An existence theorem for Brakke flow with fixed boundary conditions

Consider an arbitrary closed, countably $n$-rectifiable set in a strictly convex $(n+1)$-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 $t \uparrow \infty$, the flow sequentially converges to non-trivial solutions of Plateau's problem in the setting of stationary varifolds.


Introduction
A time-parametrized family {Γ(t)} t≥0 of n-dimensional surfaces in R n+1 (or in an open domain U ⊂ R n+1 ) is called a mean curvature flow (abbreviated hereafter as MCF) if the velocity of motion of Γ(t) is equal to the mean curvature of Γ(t) at each point and time. The aim of the present paper is to establish a global-in-time existence theorem for the MCF {Γ(t)} t≥0 starting from a given surface Γ 0 while keeping the boundary of Γ(t) fixed for all times t ≥ 0. In particular, we are interested in the case when the initial surface Γ 0 is not smooth. Typical MCF under consideration in this setting may look like a moving network with multiple junctions for n = 1, or a moving cluster of bubbles for n = 2, and they may undergo various topological changes as they evolve. Due to the presence of singularities, we work in the framework of the generalized, measure-theoretic notion of MCF introduced by Brakke and since known as the Brakke flow [2,34]. A global-in-time existence result for a Brakke flow without fixed boundary conditions was established by Kim and the secondnamed 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 Γ 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 .
(A1) U ⊂ R n+1 is a strictly convex bounded domain with boundary ∂U of class C 2 .
Since N ≥ 2, we implicitly assume that U \ Γ 0 is not connected. When n = 1, Γ 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 \ Γ 0 . If one defines F i := (clos E 0,i ) \ (U ∪ ∂Γ 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 ∈ Z 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, Γ(t) remains within the convex hull of Γ 0 ∪ ∂Γ 0 .
More precisely, {Γ(t)} t≥0 is a MCF in the sense that Γ(t) coincides with the space-time support of a Brakke flow {V t } t≥0 starting from Γ 0 . The method adopted to produce the evolving generalized surfaces Γ(t) actually gives us more. Indeed, we show the existence of It is reasonable to expect that the flow Γ(t) converges, as t → ∞, to a minimal surface in U with boundary ∂Γ 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 = ∂Γ 0 . See Corollary 2.4 for a precise statement. The limit V ∞ is a solution to Plateau's problem with boundary ∂Γ 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 Section 7 for further discussion on these points.
Next, we discuss closely related results. While there are several works on the global-in-time existence of MCF, there are relatively few results on the existence of MCF with fixed boundary conditions. When Γ 0 is a smooth graph over a bounded domain Ω in R n , global-in-time existence follows from the classical work of Lieberman [22]. Furthermore, under the assumption that Ω is mean convex, convergence of the flow to the unique solution to the minimal surfaces equation in Ω with the prescribed boundary was established by Huisken in [16]; see 2. Definitions, Notation, and Main Results 2.1. 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 .

2.2.
Varifolds. The symbol G(n + 1, k) will denote the Grassmannian of (unoriented) kdimensional 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 kdimensional varifolds in U , namely the space of Radon measures on G k (U ) := U × G(n + 1, k) (see [1,30] 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 Γ ⊂ 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 Γ is (locally) H k -rectifiable if it is H k -measurable, countably k-rectifiable, and H k (Γ) is (locally) finite. If Γ ⊂ U is locally H k -rectifiable, and θ ∈ L 1 loc (H k Γ ) is a positive function on Γ, then there is a k-varifold canonically associated to the pair (Γ, θ), namely the varifold var(Γ, θ) defined by for every ϕ ∈ C c (G k (U )) , (2.1) where T x Γ denotes the approximate tangent plane to Γ at x, which exists H k -a.e. on Γ. 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(Γ, θ) is rectifiable and θ(x) is an integer at H k -a.e. x ∈ Γ, 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 |Λ k ∇f (x) • S| := |∇f (x) · v 1 ∧ . . . ∧ ∇f (x) · v k | for any orthonormal basis {v 1 , . . . , v k } of S 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 Φ t (·) = Φ(t, ·) is any one-parameter family of diffeomorphisms of U defined for sufficiently small |t| such that Φ 0 = id U and ∂ t Φ(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 a given φ ∈ C 1 (U ; R + ) and V ∈ V k (U ), let (V, φ) ∈ V k (U ) be defined as the product φV . Then, one can define δ(V, φ)(g) as in (2.3) and obtain as in (2.4) that for every g ∈ C 1 c (U ; R n+1 ) This may be seen as a φ-weighted first variation of V . Using φ∇g = ∇(φg) − g ⊗ ∇φ in (2.7) and (2.4), we obtain If δV is absolutely continuous with respect to V , then we may use (2.5) in (2.8) to obtain 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 (2.9) to define for φ ∈ C 1 Observe that here we have used (2.6) in order to replace the term h( In fact, the "only if" part holds with equality in place of inequality. For a more comprehensive treatment of the Brakke flow, see [34,Chapter 2]. Formally, if ∂Γ(t) ⊂ ∂U is fixed in time, with φ = 1, we also obtain 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 (2.11) and (2.12), 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 (2.12) from 0 to T , we obtain the analogue of (2.13). By integrating (2.11) from t 1 to t 2 , we also obtain the analogue of (2.14) via the expression (2.10). 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 Σ".
Since we are assuming that ∂Γ 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 Γ 0 modulo H n -negligible sets (note that the assumptions (A2) and (A3) in Assumption 1.1 imply that Γ 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 [2,19,33] 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 . (11) Let µ be the product measure of V t and dt defined on U × R + , i.e. dµ := d V t dt.

Theorem 2.3. Under the same assumption of Theorem 2.2 and in addition to
Then, ∀t > 0, we have The claims (1)-(4) imply that {E i (t)} N i=1 is an L n+1 -partition of U , and that Γ(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 Γ(t) has the fixed boundary ∂Γ 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.
4 is a solution to Plateau's problem in U in the class of stationary varifolds satisfying the topological constraint (clos (spt V )) \ U = ∂Γ 0 . This is an interesting byproduct of our construction, above all considering that ∂Γ 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 [14,7] 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 Section 7.
2.6. 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 [20,2]. There, one constructs a time-parametrized 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 ∂Γ 0 . The rough idea to achieve this is to perform an "exponentially small" truncation of the approximate mean curvature vector near ∂Γ 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 ∂Γ 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 Section 4. In Section 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 Section 6, which therefore also contains the proofs of Theorems 2.2 and 2.3. Finally, Section 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 : 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 Γ = ∪ N i=1 ∂E i , we have ∂E = H n Γ , and where T x Γ ∈ G(n + 1, n) is the approximate tangent plane to Γ at x, which exists and is unique at H n -a.e. x ∈ Γ because of Definition 3.2(c).
it is Lipschitz continuous and satisfies the following.
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: 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 ψ(x) = 1 for |x| ≤ 1/2 , ψ(x) = 0 for |x| ≥ 1 , and we define, for each ε ∈ (0, 1), where the constant c(ε) is chosen in such a way that The function Φ ε will be adopted as a convolution kernel for the definition of the smoothing of a varifold. We record the properties of Φ 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 Φ 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 Φ ε is defined to be the Radon measure Φ ε * µ given by (3.15) 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 (Φ 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.
Observe that, given a Radon measure µ on R n+1 , one can identify the measure Φ ε * µ with a C ∞ function by means of the Hilbert space structure of L 2 (R n+1 ) = L 2 (L n+1 ). Indeed, for any φ ∈ C c (R n+1 ) we have that These considerations suggest the following definition for the smoothing of the first variation of a varifold. (3.17) 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 ϕ ∈ C c (G n (R n+1 )) with ϕ 0 ≤ 1, setting τ z (x) := x − z, it holds: Taking the supremum among all functions ϕ ∈ C c (G n (R n+1 )) with ϕ 0 ≤ 1 completes the proof.
We will often make use of [20,Lemma 5.1] with Ω ≡ 1 (and c 1 = 0). For the reader's convenience, we provide here the statement.
3.6. The cut-off functions η 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 timediscrete approximate flows which almost preserve the boundary ∂Γ 0 .
We shall also adopt the convention that (E) 0 = E.
Let U and Γ 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 Definition 3.15. Let ψ : (0, ∞) → R be a smooth function satisfying the following properties: Let {φ ρ } ρ , ρ > 0, be a standard family of mollifiers: precisely, let Lemma 3.16. There exists J = J(n) such that the following properties hold for all j ≥ J: Hence, η j (x) = ψ(e) = 1 because of property (a) of ψ in Definition 3.15.
, 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 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 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.
3.7. 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 Γ 0 , and we will suppose that j ≥ J(n), in such a way that the conclusions of Lemma 3.16 are satisfied.
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 Ω, respectively. We note that g η j satisfies |(g η j )(x)| ≤ jη j (x) and ∇(g η j )(x) ≤ 2 j 7 /4 η j (x): 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 any 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).
3.8. Curvature of limit varifold. 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 ℓ 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 ℓ, we have g η j ℓ = g due to Lemma 3.16(1), (3.27) and (3.26). Moreover, we may assume that g η j ℓ ∈ B j ℓ due to Lemma 3.16 (3). Then, by (3.35), (2) and (3), we have (3.37) Since η j ℓ ∈ A j ℓ 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 ) satisfieŝ 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 ℓ, we have g η j ℓφ ∈ B j ℓ and η j ℓφ ∈ A j ℓ (we may assume |φ| < 1 without loss of generality). Thus the same computation above with g η j ℓφ yieldŝ We let then i → ∞ in (3.40) to replaceφ by φ, and finally we approximate h(·, V ) by g to obtain (3.36).
3.9. 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: (2) j ≥ J(n) and η j is as in Definition 3.15; Then, for every φ ∈ A j we have the following estimates.
Proof. We want to estimate the following quantity which can be written as 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 (3.46) 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 suitably restricting ε. 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 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 |Λ n ∇f (y) • S| ≤ 1 + ε κ−5 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 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 Φ ε , we deduce that for l = 0, 1, 2. We can conclude using (3.59), (3.45)-(3.49) and suitable interpolations that:

Existence of limit measures
4.1. The construction of the approximate flows. Suppose U and Γ 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 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  Given an open set W , a function φ ∈ C 1 (R n+1 ; R + ) is said to be non decreasing in W along the fibers of the normal bundle of ∂U oriented by ν U , or simply (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 Γ 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 and such that, setting ∆t j := 2 −p j , and defining Γ j,k :

Proof of Proposition 4.2. Set
let κ = 3n+20 as in Proposition 3.21, and consider the following set of conditions for ε ∈ (0, 1): Notice that the conditions in (4.7) are compatible for large j, namely there exists j 0 with the property that for every j ≥ j 0 the set of ε ∈ (0, 1) satisfying (4.7) is not empty. Letting J(n) be the number provided by Lemma 3.16, for every j ≥ max{j 0 , J(n)} we choose ε j ∈ (0, 1) such that all conditions in (4.7) are met. Observe that lim j→∞ ε j = 0. Then, we choose p j ∈ N such that The argument is constructive, and it proceeds by means of an induction process on k ∈ {0, 1, . . . , j 2 p j }. We set U j,0 := U and E j,0 := E 0 . Properties (1), (2), (3), as well as the estimate in (4.3) are then trivially satisfied, given the definition of M and since U j,0 = U , (1), (2), (3), and (4.3) with k − 1 in place of k. We will now produce U j,k and E j,k = {E j,k,i } N i=1 satisfying the same conditions with k. At the same time, we will also show that each inductive step satisfies (4.4) and (4.5). Before proceeding, let us record the inductive assumptions for U j,k−1 and Step 1: area reducing Lipschitz deformation. First notice that D j,k−1 ⊂ U j,k−1 . Indeed, the definition of D j,k−1 , (4.9), and the choice of ε j imply that D j,k−1 ∩ (U j,k−1 △U ) = ∅, so that our claim readily follows from D j,k−1 ⊂ U . In particular, D j ⊂ D j,k−1 ⊂ U j,k−1 . Hence, we can choose f 1 ∈ E(E j,k−1 , D j , j) such that, setting E ⋆ j,k := (f 1 ) ⋆ E j,k−1 (∈ OP N (U j,k−1 ) by Lemma 3.5), we have and Step 2: retraction. Outside of D j,k−1 , we perform a suitable retraction procedure so and observe that We claim the validity of the following Lemma 4.3. We have A j,k ∩ Γ ⋆ j,k = ∅. Moreover, for any x ∈ ∂A j,k (the boundary as a subset of ∂(D j,k−1 ) j −10 ), we have dist (x, Γ ⋆ j,k ) ≥ j −10 .
Proof. By the discussion above, where the last inequality follows from k ≤ j 2 p j ≤ 2 j ε −κ j and the choice of ε j . By (4.16), we need to have somex ∈ Γ 0 ∩D j such that |x−x| < (k−1) exp(−j 1 /8 ). On the other hand, by the definitions of D j,k−1 and D j , |x −x| ≥ dist(A j,k , D j ) > 1/j 1/4 , and we have reached a contradiction. Thus the first claim follows. For the second claim, such On the other hand, dist (∂(D j,k−1 ) j −10 , D j ) > 1/(j 1/4 ), which is a contradiction. Thus we have the second claim.
Next, for each point x ∈ ∂(D j,k−1 ) j −10 , let r 0 (x) ∈ ∂D j,k−1 be the nearest point projection of x onto ∂D j,k−1 , and set r s (x) := sx + (1 − s)r 0 (x) for s ∈ (0, 1). With this notation, define 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)}. contains A j,k,l ∪ (∂A j,k,l ) j −10 . For each index l, let i(l) be this correspondence. We define for each i = 1, . . . , NẼ . By Lemma 4.4, x ∈ Ret j,k , which is a contradiction. This proves the first claim. The second claim follows from the definition ofΓ j,k , in the sense that the new partition has no boundary in Ret j,k , while Γ ⋆ 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. Proof. Note thatΓ j,k △Γ ⋆ j,k ⊂ (∂D j,k−1 ∩Ret j,k )∪Ret j,k , and thatΓ j,k ∩Ret j,k = ∅. Let Ret j,k,l and E ⋆ j,k,i(l) be as before. For any ). 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.
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
namely (4.10) with k in place of k − 1 holds true.
namely (4.11) with k in place of k − 1 holds true.
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  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 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}.
Therefore, by a diagonal argument we can choose a subsequence {j ℓ } 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 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 ℓ for every j ℓ ≥ j * . Hence, for every j ℓ ≥ j * we can apply (4.5) repeatedly with φ = φ q + i −1 ∈ A j ℓ ∩ R j ℓ and (4.38) again with φ = φ q + i −1 so that η j ℓ φ ∈ A j ℓ in order to deduce (4.39) As we let ℓ → ∞, the left-hand side of (4.39) can be bounded from below using (4.31) and (4.32) as (4.40) 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 ℓ ≤ 1, let ℓ → ∞ 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 B := t ∈ R + : lim By the monotonicity of each g q , B is a countable subset of R + , and for every t ∈ R + \ (B ∪ 2 Q ) we can define µ t (φ q ) for every φ q ∈ Z by µ t (φ q ) := lim for every t ∈ R + \ (B ∪ 2 Q ) and φ q ∈ Z . (4.44) Indeed, due to the definition of ∂E j ℓ (t), there exists a sequence {t ℓ } ∞ ℓ=1 ⊂ 2 Q such that lim ℓ→∞ t ℓ = t+ and ∂E j ℓ (t) = ∂E j ℓ (t ℓ ). For any s ∈ 2 Q with s > t, and for all suffciently large ℓ so that s > t ℓ , we deduce from (4.39) that (4.45) Taking the lim inf ℓ→∞ 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 ∂E j ℓ (t ℓ ) = ∂E j ℓ (t) yield 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 ℓ (t)} ∞ ℓ=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 for any φ ∈ C c (U ; R + ).
Before proving Proposition 5.1, we need to state two important results, which are obtained by suitably modifying [20, Theorem 7.3 & Theorem 8.6], respectively.
Proof. The existence of a subsequence {∂E j ′ ℓ } ∞ ℓ=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 ℓ ⊂ clos U j ℓ 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 ′ ℓ (U r (x)) = 0 for all sufficiently large ℓ, 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 ℓ so that

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 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 ( converges, as ℓ → ∞, 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 ℓ→∞ ∂E j ′ ℓ (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 ′ ℓ } ∞ ℓ=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 ′ ℓ (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 ′ ℓ (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 ℓ→∞ ∂E j ℓ (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 ℓ be such that j ℓ ≥ m and j ℓ ≥ t 2 , so that ∂E j ℓ (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 ℓ , 2 ∆t j ℓ , . . . , j ℓ 2 p j ℓ ∆t j ℓ . Since ∆t j ℓ → 0 as ℓ → ∞, we can assume without loss of generality that ∆t j ℓ < t 2 − t 1 , so that there exist k 1 , (5.10) Sinceφ = φ + i −1 , we can estimate the left-hand side of (5.10) from below as (5.11) so that when we let ℓ → ∞ 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 ℓ = ∂E j ℓ (t) and h ε j ℓ = h ε j ℓ (·, ∂E j ℓ ), we proceed as in (4.35) writing where we have used that ∇φ = ∇φ. Since η j ℓφ ∈ A j ℓ , 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 δ(∂E j ℓ (t),φ)(η j ℓ h ε j ℓ (·, ∂E j ℓ (t))) ≤ c , (5.17) where c depends only on φ C 2 and ∂E 0 (R n+1 ). Using (5.17) together with the definition of ∂E j ℓ (t) and Fatou's lemma, one can readily show that, when we take the lim sup as ℓ → ∞, the right-hand side of (5.10) can be bounded by By the identity in (5.13), we also have that along the same subsequence 20) where once again ∂E j ℓ = ∂E j ℓ (t) and h ε j ℓ = h ε j ℓ (·, ∂E j ℓ ). Using (5.14) and (5.16), we see that the right-hand side of (5.20) can be bounded from above by lim inf ℓ→∞ 2 b j ℓ + c, whereas the left-hand side can be bounded from below by lim sup ℓ→∞ where c depends on φ C 2 and ∂E 0 (R n+1 ). As a consequence, along any subsequence {j ′ ℓ } satisfying (5.19) one has that lim sup 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 ′ ℓ : hence, the whole sequence {∂E j ′ ℓ (t)} ∞ ℓ=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 ℓ→∞ˆR n+1 Using (5.19), (5.13), (5.14),φ > φ, and Proposition 5.
where we have also used that, as ℓ → ∞, η j ′ ℓ = 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 We proceed estimating each term of (5.26). Using that η j ′ ℓ = 1 on {∇φ = 0} for all ℓ sufficiently large, the Cauchy-Schwarz inequality gives that for all ℓ sufficiently large. Since (x, S) → |S ⊥ (∇φ(x)) − g(x)| 2 ∈ C c (G n (U )), we have that Analogously, since η j ′ ℓ = 1 on {g = 0} for all ℓ sufficiently large, we have that (3.35) and (5.22).
Next, by varifold convergence of ∂E j ′ ℓ to V t on U , given that g has compact support in U , we also have lim 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 ℓ defined bŷ The error one makes in order to putφ back into (5.18) in place ofφ j ℓ is then given by the product of ∆t j ℓ times some negative powers of ε j ℓ ; nonetheless, this error converges to 0 uniformly as ℓ ↑ ∞ by the choice of ∆t j ℓ , 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 + ).

Boundary behavior and proof of main results
6.1. 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 \ ∂Γ 0 . This is a preliminary result, and using the Brakke's inquality, we eventually prove that they actually vanish outside the convex hull of Γ 0 ∪ ∂Γ 0 in Proposition 6.4. Proposition 6.1. Forx ∈ ∂U \ ∂Γ 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 (6.1)

Remark 6.2.
Due to the definition of ∂Γ 0 and the strict convexity of U , note that there exists such an affine hyperplane A for any givenx ∈ ∂U \ ∂Γ 0 . For example, we may choose a hyperplane A which is parallel to the tangent space of ∂U atx and which passes througĥ x − ν U (x)c. 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 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/(4j 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, 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 havê 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 B 1 (y) \ B √ ε j (y), 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, Using this, we can estimate In view of (6.5), this shows that (6.7) can be absorbed as a small error term. Continuing from (6.6) with ∇φ(y) replacing ∇φ(x), (6.10) The last term of (6.10) may be estimated as Here, we used the fact that the integrand is 0 far away from U , for example, outside of (U ) 2 . The last term of (6.11) can be absorbed as a small error since j ≤ ε We replace ∇ 2 φ(y) by ∇ 2 φ(x), with the resulting error being estimated, for instance, by ≤ M ε 1 /2 j using standard methods as above. Then, we have Thus, combining (6.4)-(6.12) and recovering the notations, we obtain for all sufficiently large j. By (6.3), we have 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).
Proof. Suppose that A ⊂ R n+1 is a hyperplane such that, using the notation in the statement of Proposition 6.1, Γ 0 ∪ ∂Γ 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 \ ∂Γ 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 V t ((∂U ) r 0 ∩ A + ) = 0 (6.16) for all t ≥ 0. Let ψ ∈ C ∞ c (U ; R + ) be such that ψ = 1 on U \ (∂U ) r 0 and ψ = 0 on (∂U ) r 0 2 . We next use φ = ψ d 4 A in (5.8) with t 1 = 0 and an arbitrary t 2 = t > 0 to obtain S · ∇ 2 φ dV s (·, S) ds.
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, namelŷ (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 ℓ 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]. for each t and i the volumes L n+1 (E j ℓ ,i (t)) are uniformly bounded in ℓ. Furthermore, by the mass estimate in (4.31) we also have that ∇χ E j ℓ ,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 }, where E i (t) is a set of locally finite perimeter in R n+1 . Moreover, using that E j ℓ ,i (t) ⊂ (U ) 1/(4 j 1 /4 ℓ ) (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 ∪ (Γ 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 Γ 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 ℓ ,i(x,0) (0) for all ℓ ∈ 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 ∪ (Γ 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, By examining the definition, one obtains Γ(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 Γ(t) has empty interior, which shows (4). The claims (1) and (2) 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 ∂Γ(t) ⊂ ∂Γ 0 for all t > 0. Later in Proposition 6.9, we prove (clos (spt V t ))\U = ∂Γ 0 and ∂Γ 0 ⊂ ∂Γ(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 L 1 (R n+1 ) as ℓ ↑ ∞, for every t ∈ R + . 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 ℓ (t) (see e.g. [23,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 (Γ 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 This is a contradiction. Thus, we have L n+1 (E i (t) \Ẽ i ) = 0 for all i = 1, . . . , N . Since {Ẽ 1 , . . . ,Ẽ N } is a partion of U , we have L n+1 (E i (t)△Ẽ i ) = 0 for all i. 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. Proposition 6.9. For all t ≥ 0, it holds (clos (spt V t )) \ U = ∂Γ 0 .
Proof. Let x ∈ (clos (spt V t ))\U , and let {x k } ∞ k=1 be a sequence with x k ∈ spt V t such that x k → x as k ↑ ∞. If x / ∈ ∂Γ 0 , then by Proposition 6.1 there is r > 0 such that V t (B r (x) ∩ U ) = 0. For all suitably large k so that |x − x k | < r we then have V t (B r−|x−x k | (x k ) ∩ U ) = 0, which contradicts the fact that x k ∈ spt V t .
Conversely, let x ∈ ∂Γ 0 , and suppose for a contradiction that x / ∈ clos (spt V t ). Let Z be as in Assumption 1.1 (A4). Since Z is dense in ∂Γ 0 , we may as well assume that x ∈ Z ∩∂Γ 0 \clos (spt V t ), and there is a radius r > 0 with the property that B r (x)∩spt V t = ∅. Then, Theorem 2.3 (8) implies that ∇χ E i (t) (B r (x) ∩ U ) = 0 for every i ∈ {1, . . . , N }. Since B r (x) ∩ U is connected by the convexity of U , every χ E i (t) is either constantly equal to 0 or 1 on B r (x) ∩ U , namely for some ℓ ∈ {1, . . . , N } . (6.23) 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 Let z denote any of the points x j or x ′ j , and observe that the above condition guarantees that z ∈ ∂U \ ∂Γ 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, necessarily, 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 Γ 0 .
The proof of Theorem 2.2 is now complete: {V t } t≥0 is a Brakke flow with fixed boundary ∂Γ 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.

Applications to the problem of Plateau
As anticipated in the introduction, an interesting byproduct of our global existence result for Brakke flow is the existence of a stationary integral varifold V in U satisfying the topological boundary constraint clos(spt V ) \ U = ∂Γ 0 . This is the content of Corollary 2.4, which we prove next.
Proof of Corollary 2.4. By the estimate in (5.7), the function Again by (5.7), we have that Furthermore, combining (2.5) with (7.2) yields, via the Cauchy-Schwarz inequality, that in L 1 (U ) and pointwise L n+1 -a.e. as k → ∞ , (7.5) where E i ⊂ U are sets of finite perimeter. Since, by Theorem 2.
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 semicontinuous θ : U → R + with θ(x) ≥ 1 at each x ∈ spt V . In particular, setting Γ := 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(Γ 0 ∪ ∂Γ 0 ). We have not excluded the possibility that H n (Γ) = 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 (Γ) > 0 and this completes the proof of (3). In order to conclude the proof, we are just left with the boundary condition (4), namely (clos (spt V )) \ U = ∂Γ 0 . (7.8) Towards the first inclusion, suppose that x ∈ (clos (spt V )) \ U , and let ∈ ∂Γ 0 then Proposition 6.1 implies that there exists r > 0 such that 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 ∈ ∂Γ 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 ∂Γ 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 = ∂Γ 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 [6,5]; 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 Σ be a closed subset of R n+1 . Consider the family C Σ := γ : S 1 → R n+1 \ Σ : γ is a smooth embedding of S 1 into R n+1 \ Σ . (7.10) A subfamily C ⊂ C Σ 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 \ Σ. Given a homotopically closed C ⊂ C Σ , a relatively closed subset K ⊂ R n+1 \ Σ is C-spanning Σ if 2 K ∩ γ = ∅ for every γ ∈ C . (7.11) Remark 7.2. If C ⊂ C Σ 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 \ Σ strongly homotopically spans Σ if it C-spans Σ for every homotopically closed family C ⊂ C Σ which does not contain any homotopically trivial curve. Namely, if K ∩ γ = ∅ for every γ ∈ C Σ such that [γ] = 0 in π 1 (R n+1 \ Σ).
We can prove the following proposition, whose proof is a suitable adaptation of the argument in [7, Lemma 10]. Proposition 7.4. Let n ≥ 2, and let U, Γ 0 , E 0 be as in Assumption 1.1. Suppose that the initial partition E 0 satisfies the following additional property: Given any two connected components S 1 and S 2 of ∂U \ ∂Γ 0 , there are two indices i, j ∈ {1, . . . , N } with i = j such that S 1 ⊂ clos E 0,i and S 2 ⊂ clos E 0,j .
Proof. Let γ : S 1 → R n+1 \ ∂Γ 0 be a smooth embedding that is not homotopically trivial in R n+1 \ ∂Γ 0 . The goal is to prove that, for every t ∈ [0, ∞], Γ(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 )), and γ ∩ (∂U \ ∂Γ 0 ) = h {γ(a h ), γ(b h )}. If there is h such that γ(a h ) and γ(b h ) belong to two distinct connected components of ∂U \ ∂Γ 0 , then (⋄) implies that the arc σ h := γ| (a h ,b h ) must intersect U ∩ ∂E i (0) for some i = 1, . . . , N . In fact, since the labeling of the open partition at the boundary of U is invariant along the flow, the same conclusion holds for every t ∈ [0, ∞].
In particular, in this case γ intersects i (∂E i (t) ∩ U ) = Γ(t) for every t ∈ [0, ∞]. Hence, if by contradiction γ has empty intersection with Γ(t), then necessarily for every h there is a connected component 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 Π, 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 Φ s : R n+1 → R n+1 be the smooth diffeomorphism of R n+1 defined by Φ s (x) := x + ϕ s (ρ U (x)) ν U (Π(x)) , (7.12) where is the signed distance function from ∂U , and ϕ s = ϕ s (t) is a smooth function such that ϕ s (t) = 0 for all |t| ≥ 2s , and ϕ s (s) = s .
Example 7.5. Suppose that U = U 1 (0) ⊂ R 3 , and ∂Γ 0 is the union of two parallel and coaxial circles contained in S 2 = ∂U . Even though both the stable and the unstable catenoid with boundary ∂Γ 0 are stationary varifolds V satisfying (clos(spt V ))\U = ∂Γ 0 , if the initial partition satisfies (⋄), then they 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 \∂Γ 0 having empty intersection with each of them.
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 ∂Γ 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 Σ ⊂ ∂U such that there are N ≥ 2 and E 0 , Γ 0 as in Assumption 1.1 such that Σ = ∂Γ 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 Σ ⊂ ∂U 1 (0) in R 2 with H 1 (Σ) > 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 Σ to be admissible. Remark 7.7. Let us explicitly observe that, even in the case when Γ 0 (or more precisely var(Γ 0 , 1)) is stationary, it is false in general that V t = var(Γ 0 , 1) for t > 0. In other words, the approximation scheme which produces the Brakke flow V t may move the initial datum var(Γ 0 , 1) even when the latter is stationary. A simple example is a set consisting of two line segments with a crossing. If Γ 0 is a smooth minimal surface with smooth boundary ∂Γ 0 , the uniqueness theorem for classical MCF should allow Γ t ≡ Γ 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 Γ 0 other than the time-independent solution. It would be interesting to characterize the regularity properties of those stationary Γ 0 with E 0,1 , . . . , E 0,N satisfying Assumption 1.1 and H n (Γ 0 \∪ N i=1 ∂ * E 0,i ) = 0 which do not allow any non-trivial Brakke flow. We expect that such a Γ 0 should have some local measure minimizing properties.
and Γ be as in Corollary 2.4 obtained as t k → ∞ along a Brakke flow. Since V is integral and stationary, V = var(Γ, θ) for some H n -measurable function θ : Γ → N. One can check that Γ 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(Γ, 1) may not be stationary, and the Brakke flow starting from non-stationary var(Γ , 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.