Flat flow solution to the mean curvature flow with volume constraint

In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai-Seis-Spadaro (2016). This method is based on the gradient flow approach due to Almgren-Taylor-Wang (1993) and Luckhaus-Strurzenhecker (1995) and my aim is to replace the volume penalization by implementing the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on the second variation condition of the energy.


Introduction
A smooth family of set (E t ) t≥0 is said to evolve according to volume preserving mean curvature flow if the normal velocity V t is proportional to the mean curvature H Et as where HEt = ffl ∂Et H Et dH n .Such a geometric equation has been proposed in the physical literature to model coarsening phenomena, where the system consisting on several subdomains evolve such that it decreases the interfacial area while keeping the total volume unchanged [7,18].From purely mathematical point of view the equation (1.1) can be seen as the L 2 -gradient flow of the surface area under the volume constraint [18].One has to be careful in this interpration as the Riemannian distance between two sets is in general degenerate [16].In order to overcome this one may use the idea due to Almgren-Taylor-Wang [3] and Luckhaus-Strürzenhecker [14] and to view (1.1) as the gradient flow of the surface area with respect to a different, non-degerate, distance.Using the gradient flow structure, one may then construct a discrete-in-time approximation to the solution of (1.1) via the Euler implicit method, also known as the minimizing movements scheme.By letting the time step to zero, one then obtains a candidate for a weak solution of (1.1) called flat flow, as the convergence is measured in terms of "flat norm".This method is implemented to the volme preserving setting in [19].
In [19] the authors observe that from technical point of view it is easier to replace the volume constraint of the problem with a volume penalization, as this simplifies certain regularity issues at the level of the discrete approximation.My aim here is to show that one may construct the flat flow Date: January 25, 2023.Let me quickly recall the discrete minimizing movements scheme for (1.1).One defines a sequence of sets (E h k ) k , with fixed time step h > 0, iteratively such that E h 0 = E 0 , where E 0 is the given initial set, and E h k+1 is a minimizer of the functional Here P (E) denotes the perimeter (generalized surface area) of the set E and dF is the signed distance function of the set F (see next section).One then defined an approximative flat flow solution to (1.1) (E h t ) t≥0 from the previous sequence by Any cluster point of (E h t ) t≥0 is then defined as flat flow solution to (1.1).The advantage is that such a solution is defined for all times and for rough initial data.The main result in the paper is the existence of a flat flow solution.
Theorem 1. Assume that E 0 ⊂ R n+1 is an open and bounded set with finite perimeter and let (E h t ) t≥0 be an approximative flat flow solution to (1.1) stating from E 0 (see Definition 2.1).Then there exists a family of bounded sets of finite perimeter (E t ) t≥0 and a subsequence h k → 0 such that and for every 0 < t < s it holds E t = E 0 , P (E t ) ≤ P (E 0 ) and where C depends on the dimension and on E 0 .Moreover, if the initial set E 0 is C 1,1 -regular, then any such limit flow (E t ) t≥0 agrees with the unique classical solution of (1.1) as long as the latter exists.
The above theorem thus provides the existence of a flat flow solution and guarantees that this notion is consistence with the classical solution when the initial set is regular enough.The disadvantage of the flat flow is that it is not clear if it provides a solution to the original equation (1.1) in any weak sense.However, the conditional result in the spirit of Luckhaus-Strürzenhecker [14] holds also in this case.
Theorem 2. Let (E h t ) t≥0 be an approximative flat flow solution to (1.1) and let (E h k t ) t≥0 be the converging subsequence in Theorem 1. Assume further that it holds lim Then for n ≤ 6 the flat flow (E t ) t≥0 is a distributional solution to (1.1) (see Definition 4.5).
One may also try to view the equation (1.1) as a mean curvature flow with forcing, where the forcing term depends on the flow itself.In this way one may try use different methods to construct a solution to the equation see e.g.[5,6] .I also refer the recent work [13] for a weak-strong uniqueness result related to (1.1).
As I already mentioned, the flat flow is defined for all times and one may study its asymptotical behavior.Indeed, by using the metods from [10,11] one may deduce the convergence of the flow in low dimensions.
Remark 1.1.Assume that E 0 ⊂ R n+1 , with n ≤ 2, is as in Theorem 1 and let (E t ) t≥0 be a limit flat flow.When n = 1, the flow E t converges to a union of disjoint balls exponentially fast and when n = 2 the flow converges to a union of disjoint balls up to a possible translation of the components.
The main technical challenge in proving Theorem 1 is to obtain the sharp density estimate for the discrete flow.This is also the main technical novelty of this paper.There are several techniques to deal with the volume constraint in variational problems, e.g. by using the argument from [2] (see also [15,Lemma 17.21]) or from [8] (see also [4,9]).However, due to the presence of the dissipation term in the energy it is not obvious how to apply these arguments in order to obtain sharp density estimates in terms of the time step h.I will use an argument which is based on the second variation condition of the energy to prove the density estimate in Proposition 3.1.After this the proof of Theorem 1 follows exactly as in [14,19] and the consistency follows almost directly using the argument in [12].The proof also provides the dissipation inequality and therefore the results in [10,11] hold and one obtains the result stated in Remark 1.1.Finally I would like to point out that this article is not self-consistence as several arguments are well-known, in particular, in Section 4.

Preliminaries
In this section I will briefly introduce the notation, the definition of the flat flow solution and recall some of its basic properties.
Given a set E ⊂ R n+1 the distance function dist(⋅, E) ∶ R n+1 → [0, ∞) is defined, as usual, as dist(x, E) ∶= inf y∈E x−y and denote the signed distance function by dE ∶ R n+1 → R, Then clearly it holds dist(⋅, ∂E) = dE .I denote the ball with radius r centered at x by B r (x) and by B r if it is centered at the origin.For a measurable set E ⊂ R n+1 the perimeter in an open set U ⊂ R n+1 is defined by and write P (E) = P (E, R n+1 ).If P (E) < ∞, then E is called a set of finite perimeter.For an introduction to the topic I refer to [15].
is defined via the divergence thereon such that for every test vector field I will consider a flat flow solution to (1.1) in the spirit of Almgren-Taylor-Wang [3] and Luckhaus-Stürzenhecker [14].To this this aim for a fixed h ∈ (0, 1) and a given (open) set F ⊂ R n+1 I define the functional The flat flow solution is defined analogously as in [19].
be an open and bounded set of finite perimeter and fix h ∈ (0, 1).Define the sequence of sets Moreover, define an approximative flat flow (E h t ) t≥0 for (1.1) starting from E 0 as One has to be carefull in the definition of the functional (2.1) if the set F is merely a set of finite perimeter as its value depends on the choice of the representative of F .One may overcome this by choosing a proper representative of the set F .In my case this is not necessary, as the regularity theorem below implies that one may in fact assume the sets E h k to be open.The difference in the Definition 2.1 to the scheme in [19] is that here the minimizing problem is under volume constraint.On one hand this makes the minimization problem more natural, but on the other hand, it makes the quantitative density estimates more difficult to prove.
For a given open and bounded set where F h (⋅, F ) is defined in (2.1).One may use an argument similar to [9] or [15,Lemma 17.21] to get rid of the volume constraint in (2.2) and deduce that a minimizer of (2.2) is a minimizer also for when Λ is chosen large.Note that the constant Λ may have unoptimal dependence on E and on h.However, the property (2.3) is enough deduce qualitative regularity properties since it implies that the minimizer inherits the regularity from the theory of the perimeter minimizers [15].One may also write the Euler-Lagrange equation and by standard calculations (see e.g.[1]) we have the second variation condition.We state this in the following proposition.
Proposition 2.2.Let F ⊂ R n+1 be an open and bounded set, fix h ∈ (0, 1) and let E be a minimizer of (2.2).Then E can be chosen to be open, which topological boundary is C 2,α -regular up to a relatively closed singular set which Hausdorff dimension is at most n − 7.In fact, the regular part is exactly the reduced boundary ∂ * E.
The Euler-Lagrange equation where λ ∈ R is the Lagrange-multiplier, holds point wise on ∂ * E and in a distributional sense on ∂E.The quadratic form associated with the second variation of the energy is non-negative, i.e., for all ϕ ∈ H 1 (∂ * E) with where B E (x) denotes the second fundamental form at x ∈ ∂ * E.
Proof.Since the argument is standard, I will only give the outline.As I already mentioned, the minimizer E is also a minimizer of the problem (2.3) for some large constant Λ, which depend on h and on E itself.This implies that the set E is a Λ-minimizer of the perimeter and thus the reduced boundary ∂ * E is relatively open, C 1,α -regular hypersurface and the singular set ∂E ∖ ∂ * E has dimension at most n − 7 [15].The C 2,α -regularity then follows from the Euler-Lagrange equation and from standard Schauder-estimates for elliptic PDEs.
One may obtain the second variation condition (2.5) by using the argument from [1].Indeed, given a function Then the inequality follows from the minimality of E as ∂ 2 ∂t 2 t=0 F h (Φ t (E), F ) ≥ 0 and following the standard calculation of the second variation (see e.g.[1]).Finally one obtains (2.5) for all ϕ ∈ H 1 (∂ * E) by approximation argument and by the fact that the singular set has zero capacity.

Density estimates
This section is the theoretical core of the paper.The aim is to prove the following density estimate.Proposition 3.1.Let F ⊂ R n+1 be an open and bounded set of finite perimeter, fix h ∈ (0, 1) and let E be a minimizer of (2.2).Then there is a constant c > 0, which depends on the dimension n, F and on P (F ) such that for all r ≤ √ h and all x ∈ ∂E it holds and for all r ≤ C 0 √ h, where C 0 ≥ 1, it holds where C 1 depends also on C 0 .Moreover, the following holds It is interesting that in [19,Corollary 3.3] the authors obtain similar result for their scheme for a constant which is independent of P (F ).
I need several lemmas in order to prove Proposition 3.1 and therefore I postpone its proof to the end of the section.Before proceeding to technical details, I state a useful consequence of Proposition 3.1.Proposition 3.2.Let F, E ⊂ R n+1 be as in Proposition 3.1.Then there are constants C ≥ 1, c > 0 and h 0 > 0, depending on the dimension, F and P (F ) such that E is (Λ, r)-minimizer of the perimeter for for h < h 0 .To be more precise, for sets Proof.The argument is standard but I recall it for the reader's convenience.
Let me first show that there is x ∈ E and c > 0 such that for ρ = c√ h it holds B ρ (x) ⊂ E. Fix ρ and apply Besicovich covering theorem to find disjoint balls I claim that for some i = 1, 2, . . ., N it holds B ρ 2 (x i ) ⊂ E. Indeed, if this is not the case then Proposition 3.1 implies for all i.Since the balls are disjoint, one has by the above and by (3.1) This is a contradiction when h is small enough.Fix x 0 ∈ ∂E and G as in the claim.Note that in general the set G does not have the same measure as E and one needs to modify it to G with G = E e.g. by using the argument from [9] as follows.Assume that G < E (the case G > E follows from similar argument).Since B ρ (x) ⊂ E, then by decreasing ρ and r if needed, it holds B ρ (x) ⊂ G.By continuity there is Then by the minimality of E and Proposition 3.1 it holds Arguing as in [9] one then deduces and the claim follows as G∆E ≤ 2 G∆E .
The first technical result which I need is the classical density estimate which can be found e.g. in [20].Lemma 3.3.Assume E ⊂ R n+1 is a set of finite perimeter with distributional mean curvature H E which satisfies H E L ∞ (B 2R (x 0 )) ≤ Λ.Then for all x ∈ B R (x 0 ) and r ≤ min{R, Λ −1 } it holds For a minimizer of (2.2) it holds the inverse of the isoperimetric inequality.Lemma 3.4.Let F and E be as in Proposition 3.1.Then for all x ∈ ∂E and r ≤ C 0 √ h it holds for a constant which depends on the dimension and on C 0 > 0. In, particular it holds P (E, B r (x)) ≤ Cr n .
Proof.Fix h ∈ (0, 1), x and r > 0 as in the claim and without loss of generality assume that x = 0.One may also consider only the case E ∩ B 2r ≤ B 2r ∖ E as the other case is similar.In particular, then it holds Consider first the set E 1 = E ∖ Bρ .In order to have a competing set with the volume of E, define ρ < ρ to be a radius such that B ρ = E ∩ B ρ and define Then it holds by construction that E 2 = E .By the minimality of E we have Estimate the perimeter of E 2 using (3.2) as Use then E∆E 2 ⊂ B 2r , E = E 2 and the fact that the signed distance function is 1-Lipschitz to estimate Therefore one obtains by combining the three above inequalities and r ≤ JULIN By Lemma 3.3 and Lemma 3.4 it is clear that for Proposition 3.1 it is crucial to prove the curvature estimate . The next lemma is a step towards this.Lemma 3.5.Let F, E and h be as in Proposition 3.1.Then it holds where the constant C 1 depends on the dimension and on F and P (F ).
Proof.The proof relies on the second variation inequality in Proposition 2.2.I would like to point out that in the case of the mean curvature flow, when there is no volume constraint, the proof is considerable easier as one could choose constant function in (2.5).In the volume preserving case I will choose a cut-off function for a test function.
To this aim use first [11,Proposition 2.3] (see also [17,Lemma 2.1]) to find a point x 0 ∈ R n+1 and a radius r ∈ (c, 1), where c = c(n, F , P (F )), such that Note that the minimality of E yields P (E) ≤ P (F ) ≤ C.Moreover, by the isoperimetric inequality it holds ≥ c.These estimates are used repeatedly from now on without mentioning.Without loss of generality assume that Since We repeat the same argument by defining a cut-off function

5) and arguing as above yields
and the claim follows.
The last lemma I need is a bound on the Lagrange multiplier in the Euler-Lagrange equation (2.4).Lemma 3.6.Let F, E and h be as in Proposition 3.1.Then for the Lagrange multiplier in (2.4), i.e., dF where the constant C 2 depends on the dimension, on F and on P (F ).
Proof.Let Λ ≥ 0 be such that λ = Λ √ h . Below all the constant depend on n, F and P (F ).I only treat the case when λ is positive as in the negative case the proof is the similar.Define the set I claim that we may choose Ĉ > 2 such that it depends on n, F , P (F ) and on C 1 from Lemma 3.5 and it holds Indeed, by the Euler-Lagrange equation (2.4) , by Lemma 3.5 and by Ĉ > 2 it holds Ĉ2 By choosing Ĉ large enough one then obtains H n (∂ * E ∖ Σ) < P (E) 2 and (3.6) follows.
By Besikovich covering theorem one finds disjoint balls of radius √ By the Euler-Lagrange equation (2.4) and by the definition of the set Σ it holds for all Therefore by Lemma 3.3 it holds On the other hand applying Lemma 3.4 first with r = √ h 2 yields and then with r = √ for all balls in the cover.The minimality of E implies 1 Note that by (3.8), by λ = Λ √ h and by the Euler-Lagrange equation (2.4) we have for all Therefore either Λ ≤ 4 Ĉ, in which case the claim follows trivially, or I assume the latter and show that even in this case Λ is bounded.Indeed, by the above discussion the balls B √ h (x i ) are in the exterior of F .Therefore we estimate by (3.7) and (3.9) that , the above and (3.10) gives a bound for Λ and the claim follows.
Here is the proof of the density estimate.
Proof of Proposition 3.1.By Lemma 3.3, Lemma 3.4, Lemma 3.6 and by the Euler-Lagrange equation (2.4) Argue by contradiction and assume that there is Without loss of generality assume that x 0 = 0 and consider only the case dF (x 0 ) > 0 as the other case is similar.By the Euler-Lagrange equation (2.4), by Lemma 3.3 and Lemma 3.6 it holds for r 0 = √ h 2Λ that P (E, B r 0 ) ≥ cr n 0 . (3.12) Define radii r k = k √ h + r 0 for k = 0, 1, . . ., n + 1.For every k = 0, 1, . . ., n + 1 choose a cut-off function as a test function in the second variation condition (2.5), use ⟨∇ dF , ν E ⟩ ≤ 1 and (3.5), and obtain (3.13) Since ζ k = 0 outside B r k+1 , one may estimate Moreover, since 0 ≤ ζ k ≤ 1 it holds When Combining the three above estimates with (3.13) yields For Λ large enough this implies

Existence of the flat flow
Now that the density estimates are proven the proof of Theorem 1 follows from the arguments from [14,19] without major changes.In this section I consider the approximative flat flow (E h t ) t≥0 and the associated sequence (E h k ) k≥0 as in the Definition 2.1 starting from an open and bounded set of finite perimeter E 0 .The proof for the following "interpolation" result can be found in [14, Using (4.2) and the argument [14, Lemma 2.1] (see also [19,Lemma 3.6]) one obtains the following dissipation inequality.Lemma 4.2.Let (E h t ) t≥0 be an approximative flat flow starting from E 0 and fix h ∈ (0, 1).Then for all T 2 > T 1 ≥ h it holds The constant depends on the dimension, E 0 and P (E 0 ).
Proof.I will only sketch the proof.Let (E h k ) be the sequence of sets associated with Once (4.3) is obtained, then the convergence of a subsequence E h k t → E t in measure follows as in [14,19].
Let j, k be such that s ∈ [jh, (j + 1)h) and t ∈ [(j + k)h, (j + k + 1)h).Then by applying Lemma 4.1 for l = h √ s−t and by (4.2) one obtains Since kh ≤ 2(s − t), one obtains (4.3).The proof of the consistency principle for C 1,1 -regular initial sets follows using the arguments in [12].The volume penalization is used only in [12, Lemma 3.2], but one may overcome this by using the lemma below.Lemma 4.4.Let F ⊂ R n+1 be an open and bounded set which satisfies interior and exterior ball condition with radius r 0 > 0 and let E be a minimizer of (2.2).There are ρ 0 and h 0 with the property that if G is a set of finite perimeter such that G∆E ⊂ B ρ (x) ∩ N C 0 h (∂F ), for ρ ≤ ρ 0 and h ≤ h 0 where N C 0 h (∂F ) = {x ∶ dist(x, ∂F ) < C 0 h}, then it holds P (E) ≤ P (G) + Cρ n+1 .
Above the constant depends on the dimension, on r 0 , C 0 , F and P (F ).
Proof.By approximation one may assume G to be smooth.Since F satisfies interior and exterior ball condition, then by [ The paper concludes with Theorem 2. To this aim I recall the definition of a distributional solution of (1.1) from [14].Definition 4.5.Family of sets of finite perimeter (E t ) t≥0 is a distributional solution to (1.1) starting from E 0 ⊂ R n+1 if the following holds: (1) for almost every t > 0 the set E t has mean curvature H Et in a distributional sense and for every T > 0

1 2 JULIN
solution to (1.1) by implementing the volume constraint in the minimizing movements scheme directly and thus avoid the volume penalization.

12 ,
Lemma 3.1] it holds max x∈E∆F dist(x, ∂F ) ≤ Ch (4.4)when h ≤ h 0 .As in the proof of Proposition 3.2 the set G does not have the same measure as E and modifies it to G with G = E .Assume again that G < E .Since F satisfies interior ball condition with radius r 0 , there is y ∈ G such that B r 0 2 (y) ⊂ G.By continuity there is z ∈ R n+1 such that z−x ≥ 2ρ 0 and G∪B r 0 2 (z) = E when ρ 0 is small.Define G = G∪B r 0 2 (z).Then by the minimality of E and by (4.4) and by the assumptionG∆E ⊂ B ρ (x) ∩ N C 0 h (∂F ) it holds P (E) ≤ P ( G) + C G∆E ≤ P ( G) + Cρ n+1 .Arguing as in[9] one then deducesP ( G) − P (G) ≤ H n (∂B r 0 2 (z) ∖ G) − H n (∂G ∩ B r 0 2 (z)) ≤ 2(n + 1) B 1 r 0 B r 0 2 (z) ∖ G ≤ C G∆E ≤ Cρ n+1 .and the claim follows.
Λ is large enough, then for all x ∈ ∂ * E ∩ B r k+1 and all k ≤ n + 1 it holds dF (x) ≥ Λ Lemma 1.5].Lemma 4.1.Let (E h t ) t≥0 be an approximative flat flow starting from E 0 and fix h ∈ (0, 1) and t > h.Then for all l ≤ √ E h t and in a distributional sense on ∂E h t .Here λ t,h is the Lagrange multiplier.Using the minimality of E h t against the previous set E h t−h one obtains the important inequality * [19,f of Theorem 2. The proof is exactly the same as[19, Theorem 2.3].Note that Proposition 3.2 implies that the sets E h t are (Ch −1 2 , c √ h)-minimizers of the perimeter, i.e., for every F with F ∆E h t ⊂ B c