite time t C > 0. By Theorem 5.5, the i -th phase M i of any M ∈ GM M (F, G) starting from the i -th phase G i of G does not disappear in the time-interval (0, t C i ) for any i ∈ {1, . . . , N }. Analogously, Theorem 5.5 implies that (N + 1) -th phase of M becomes empty, i.e., R n \ M N +1 (t) = ∅ if t ≥ t C N+1 .