Global solutions to the volume-preserving mean-curvature flow

In this paper, we construct global distributional solutions to the volume-preserving mean-curvature flow using a variant of the time-discrete gradient flow approach proposed independently by Almgren, Taylor and Wang (SIAM J. Control Optim. 31(2): 387- 438, 1993) and Luckhaus and Sturzenhecker (Calc. Var. Partial Differential Equations 3(2): 253-271, 1995).


Introduction
A family of open sets with smooth boundary {E t } 0≤t<T in R n is said to move according to volume-preserving mean-curvature flow if the motion law, expressed as an evolution equation for the boundaries ∂E t , takes the form is the scalar mean curvature (with the sign convention that H is positive for balls, see the next section), and the brackets · denote the average of a quantity over the boundary of E t .
It is immediately verified that the volume of the sets E t (i.e., its n-dimensional Lebesgue measure, denoted by |E t |) is indeed preserved under the smooth flow And thus, upon rescaling variables, we may assume that |E t | = |E 0 | = 1 for any t ∈ [0, T ]. Moreover, the perimeter of the sets E t is decreasing because During a typical evolution, a volume-preserving mean-curvature flow exhibits singularities of different kinds, even in the case of smooth initial data. These singularities correspond to changes in the topology of the configuration and include shrinkage of islands to points and disappearance, collisions and merging of neighboring islands, pinch-offs etc. . . . In the moment of a topological change, the boundary of the evolving set looses regularity and, as a consequence, the formulation (1.1) of the evolution law is inadequate. The goal of the present work is the construction of a notion of a weak solution to the volume-preserving mean-curvature flow that is global in time and thus overcomes these singular moments.
Several solutions to volume-preserving mean-curvature flow have been proposed in the literature: existence and uniqueness of a global in time smooth solution and its convergence to a sphere is shown in [13,18] for smooth convex initial data and in [11,20] for initial data close to a sphere (for further related results see [3,4,5] and the references therein).
In principle, these results also yield local in time existence and uniqueness of smooth solutions. In [26] and [27], the authors consider level-set and diffusiongenerated solutions for the purpose of numerical studies. However, to the best of our knowledge, there are no rigorous results regarding existence or uniqueness of global in time weak solutions with arbitrary initial configuration available in the mathematical literature.
It is well-known that volume-preserving mean-curvature can be (formally) interpretated as the L 2 -gradient flow of the perimeter functional for configurations with a fixed volume, see, e.g., [25,Sec. 2]. This gradient flow structure, however, is for the purpose of well-posedness results impracticle, since the L 2 (geodesic) distance is degenerate in the sense that two well separated configurations may have zero L 2 distance [24]. In the present manuscript, we follow the method proposed independently by Almgren, Taylor & Wang [1] and Luckhaus & Sturzenhecker [22] in the study of (forced) mean-curvature flows to bypass this difficulty. The authors consider an implicit time-discretization of the flow, which comes as a gradient flow of the perimeter functional with respect to a new non-degenerate distance function that approximates the L 2 distance. The main difference between the present work and [1,22] relies on the non-locality of the volume-preserving mean-curvature flow. As an immediate consequence, there is no maximum-principle available for (1.1). A more detailed discussion on the different features of the flows in [1,22] and the one considered in the present manuscript will follow in Section 3 below.
We conclude this subsection with a short discussion on the background of this evolution. Volume-preserving mean-curvature flow can be considered as a simplified model for attachment-limited kinetics, and as such it plays an important role in the study of solidification processes, where solid islands grow in an undercooled liquid of the same substance. In such situations, solid particles melt at high-curvature regions and simultaneously precipitate at low-curvature regions, while the total mass of the solid remains essentially constant [29,8,28]. In this way, the total surface area of solid islands is decreasing, and thus, this process leads to the growth of larger islands at the expense of smaller ones: a phenomenon called coarsening [25]. In general, solidification processes are mathematically often modeled by Mullins-Sekerka equations (or a Stefan problem), where the Gibbs-Thompson relation is modified by a kinetic drift term [21,17], and Volume-preserving Mean-curvature Flow 3 their phase-field counterparts respectively [6]. This model allows for both attachment kinetics (kinetic drift) and bulk diffusion (Mullins-Sekerka/Stefan). It turns out that attachment kinetics is the relevant mass transport mechanism in earlier stages of the evolution while bulk diffusion predominates the later stages [9]. In a certain sense, volume-preserving mean-curvature flow naturally arises as the singular limit of this more general solidification model in the regime of vanishing bulk diffusion. More recently, variants of volume-preserving mean curvature flow were also applied in the context of shape recovery in image processing [7].
The article is organized as follows: in Section 2 we fix the notation and state the main results of the paper, which are then proved in Sections 3 and 4 and are the existence of flat volume-preserving mean-curvature flows and the existence of distributional solutions, respectively.

Acknowledgements
CS acknowledges the kind hospitality of the Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig, Germany, where part of this work was performed.

Statements of the main results
2.1. Notation. For any Lebesgue measurable set E ⊂ R n , we denote by |E| the n-dimensional Lebesgue measure of E and by χ E the characteristic function of E, i.e. χ E (x) = 0 if x / ∈ E and χ E (x) = 1 if x ∈ E. The perimeter of E in an open set Ω ⊂ R n is defined as and we write Per(E) := Per(E, R n ). If the latter quantity is finite, we will call E a set of finite perimeter. In the case that E is an open set with ∂E of class C 1 , we simply have Per(E, Ω) = H n−1 (∂E ∩ Ω) and Per(E) = H n−1 (∂E). The reduced boundary of a set of finite perimeter E is denoted by ∂ * E, cp. [12,Sec. 5.7], and for the unit outer normal to E we write ν E . The tangential divergence of a vector field Ψ ∈ C 1 (R n ; R n ) with respect to ∂E is defined by div ∂E Ψ := divΨ − ν E · ∇Ψν E . We say that a set of finite perimeter E has a (generalized) mean-curvature H E ∈ L 1 (∂ * E, dH n−1 ) provided that Observe that with this sign convention it is H B R ≡ n−1 R .
We write sd F for the signed distance from a Lebesgue measurable set F , with the convention that sd F is negative inside F , i.e., Here, F c := R n \ F denotes the complement set of F , and the distance from a set F is by definition dist(x, F ) = inf y∈F |x − y|.
We will sometimes use the notation d F = |sd F |.
By [t] we denote the integer part of a real number t, that is the biggest integer m such that m ≤ t.
Finally, we denote by c n , C n positive constants that depend on the space dimension only. Moreover, c n,0 , C n,0 , and C n,0,T are constants that may additionally depend on the initial data or the time T > 0. During the computations, the value of these constants may change from line to line. However, for the sake of clarity we need to keep track of the dimensional constant in Proposition 3.2.1, therefore we make an exception to the above convention and denote it by γ n . The volume of the n-dimensional unit ball will be denoted by ω n , and thus its surface area is nω n .

2.2.
Approximate solutions. In this paper we introduce a notion of global flat solution to the volume-preserving mean-curvature flow which is based on the implicit time discretization of (1.1) in the spirit of Almgren, Taylor & Wang [1] and Luckhaus & Sturzenhecker [22]. That is, we consider a time-discrete gradient flow for the perimeter functional. For this purpose, we define for any two sets of finite perimeter E and F in R n . Here, h is a positive small number that plays the role of the time step of approximate solutions. The second term in the above functional approximates the degenerate L 2 geodesic distance on the configuration space of hypersurfaces. The functional differs from the one considered in the original papers [1,22] only in the last term: a weak penalization that favors unit-volume of minimizing sets.
We furthermore define for any t ∈ [kh, (k + 1)h), t } t≥0 an approximate flat solution to the volume-preserving meancurvature flow with initial datum E 0 .
Volume-preserving Mean-curvature Flow

5
The existence of minimizers E (h) kh and thus the existence of an approximate solution is guaranteed by Lemma 3.1.1 below. Incorporating the volume constraint in a soft way into the energy functional rather than imposing a hard constraint on the admissible sets has the advantage that we are free to chose arbitrary competitors, most notably in the derivation of density estimates. Thanks to the penalizing factor 1/ √ h, the constraint becomes active in the limit h ↓ 0. Even more can be shown: the number of time steps in which approximate solutions violate the volume constraint |E (h) t | = 1 can be bounded uniform in h, cf. Corollary 3.4.5 below. A similar functional including a soft volume constraint was recently considered by Goldman & Novaga [16] in the study of a prescribed curvature problem.

Main results.
We can now state our main results. The first one is a convergence result for approximate solutions. t } t≥0 be an approximate solution to the volume-preserving mean-curvature flow with initial datum E 0 . Then, there exists a family of sets of finite perimeter {E t } t≥0 and a subsequence h k ↓ 0 such that and, for a.e. 0 ≤ s ≤ t, Our next statement is the existence of a distributional solution in the sense of Luckhaus & Sturzenhecker [22] to the volume-preserving mean-curvature flow under the hypothesis that the perimeters of the approximate solutions converge to the perimeter of the limiting solutions identified in the previous theorem.

Theorem
then {E t } 0≤t<T is a distributional solution to the volume-preserving meancurvature flow with initial datum E 0 in the following sense: (1) for almost every t ∈ [0, T ) the set E t has (generalized) mean curvature in the sense of (2.1) satisfyinĝ In the second part of the theorem, L 2 0 is the set of all L 2 functions with zero mean.
Note that (2.4) is a weak formulation of (1.1), while (2.5) establishes the link between v and the velocity of the boundaries of E t . It is straightforward to check that smooth solutions of (1.1) satisfy (2.4) and (2.5).

Flat volume-preserving mean-curvature flows
In this section we prove the first main result in Theorem 2.3.1. We follow quite closely Luckhaus & Sturzenhecker [22], providing all the details for the readers' convenience.

3.1.
Existence of approximate solutions. We start remarking that The existence of the approximate solutions is guaranteed by the following lemma.
3.1.1. Lemma. Let F ⊂ R n be a bounded set of finite perimeter. For every h > 0, there exists a minimizer E of F h ( · , F ) and, moreover, E satisfies the discrete dissipation inequality Proof. Since F is an admissible competitor, we obtain by (3.1) that , there exists a subsequence (not relabeled) that converges weakly to a function χ in BV (B R ), and thus strongly in L 1 (B R ). In particular, χ is the characteristic function of some set of finite perimeter E. Since χẼ →´Ẽ sd F dx is continuous and the perimeter is lower semi-continuous with respect to L 1 convergence, it follows that Therefore, E minimizes F h ( · , F ) and (3.2) follows from (3.3).
By standard results on minimal surfaces (see [23]), it holds that the minimizers E of F h ( · , F ) can be chosen to be a closed subsets with ∂E of class C 1,α up to a (relatively closed) singular set of dimension at most n − 7. Using the Euler-Lagrange equation for F h ( · , F ), one can also show that the regular part of the boundary ∂E is actually C 2,κ (cp. Lemma 3.4.2).
3.2. L ∞ and L 1 -estimates. Our next statement gives a uniform bound on the distance between the boundary of the minimizing set and the boundary of the reference set.
There exists a dimensional constant γ n > 0 with the following property. Let F ⊂ R n be a bounded set of finite perimeter and let E be a minimizer of F h ( · , F ). Then, Proof. The proof of this proposition is based on the density estimates for one-side minimizing set which for readers' convenience we prove in the Appendix A. We claim indeed that the statement holds with where c n is the dimensional constant in Lemma A.0.1. The argument is by contradiction. Let c > max 3, 4nωn cn and let x 0 ∈ F ∆E contradict (3.4). Without loss of generality, we can assume that x 0 ∈ F \ E: the other case is at all analogous. We then have that We use (3.5) and r ≤ c Then (3.6) and (3.7) yield By assumption c > 3 and we can apply Lemma A.0.1 with µ = 0 and obtain On the other hand, from (3.8) we deduce also that for a.e. 0 < r < c Combining (3.9) and (3.10), we get that for almost all 0 < r < c √ h 2 , which gives the desired contradiction to choice of c as soon r ↑ c √ h 2 . The following density estimates are now an immediate consequence.

Corollary.
Let F ⊂ R n be a bounded set of finite perimeter and let E be a minimizer of F h ( · , F ). Then, for every r ∈ (0, γ n √ h) and for every Estimating the second term via Proposition 3.2.1, we obtain A similar analysis shows that Therefore, by Lemma A.0.1 we deduce (by possibly redifining c n ) The first inequality in (3.12) is now an immediate consequence of the relative isoperimetric inequality (cf. [15,Cor. 1.29]). For the second inequality, we rewrite (3.13) as Since r < γ n √ h, the upper bound is obvious.
Next we prove an estimate on the volume of the symmetric difference of two consecutive sets of the approximate solutions. (L 1 -estimate). Let F ⊂ R n be a bounded set of finite perimeter and let E be a minimizer of F h ( · , F ). Then,

Proposition
(3.14) Proof. In order to estimate E∆F , we split it into two parts: The second term is easily estimated by To estimate the first term, we use a simple covering argument to find a collection of disjoint balls {B (x i )} i∈I with x i ∈ ∂ * E and I ⊂ N a finite set such that ∂ * E ⊂ ∪ i∈I B 2 (x i ). Note that by (3.11) Note finally that the set {x ∈ E∆F : d F (x) ≤ } is covered by {B 3 (x i )} i∈I . Summing over i and the choice of the balls {B (x i )} i∈I yields ≤ C n Per(E).
3.3.1. Proposition (C 1/2 regularity in time). Let h ≤ 1 and let {E t } t≥0 be an approximate flat flow. Then it holds Proof. Clearly it is enough to consider the case s − t ≥ √ h. Let j ∈ N and k ∈ N \ {0} be such that t ∈ [jh, (j + 1)h) and s ∈ [(j + k)h, (j + k + 1)h). Then, we can use Proposition 3.2.3 with = γ n h /|t − s| 1/2 (note that ≤ γ n √ h by the assumption s − t ≥ √ h) and (3.16), and estimate in the following way: By using (3.15) we estimate the sum above by Therefore, by (3.16) and (3.17), we get where we used kh ≤ |t − s| + h ≤ 2|t − s|, thus concluding the proof of the proposition.

First variations and first consequences.
We now introduce the timediscrete normal velocity: for all t ≥ 0 and x ∈ R n , we set (3.20) Proof. We fix t ∈ [h, +∞) and consider for every ∈ Z with 2 ≤ γn / √ h the sets so that R n = K( ). It follows from 2 −1 h ≤ γ n √ h/2 and from Corollary 3.2.2 that, for every x ∈ ∂E Hence, combining these two estimates, we havê We now show how the above estimate implies (3.20). In view of (3.15) we havê Integrating in time and using |E 0 | = 1, we obtain from which, by a simple limit for T → +∞, (3.20) follows.
We now derive the Euler-Lagrange equations which constitute the weak motion law for the time-discrete evolution.

Lemma (Euler-Lagrange equations). For every t ∈ [h, +∞) and Ψ
As we shall see in the proof below, the constants λ (h) (t) defined in (3.25) are Lagrange multipliers corresponding to the volume constraint, whenever it is active. Since this constraint is satisfied up to a finite number of times (uniformly in h) by Corollary 3.4.5 below, by a slight abuse of terminology, we call λ (h) (t) a Lagrange multiplier, even if the volume constraint is not active.

Proof. If |E
(h) t | = 1, it is very simple to compute the variations of F h (·, E (h) t−h ) along the vector field Ψ ∈ C 1 c (R n ; R n ) and see that they are given by (3.24) Hence, performing variations of within the class of sets of unit volume, for every Ψ ∈ C 1 c (R n ; R n ), we again find (3.24), where λ (h) (t) is the Lagrange multiplier related to the constraint |F | = 1. Observe that in both cases, we can choose a sequence of Ψ ∈ C 1 c (R n ; R n ) approximating ν E (h) t on ∂E (h) t , and conclude that λ (h) (t) is given by (3.25).

13
It is now clear that the regular part of ∂ * E (h) t is of class C 2,κ . Indeed, by choosing a suitable system of coordinates, ∂ * E (h) t can be written in a neighbourhood of any regular point as the graph of a C 1,κ function u solving the following equation in the sense of distributions: div ∇u Since v (h) is Lipschitz continuous, by standard elliptic regularity theory (cp., e.g., [14]), one then deduce that u ∈ C 2,κ for every κ ∈ (0, 1).
Next we prove that the whole family of sets defining the discrete flow up to time T > 0 is contained in a large ball, whose radius does not depend on the discrete time-step h but may depend on the T .

Lemma (Boundedness of minimizing sets). Let {E
Proof. We fix h > 0, and for every t ∈ [0, T ) we let We noticeB rt ∩ ∂E

By this observation and the Euler-Lagrange equation
Iterating the above estimate, we then deduce that (3.26) To get some control on λ (h) (t) we consider Ψ ∈ C 1 c (R n ; R n ) such that Ψ(x) = x in B rt , and, since for h small enough |E (h) t | ≥ 1 2 by (3.17), using Ψ as test in (3.24), we obtain t . Integrating in time and using the Cauchy-Schwarz inequality together with (3.16) and (3.20), we obtain Combining (3.26) and (3.28), it follows that The remainder of the proof is a standard ODE argument. Indeed, squaring both sides of the equation and redefining C n,0 yields Using the triangular inequality and Proposition 3.3.1, we deduce that Now a simple continuity argument implies that the sequence E (h k ) t converges to sets E t for all times t ≥ 0 and satisfies (3.30) for all s, t ∈ [0, +∞). Finally, note that passing to the limit in (3.15) yields that |E t | = 1 and Per(E t ) ≤ Per(E 0 ) (cf. [12, Sec. 5.2.1]) for a.e. t ∈ (0, T ).
3.6. Remark. It is also possible to show that the sequence of characteristic functions are precompact in L 1 ((0, T ) × R n ) for every T > 0, thus giving an alternative proof of the theorem.

Distributional solutions
In this section we prove Theorem 2.3.2 on the existence of distributional solutions. The two main ingredients of the proof besides the estimates of the previous section are the hypothesis (2.2) on the continuity of the perimeters of the approximate solutions and the following proposition which links the discrete velocity to the distributional time derivative of the flat flow.
Assuming the proposition we give a proof of the theorem. weakly converges to H n−1 ∂ * E t for almost every t ∈ [0, +∞). In particular this implies that, for almost every t ∈ [0, +∞), the boundaries ∂ * E (h k ) t converge to ∂ * E t in the sense of varifolds: namely, for a.e. t ∈ [0, +∞) it holds for every F ∈ C c (R n × R n ). Indeed, by a simple approximation argument it is easy to verify that it is enough to consider F ∈ C 1 c (R n × R n ). Then, for every ε > 0 we pick a continuous function ν ε : R n → R n such that and estimate as follows where we used (2.2) and the weak convergence of the the vector valued measures ν In particular, testing (4.2) with F (x, ν) := divΨ − ν · ∇Ψν for some Ψ ∈ C 1 c (R n ; R n ) and using (4.5), by a simple approximation argument we conclude that, for a.e. t ∈ [0, +∞), We need only to show (2.5). To this aim we use Proposition 4.0.1. For every φ ∈ C 1 c ([0, +∞) × R n ), by a change of variables we have that Therefore it follows by a simple convergence argument that In view of (4.3) and (4.1), we conclude (2.5) straightforwardly.

4.2.
Tilting of the tangent planes. In this subsection and in the next one we give the proof of Proposition 4.0.1. We follow closely the arguments in [22] and for the sake of completeness we provide a detailed proof in different steps. This subsection is devoted to the estimate of the tilting of the normals around points of small curvature. We recall that we assume in this section n ≤ 7 (in Moreover, using the lower semicontinuity of the perimeter with respect to L 1 convergence and β > 1 2 , we deduce that E β 1 and E β 2 are locally minimizing the perimeter. By the assumption n ≤ 7 and a Bernstein theorem (see [15,Theorem 17.3]), E β 1 , E β 2 are half-spaces. Moreover, by hypothesis it holds thus implying that E β 1 = E β 2 , and by the fact that both are hyperplanes there exists ν ∈ R n with |ν| = 1 such that To reach the conclusion of the lemma we need only to invoke the regularity theory of Λ-minimizing set (cp. [23,Theorem 26.3]) and conclude that ∂E Then, there exists a dimensional constant C > 0 such that ˆC Proof. From Lemma 4.2.1 we know that, for h sufficiently small, ∂E can both be written as graphs of functions of class C 1,α . Namely, by an affine change of coordinates we can assume without loss of generality that x 0 = 0 and ν = e n , and for simplicity we set C := C h β /2 (0, e n ). With this assumption we then have that for s = t, t − h (4.12) In view of Fubini's theorem it is then clear that Moreover, from (4.12) it follows that there exists a geometric constant C > 0 such that, for every y ∈ B h β /2 , Therefore, one infers (4.11) as follows where we used that (4.12).
We are finally ready for the proof of Proposition 4.0.1.

4.3.
Proof of Proposition 4.0.1. We fix any time t ∈ [0, +∞). For every t , we fix α ∈ 1 2 , n+2 2(n+1) and consider the following open set A x 0 defined as follows: We estimate the contribution of the integrals in (4.1) in every A x 0 with x 0 ∈ I in two steps, depending on whether (i) above applies or (ii).
Estimate in case (i). We use Corollary 4.2.2 and deduce that ˆA where we used the fact that A x 0 = C h β /2 (x 0 , ν).
Estimate in case (ii). By assumption there exists a point y 0 t−h ) such that |v (h) (t, y 0 )| > h α−1 . Without loss of generality we can assume that y 0 ∈ E (h) t (the other case can be treated analogously and we leave the details to the reader). It is then clear that B h α−1 /2 (y 0 ) ⊂ E (h) t and v (h) (t, y) > h α−1 /2 for every y ∈ B h α−1 /2 (y 0 ). Since h α−1 /2 < γ n h − 1 /2 , we can apply the density estimate in (3.11) and deduce that (4.14) Similarly, by the density estimate in (3.12) and Proposition 3. From (4.14), (4.15) and B h α−1 /2 (y 0 ) ⊂ B γn Finally, integrating in time we get where T > 0 is such that supp(φ) ⊂ [0, T ] × R n . Recalling the definition of α and taking the limit as h goes to 0, we conclude (4.1).

Appendix A. A density lemma
We premise the following density estimate for one-sided minimizers of the perimeter. The estimate can be easily deduce from the original arguments by De Giorgi exploited for minimizers [10].
A.0.1. Lemma. There exists a dimensional constant c n > 0 with this property. Let E ⊂ R n be a set of finite perimeter, R, µ > 0 and x 0 ∈ R n be such that Then, c n r n ≤ |B r (x 0 ) \ E| ∀ 0 < r < min R, µ −1 .

(A.2)
Proof. Without loss of generality, we can assume x 0 = 0. We use the following indentity which are true a.e. r > 0: Per(E ∪ B r ) = H n−1 (∂B r \ E) + Per(E, R n \ B r (x)), Finally, note that f (r) ≤ ω n r n , from which f (r) ≤ ω 1 /n n r f (r) n − 1 /n . Therefore, there exists a dimensional constant C n > 0 such that if 0 < r < min{R, C n µ −1 }, then the last term in (A.8) can be absorbed in the left hand side and deduce that f (r) n−1 n ≤ C f (r).