An existence theorem for Brakke flow with fixed boundary conditions

Consider an arbitrary closed, countably n-rectifiable set in a strictly convex (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}-dimensional domain, and suppose that the set has finite n-dimensional Hausdorff measure and the complement is not connected. Starting from this given set, we show that there exists a non-trivial Brakke flow with fixed boundary data for all times. As t↑∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \uparrow \infty $$\end{document}, the flow sequentially converges to non-trivial solutions of Plateau’s problem in the setting of stationary varifolds.

named author in [20] by reworking [2] thoroughly. The major challenge of the present work is to devise a modification to the approximation scheme in [20] which preserves the boundary data.
Though somewhat technical, in order to clarify the setting of the problem at this point, we state the assumptions on the initial surface 0 and the domain U hosting its evolution. Their validity will be assumed throughout the paper. Assumption 1.1 Integers n ≥ 1 and N ≥ 2 are fixed, and clos A denotes the topological closure of A in R n+1 .
Since N ≥ 2, we implicitly assume that U \ 0 is not connected. When n = 1, 0 could be for instance a union of Lipschitz curves joined at junctions, with "labels" from 1 to N being assigned to each connected component of U \ 0 . If one defines F i := (clos E 0,i ) \ (U ∪ ∂ 0 ) for i = 1, . . . , N , one can check that each F i is relatively open in ∂U , F 1 , . . . , F N are mutually disjoint, and The assumption (A4) is equivalent to the requirement that each x ∈ ∂ 0 is in ∂ F i 1 ∩ ∂ F i 2 for some indices i 1 = i 2 . The main result of the present paper can then be roughly stated as follows. For all t > 0, (t) remains within the convex hull of 0 ∪ ∂ 0 .
More precisely, { (t)} t≥0 is a MCF in the sense that (t) coincides with the slice, at time t, of the space-time support of a Brakke flow {V t } t≥0 starting from 0 . The method adopted to produce the evolving generalized surfaces (t) actually gives us more. Indeed, we show the existence of N families {E i (t)} t≥0 (i = 1, . . . , N ) of evolving open sets such that E i (0) = E 0,i for every i, and (t) = U \ ∪ N i=1 E i (t) for all t ≥ 0. At each time t ≥ 0, the sets E 1 (t), . . . , E N (t) are mutually disjoint and form a partition of U . Moreover, for each fixed i the Lebesgue measure of E i (t) is a continuous function of time, so that the evolving (t) do not exhibit arbitrary instantaneous loss of mass. See Theorems 2.2 and 2.3 for the full statement.
It is reasonable to expect that the flow (t) converges, as t → ∞, to a minimal surface in U with boundary ∂ 0 . We are not able to prove such a result in full generality; nonetheless, we can show the following Theorem B There exists a sequence of times {t k } ∞ k=1 with lim k→∞ t k = ∞ such that the corresponding varifolds V k := V t k converge to a stationary integral varifold V ∞ in U such that (clos (spt V ∞ )) \ U = ∂ 0 .
See Corollary 2.4 for a precise statement. The limit V ∞ is a solution to Plateau's problem with boundary ∂ 0 , in the sense that it has the prescribed boundary in the topological sense specified above and it is minimal in the sense of varifolds. We warn the reader that V ∞ may not be area-minimizing. Furthermore, the flow may converge to different limit varifolds along different diverging sequences of times in all cases when uniqueness of a minimal surface with the prescribed boundary is not guaranteed. The possibility to use Brakke flow in order to select solutions to Plateau's problem in classes of varifolds seems an interesting byproduct of our theory. See Sect. 7 for further discussion on these points.
Next, we discuss closely related results. While there are several works on the global-intime existence of MCF, there are relatively few results on the existence of MCF with fixed boundary conditions. When 0 is a smooth graph over a bounded domain in R n , globalin-time existence follows from the classical work of Lieberman [25]. Furthermore, under the assumption that is mean convex, convergence of the flow to the unique solution to the minimal surfaces equation in with the prescribed boundary was established by Huisken in [16]; see also the subsequent generalizations to the Riemannian setting in [31,34]. The case of network flows with fixed endpoints and a single triple junction was extensively studied in [28,30]. For other configurations and related works on the network flows, see the survey paper [29] and references therein. In the case when N = 2 (which does not allow triple junctions in general), a powerful approach is the level set method [4,10]. Existence and uniqueness in this setting were established in [35], and the asymptotic limit as t → ∞ was studied in [18]. Recently, White [39] proved the existence of a Brakke flow with prescribed smooth boundary in the sense of integral flat chains mod (2). The proof uses the elliptic regularization scheme discovered by Ilmanen [17], which allows one to obtain a Brakke flow with additional good regularity and compactness properties; see also [32] for an application of elliptic regularization within the framework of flat chains with coefficients in suitable finite groups to the long-time existence and short-time regularity of unconstrained MCF starting from a general surface cluster. Observe that the homological constraint used by White prevents the flow to develop interior junction-type singularities of odd order (namely, junctions which are locally diffeomorphic to the union of an odd number of half-hyperplanes), because these singularities are necessarily boundary points mod (2). As a consequence, the flows obtained in [39] may differ greatly from those produced in the present paper. This is not surprising, as solutions to Brakke flow may be highly non-unique. A complete characterization of the topological changes that the evolving surfaces can undergo with either of the two approaches is, in fact, an interesting open question. It is worth noticing that analogous generic nonuniqueness holds true also for Plateau's problem: in that context, different definitions of the key words surfaces, area, spanning in its formulation lead to solutions with dramatically different regularity properties, thus making each model a better or worse predictor of the geometric complexity of physical soap films; see e.g. the survey papers [6,15] and the references therein, as well as the more recent works [7][8][9][22][23][24]27]. It is then interesting and natural to investigate different formulations for Brakke flow as well.

Basic notation
The ambient space we will be working in is Euclidean space R n+1 . We write R + for [0, ∞). For A ⊂ R n+1 , clos A (or A) is the topological closure of A in R n+1 (and not in U ), int A is the set of interior points of A and conv A is the convex hull of A. The standard Euclidean inner product between vectors in R n+1 is denoted x · y, and |x| := √ x · x. If L, S ∈ L (R n+1 ; R n+1 ) are linear operators in R n+1 , their (Hilbert-Schmidt) inner product is L · S := trace(L T • S), where L T is the transpose of L and • denotes composition.
The corresponding (Euclidean) norm in L (R n+1 ; R n+1 ) is then |L| := √ L · L, whereas the operator norm in L (R n+1 ; R n+1 ) is L := sup |L(x)| : x ∈ R n+1 with |x| ≤ 1 . If u, v ∈ R n+1 then u ⊗ v ∈ L (R n+1 ; R n+1 ) is defined by (u ⊗ v)(x) := (x · v) u, so that u ⊗ v = |u| |v|. The symbol U r (x) (resp. B r (x)) denotes the open (resp. closed) ball in R n+1 centered at x and having radius r > 0. The Lebesgue measure of a set A ⊂ R n+1 is denoted L n+1 (A) or |A|. If 1 ≤ k ≤ n + 1 is an integer, U k r (x) denotes the open ball with center x and radius r in R k . We will set ω k := L k (U k 1 (0)). The symbol H k denotes the k-dimensional Hausdorff measure in R n+1 , so that H n+1 and L n+1 coincide as measures.
A Radon measure μ in U ⊂ R n+1 is always also regarded as a linear functional on the space C c (U ) of continuous and compactly supported functions on U , with the pairing denoted μ(φ) for φ ∈ C c (U ). The restriction of μ to a Borel set A is denoted μ A , so that (μ A )(E) := μ(A ∩ E) for any E ⊂ U . The support of μ is denoted spt μ, and it is the relatively closed subset of U defined by spt μ := {x ∈ U : μ(B r (x)) > 0 for every r > 0} .
The upper and lower k-dimensional densities of a Radon measure μ at x ∈ U are θ * k (μ, x) := lim sup respectively. If θ * k (μ, x) = θ k * (μ, x) then the common value is denoted θ k (μ, x), and is called the k-dimensional density of μ at x. For 1 ≤ p ≤ ∞, the space of p-integrable (resp. locally p-integrable) functions with respect to μ is denoted L p (μ) (resp. L p loc (μ)). For a set E ⊂ U , χ E is the characteristic function of E. If E is a set of finite perimeter in U , then ∇χ E is the associated Gauss-Green measure in U , and its total variation ∇χ E in U is the perimeter measure; by De Giorgi's structure theorem, ∇χ E = H n ∂ * E , where ∂ * E is the reduced boundary of E in U .

Varifolds
The symbol G(n + 1, k) will denote the Grassmannian of (unoriented) k-dimensional linear planes in R n+1 . Given S ∈ G(n + 1, k), we shall often identify S with the orthogonal projection operator onto it. The symbol V k (U ) will denote the space of k-dimensional varifolds in U , namely the space of Radon measures on G k (U ) := U × G(n + 1, k) (see [1,33] for a comprehensive treatment of varifolds). To any given V ∈ V k (U ) one associates a Radon measure V on U , called the weight of V , and defined by projecting V onto the first factor in G k (U ), explicitly: for every φ ∈ C c (U ) .
A set ⊂ R n+1 is countably k-rectifiable if it can be covered by countably many Lipschitz images of R k into R n+1 up to a H k -negligible set. We say that is (locally) H k -rectifiable if it is H k -measurable, countably k-rectifiable, and H k ( ) is (locally) finite. If ⊂ U is locally H k -rectifiable, and θ ∈ L 1 loc (H k ) is a positive function on , then there is a k-varifold canonically associated to the pair ( , θ ), namely the varifold var( , θ ) defined by where T x denotes the approximate tangent plane to at x, which exists H k -a.e. on . Any varifold V ∈ V k (U ) admitting a representation as in (2.1) is said to be rectifiable, and the space of rectifiable k-varifolds in U is denoted by RV k (U ). If V = var( , θ ) is rectifiable and θ(x) is an integer at H k -a.e. x ∈ , then we say that V is an integral k-dimensional varifold in U : the corresponding space is denoted IV k (U ).

First variation of a varifold
If V ∈ V k (U ) and f : U → U is C 1 and proper, then we let f V ∈ V k (U ) denote the push-forward of V through f . Recall that the weight of f V is given by where is the Jacobian of f along S ∈ G(n + 1, k). Given a varifold V ∈ V k (U ) and a vector field g ∈ C 1 c (U ; R n+1 ), the first variation of V in the direction of g is the quantity where t (·) = (t, ·) is any one-parameter family of diffeomorphisms of U defined for sufficiently small |t| such that 0 = id U and ∂ t (0, ·) = g(·). TheŨ is chosen so that closŨ ⊂ U is compact and spt g ⊂Ũ , and the definition of (2.3) does not depend on the choice ofŨ . It is well known that δV is a linear and continuous functional on C 1 c (U ; R n+1 ), and in fact that where, after identifying S ∈ G(n + 1, k) with the orthogonal projection operator R n+1 → S, If δV can be extended to a linear and continuous functional on C c (U ; R n+1 ), we say that V has bounded first variation in U . In this case, δV is naturally associated with a unique R n+1 -valued measure on U by means of the Riesz representation theorem. If such a measure is absolutely continuous with respect to the weight V , then there exists a V -measurable and locally V -integrable vector field h(·, V ) such that by the Lebesgue-Radon-Nikodým differentiation theorem. The vector field h(·, V ) is called the generalized mean curvature vector of V . In particular, if δV (g) = 0 for all g ∈ C 1 c (U ; R n+1 ), V is called stationary, and this is equivalent to h(·, V ) = 0 V -almost everywhere. For any V ∈ IV k (U ) with bounded first variation, Brakke's perpendicularity theorem [2,Chapter 5] says that Here, S ⊥ is the projection onto the orthogonal complement of S in R n+1 . This means that the generalized mean curvature vector is perpendicular to the approximate tangent plane almost everywhere.
Other than the first variation δV discussed above, we shall also use a weighted first variation, defined as follows. For where t denotes the one-parameter family of diffeomorphisms of U induced by g as above.
Proceeding as in the derivation of (2.4), one then obtains the expression If δV has generalized mean curvature h(·, V ), then we may use (2.5) in (2.9) to obtain (2.10) The definition of Brakke flow requires considering weighted first variations in the direction of the mean curvature. Suppose V ∈ IV k (U ), δV is locally bounded and absolutely continuous with respect to V and h(·, V ) is locally square-integrable with respect to V . In this case, it is natural from the expression (2.10) to define for φ ∈ C 1 Observe that here we have used (2.6) in order to replace the term h(

Brakke flow
To motivate a weak formulation of the MCF, note that a smooth family of k-dimensional surfaces { (t)} t≥0 in U is a MCF if and only if the following inequality holds true for all −φ |h(·, (t))| 2 + ∇φ · h(·, (t)) + ∂φ ∂t dH k . (2.12) In fact, the "only if" part holds with equality in place of inequality. For a more comprehensive treatment of the Brakke flow, see [38,Chapter 2]. Formally, if ∂ (t) ⊂ ∂U is fixed in time, with φ = 1, we also obtain |h(x, (t))| 2 dH k (x) , (2.13) which states the well-known fact that the L 2 -norm of the mean curvature represents the dissipation of area along the MCF. Motivated by (2.12) and (2.13), and for the purposes of this paper, we give the following definition.
In this paper, we are interested in the n-dimensional Brakke flow in particular. Formally, by integrating (2.13) from 0 to T , we obtain the analogue of (2.14). By integrating (2.12) from t 1 to t 2 , we also obtain the analogue of (2.15) via the expression (2.11). We recall that the closure is taken with respect to the topology of R n+1 while the support of V t is in U . Thus (e) geometrically means that "the boundary of V t (or V t ) is ".

Main results
The main existence theorem of a Brakke flow with fixed boundary is the following.
Since we are assuming that ∂ 0 = ∅, we have V t = 0 for all t > 0. If the union of the reduced boundaries of the initial partition in U coincides with 0 modulo H n -negligible sets (note that the assumptions (A2) and (A3) in Assumption 1.1 imply that 0 = U ∩ N i=1 ∂ E 0,i ), then the claim is that the initial condition is satisfied continuously as measures. Otherwise, an instantaneous loss of measure may occur at t = 0. As far as the regularity is concerned, under the additional assumption that {V t } t>0 is a unit density flow, partial regularity theorems of [2,19,37] show that V t is a smooth MCF for a.e. time and a.e. point in space, just like [20], see [20,Theorem 3.6] for the precise statement. No claim of the uniqueness is made here, but the next Theorem 2.3 gives an additional structure to V t in the form of "moving partitions" starting from E 0,1 , . . . , E 0,N .
Then, ∀t > 0, we have The claims (1) is an L n+1 -partition of U , and that (t) has empty interior in particular. The claim (5) is an expected property for the MCF, and, by (11), spt V t is also in the same convex hull. (7) says that (t) has the fixed boundary ∂ 0 . In general, the reduced boundary of the partition and V t may not match, but the latter is bounded from below by the former as in (8). By (10), the Lebesgue measure of each E i (t) changes continuously in time, so that arbitrary sudden loss of measure of V t is not allowed. The statement in (11) says that the time-slice of the support of μ at time t contains the support of V t and is equal to the topological boundary of the moving partition.
As a corollary of the above, we deduce the following.

Corollary 2.4
There exist a sequence {t k } ∞ k=1 with lim k→∞ t k = ∞ and a varifold V ∈ IV n (U ) such that V t k → V in the sense of varifolds. The varifold V is stationary. Furthermore, there is a mutually disjoint family The varifold V in Corollary 2.4 is a solution to Plateau's problem in U in the class of stationary varifolds satisfying the topological constraint (clos (spt V ))\U = ∂ 0 . This is an interesting byproduct of our construction, above all considering that ∂ 0 enjoys in general rather poor regularity (in particular, it may have infinite (n − 1)-dimensional Hausdorff measure, and also it may not be countably (n − 1)-rectifiable). Even though the topological boundary condition specified above seems natural in this setting, other notions of spanning may be adopted: for instance, in Proposition 7.4 we show that a strong homotopic spanning condition in the sense of [7,14] is preserved along the flow and in the limit if it is satisfied at the initial time t = 0. We postpone further discussion and questions concerning the application to Plateau's problem to Sect. 7.

General strategy and structure of the paper
The general idea behind the proof of Theorems 2.2 and 2.3 is to suitably modify the time-discrete approximation scheme introduced in [2,20]. There, one constructs a timeparametrized flow of open partitions which is piecewise constant in time. We will call epoch any time interval during which the approximating flow is constant. The open partition at a given epoch is constructed from the open partition at the previous epoch by applying two operations, which we call steps. The first step is a small Lipschitz deformation of partitions with the effect of "regularizing singularities" by "locally minimizing the area of the boundary of partitions" at a small scale. This deformation is defined in such a way that, if the boundary of partitions is regular (relative to a certain length scale), then the deformation reduces to the identity. The second step consists of flowing the boundary of partitions by a suitably defined "approximate mean curvature vector". The latter is computed by smoothing the surface measures via convolution with a localized heat kernel. Note that, typically, the boundary of open partitions has bounded n-dimensional measure, but the unit-density varifold associated to it may not have bounded first variation. In [20], a time-discrete approximate MCF is obtained by alternating these two steps, epoch after epoch. In the present work, we need to fix the boundary ∂ 0 . The rough idea to achieve this is to perform an "exponentially small" truncation of the approximate mean curvature vector near ∂ 0 , so that the boundary cannot move in the "polynomial time scale" defining an epoch with respect to a certain length scale. We also need to make sure that the time-discrete movement does not push the boundary of open partitions to the outside of U . To prevent this, in addition to the two steps (Lipschitz deformation and motion by smoothed and truncated mean curvature vector), we add another "retraction to U " step to be performed in each epoch. All these operations have to come with suitable estimates on the surface measures, in order to have convergence of the approximating flow when we let the epoch time scale approach zero. The final goal is to show that this limit flow is indeed a Brakke flow with fixed boundary ∂ 0 as in Definition 2.1.
The rest of the paper is organized as follows. Section 3 lays the foundations to the technical construction of the approximate flow by proving the relevant estimates to be used in the Lipschitz deformation and flow by smoothed mean curvature steps, and by defining the boundary truncation of the mean curvature. Both the discrete approximate flow and its "vanishing epoch" limit are constructed in Sect. 4. In Sect. 5 we show that the one-parameter family of measures obtained in the previous section satisfies conditions (a) to (d) in Definition 2.1. The boundary condition (e) is, instead, proved in Sect. 6, which therefore also contains the proofs of Theorems 2.2 and 2.3. Finally, Sect. 7 is dedicated to the limit t → ∞: hence, it contains the proof of Corollary 2.4, as well as a discussion of related results and open questions concerning the application of our construction to Plateau's problem.

Preliminaries
In this section we will collect the preliminary results that will play a pivotal role in the construction of the time-discrete approximate flows. Some of the results are straightforward adaptations of the corresponding ones in [20]: when that is the case, we shall omit the proofs, and refer the reader to that paper.

Classes of test functions and vector fields
Define, for every j ∈ N, the classes A j and B j as follows: The properties of functions φ ∈ A j and vector fields g ∈ B j are precisely as in [20,Lemma 4.6,Lemma 4.7], and we record them in the following lemma for future reference. Lemma 3.1 Let x, y ∈ R n+1 and j ∈ N. For every φ ∈ A j , the following properties hold: Also, for every g ∈ B j :

Open partitions and admissible functions
LetŨ ⊂ R n+1 be a bounded open set. Later,Ũ will be an open set which is very close to U in Assumption 1.1.
The set of all open partitions ofŨ of N elements will be denoted OP N (Ũ ).
Note that some of the E i may be empty. Condition (b) implies that and thus that N i=1 ∂ E i is H n -rectifiable and each E i is in fact an open set with finite perimeter inŨ . By De Giorgi's structure theorem, the reduced boundary ∂ * E i is H n -rectifiable: nonetheless, the reduced boundary ∂ * E i may not coincide in general with the topological boundary ∂ E i , which makes condition (c) not redundant. We keep the following for later use. The proof is straightforward.
Notation Given E ∈ OP N (Ũ ), we will set Here, to avoid some possible confusion, we emphasize that we want to consider ∂E as a varifold on R n+1 when we construct approximate MCF. On the other hand, note that we still consider the relative topology ofŨ , as ∂ E i ⊂Ũ here. In particular, writing = ∪ N i=1 ∂ E i , we have ∂E = H n , and where T x ∈ G(n + 1, n) is the approximate tangent plane to at x, which exists and is unique at H n -a.e. x ∈ because of Definition 3.2(c).

Definition 3.4 Given
be an open partition ofŨ in N elements, C ⊂⊂Ũ , and let f be E-admissible in C. If we defineẼ : Proof We check thatẼ satisfies properties (a)-(c) in Definition 3.2. By Definition 3.4(a) and (b), it is clear thatẼ 1 , . . . ,Ẽ N are open and mutually disjoint subsets ofŨ , which gives (a). In order to prove (b), we use Definition 3.4(c) and the area formula to compute: where we have used Definition 3.2(b) and (3.7). This also showsŨ Since any subset of a countably n-rectifiable set is countably n-rectifiable, also N i=1 ∂Ẽ i is countably n-rectifiable.
then the open partitionẼ ∈ OP N (Ũ ) will be denoted f E.

Area reducing Lipschitz deformations
, j ∈ N and a closed set C ⊂⊂Ũ , define E(E, C, j) to be the set of all E-admissible functions f in C such that: is the symmetric difference of the sets E and F; and ∂E is the weight of the multiplicity one varifold associated to the open partition E.
The set E(E, C, j) is not empty, as it contains the identity map. Definition 3.7 Given E ∈ OP N (Ũ ) and j, and given a closed set C ⊂⊂Ũ , we define Observe that it always holds j ∂E (C) ≤ 0, since the identity map f (x) = x belongs to E(E, C, j). The quantity j ∂E (C) measures the extent to which ∂E can be reduced by acting with area reducing Lipschitz deformations in C.

Smoothing of varifolds and first variations
We let ψ ∈ C ∞ (R n+1 ) be a radially symmetric function such that and we define, for each ε ∈ (0, 1), where the constant c(ε) is chosen in such a way that The function ε will be adopted as a convolution kernel for the definition of the smoothing of a varifold. We record the properties of ε in the following lemma (cf. [20,Lemma 4.13]).

Lemma 3.8
There exists a constant c = c(n) such that, for ε ∈ (0, 1), we have: Next, we use the convolution kernel ε in order to define the smoothing of a varifold and its first variation. Recall that, given a Radon measure μ on R n+1 , the smoothing of μ by means of the kernel ε is defined to be the Radon measure ε * μ given by The definition of smoothing of a varifold V is the equivalent of (3.15) when regarding V as a Radon measure on G n (R n+1 ), keeping in mind that the operator ( ε * ) acts on a test function ϕ ∈ C c (G n (R n+1 )) by convolving only the space variable. Explicitly, we give the following definition.

Definition 3.9 Given
Observe that, given a Radon measure μ on R n+1 , one can identify the measure ε * μ with a C ∞ function by means of the Hilbert space structure of These considerations suggest the following definition for the smoothing of the first variation of a varifold.

Definition 3.10 Given
in such a way that Proof The identities (3.19) and (3.20) are proved in [20,Lemma 4.16]. Concerning (3.21), we observe that for any Taking the supremum among all functions ϕ ∈ C c (G n (R n+1 )) with ϕ 0 ≤ 1 completes the proof.

Smoothed mean curvature vector
(3.22) We will often make use of [20, Lemma 5.1] with ≡ 1 (and c 1 = 0). For the reader's convenience, we provide here the statement.

The cut-off functions Á j
In this subsection we construct the cut-off functions which will later be used to truncate the smoothed mean curvature vector in order to produce time-discrete approximate flows which almost preserve the boundary ∂ 0 . Given a set E ⊂ R n+1 and s > 0, (E) s denotes the s-neighborhood of E, namely the open set We shall also adopt the convention that (E) 0 = E.
Let U and 0 be as in Assumption 1.1.
Definition 3.14 We define for j ∈ N: Observe that D j is not empty for all j sufficiently large (depending on U ). Also, we define the sets
Next, we prove (2). Let x ∈K j , so that there exists z ∈ 0 \ D j such that |x − z| < 2 j − 1 /4 . If y ∈ B ρ j (x), then |y − z| < 3 j − 1 /4 by the definition of ρ j , and thus, for j suitably large, Hence, by property (a) of ψ in Definition 3.15: In particular, up to taking larger values of j, we see that Finally, we prove (3). To this aim, we compute the gradient of η j : at any point x, we have Using that t = ψ(t) for 0 ≤ t ≤ 1/2, ψ (t) = 0 for t ≥ 3/2, and that |t| = t ≤ 2 ψ(t) for t ∈ [1/2, 3/2], together with the fact that |ψ | ≤ 1, we can estimate where we have used that ∇d j (x) = φ ρ j * ∇d j (x), so that In particular, |∇η j | ≤ j 3 /4 η j as soon as j ≥ 4. Next, we compute the Hessian of η j from which we estimate Now, observe that Hence, recalling that ρ j = j − 1 /4 , we conclude the estimate for a constant C depending only on n. Thus, we conclude η j ∈ A j 3 /4 for j sufficiently large.

L 2 approximations
In this subsection, we collect a few estimates of the error terms deriving from working with smoothed first variations and smoothed mean curvature vectors. They will be critically important to deduce the convergence of the discrete approximation algorithm. The first estimate is a modification of [20,Proposition 5.3]. We let η j be the cut-off function as in Definition 3.15, corresponding to U and 0 , and we will suppose that j ≥ J (n), in such a way that the conclusions of Lemma 3.16 are satisfied.

Proposition 3.17
For every M > 0, there exists ε 2 ∈ (0, 1) depending only on n and M such that the following holds. For Given the validity of (3.18), we see that (3.32) measures the deviation from the identity (2.5). The difference with [20,Proposition 5.3] is that there, in place of η j g (left-hand side of (3.32)) and η j (right-hand side of (3.32)), we have g and , respectively. We note that g η j : using these, the modification of the proof is straightforward, and thus we omit the details.

Proposition 3.18
There exists a constant ε 3 ∈ (0, 1) depending only on n and M with the following property. Given Note that formula (3.33) estimates the deviation from the identity (2.5) with g = h(·, V ). The next statement is [20,Proposition 5.5]. The proof is a straightforward modification, using (3.32).

Proposition 3.19
For every M > 0, there exists ε 4 ∈ (0, 1) depending only on n and M with the following property. For (3.35)

Curvature of limit varifolds
The next Proposition 3.20 corresponds to [20,Proposition 5.6] when there is no boundary.
Proof By (1), we may choose a (not relabeled) subsequence V j converging to V as varifolds on R n+1 , and we may assume that the integrals in (2) for this subsequence converge to the lim inf of the original sequence. Fix g ∈ C 2 c (U ; R n+1 ). For all sufficiently large , we have g η j = g due to Lemma 3.16 (1), (3.27) and (3.26). Moreover, we may assume that g η j ∈ B j due to Lemma 3.16 (3). Then, by (3.35), (2) and (3), we have Since η j ∈ A j in particular, by the Cauchy-Schartz inequality and (3.34), we have This shows that δV is absolutely continuous with respect to V on U and h(·, V ) satisfies Given φ ∈ C 2 c (U ; R + ) (C c case is by approximation), let i ∈ N be arbitrary and consider φ := φ + i −1 . For all sufficiently large , we have g η j φ ∈ B j and η j φ ∈ A j (we may assume |φ| < 1 without loss of generality). Thus the same computation above with g η j φ yields We let then i → ∞ in (3.40) to replaceφ by φ, and finally we approximate h(·, V ) by g to obtain (3.36).

Motion by smoothed mean curvature with boundary damping
We aim at proving the following proposition: it contains the perturbation estimates for a varifold V which is moved by a vector field consisting of a boundary damping of its smoothed mean curvature for a time t. Proposition 3.21 There exists ε 5 ∈ (0, 1), depending only on n, M and U such that the following holds. Suppose that: Then, for every φ ∈ A j we have the following estimates.
Proof. We want to estimate the following quantity which can be written as with Choose ε 5 ≤ min{ε 1 , ε 3 }, so that the conclusions of Lemma 3.13 and Proposition 3.18 hold with ε ∈ (0, ε 5 ). In order to estimate the size of the various integrands appearing in the definition of I 1 , I 2 and I 3 , we first observe that, by (3.23) and our assumption on t, (3.45) Furthermore, using (3.23), (3.24), (3.31), and the fact that η j ∈ A j we obtain Since φ ∈ A j , we can use the results of Lemma 3.1 to estimate: for any orthonormal basis {v 1 , . . . , v n } of S, we can Taylor expand the tangential Jacobian and deduce the estimates modulo choosing a smaller value of ε if necessary. Putting all together, we can finally conclude the proof of (3.41): In order to prove (3.42), we use (3.41) with φ(x) ≡ 1, which implies that On the other hand, since η j ∈ A j we can apply (3.33) to further estimate so that (3.42) follows by choosing ε so small that 1 − ε 1 /4 ≥ 1/4. Finally, we turn to the proof of (3.43) and (3.44). In order to simplify the notation, let us writeV instead of f V . Using the same strategy as in [20, Proof of Proposition 5.7], we can estimate The first term can be estimated by observing that for some pointŷ on the segment and using that because of (3.49), so that Concerning the second term in the sum, we can use (3.49) again to estimate Putting the two estimates together, we see that (3.54) Analogous calculations lead to The rough estimates also give The estimates (3.54), (3.55), and (3.56) immediately yield as well as (3.58) Observe that, since spt V ⊂ (U ) 1 , the right-hand side of estimates (3.57) and (3.58) is zero whenever dist(x, clos(U )) > 3. Hence, (3.58) and the monotonicity of the mass by possibly choosing a smaller value of ε (depending on U and M). This proves (3.44). Finally, we prove (3.43). By (3.22), (3.57), and the properties of ε , we deduce that for l = 0, 1, 2. We can conclude using (3.59), (3.45)-(3.49) and suitable interpolations that:

The construction of the approximate flows
Suppose U and 0 are as in Assumption 1.1. Together with the sets D j , K j ,K j ,K j introduced in Definition 3.14, for k = 0, 1, . . ., we set Once again, here the indices j and k are chosen in such a way that the corresponding sets D j,k are non-empty proper subsets of U . Observe that we have the elementary inclusions D j,0 ⊂ D j,k ⊂ D j,k for every 0 ≤ k ≤ k , and that D j ⊂ D j,k for every k.
Before proceeding with the construction of the time-discrete approximate flows, we need to introduce a suitable new class of test functions. Since U is an open and bounded convex domain with boundary ∂U of class C 2 , there exists a neighborhood (∂U ) s 0 such that, denoting is monotone non decreasing for t such that (4.1) The following proposition and its proof contain the constructive algorithm which produces the time-discrete approximations of our Brakke flow with fixed boundary.
, and 0 be as in Assumption 1.1. There exists a positive integer J = J (n) with the following property. For every j ≥ J (n), there exist ε j ∈ (0, 1) satisfying (3.31), p j ∈ N, and, for every

2)
and such that, setting t j := 2 − p j , and defining j,k := U j,k \ N i=1 E j,k,i , the following holds true: Moreover, we have: for every k ∈ {1, . . . , j 2 p j } and φ ∈ A j ∩ R j .
The set A j,k is a relatively open subset of ∂(D j,k−1 ) j −10 . Let A j,k,l ⊂ A j,k be any of the (at most countably many) connected components of A j,k and define Ret j,k,l := {r s (x) : x ∈ A j,k,l , s ∈ (0, 1)}.
Proof The claim follows directly from Lemma 4.3.
Lemma 4.5 implies that for each l there exists some i(l) ∈ {1, . . . , N } such that E j,k,i(l) contains A j,k,l ∪ (∂ A j,k,l ) j −10 . For each index l, let i(l) be this correspondence. We define

In other words, when
Proof Note that˜ j,k ∩ Ret j,k \ D j,k−1 = ∅ since ∂Ret j,k \ D j,k−1 is contained in some open partition by Lemma 4.5 and˜ j,k ∩ Ret j,k = ∅. If there exists x ∈˜ j,k \ (K j ∪ D j,k−1 ), then x / ∈ Ret j,k and thus x ∈ j,k \ (K j ∪ D j,k−1 ) = j,k−1 \ (K j ∪ D j,k−1 ). By (4.11), x ∈ (D j,k−1 ) j −10 \ (K j ∪ D j,k−1 ). By Lemma 4.4, x ∈ Ret j,k , which is a contradiction. This proves the first claim. The second claim follows from the definition of˜ j,k , in the sense that the new partition has no boundary in Ret j,k , while j,k \ (D j,k−1 ∪ Ret j,k ) is kept intact. The identity in (4.14) is also used to obtain the last equality.

Lemma 4.7
For any φ ∈ R j we have: Proof Note that˜ j,k j,k ⊂ (∂ D j,k−1 ∩ Ret j,k ) ∪ Ret j,k , and that˜ j,k ∩ Ret j,k = ∅. Let Ret j,k,l and E j,k,i(l) be as before. For any x ∈˜ j,k ∩ Ret j,k,l ⊂ ∂ D j,k−1 , consider x ∈ ∂(D j,k−1 ) j −10 such that r 0 (x) = x. Note thatx = r 1 (x) ∈ E j,k,i(l) . If r s (x) / ∈ j,k for all s ∈ [0, 1), then r 0 (x) = x ∈ E j,k,i(l) and we have x ∈Ẽ j,k,i(l) , which is a contradiction to x ∈˜ j,k . Thus there exists s ∈ [0, 1) such that r s (x) ∈ j,k . In particular, we see that j,k ∩Ret j,k is in the image of j,k ∩Ret j,k through the normal nearest point projection onto ∂ D j,k−1 . Furthermore, since r s (x) = x + s |x − x| ν U (x), and since φ is ν U -non decreasing in R n+1 \ D j , it holds φ(x) ≤ φ(r s (x)). Given that the normal nearest point projection onto ∂ D j,k−1 is a Lipschitz map with Lipschitz constant = 1, the desired estimate follows from the area formula.
Note that, as a corollary of Lemma 4.7, we have that, settingẼ j, (4.20) Step 3: motion by smoothed mean curvature with boundary damping. LetṼ j,k = ∂Ẽ j,k as defined in (3.8), and compute h ε j (·) := h ε j (·,Ṽ j,k ). Also, let η j ∈ A j 3 /4 be the cutoff function defined in Definition 3.15. Observe that j has been chosen so that the conclusions of Lemma 3.16 hold. Define the smooth diffeomorphism f j,k (x) Observe that the induction hypothesis (4.12), together with (4.15) and (4.20), implies that Ṽ j,k (R n+1 ) ≤ M as defined in (4.6). Hence, by Lemma 3.16, and using (3.23) and the definition of t j , we can conclude that |η j h ε t j | ≤ exp(− j 1 /8 ) onK j . By the choice of ε j , we also have that |η j h ε t j | ≤ j −10 everywhere. Set

Lemma 4.8 We have
namely (4.9) with k in place of k − 1 holds true.

Lemma 4.9
We have Proof Suppose, towards a contradiction, that x ∈ f j,k (D j,k−1 ) ∩ (K j \ D j,k ). Since | t j η j h ε j | 1/ j 1/4 for all points,x := f −1 j,k (x) is inK j in particular. Then, |η j (x) h ε j (x) t j | ≤ exp(− j 1 /8 ). This means that |x −x| ≤ exp(− j 1 /8 ). Since x / ∈ D j,k , we need to havex / ∈ D j,k−1 by the definition of these sets. But this is a contradiction since x = f j,k (x) ∈ f j,k (D j,k−1 ) and f j,k is bijective. Lemma 4. 10 We have (4.22) namely (4.10) with k in place of k − 1 holds true.

Lemma 4.11
We have namely (4.11) with k in place of k − 1 holds true.
Proof If x ∈ j,k \ K j , then there isx ∈˜ j,k such that x = f j,k (x). Ifx / ∈ K j , then x ∈ D j,k−1 ⊂ D j,k by Lemma 4.6, and since |x −x| < j −10 by the properties of the diffeomorphism f j,k our claim holds true. Hence, suppose thatx ∈ K j . Since in this case |x −x| ≤ exp(− j 1 /8 ), ifx ∈ D j,k−1 then evidently x ∈ D j,k , and the proof is complete. On the other hand, we claim that it has to bex ∈ D j,k−1 . Indeed, otherwise we would havẽ x ∈˜ j,k ∩ K j \ D j,k−1 , and thus, again by Lemma 4.6,x ∈ j,k ∩ K j \ D j,k−1 = j,k−1 ∩ K j \ D j,k−1 . But then, by (4.10), there exists y ∈ 0 such that |x − y| . But this contradicts the fact that x / ∈ K j and completes the proof.
Conclusion. Together, Lemmas 4.8, 4.10 and 4.11 complete the induction step from k − 1 to k for properties (1), (2), (3). Concerning (4.3), first we observe that, since f j,k is a diffeomorphism, (4.24) We can then use (3.42) with V = ∂Ẽ j,k , M as defined in (4.6), ε = ε j , and t = t j in order to conclude that Combining (4.25) with (4.15) and (4.20), and using that 2 ε We are now in a position to define an approximate flow of open partitions. As anticipated in the introduction, the flow is piecewise constant in time; the parameter t j defined in (4.8) is the epoch length, namely the length of the time intervals in which the flow is set to be constant.

Convergence in the sense of measures
for all φ ∈ C c (U ) and t ∈ R + . The limits lim s→t+ μ s (φ) and lim s→t− μ s (φ) exist and satisfy

29)
and for a.e. t ∈ R + it holds Proof Let 2 Q be the set of all non-negative numbers of the form i 2 j for some i, j ∈ N ∪ {0}. 2 Q is countable and dense in R + . For each fixed T ∈ N, the mass estimate in (4.3) implies that lim sup (4.31) Therefore, by a diagonal argument we can choose a subsequence { j } and a family of Radon Furthermore, with (4.31), we also deduce that Next, let Z := {φ q } q∈N be a countable subset of C 2 c (U ; R + ) which is dense in C c (U ; R + ) with respect to the supremum norm. We claim that the function is monotone non-increasing. To see this, first observe that since φ q has compact support, and since the definition in (4.34) depends linearly on φ q , we can assume without loss of generality that φ q < 1. For convenience, for t ≤ 0, we define g q (t) := μ 0 (φ q ) = ∂E 0 (φ q ). Next, given any j ≥ J (n) as in Proposition 4.2, for every positive function φ such that η j φ ∈ A j we can compute for every t ∈ [0, j], and where h ε j (·) = h ε j (·, ∂E j (t)). By the choice of ε j , and since η j φ ∈ A j , we can use (3.33) to estimate whereas Young's inequality together with (3.34) yields (4.37) Plugging (4.36) and (4.37) into (4.35), we obtain for every t ∈ [0, j] and for every positive function φ such that η j φ ∈ A j . Now, for every T ∈ N, for every φ q ∈ Z with φ q < 1, and for every sufficiently large i ∈ N, choose j * ≥ max{T , J (n)} so that for every j ≥ j * . Using that η j ∈ A j 3 /4 for every j ≥ J (n) and that φ q = 0 outside some compact set K ⊂ U , it is easily seen that the two conditions above can be met by choosing j * sufficiently large, depending on i, φ q C 2 , and K . In particular, j * is so large that φ q ≡ 0 on (∂U ) − s 0 \ D j * , so that φ q + i −1 is trivially ν U -non decreasing in R n+1 \ D j * because it is constant in there. For any fixed t 1 , t 2 ∈ [0, T ] ∩ 2 Q with t 2 > t 1 , choose a larger j * , so that both t 1 and t 2 are integer multiples of 1/2 p j * . Then, both t 2 and t 1 are integer multiples of t j for every j ≥ j * . Hence, for every j ≥ j * we can apply (4.5) repeatedly with φ = φ q + i −1 ∈ A j ∩ R j and (4.38) again with φ = φ q + i −1 so that η j φ ∈ A j in order to deduce (4.39) As we let → ∞, the left-hand side of (4.39) can be bounded from below, using (4.31) and (4.32), as follows: In order to estimate the right-hand side of (4.39), we note that so that if we plug (4.41) in (4.39), use that η j ≤ 1, let → ∞ by means of (4.31), and finally let i → ∞ we conclude for every t 1 , t 2 ∈ [0, T ] ∩ 2 Q with t 2 > t 1 and for any φ q ∈ Z with φ q < 1, thus proving that the function defined in (4.34) is indeed monotone non-increasing on [0, T ]. Since T is arbitrary, the same holds on R + . Define now By the monotonicity of each g q , B is a countable subset of R + , and for every t for every t ∈ R + \ (B ∪ 2 Q ) and φ q ∈ Z . (4.44) Indeed, due to the definition of ∂E j (t), there exists a sequence {t } ∞ =1 ⊂ 2 Q with t > t such that lim →∞ t = t and ∂E j (t) = ∂E j (t ). For any s ∈ 2 Q with s > t, and for all suffciently large so that s > t , we deduce from (4.39) that (4.45) Taking the lim inf →∞ and then the lim i→∞ on both sides of (4.45) we obtain that so that when we let s → t+ the definition of μ t and the fact that An analogous argument provides, at the same time, so that (4.47) and (4.48) together complete the proof of (4.44). Since Z is dense in C c (U ; R + ), (4.44) determines the limit measure uniquely, and the convergence holds for every φ ∈ C c (U ) at every t ∈ R + \ B. On the other hand, since B is countable we can extract a further subsequence of {∂E j (t)} ∞ =1 converging to a Radon measure μ t in U for every t ≥ 0. The continuity of μ t (φ) on R + \ B follows from the definition of B and a density argument. The existence of limits and the inequalities (4.28) can be also deduced from (4.42) in the case φ = φ q , and by density for φ ∈ C c (U ; R + ). This completes the proof of the first part of the statement.
The claim in (4.29) follows from (4.4). Finally, (4.29) implies that for each T > 0 where in the last identity we have used that given the definition of κ and the fact that ε j satisfies (3.31). The proof is now complete.

Brakke's inequality, rectifiability and integrality of the limit
In the next proposition we deduce further information concerning the family {μ t } t≥0 of measures in U introduced in Proposition 4.13. (1) For a.e. t ∈ R + the measure μ t is integral, namely there exists an integral varifold then ∂E j (t) converges to V t ∈ IV n (U ) as varifolds in U as → ∞, namely for any φ ∈ C c (U ; R + ). (2) lim →∞ j 4 ε j = 0 and j ≤ ε Here, c 0 is a constant depending only on n. Furthermore, V G n (U ) ∈ RV n (U ).
Proof The existence of a subsequence {∂E j } ∞ =1 converging in the sense of varifolds to V ∈ V n (R n+1 ) follows from the compactness theorem for Radon measures using assumption (3). The limit varifold V satisfies spt V ⊂ clos U because of assumption (1). Indeed, since spt ∂E j ⊂ clos U j by definition of open partition, if x ∈ R n+1 \ clos U then (1) implies that there is a radius r > 0 such that ∂E j (U r (x)) = 0 for all sufficiently large , which in turn gives V (U r (x)) = 0. Furthermore, the validity of (2), (3), and (4) allows us to apply Proposition 3.20 in order to deduce that δV U is a Radon measure. Hence, the rectifiability of the limit varifold in U is a consequence of Allard's rectifiability theorem [1, Theorem 5.5(1)] once we prove (5.4). In turn, the latter can be obtained by repeating verbatim the arguments in [20,Theorem 7.3]. Indeed, the proof in there is local, and for a given x 0 ∈ U it can be reproduced by replacing B 1 (x 0 ) in [20,Theorem 7.3] by B ρ (x 0 ) for sufficiently small ρ > 0 and large so that B ρ (x 0 ) ⊂ D j and η j = 1 on B ρ (x 0 ).

Theorem 5.3 (Integrality Theorem). Under the same assumptions of Theorem 5.2, if the stronger
Just like Theorem 5.2, the claim is local in nature and the proof is the same as [20,Theorem 8.6].

Proof of Proposition 5.1 First, observe that by (4.29) and Fatou's lemma we have lim inf
for a.e. t ∈ R + . Furthermore, from (4.3) and the definition of ∂E j (t) we also have that for Let t ∈ R + be such that ( , ∂E j (t) converges, as → ∞, to a varifold V t ∈ V n (R n+1 ) with spt V t ⊂ clos U and such that V t G n (U ) ∈ IV n (U ). Since the convergence is in the sense of varifolds, the weights converge as Radon measures, and thus lim →∞ ∂E j (t) = V t : (4.27) then readily implies that V t U = μ t as Radon measures on U , thus proving (1). Concerning the statement in (2), let { j } ∞ =1 be a subsequence along which (5.1) holds. Then, any converging further subsequence must converge to a varifold satisfying the conclusion of Theorem 5.3. A priori, two distinct subsequences may converge to different limits. On the other hand, each subsequential limit V t is a rectifiable varifold when restricted to the open set U , and furthermore it satisfies V t U = μ t . Since rectifiable varifolds are uniquely determined by their weight, we deduce that the limit in U is independent of the particular subsequence, and thus (5.1) forces the whole sequence ∂E j (t) to converge to a uniquely determined integral varifold V t in U . Finally, (3) follows from Proposition 3.20.
A byproduct of the proof of Proposition 5.1 is the existence of a (uniquely defined) integral varifold V t ∈ IV n (U ) with weight V t = μ t for every t ∈ R + \ Z , where L 1 (Z ) = 0. Such a varifold V t is the limit on U of any sequence ∂E j (t) along which (5.1) holds true. We can now extend the definition of V t to t ∈ Z so to have a one-parameter family {V t } t∈R + ⊂ V n (U ) of varifolds satisfying V t = μ t for every t ∈ R + . Such an extension can be defined in an arbitrary fashion: for instance, if t ∈ Z then we can set V t (ϕ) := ϕ(x, S) dμ t (x) for every ϕ ∈ C c (G n (U )), where S is any constant plane in G(n + 1, n).
In the next theorem, we show that the family of varifolds {V t } is indeed a Brakke flow in U . The boundary condition and the initial condition will be discussed in the following section.

Theorem 5.4 (Brakke's inequality). For every T > 0 we have
Furthermore, for any φ ∈ C 1 c (U × R + ; R + ) and 0 ≤ t 1 < t 2 < ∞ we have: Proof In order to prove (5.7), we use (4.5) with φ = 1 which belongs to A j ∩ R j for all j. Assume T ∈ 2 Q first. By summing over the index k and for all sufficiently large j, we have By (3.33) and (5.3) as well as V T (U ) ≤ lim inf →∞ ∂E j (T ) (U ), we obtain (5.7). For T / ∈ 2 Q , use (4.28) to deduce the same inequality. We now focus on proving the validity of Brakke's inequality (5.8).
Step 1. We will first assume that φ is independent of t, and then extend the proof to the more general case. By an elementary density argument, we can assume that φ ∈ C ∞ c (U ; R + ). Moreover, since the support of φ is compact and (5.8) depends linearly on φ, we can also normalize φ in such a way that φ < 1 everywhere. Then, for all sufficiently large i ∈ N, alsô φ := φ + i −1 < 1 everywhere. Arguing as in the proof of Proposition 4.13, we can choose m ∈ N so that m ≥ J (n) (see Lemma 3.16) and furthermore for all j ≥ m. Next, fix 0 ≤ t 1 < t 2 < ∞, and let be such that j ≥ m and j ≥ t 2 , so that ∂E j (t) is certainly well defined for t ∈ [t 1 , t 2 ]. By the condition (i) above, we can apply (4.5) withφ and deduce for every t = t j , 2 t j , . . . , j 2 p j t j . Since t j → 0 as → ∞, we can assume without loss of generality that t j < t 2 − t 1 , so that there exist k 1 , k 2 ∈ N with k 1 < k 2 such that t 1 ∈ (k 1 − 2) t j , (k 1 − 1) t j and t 2 ∈ (k 2 − 1) t j , k 2 t j . If we sum (5.9) on t = k t j for k ∈ [k 1 , (5.10) we can estimate the left-hand side of (5.10) from below as (5.11) so that when we let → ∞ we conclude where we have used (4.27) together with Proposition 5.1(1). Next, we estimate the right-hand side of (5.10) from above. Setting ∂E j = ∂E j (t) and h ε j = h ε j (·, ∂E j ), we proceed as in (4.35) writing (5.13) where we have used that ∇φ = ∇φ. Since η j φ ∈ A j , we can apply (3.33) in order to obtain that where we have set for simplicity Concerning the second summand in (5.13), we use the Cauchy-Schwarz inequality to estimate where c depends only on φ C 2 , and where we have used (3.34). Using (5.14), (5.16) and (4.3), we can then conclude that where c depends only on φ C 2 and ∂E 0 (R n+1 ). Using (5.17) together with the definition of ∂E j (t) and Fatou's lemma, one can readily show that, when we take the lim sup as → ∞, the right-hand side of (5.10) can be bounded by Now, fix t ∈ [t 1 , t 2 ] such that lim inf →∞ b j < ∞ (which holds for a.e. t), and let { j } ⊂ { j } be a subsequence which realizes the lim sup, namely with ) .

(5.19)
By the identity in (5.13), we also have that along the same subsequence (5.20) where once again ∂E j = ∂E j (t) and h ε j = h ε j (·, ∂E j ). Using (5.14) and (5.16), we see that the right-hand side of (5.20) can be bounded from above by lim inf →∞ 2 b j +c, whereas the left-hand side can be bounded from below by lim sup →∞ 1 2 b j − c, where c depends on φ C 2 and ∂E 0 (R n+1 ). As a consequence, along any subsequence { j } satisfying (5.19) one has that lim sup where ∂E j = ∂E j (t). Let us denote the right-hand side of (5.21) as B(t). Sinceφ ≥ i −1 , and thanks to (5.21), if B(t) < ∞ then the assumption (5.1) of Proposition 5.1 is satisfied along j : hence, the whole sequence {∂E j (t)} ∞ =1 converges to V t ∈ IV n (U ) as varifolds in U . Furthermore, using one more time thatφ ≥ i −1 we deduce that lim sup Using (5.19), (5.13), (5.14),φ > φ, and Proposition 5.
where we have also used that, as → ∞, η j = 1 on {∇φ = 0} ⊂⊂ U . Now, recall that V t ∈ IV n (U ). Therefore, there is an H n -rectifiable set M t ⊂ U such that for any ε > 0 there are a vector field g ∈ C ∞ c (U ; R n+1 ) and a positive integer m such that g ∈ B m and In order to estimate the lim sup in the right-hand side of (5.23), we can now compute, for ∂E j = ∂E j (t): We proceed estimating each term of (5.26). Using that η j = 1 on {∇φ = 0} for all sufficiently large, the Cauchy-Schwarz inequality gives that for all sufficiently large. Since (x, S) → |S ⊥ (∇φ(x)) − g(x)| 2 ∈ C c (G n (U )), we have that (5.25) ≤ ε 2 .
Analogously, since η j = 1 on {g = 0} for all sufficiently large, we have that by (3.35) and (5.22). Next, by varifold convergence of ∂E j to V t on U , given that g has compact support in U , we also have Finally, letting ψ be any function in C c (U ; R + ) such that ψ = 1 on {g = 0} ∪ {∇φ = 0} and 0 ≤ ψ ≤ 1, the Cauchy-Schwarz inequality allows us to estimate where we have also used (2.6).
We can now combine (5.10), (5.12), (5.18), (5.23), and (5.33) to deduce that We use the Cauchy-Schwarz inequality one more time, and combine it with the definition of B(t) as the right-hand side of (5.21) and with Fatou's lemma to obtain the bound which is finite (depending on t 2 ) by (4.29) (recall thatφ ≤ 1 everywhere). Brakke's inequality (5.8) for a test function φ which does not depend on t is then deduced from (5.34) after letting ε ↓ 0 and then i ↑ ∞.
Step 2. We consider now the general case of a time dependent test function φ ∈ C 1 c (U × R + ; R + ). We can once again assume that φ is smooth, and then conclude by a density argument. The proof follows the same strategy of Step 1. We defineφ analogously, and then we apply (4.5) with φ =φ(·, t). In place of (5.9), we then obtain a formula with one extra term, namely Similarly, the inequality in (5.10) needs to be replaced with an analogous one containing, in the right-hand side, also the term (5.37) Using the regularity of φ and the estimates in (4.3) and (4.4), we may deduce that where the last identity is a consequence of (4.27), Proposition 5.1(1), and Lebesgue's dominated convergence theorem. The remaining part of the argument stays the same, modulo the following variation. The identity in (5.18) remains true ifφ is replaced by the piecewise constant functionφ j defined bŷ The error one makes in order to putφ back into (5.18) in place ofφ j is then given by the product of t j times some negative powers of ε j ; nonetheless, this error converges to 0 uniformly as ↑ ∞ by the choice of t j , see (4.8). This allows us to conclude the proof of (5.8) precisely as in the case of a time-independent φ whenever φ ∈ C ∞ c (U × R + ; R + ), and in turn, by approximation, also when φ ∈ C 1 c (U × R + ; R + ).

Vanishing of measure outside the convex hull of initial data
First, we prove that the limit measures V t vanish uniformly in time near ∂U \ ∂ 0 . This is a preliminary result, and using the Brakke's inquality, we eventually prove that they actually vanish outside the convex hull of 0 ∪ ∂ 0 in Proposition 6.4. Proposition 6.1 Forx ∈ ∂U \ ∂ 0 , suppose that an affine hyperplane A ⊂ R n+1 withx / ∈ A has the following property. Let A + and A − be defined as the open half-spaces separated by A, i.e., R n+1 is a disjoint union of A + , A and A − , withx ∈ A + . Define d A (x) := dist (x, A − ), and suppose that Then for any compact set C ⊂ A + , we have (6.1)

Remark 6.2
Due to the definition of ∂ 0 and the strict convexity of U , note that there exists such an affine hyperplane A for any givenx ∈ ∂U \ ∂ 0 . For example, we may choose a hyperplane A which is parallel to the tangent space of ∂U atx and which passes througĥ By the strict convexity of U and the C 1 regularity of ν U , for all sufficiently small c > 0, one can show that such A satisfies the above (1) and (2).

Remark 6.3
In the following proof, we adapted a computation from [17, p.60]. There, the object is the Brakke flow, but the basic idea here is that a similar computation can be carried out for the approximate MCF with suitable error estimates.

Proof
We may assume after a suitable change of coordinates that A = {x n+1 = 0} and A + = {x n+1 > 0}. With this, we have clos 0 ⊂ {x n+1 < 0} and d A (x) = max{x n+1 , 0} is ν U -non decreasing in {x n+1 > 0}. Let s > 0 be arbitrary, and define for some β ≥ 3 to be fixed later. Then φ ∈ C 2 (R n+1 ; R + ), and letting {e 1 , . . . , e n+1 } denote the standard basis of R n+1 , we have With s > 0 fixed, we choose sufficiently large j so that φ ∈ A j . Actually, the function φ as defined in (6.2) is unbounded. Nonetheless, since we know that spt ∂E j (t) ⊂ (U ) 1/(4 j 1 /4 ) , we may modify φ suitably away from U by multiplying it by a small number and truncating it, so that φ ≤ 1. We assume that we have done this modification if necessary. We also choose j so large that η j = 1 on {x n+1 ≥ 0}. This is possible due to Lemma 3.16 (1). Additionally, since d A is ν U -non decreasing in A + , and since φ is constant in R n+1 \ A + , we have φ ∈ R j . Thus, by (4.5), we have for ∂E j,k =: V and ∂E j,k−1 =:V with k ∈ {1, . . . , j2 p j } For all sufficiently large j, we also have η j φ ∈ A j , thus we may proceed as in (4.35) and estimate (6.5) Here we have used that η j = 1 when ∇φ = 0. In the present proof, we omit the domains of integration, which are either R n+1 or G n (R n+1 ) unless specified otherwise. We use (3.34) to proceed as: We prove that the last term gives a good negative contribution. We have Here we replace ∇φ(x) by ∇φ(y) and estimate the error To estimate (6.7), since η j φ ∈ A j , (3.1) and (3.3) imply By separating the integration to B √ ε j (y) and y)).

(6.8)
Let us denote c ε j := c(n)ε −n−1 j j exp( j −(2ε j ) −1 ) and note that it is exponentially small (say, for all sufficiently large j. By (6.3), we have (6.14) where in the last identity we have used that S is the matrix representing an orthogonal projection operator, so that S is symmetric and S 2 = S, whence In particular, the quantity in (6.14) can be made negative if β = 4, for example. This shows that (6.13) is less than 2ε 1 /8 j . By summing over k = 1, . . . , j 1 /2 /( t j ) and using that We use this in (6.15), and we let first j → ∞ and then s → 0 in order to obtain (6.1).

Proposition 6.4 For all t
Proof Suppose that A ⊂ R n+1 is a hyperplane such that, using the notation in the statement of Proposition 6.1, 0 ∪ ∂ 0 ⊂ A − . If d A is ν U -non decreasing in A + , then (6.1) proves immediately that V t (A + ) = 0 for all t ≥ 0. Thus, suppose that d A does not satisfy this property. Still, due to Proposition 6.1, for each x ∈ ∂U \ ∂ 0 , there exists a neighborhood B r (x) such that V t (B r (x) ∩ U ) = 0 for all t ≥ 0. In particular, there exists some r 0 > 0 such that We next use φ = ψ d 4 A in (5.8) with t 1 = 0 and an arbitrary t 2 = t > 0 to obtain (6.17) By (6.16), φ = d 4 A on the support of V s . Since S · ∇ 2 d 4 A ≥ 0 for any S ∈ G(n + 1, n) (see (6.14)), the right-hand side of (6.17) is ≤ 0. Since V 0 (φ) = 0, we have V t (A + ) = 0 for all t > 0. This proves the claim.
In the following, we list results from [20,Section 10]. The results are local in nature, thus even if we are concerned with a Brakke flow in U instead of R n+1 , the proofs are the same. We recall the following (cf. Theorem 2.3(11)): Definition 6.5 Define a Radon measure μ on U × R + by setting dμ := d V t dt, namely for every φ ∈ C c (U × R + ) . (6.18) Lemma 6. 6 We have the following properties for μ and {V t } t∈R + .
The next Lemma (see [20,Lemma 10.10 and 10.11]) is used to prove the continuity of the labeling of partitions.
denote the open partitions for each j and t ∈ R + , i.e., Then for all t ∈ (t − r 2 , t + r 2 ], we have Then for all t ∈ (0, r 2 ], we have The following is from [2, 3.7]. Lemma 6.8 Suppose that V t (U r (x)) = 0 for some t ∈ R + and U r (x) ⊂⊂ U . Then, for every t ∈ t, t + r 2 2n it holds V t (U √ r 2 −2n (t −t) (x)) = 0. 1 , for each t and i the volumes L n+1 (E j ,i (t)) are uniformly bounded in . Furthermore, by the mass estimate in (4.31) we also have that ∇χ E j ,i (t) (R n+1 ) are uniformly bounded. Hence, we can use the compactness theorem for sets of finite perimeter in order to select a (not relabeled) subsequence with the property that, for each fixed i ∈ {1, . . . , N },

Proof of Theorem 2.3 Let
where E i (t) is a set of locally finite perimeter in R n+1 . Moreover, using that E j ,i (t) ⊂ (U ) 1/(4 j 1 /4 ) (see Proposition 4.2 and (4.7)) we see that L n+1 (E i (t) \ U ) = 0. Since sets of finite perimeter are defined up to measure zero sets, we can then assume without loss of generality that E i (t) ⊂ U . Hence, since H n (∂U ) < ∞, E i (t) is in fact a set of finite perimeter in R n+1 .
Next, consider the complement of spt μ ∪ ( 0 × {0}) in U × R + , which is relatively open in U × R + , and let S be one of its connected components. For any point (x, t) ∈ S there exists r > 0 such that either We first consider the case t = 0. Since B 2 r (x) lies in the complement of 0 , there exists i(x, 0) ∈ {1, . . . , N } such that B 2 r (x) ⊂ E 0,i(x,0) , and thus B 2 r (x) ⊂ E j ,i(x,0) (0) for all ∈ N. Since also μ(B 2 r (x) × 0, r 2 ) = 0, we can apply Lemma 6.7 (2) and conclude that Similarly, if t > 0, since μ(B 2 r (x) × t − r 2 , t + r 2 ) = 0, we can apply Lemma 6.7(1) to conclude that there is a unique i(x, t) ∈ {1, . . . , N } such that Now, observe that if S is any connected component of the complement of spt μ∪( 0 ×{0}) in U × R + , then by (6.20) and (6.21), and since S is connected, for any two points (x, t) and (y, s) in S it has to be i(x, t) = i(y, s). For every i ∈ {1, . . . , N }, we can then let S(i) denote the union of all connected components S such that i(x, t) = i for every (x, t) ∈ S. It is clear that S(i) are open sets, and that E 0,i = {x ∈ U : (x, 0) ∈ S(i)} (notice that if x ∈ E 0,i then (x, 0) / ∈ spt μ as a consequence of Lemma 6.8), so that each S(i) is not empty. Furthermore, . For every t ∈ R + , we can thus define By examining the definition, one obtains (t) = {x ∈ U : (x, t) ∈ spt μ} for all t > 0. Combined with Lemma 6.6(1), we have (11). By Lemma 6.6(2), we have (3), and this also proves that (t) has empty interior, which shows (4). The claims (1) and (2) hold true by construction. (5) is a consequence of Proposition 6.4 and the definition of μ being the product measure. (6) is similar: if x ∈ U \ conv( 0 ∪ ∂ 0 ) then the half-line t ∈ R + → γ x (t) := (x, t) ∈ U × R + must be contained in the same connected component of (U × R + ) \ (spt μ ∪ ( 0 × {0})), for otherwise there would be t > 0 such that (x, t) ∈ spt μ, thus contradicting (5). For (7), by the strict convexity of U and (5), we have ∂ (t) ⊂ ∂ 0 for all t > 0. Later in Proposition 6.9, we prove (clos (spt V t )) \ U = ∂ 0 and ∂ 0 ⊂ ∂ (t) follows from this and (11). Coming to (8), we use (6.21) together with the conclusions in Proposition 4.2(1) to see that In particular, the lower semi-continuity of perimeter allows us to deduce that for any φ ∈ C c (U ; R + ) thus proving ∇χ E i (t) ≤ V t of (8). Using the cluster structure of each ∂E j (t) (see e.g. [26,Proposition 29.4]), we have in fact that for every φ as above , which shows the other statement N i=1 ∇χ E i (t) ≤ 2 V t in (8). Since the claim of (9) is interior in nature, the proof is identical to the case without boundary as in [20,Theorem 3.5(6)]. For the proof of (10), fort ≥ 0, we prove that χ E i (t) → χ E i (t) in L 1 (U ) as t →t for each i = 1, . . . , N . Since ∇χ E i (t) (U ) ≤ V t (U ) ≤ H n ( 0 ), for any t k →t, there exists a subsequence (denoted by the same index) andẼ i ⊂ U such that χ E i (t k ) → χẼ i in L 1 (U ) and L n+1 a.e. by the compactness theorem for sets of finite perimeter. We also have L n+1 (Ẽ i ∩Ẽ j ) = 0 for i = j and L n+1 (U \ ∪ N i=1Ẽ i ) = 0. For a contradiction, assume that L n+1 (E i (t) \Ẽ i ) > 0 for some i. Then, there must be U r (x) ⊂⊂ E i (t) such that L n+1 (U r (x) \Ẽ i ) > 0. We then use Theorem 2.3(9) with g(t) = L n+1 (E i (t) ∩ U r (x)), which gives lim t→t g(t) = g(t) = L n+1 (E i (t) ∩U r (x)) = L n+1 (U r (x)). On the other hand, This proves (9), and finishes the proof of (1)-(11) except for (7), which is independent and is proved once we prove Proposition 6.9.
Conversely, let x ∈ ∂ 0 , and suppose for a contradiction that x / ∈ clos (spt V t ), so that there is a radius r > 0 with the property that B r (x) ∩ spt V t = ∅. Then, Theorem 2.3 (8) If t = 0, since E i (0) = E 0,i for every i = 1, . . . , N , the conclusion in (6.23) is evidently incompatible with (A4), thus providing the desired contradiction. We can then assume t > 0. By (A4), there are at least two indices i = i ∈ {1, . . . , N } and sequences of balls such that x j , x j ∈ ∂U , lim j→∞ x j = lim j→∞ x j = x and B r j (x j ) ∩ U ⊂ E 0,i whereas B r j (x j ) ∩ U ⊂ E 0,i . Let z denote any of the points x j or x j , and observe that the above condition guarantees that z ∈ ∂U \ ∂ 0 . In turn, by arguing as in Remark 6.2 we deduce that there is a neighborhood B ρ (z)∩U such that V t (B ρ (z)∩U ) = 0 for all t ≥ 0, and thus also ∇χ E l (t) (B ρ (z) ∩ U ) = 0 for every t ≥ 0 and for every l ∈ {1, . . . , N }. Since B ρ (z) ∩ U is connected this implies that B ρ (z) ∩ U ⊂ E l (t) for some l. Applying this argument with z = x j and z = x j we then find radii ρ j and ρ j such that, for all t ≥ 0. Since x j → x and x j → x this conclusion is again incompatible with (6.23), thus completing the proof. Proposition 6. 10 We have for each φ ∈ C c (U ; R + ) .
where we also used Theorem 2.3 (8) and (10). This proves the first inequality. The second equality and the third inequality follow from (4.28), μ t = V t and V 0 = H n 0 .
The proof of Theorem 2.2 is now complete: {V t } t≥0 is a Brakke flow with fixed boundary ∂ 0 due to Proposition 5.1(1), Theorem 5.4 and Proposition 6.9. Proposition 6.10 proves the claim on the continuity of measure at t = 0.
where E i ⊂ U are sets of finite perimeter. Since, by Theorem 2.3 (3) The validity of Theorem 2.3(8) implies conclusion (1), namely that in the sense of Radon measures in U . As a consequence of (7.6), we have that spt ∇χ E i ⊂ spt V for every i = 1, . . . , N . Since V is a stationary integral varifold, the monotonicity formula implies that spt V is H n -rectifiable, and V = var(spt V , θ) for some upper semi-continuous θ : U → R + with θ(x) ≥ 1 at each x ∈ spt V . In particular, setting := spt V , we have where the last inequality is a consequence of (5.7) and the lower semicontinuity of the weight with respect to varifold convergence. → χ E i now holds pointwise on U \conv( 0 ∪∂ 0 ). We have not excluded the possibility that H n ( ) = 0. But this should imply V = 0 by (7.7), and ∇χ E i = 0 for every i ∈ {1, . . . , N } by (7.6), which is a contradiction to (2). Thus we have necessarily H n ( ) > 0 and this completes the proof of (3). In order to conclude the proof, we are just left with the boundary condition (4), namely Towards the first inclusion, suppose that x ∈ (clos (spt V )) \ U , and let {x h } ∞ h=1 be a sequence with x h ∈ spt V such that x h → x as h → ∞. If x / ∈ ∂ 0 then Proposition 6.1 implies that there exists r > 0 such that lim sup By the lower semi-continuity of the weight with respect to varifold convergence, we deduce then that V (U ∩ U r (x)) = 0. For h large enough so that |x − x h | < r we then have V (U ∩U r −|x−x h | (x h )) = 0, thus contradicting that x h ∈ spt V . For the second inclusion, let x ∈ ∂ 0 , and suppose towards a contradiction that x / ∈ clos(spt V ) \ U . Then, there exists a radius r > 0 such that U r (x) ∩ spt V = ∅. In particular, ∇χ E i (U ∩ U r (x)) = 0 for every i ∈ {1, . . . , N }. Since U is convex, U ∩ U r (x) is connected, and thus every χ E i is either identically 0 or 1 in U r (x) ∩ U , namely If z denotes any of the points x j or x j , Proposition 6.1 and Remark 6.2 ensure the existence of ρ such that V t (B ρ (z) ∩ U ) = 0 for all t ≥ 0. Again by lower semicontinuity of the weight with respect to varifold convergence, Since both x j → x and x j → x, this conclusion is incompatible with (7.9). This completes the proof.
The stationary varifold V from Corollary 2.4 is a generalized minimal surface in U , and for this reason it can be thought of as a solution to Plateau's problem in U with the prescribed boundary ∂ 0 . Brakke flow provides, therefore, an interesting alternative approach to the existence theory for Plateau's problem compared to more classical methods based on mass (or area) minimization. Another novelty of this approach is that the structure of partitions allows to prescribe the boundary datum in the purely topological sense, by means of the constraint (clos (spt V )) \ U = ∂ 0 . This adds to the several other possible interpretations of the spanning conditions that have been proposed in the literature: among them, let us mention the homological boundary conditions in Federer and Fleming's theory of integral currents [12] or of integral currents mod( p) [11] (see also Brakke's covering space model for soap films [3]); the sliding boundary conditions in David's sliding minimizers [5,6]; and the homotopic spanning condition of Harrison [13], Harrison-Pugh [14] and De Lellis-Ghiraldin-Maggi [7].
Concerning the latter, we can actually show that, under a suitable extra assumption on the initial partition E 0 , a homotopic spanning condition is satisfied at all times along the flow. Before stating and proving this result, which is Proposition 7.4 below, let us first record the definition of homotopic spanning condition after [7]. Definition 7.1 (see [7,Definition 3]). Let n ≥ 2, and let be a closed subset of R n+1 . Consider the family C := γ : S 1 → R n+1 \ : γ is a smooth embedding of S 1 into R n+1 \ . (7.10) A subfamily C ⊂ C is said to be homotopically closed if γ ∈ C implies thatγ ∈ C for everỹ γ ∈ γ , where γ is the equivalence class of γ modulo homotopies in R n+1 \ . Given a homotopically closed C ⊂ C , a relatively closed subset K ⊂ R n+1 \ is C-spanning if 2 K ∩ γ = ∅ for every γ ∈ C . (7.11)

Remark 7.2
If C ⊂ C contains a homotopically trivial curve, then any C-spanning set K will necessarily have non-empty interior (and therefore infinite H n measure). For this reason, we are only interested in subfamilies C with γ = 0 for every γ ∈ C. Definition 7. 3 We will say that a relatively closed subset K ⊂ R n+1 \ strongly homotopically spans if it C-spans for every homotopically closed family C ⊂ C which does not contain any homotopically trivial curve. Namely, if K ∩ γ = ∅ for every γ ∈ C such that γ = 0 in π 1 (R n+1 \ ).
We can prove the following proposition, whose proof is a suitable adaptation of the argument in [7, Lemma 10]. Then, the set (t) strongly homotopically spans ∂ 0 for every t ∈ [0, ∞].
Proof Let γ : S 1 → R n+1 \ ∂ 0 be a smooth embedding that is not homotopically trivial in R n+1 \ ∂ 0 . The goal is to prove that, for every t ∈ [0, ∞], (t) ∩ γ = ∅. First observe that it cannot be γ ⊂ U , for otherwise γ would be homotopically trivial. For the same reason, since the ambient dimension is n + 1 ≥ 3 also γ ⊂ R n+1 \ clos U is incompatible with the properties of γ . Hence, we conclude that γ must necessarily intersect ∂U . We first prove the result under the additional assumption that γ and ∂U intersect transversally. We can then find finitely many closed arcs I h = [a h , b h ] ⊂ S 1 with the property that γ ∩U = h γ ((a h , b h ) Furthermore, this can be achieved under the additional condition that τ h (I h ) ∩ τ h (I h ) = ∅ for every h = h . We can then define a piecewise smooth embeddingγ of S 1 into R n+1 \ ∂ 0 such thatγ | I h := τ h | I h for every h, andγ = γ on the open set S 1 \ h I h . We have γ = γ in π 1 (R n+1 \ ∂ 0 ). We can then construct a smooth embeddingγ : S 1 → R n+1 \ ∂ 0 such that γ = γ in π 1 (R n+1 \ ∂ 0 ), and withγ ⊂ R n+1 \ ∂U . Since n + 1 ≥ 3 this contradicts the assumption that γ = 0 and completes the proof if γ and ∂U intersect transversally. Finally, we remove the transversality assumption. Let δ = δ(∂U ) > 0 be such that the tubular neighborhood (∂U ) 2δ has a well-defined smooth nearest point projection , and consider, for |s| < δ, the open sets U s having boundary ∂U s = {x − s ν U (x) : x ∈ ∂U }, where ν U is the exterior normal unit vector field to ∂U . Since γ is smooth, by Sard's theorem γ intersects ∂U s transversally for a.e. |s| < δ. Fix such an s ∈ (0, δ), and let s : R n+1 → R n+1 be the smooth diffeomorphism of R n+1 defined by s (x) := x + ϕ s (ρ U (x)) ν U ( (x)) , (7.12) where In particular, s maps ∂U s diffeomorphically onto ∂U , and furthermore s → id uniformly on R n+1 as s → 0 + . (7.13) Since γ intersects ∂U s transversally, the curve s • γ intersects ∂U transversally. Furthermore, since γ and ∂ 0 are two compact sets with empty intersection, (7.13) implies that if we choose s sufficiently small then also ( s • γ ) ∩ ∂ 0 = ∅. Since s • γ = γ = 0 in π 1 (R n+1 \ ∂ 0 ), the first part of the proof guarantees that for every t ∈ [0, ∞] we have (t)∩( s •γ ) = ∅. For every t we then have points z s (t) ∈ (t)∩ s •γ . Along a sequence s h → 0+, then, by compactness, (7.13), and the fact that each set (t) is closed, we have that the points z s h (t) converge to a point z 0 (t) ∈ (t) ∩ γ . The proof is now complete. Example 7.5 Suppose that U = U 1 (0) ⊂ R 3 , and ∂ 0 is the union of two parallel circles contained in S 2 = ∂U at distance 2h from one another, with h ∈ (0, 1). Then, ∂U \ ∂ 0 consists of the union of three connected components S u ∪ S l ∪ S d (here u, l, d stand for up, lateral, and down, respectively). If h is suitably small, then there are two smooth minimal catenoidal surfaces C 1 ⊂ U and C 2 ⊂ U , one stable and the other unstable, satisfying clos(C j ) \ U = ∂ 0 . Nonetheless if the initial partition {E 0,i } i satisfies ( ), then, as a consequence of Proposition 7.4, both C 1 and C 2 are not admissible limits of Brakke flow as in Corollary 2.4, since there exists a smooth and homotopically non-trivial embedding γ : S 1 → R 3 \ ∂ 0 having empty intersection with each of them. For instances, if N = 3 and the initial partition is such that S u ⊂ clos E 0,1 , S l ⊂ clos E 0,2 , and S d ⊂ clos E 0,3 , then the corresponding Brakke flows will converge, instead, to a singular minimal surface in U consisting of the union =C 1 ∪C 2 ∪ D, whereC j are pieces of catenoids, and D is a disc contained in the plane {z = 0}, which join together forming 120 • angles along the "free boundary" circle = ∂ D; see Fig. 1.
We will conclude the section with three remarks containing some interesting possible future research directions. Remark 7.6 First, we stress that the requirements on ∂ 0 are rather flexible, above all in terms of regularity. It would be interesting to characterize, for a given strictly convex domain U ⊂ R n+1 , all its admissible boundaries, namely all subsets ⊂ ∂U such that there are N ≥ 2 and E 0 , 0 as in Assumption 1.1 such that = ∂ 0 . A first observation is that admissible boundaries do not need to be countably (n −1)-rectifiable, or to have finite (n −1)dimensional Hausdorff measure: for example, it is not difficult to construct an admissible ⊂ ∂U 1 (0) in R 2 with H 1 ( ) > 0, essentially a "fat" Cantor set in S 1 . The assumption (A4) requires any admissible boundary to have empty interior. It is unclear whether this condition is also sufficient for a subset to be admissible.

Remark 7.7
Let us explicitly observe that, even in the case when 0 (or more precisely V 0 := var( 0 , 1)) is stationary, it is false in general that V t = V 0 for t > 0. In other words, the approximation scheme which produces the Brakke flow V t may move the initial datum V 0 even when the latter is stationary. A simple example is a set consisting of two line segments with a crossing, for which multiple non-trivial solutions (depending on the choice of the initial partition) are possible; see Fig. 2. In fact, one can prove that such one-dimensional configuration cannot stay time-independent with respect to the Brakke flow constructed in the present paper: [21, Theorem 2.2], indeed, shows that one-dimensional Brakke flows obtained in the present paper and in [20] necessarily satisfy a specific angle condition at junctions for a.e. time, with the only admissible angles being 0, 60, or 120 degrees. Thus, depending on the initial labeling of domains, one of the two evolutions depicted in Fig. 2 has to occur instantly.
If 0 is a smooth minimal surface with smooth boundary ∂ 0 , the uniqueness theorem for classical MCF should allow t ≡ 0 as the unique solution, even if the latter is unstable (i.e. the second variation is negative for some direction). In other words, in the smooth case we expect that there is no other Brakke flow starting from 0 other than the time-independent solution (notice, in passing, that both the area-reducing Lipschitz deformation step and the motion by smoothed mean curvature step in our time-discrete approximation of Brakke flow trivialize in this case -at least locally -, because smooth minimal surfaces are already locally area minimizing at suitably small scales around each point).
On the other hand, in [36] we show that time-dependent solutions may arise even from the existence, on 0 , of singular points at which V 0 has a flat tangent cone, that is a tangent cone which is a plane T with multiplicity Q ≥ 2. It would be interesting to characterize the regularity properties of those stationary 0 with E 0,1 , . . . , E 0,N satisfying Assumption 1.1 and H n ( 0 \ ∪ N i=1 ∂ * E 0,i ) = 0 which do not allow any non-trivial Brakke flows (dynamically stable stationary varifolds, in the terminology introduced in [36]). We expect that such a 0 should have some local measure minimizing properties.

Remark 7.8 Let
and be as in Corollary 2.4 obtained as t k → ∞ along a Brakke flow. Since V is integral and stationary, V = var( , θ ) for some H n -measurable function θ : → N. One can check that and {E i } N i=1 (after removing empty E i 's if necessary) again satisfy the Assumption 1.1, thus we may apply Theorem 2.2 and obtain another Brakke flow with the same fixed boundary. Note that if we have V ({x : θ(x) ≥ 2}) > 0, then var( , 1) may not be stationary, and the Brakke flow starting from non-stationary var( , 1) is genuinely time-dependent. We then obtain another stationary varifold as t → ∞ by Corollary 2.4. It is likely that, after a finite number of iterations, this process produces a unit density stationary varifold which does not move anymore. The other possibility is also interesting, in that we would have infinitely many different integral stationary varifolds with the same boundary condition, each having strictly smaller H n measure than the previous one.