# Convergence of Allen–Cahn equations to De Giorgi’s multiphase mean curvature
flow

Pascal<EMAIL_ADDRESS>Institute for Applied Mathematics,
University of Bonn, Endenicher Allee 60, D-53115 Bonn / Germany

###### Abstract

This paper presents a conditional convergence result of solutions to the
Allen–Cahn equation with arbitrary potentials to a De Giorgi type
$\operatorname{BV}$-solution to multiphase mean curvature flow. Moreover we
show that De Giorgi type $\operatorname{BV}$-solutions are De Giorgi type
varifold solutions, and thus our solution is unique in a weak-strong sense.
Keywords: Gradient flows, Mean curvature flow, Allen–Cahn equation
Mathematical Subject Classification: 35A15, 35K57, 35K93, 53E10, 74N20

##  1 Introduction

###  1.1 History and main results

Multiphase mean curvature flow is an important geometric evolution equation
which has been studied for a long time, bearing not only mathematical
importance, but also for the applied sciences. Originally it was proposed to
study the evolution of grain boundaries in annealed recrystallized metal, as
described by Mullins in [Mul56], who cites Beck in [Bec52] as already having
observed such a behaviour in 1952.

Over the years numerous different solution concepts for multiphase mean
curvature flow have been proposed. Classically we have smooth solutions, where
we require the evolution of the interfaces to be smooth, for example described
by Huisken in [Hui90]. Another description of smoothly evolving mean curvature
flow can be found in the work of Gage and Hamilton [GH86], who proved the
“shrinking conjecture” for convex planar curves. Brakke describes in his book
[Bra78] the motion by mean curvature using varifolds, which yields a quite
abstract and general notion for mean curvature flow and is based on the
gradient flow structure of mean curvature flow. Luckhaus and Sturzenhecker
introduced a distributional solution concept for mean curvature flow in their
work [LS95]. Another approach is the viscosity solution concept presented in
[CGG91] and [ES91], where it is shown that solutions of a certain parabolic
equation have the property that if they are smooth, the corresponding level
sets move by mean curvature.

For some smooth potential $W\colon\mathbb{R}^{N}\to[0,\infty)$ with finitely
many zeros $\alpha_{1},\dotsc,\alpha_{P}$, the Allen–Cahn equation

$\partial_{t}u_{\varepsilon}=\Delta
u_{\varepsilon}-\frac{1}{\varepsilon^{2}}\nabla W(u_{\varepsilon})$ (1)

is commonly used as a phase-field approximation for mean curvature flow and
was first proposed by Allen and Cahn in their paper [AC79]. Since then it has
been an extensively researched topic, both from a numerical and analytical
standpoint.

If we consider the two-phase case, which corresponds to setting $N=1$ and
$P=2$, then the behaviour of the solutions to (1) as $\varepsilon$ tends to
zero are thoroughly researched. The authors Bronsard and Kohn showed in [BK91]
the compactness of solutions and regularity for the limit by exploiting the
gradient flow structure of (1), more precisely by mainly utilizing the
corresponding energy dissipation inequality. Moreover they studied the
behaviour for radially symmetric initial data and showed that the limit moves
by mean curvature in this case. Chen has proven in [Che92] that as long as the
interface of the limit evolves smoothly, we have convergence to (classical)
mean curvature flow. A similar result has been shown by De Mottoni and
Schatzmann in [DS95], where they showed short time convergence. Their strategy
was to prove through a spectral estimate that the solution $u_{\varepsilon}$
already coincides with the formal asymptotic expansion up to an error. Similar
strategies, but for nonlinear Robin boundary conditions with angle close to 90
degrees, can be found in the paper [AM22] by Abels and Moser.

Ilmanen made fundamental contributions in [Ilm93] by proving the convergence
to Brakke’s mean curvature flow as long as the initial conditions are well-
prepared by exploiting the equipartition of energies. However his methods seem
to only work in the two-phase case since he uses a comparison principle whose
generalization to the vectorial case is not clear.

The multiphase case for both the convergence of the Allen–Cahn equation and
mean curvature flow is much more involved and a topic of current research. For
the convergence of the Allen–Cahn equation, asymptotic expansions with Neumann
boundary conditions have been studied by Keller, Rubinstein and Sternberg in
[RSK89]. Even though they considered the vector-valued Allen–Cahn equation
with a multiwell potential, their analysis restricts to the parts of the
interfaces where no triple junctions appear. Bronsard and Reitich however
considered in [BR93] a formal asymptotic expansion which also takes triple
junctions into account for the three-phase case. Moreover they proved short-
time existence for the three-phase boundary problem.

The analysis of multiphase mean curvature flow has been studied for example by
Mantegazza, Novaga and Tortorelli in [MNT04], where they considered the planar
case when only a single triple junction appears, and their work has been
extended to several triple junctions in [Man+16]. Ilmanen, Neves and Schulze
furthermore proved in [INS19] that even for non-regular initial data in the
sense that Herring’s angle condition is not satisfied, we have short time
existence by approximation through regular networks. For long time existence,
it has been shown by Kim and Tonegawa in [KT17] through a modification of
Brakke’s approximation scheme that one can obtain non-trivial mean curvature
flow even with singular initial data. Recently Stuvard and Tonegawa improved
this result in [ST22], where they extended their findings to more general
initial data and also showed that their constructed Brakke flow is a
$\operatorname{BV}$-solution to mean curvature flow.

The first main goal of this paper Theorem 4.5 is to prove a conditional
convergence result of solutions to the vectorial Allen–Cahn equation (1) to a
De Giorgi type $\operatorname{BV}$-solution of multiphase mean curvature flow
in the sense of Definition 4.1. The proof is based mainly on a duality
argument and the results of Laux and Simon in [LS18]. However the strong
assumption we make here is that the Cahn–Hilliard energies of the solutions to
the Allen-Cahn equation (1) converge to the perimeter functional (9) applied
to the limit. This prevents that as $\varepsilon$ tends to zero, the
approximate interfaces collapse, which corresponds to a loss of energy in the
limit $\varepsilon\to 0$, see also the discussion in Section 4.4. In general
we are inclined to believe that this assumption could fail. For example it has
been shown by Bronsard and Stoth in [BS96] that for the volume preserving
Allen–Cahn equation and for radial-symmetric initial data, we can have any
number of higher multiplicity transition layers which are at most
$C\varepsilon^{\alpha}$ apart, at least for times of order one. Here $\alpha$
is some exponent between zero and one third. However it has been proven that
in some cases, the energy convergence automatically holds for suitable initial
data. For example Nguyen and Wang proved in [NW20] that for mean convex
initial data, we obtain the desired convergence of energies. In the context of
the Almgren-Taylor-Wang scheme for mean curvature flow, this has also been
done by De Philippis and Laux in [DL20].

The energy convergence assumption provides us with the important equipartition
of energies, whose proof under milder assumption was the main obstacle of
Ilmanen in [Ilm93]. Moreover it is the key for our localization estimates and
lets us localize on the different phases of the limit. Lastly it assures that
the differential $\nabla u_{\varepsilon}$ can locally up to an error be
written as a rank-one matrix. In fact it is the tensor of the approximate
frozen unit normal and the gradient of the geodesic distance function
associated to the majority phase evaluated at $u_{\varepsilon}$, see the proof
of [LS18, Prop. 3.1].

The second main goal is to compare De Giorgi type
$\operatorname{BV}$-solutions to De Giorgi type varifold solutions. The latter
were proposed by Hensel and Laux in [HL21]. We will show that the De Giorgi
type $\operatorname{BV}$-solution concept is stronger in the sense that every
$\operatorname{BV}$-solution is also a varifold solution. Since Hensel and
Laux have shown weak-strong uniqueness for their varifold solution concept, it
follows that we also obtain weak-strong uniqueness for our De Giorgi type
$\operatorname{BV}$-solution concept, which is the best we can expect. In fact
it has been shown on a numerical basis by Angenent, Chopp and Ilmanen in
[AIC95] that even in three dimensions, there exists a smooth hypersurface
whose evolution by mean curvature flow admits a singularity at a certain time
after which we have nonuniqueness. A rigorous analysis of this phenomenon has
been done in dimensions 4, 5, 6, 7 and 8 by Angenent, Ilmanen and Velázquez in
[AIV02].

Let us also mention some of the closely related unanswered questions. For one
Hensel and Laux have shown in [HL21] that in the two-phase case and under well
prepared initial conditions, the solutions of the Allen–Cahn equation (1)
converge to a De Giorgi type varifold solution. However their methods have no
obvious generalization to the multiphase case without the energy convergence
assumption, and one even struggles to find an approximate sequence which
constructs the desired varifolds. And even then one would have to find
suitable substitutions for localization which will be crucial for our proof.
These localization estimates are based on De Giorgi’s structure theorem and
thus only work for $\operatorname{BV}$-functions.

Another possible question would be how to generalize the results to the case
of arbitrary mobilities: Throughout the paper, and also the main background
papers [LS18], [HL21], it is always assumed that the mobilities are fixed
through the relation $\mu_{ij}=1/\sigma_{ij}$. Here $\sigma_{ij}$ denotes the
surface tension of the $(i,j)$-th interface and $\mu_{ij}$ its mobility. As
proposed by Bretin, Danescu, Penuelas and Masnou in [Bre+18], passing to
arbitrary mobilities should amount to multiplying an appropriate “mobility
matrix” $M$ onto the right-hand side of (1) and changing the metric of the
underlying space accordingly to $\langle u,v\rangle=\int\left\langle
Mu\,,\,v\right\rangle\differential{x}$. The difference in their approach is to
first uncouple their system so that they arrive at the scalar Allen–Cahn
equation and then couple the components through a Lagrange-multiplier, which
assures that the limit is a partition.

###  1.2 Structure of the paper

In Section 2 we first describe the concept of gradient flows in an easy
setting. Moreover we derive De Giorgi’s optimal energy dissipation inequality
in a simple example and discuss its usefulness for reformulating the gradient
flow equation. Further we apply our observations to (multiphase) mean
curvature flow.

Continuing with Section 3, we take a look at the Allen–Cahn equation and the
convergence of its solutions as $\varepsilon$ tends to zero to an evolving
partition by citing the results of [LS18].

Lastly in Section 4 we present new results building on the previous insights.
We start off by presenting a De Giorgi type $\operatorname{BV}$-solution
concept for multiphase mean curvature flow. This is followed by proving a
similar conditional convergence result as in the previous section. In the
final Section 4.4, we discuss the assumption of energy convergence. Moreover
we show that every De Giorgi type $\operatorname{BV}$-solution is a De Giorgi
type varifold solution in the sense of Hensel and Laux in [HL21], whose
solution concept does not rely on the assumption of energy convergence.

###  1.3 Notation

  * •

Let $u\colon\mathbb{R}^{d}\to\mathbb{R}^{N}$ be differentiable at a point
$x\in\mathbb{R}^{d}$. We write $\mathrm{D}u(x)$ for the (total) derivative at
$x$, which means that $\mathrm{D}u(x)\in\mathbb{R}^{N\times d}$ and
$\left(\mathrm{D}u(x)\right)_{ij}=\partial_{x_{j}}u^{i}(x)$. We always use the
notation $\nabla u(x)$ for the transpose of the total derivative.

  * •

If $u\colon(0,T)\times\mathbb{R}^{d}\to\mathbb{R}^{N}$, then we denote by
$\mathrm{D}u(t,x)$ respectively $\nabla u(t,x)$ only the derivatives in space,
and we always write $\partial_{t}u$ for the derivative in time.

  * •

We use the symbol $\lesssim$ if an inequality holds up to a positive constant
on the right hand side. This constant must only depend on the dimensions, the
chosen potential $W$ and possibly the given initial data.

  * •

The bracket $\left\langle\>\cdot\>\,,\,\>\cdot\>\right\rangle$ is used as the
Euclidean inner product. Depending on the situation, it acts on vectors or
matrices. If we apply $\absolutevalue{\>\cdot\>}$ to a vector or matrix, then
we always uses the norm induced by the Euclidean inner product.

  * •

We denote by $\operatorname{C}_{\mathrm{c}}$ the compactly supported
continuous functions. Note that for the flat torus
$\operatorname{\mathbb{T}}$, we have
$\operatorname{C}_{\mathrm{c}}(\operatorname{\mathbb{T}})=\operatorname{C}(\operatorname{\mathbb{T}})$.

  * •

For a locally integrable function $u\colon(0,T)\times\Omega\to\mathbb{R}$ with
$\Omega\subseteq\mathbb{R}^{d}$ being some open set or the flat torus, we
denote the total variation in space for a time $t\in(0,T)$ by

$\absolutevalue{\nabla
u(t,\cdot)}\coloneqq\sup\left\\{\int_{\Omega}u(t,x)\operatorname{div}\xi(x)\differential{x}\,\colon\,\xi\in\operatorname{C}_{\mathrm{c}}^{1}(\Omega;\mathbb{R}^{d}),\absolutevalue{\xi}\leq
1\right\\}$

and the total variation in space taken both over time and space by

$\displaystyle\absolutevalue{\nabla u}_{d+1}$ $\displaystyle\coloneqq{}$
$\displaystyle\left\\{\int_{(0,T)\times\Omega}u(t,x)\operatorname{div}_{x}\xi(t,x)\differential{(t,x)}\,\colon\,\xi\in\operatorname{C}_{\mathrm{c}}^{1}((0,T)\times\Omega;\mathbb{R}^{d}),\absolutevalue{\xi}\leq
1\right\\}.$

  * •

If $u\colon(0,T)\times\Omega\to\mathbb{R}^{N}$ is some map, then for a given
time $t\in(0,T)$, we sometimes write $u(t)$ for the map $u(t)(x)\coloneqq
u(t,x)$ by a slight abuse of notation.

  * •

To shorten notation, we always write
$\int\differential{x}=\int_{\operatorname{\mathbb{T}}}\differential{x}$, where
$\operatorname{\mathbb{T}}$ is the flat torus.

##  2 Gradient flows and mean curvature flow

###  2.1 Gradient flows

In the simplest case, a gradient flow of a given energy
$\operatorname{E}\colon\mathbb{R}^{N}\to\mathbb{R}$ with respect to the
Euclidean inner product is a solution to the ordinary differential equation

$\derivative{t}x(t)=-\nabla\operatorname{E}(x(t)),$ (2)

where we usually prescribe some initial value $x(0)=x_{0}\in\mathbb{R}^{N}$.
The central structure here is that on the right hand side of the equation, we
do not have an arbitrary vector field, but the gradient of some continuously
differentiable function. Remember that the gradient of a function always
depends on the chosen metric of our space. In fact, the gradient
$\nabla\operatorname{E}(x)$ is always the unique element of the tangent space
at $x$ which satisfies

$\left\langle\nabla\operatorname{E}(x)\,,\,y\right\rangle=\differential_{x}\operatorname{E}(y)$

for all elements $y$ in the tangent space at $x$. In this simple setting, the
tangent space at any point is exactly $\mathbb{R}^{N}$.

A solution $x$ moves in the direction of the steepest descent of the energy
$\operatorname{E}$. Moreover this allows for the following computation given a
continuously differentiable solution $x$ of (2):

$\displaystyle\derivative{t}\operatorname{E}(x(t))$
$\displaystyle=\left\langle\nabla\operatorname{E}(x(t))\,,\,x^{\prime}(t)\right\rangle=-\absolutevalue{\nabla\operatorname{E}(x(t))}^{2}=-\absolutevalue{x^{\prime}(t)}^{2}$
$\displaystyle=-\frac{1}{2}\left(\absolutevalue{x^{\prime}(t)}^{2}+\absolutevalue{\nabla\operatorname{E}(x(t))}^{2}\right).$

We especially obtain that the function $\operatorname{E}(x(t))$ is non-
increasing, which coincides with our intuition of the steepest descent. But
more precisely, we obtain from the fundamental theorem of calculus the _energy
dissipation identity_

$\operatorname{E}(x(T))+\frac{1}{2}\int_{0}^{T}\absolutevalue{x^{\prime}(t)}^{2}+\absolutevalue{\nabla\operatorname{E}(x(t))}^{2}\differential{t}=\operatorname{E}(x(0)).$
(3)

One could now raise the question if this identity already characterizes
equation (2). But as it turns out, we can go even one step further. Namely we
only ask for _De Giorgi’s optimal energy dissipation inequality_ given by

$\operatorname{E}(x(T))+\frac{1}{2}\int_{0}^{T}\absolutevalue{x^{\prime}(t)}^{2}+\absolutevalue{\nabla\operatorname{E}(x(t))}^{2}\differential{t}\leq\operatorname{E}(x(0)).$
(4)

We call this inequality optimal since as demonstrated before, we usually
expect an equality to hold if $x$ is a solution to (2). Now let us assume that
$x$ satisfies inequality (4) and is sufficiently regular. Then we can estimate
again by the fundamental theorem of calculus that

$\displaystyle\frac{1}{2}\int_{0}^{T}\absolutevalue{\nabla\operatorname{E}(x(t))+x^{\prime}(t)}^{2}\differential{t}$
$\displaystyle={}$
$\displaystyle\int_{0}^{T}\left\langle\nabla\operatorname{E}(x(t))\,,\,x^{\prime}(t)\right\rangle\differential{t}+\frac{1}{2}\int_{0}^{T}\absolutevalue{x^{\prime}(t)}^{2}+\absolutevalue{\nabla\operatorname{E}(x(t))}^{2}\differential{t}$
$\displaystyle={}$
$\displaystyle\operatorname{E}(x(T))-\operatorname{E}(x(0))+\frac{1}{2}\int_{0}^{T}\absolutevalue{x^{\prime}(t)}^{2}+\absolutevalue{\nabla\operatorname{E}(x(t))}^{2}\differential{t}$
$\displaystyle\leq{}$ $\displaystyle 0.$

Since we started with an integral over a non-negative function, this implies
that for almost every time $t$, we have that
$x^{\prime}(t)=-\nabla\operatorname{E}(x(t))$. But if $x$ and $E$ are
sufficiently regular, this already implies that
$x^{\prime}(t)=-\nabla\operatorname{E}(x(t))$ holds for all times $t$. Thus
$x$ is a gradient flow of the energy $\operatorname{E}$.

The real strength of formulating the differential equation (2) via inequality
(4) becomes clear if we want to consider gradient flows in a more complicated
setting. In order to formulate equation (2), we need to have a notion of
differentiation and a gradient in the target of $x$. Therefore we may use pre-
Hilbert spaces or smooth Riemannian manifolds as suitable substitutes for
$\mathbb{R}^{N}$.

Examples for such more complicated gradient flows include the heat equation,
which can be written as the $\operatorname{L}^{2}$-gradient flow of the
Dirichlet energy. Further examples include the Fokker-Planck equation, the
Allen–Cahn equation and mean curvature flow. The two latter of course play a
key role for us.

The observations in this section have been around for a long time and are
credited to De Giorgi and his paper [De ̵93]. Sandier and Serfaty have also
written an excellent paper [SS04] on this topic, as well as the book [AGS05]
by Ambrosio, Gigli and Savaré, to which we refer the interested reader.

###  2.2 Mean curvature flow

Mean curvature flow describes the geometric evolution of a set
$(\Omega(t))_{t\geq 0}$ respectively the evolution of its boundary
$\Sigma(t)\coloneqq\partial\Omega(t)$. It is formulated through the equation

$\frac{1}{\mu}V=-\sigma H\quad\text{on }\Sigma.$ (5)

Here $\mu>0$ is a positive constant which is called the _mobility_ and
$\sigma>0$ is a positive constant as well which we define as the _surface
tension_. By $V$ we denote the normal velocity of the set and by $H$ its mean
curvature, which is defined as the sum of the principle curvatures at a given
point. There are a lot of different notions of solutions to this equation as
already mentioned in the introduction. For us the central structure of this
equation is its gradient flow structure. Formally we want to consider the
space

$\mathcal{M}\coloneqq\left\\{\text{hypersurfaces
in}\operatorname{\mathbb{T}}\right\\},$

where the tangent space at a given $\Sigma\in\mathcal{M}$ consists of the
normal velocities on $\Sigma$. The metric tensor at a surface $\Sigma$ of two
normal velocities is then given by the rescaled $\operatorname{L}^{2}$-inner
product on $\Sigma$

$\left\langle
V\,,\,W\right\rangle_{\Sigma}\coloneqq\frac{1}{\mu}\int_{\Sigma}VW\differential{\operatorname{\mathcal{H}}^{d-1}}$
(6)

and our energy will simply be the rescaled perimeter functional

$\operatorname{E}(\Sigma)\coloneqq\sigma\operatorname{\mathcal{H}}^{d-1}(\Sigma).$

By [Mag12, Thm. 17.5] the first inner variation of the perimeter functional is
given by the mean curvature vector. Remember moreover that by the definition
of the metric (6) the gradient of the energy at a given hypersurface $\Sigma$
has to satisfy

$\frac{1}{\mu}\int_{\Sigma}\left\langle\nabla_{\Sigma}\operatorname{E}\,,\,V\right\rangle\differential{\operatorname{\mathcal{H}}^{d-1}}=\differential_{\Sigma}\operatorname{E}(V)$

for all normal vector fields $V$ on $\Sigma$. Combining these arguments the
gradient of the energy should simply be the mean curvature vector multiplied
by $\sigma\mu$. Thus the mean curvature flow equation (5) corresponds exactly
to the gradient flow equation (2). Note however that this metric tensor
induces a degenerate metric in the sense that the distance between any two
hypersurfaces is zero, which has been shown by Michor and Mumford in [MM06].

In Section 2.1 we highlighted the importance of De Giorgi’s optimal energy
dissipation inequality (4). In our setting this translates to the inequality

$\displaystyle\operatorname{E}(\Sigma(T))+\frac{1}{2}\int_{0}^{T}\left\langle
V(t)\,,\,V(t)\right\rangle_{\Sigma(t)}+\left\langle\nabla_{\Sigma(t)}E\,,\,\nabla_{\Sigma(t)}E\right\rangle_{\Sigma(t)}\differential{t}$
$\displaystyle={}$
$\displaystyle\operatorname{E}(\Sigma(T))+\frac{1}{2}\int_{0}^{T}\int_{\Sigma(t)}\frac{1}{\mu}V(t)^{2}+\sigma^{2}\mu
H(t)^{2}\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}$
$\displaystyle\leq{}$ $\displaystyle\operatorname{E}(\Sigma(0)).$

This is the main motivation for the definition of De Giorgi type solutions to
mean curvature flow in the two-phase case, see Definition 4.1 and Definition
4.6.

Far more important for us shall however be multiphase mean curvature flow,
which is of high importance in the applied sciences and mathematically quite
complex. Essentially instead of just considering the evolution of one single
set, we look at the evolution of a partition of the flat torus and require
that at each interface of the sets, the mean curvature flow equation (5) is
satisfied. Moreover we require a stability condition at points where three
interfaces meet.

More precisely we say that a partition $(\Omega_{i}(t))_{i=1,\dotsc,P}$ with
$\Sigma_{ij}\coloneqq\partial\Omega_{i}\cap\partial\Omega_{j}$ satisfies
multiphase mean curvature flow with mobilities $\mu_{ij}$ and surface tensions
$\sigma_{ij}$ if

$\displaystyle\frac{1}{\mu_{ij}}V_{ij}$
$\displaystyle=-\sigma_{ij}H_{ij}\quad$ $\displaystyle\text{on
}\Sigma_{ij}\text{ for all }i\neq j\text{ and}$ (7)
$\displaystyle\sigma_{ij}\nu_{ij}+\sigma_{jk}\nu_{jk}+\sigma_{ki}\nu_{ki}$
$\displaystyle=0$ $\displaystyle\text{at triple junctions}.$ (8)

The second equation is a stability condition when more than two sets meet and
is called _Herring’s angle condition_. Here $\nu_{ij}$ is the outer unit
normal of $\Omega_{i}$ on $\Sigma_{ij}$ pointing towards $\Omega_{j}$. One
could argue that we would also want stability conditions at for example
quadruple junctions. In two dimensions though, such quadruple junctions are
expected to immediately dissipate. In higher dimensions, quadruple junctions
might become stable, but only on lower dimensional sets, and shall thus not be
relevant for us.

As our space, we shall therefore now consider tuples of hypersurfaces in
$\operatorname{\mathbb{T}}$. The energy and metric tensor are given by

$\displaystyle\operatorname{E}(\Sigma)$
$\displaystyle\coloneqq\sum_{i<j}\sigma_{ij}\operatorname{\mathcal{H}}^{d-1}(\Sigma_{ij})$
(9) and $\displaystyle\left\langle V\,,\,W\right\rangle_{\Sigma}$
$\displaystyle\coloneqq\sum_{i<j}\frac{1}{\mu_{ij}}\int_{\Sigma_{ij}}V_{ij}W_{ij}\differential{\operatorname{\mathcal{H}}^{d-1}}.$

Of course this again produces a degenerate metric. Using a variant of the
divergence theorem on surfaces ([Mag12, Thm. 11.8]) and again the computation
for the first variation of the perimeter ([Mag12, Thm. 17.5]), we see that
multiphase mean curvature flow has the desired gradient flow structure. In
this case De Giorgi’s inequality (4) translates to

$\operatorname{E}(\Sigma(T))+\frac{1}{2}\sum_{i<j}\int_{0}^{T}\int_{\Sigma_{ij}}\frac{1}{\mu_{ij}}V_{ij}^{2}+\sigma_{ij}^{2}\mu_{ij}H_{ij}^{2}\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}\leq\operatorname{E}(\Sigma(0)).$

Furthermore if we make the simplifying assumption that the mobilities are
already determined through the surface tensions by the relation
$\mu_{ij}=1/\sigma_{ij}$, then this inequality becomes

$\operatorname{E}(\Sigma(T))+\frac{1}{2}\sum_{i<j}\sigma_{ij}\int_{0}^{T}\int_{\Sigma_{ij}}V_{ij}^{2}+H_{ij}^{2}\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}\leq\operatorname{E}(\Sigma(0)),$

which motivates Definition 4.1 and Definition 4.7.

##  3 Convergence of the Allen-Cahn equation to an evolving partition

###  3.1 Allen-Cahn equation

This chapter follows [LS18]. Let $\Lambda>0$ and define the flat torus as the
quotient
$\operatorname{\mathbb{T}}\coloneqq[0,\Lambda)^{d}=\mathbb{R}^{d}/\Lambda\mathbb{Z}^{d}$,
which means that we impose periodic boundary conditions on $[0,\Lambda)^{d}$.
Then for $u\colon[0,\infty)\times\operatorname{\mathbb{T}}\to\mathbb{R}^{N}$
and some smooth potential $W\colon\mathbb{R}^{N}\to[0,\infty)$, the
_Allen–Cahn equation_ with parameter $\varepsilon>0$ is given by

$\partial_{t}u=\Delta u-\frac{1}{\varepsilon^{2}}\nabla W(u).$ (10)

To understand this equation better, we consider the _Cahn–Hilliard energy_ ,
which assigns to $u$ for a fixed time the value

$\operatorname{E}_{\varepsilon}(u)\coloneqq\int\frac{1}{\varepsilon}W(u)+\frac{\varepsilon}{2}\absolutevalue{\nabla
u}^{2}\differential{x}.$ (11)

As it turns out the partial differential equation (10) is the
$\operatorname{L}^{2}$-gradient flow rescaled by $\sqrt{\varepsilon}$ of the
Cahn–Hilliard energy. Thus a solution to (10) can be constructed via De
Giorgi’s minimizing movements scheme ([De ̵93]).

But first we need to clarify what our potential $W$ should look like.
Classical examples in the scalar case are given by
$W(u)=\left(u^{2}-1\right)^{2}$ or $W(u)=u^{2}(u-1)^{2}$, and we call
functions like these _doublewell potentials_ , see also Figure 1.

$-2$$-1$$0$$1$$2$$0$$2$$4$$6$$8$$u$$W(u)=(u^{2}-1)^{2}$ Figure 1: Plot of the
doublewell potential $W(u)=(u^{2}-1)^{2}$

In higher dimensions, we want to accept potentials
$W\colon\mathbb{R}^{N}\to[0,\infty)$ which are smooth multiwell potentials,
meaning that they have finitely many zeros at
$u=\alpha_{1},\dotsc,\alpha_{P}\in\mathbb{R}^{N}$. Furthermore we ask for
polynomial growth in the sense that there exists some $p\geq 2$ such that for
all $u$ sufficiently large, we have

$\absolutevalue{u}^{p}\lesssim W(u)\lesssim\absolutevalue{u}^{p}$ (12)

and

$\absolutevalue{\nabla W(u)}\lesssim\absolutevalue{u}^{p-1}.$ (13)

Lastly we want $W$ to be convex up to a small perturbation in the sense that
there exist smooth functions $W_{\mathrm{conv}}$,
$W_{\mathrm{pert}}\colon\mathbb{R}^{N}\to[0,\infty)$ such that

$W=W_{\mathrm{conv}}+W_{\mathrm{pert}}\,,$ (14)

$W_{\mathrm{conv}}$ is convex and

$\sup_{x\in\mathbb{R}^{N}}\absolutevalue{\nabla^{2}W_{\mathrm{pert}}}<\infty.$
(15)

One can check that the above mentioned doublewell potentials satisfy these
assumptions. In general one may consider potentials $W$ of the form

$W(u)\coloneqq\absolutevalue{u-\alpha_{1}}^{2}\cdot\dotsc\cdot\absolutevalue{u-\alpha_{P}}^{2},$

see also Figure 2 for the case $P=3$ in two dimensions.

(a) Graph of the potential

(b) Contour lines of the potential

Figure 2: Graphics for the potential
$W(u)=\absolutevalue{u-e^{i\pi\frac{3}{6}}}^{2}\absolutevalue{u-e^{i\pi\frac{7}{6}}}^{2}\absolutevalue{u-e^{i\pi\frac{11}{6}}}^{2}$

The existence of the function $W_{\mathrm{pert}}$ can then be seen by
multiplying some suitable cutoff to $W$ which is equal to 1 on a sufficiently
large ball.

As it is custom for parabolic partial differential equations, we view
solutions of the Allen–Cahn equation (10) as maps from $[0,T]$ into some
suitable function space and thus use the following definition.

######  Definition 3.1.

We say that a function

$u_{\varepsilon}\in\mathrm{C}\left([0,T];\operatorname{L}^{2}\left(\operatorname{\mathbb{T}};\mathbb{R}^{N}\right)\right)\cap\operatorname{L}^{\infty}\left([0,T];\operatorname{W}^{1,2}(\operatorname{\mathbb{T}};\mathbb{R}^{N})\right)$

is a weak solution of the Allen–Cahn equation (10) with parameter
$\varepsilon>0$ and initial condition
$u_{\varepsilon}^{0}\in\operatorname{L}^{2}(\operatorname{\mathbb{T}};\mathbb{R}^{N})$
if

  1. 1.

the energy stays bounded, which means that

$\operatorname*{ess\,sup}_{0\leq t\leq
T}\operatorname{E}_{\varepsilon}(u_{\varepsilon}(t))<\infty,$ (16)

  2. 2.

its weak time derivative satisfies

$\partial_{t}u_{\varepsilon}\in\operatorname{L}^{2}\left([0,T]\times\operatorname{\mathbb{T}};\mathbb{R}^{N}\right),$
(17)

  3. 3.

for almost every $t\in[0,T]$ and every

$\xi\in\operatorname{L}^{p}([0,T]\times\operatorname{\mathbb{T}};\mathbb{R}^{N})\cap\operatorname{W}^{1,2}([0,T]\times\operatorname{\mathbb{T}};\mathbb{R}^{N}),$

we have

$\int\left\langle\frac{1}{\varepsilon^{2}}\nabla
W(u_{\varepsilon}(t))\,,\,\xi\right\rangle+\left\langle\nabla
u_{\varepsilon}(t)\,,\,\nabla\xi\right\rangle+\left\langle\partial_{t}u_{\varepsilon}(t)\,,\,\xi\right\rangle\differential{x}=0,$
(18)

  4. 4.

the initial conditions are attained in the sense that
$u_{\varepsilon}(0)=u_{\varepsilon}^{0}$.

######  Remark 3.2.

The exponent $p$ which appears in condition 3 is the same exponent as for the
growth assumptions (12) and (13) of $W$. Moreover we note that given our
setup, we automatically have $\nabla
W(u_{\varepsilon})(t)\in\operatorname{L}^{p^{\prime}}(\operatorname{\mathbb{T}})$.
In fact we can estimate

$\absolutevalue{\nabla W(u_{\varepsilon})}^{p/(p-1)}\lesssim
1+\absolutevalue{u_{\varepsilon}}^{p}\lesssim 1+W(u),$ (19)

which is integrable for almost every time $t$ since we assume that the energy
stays bounded, thus the integral in equation (18) is well defined.

Moreover we already obtain $1/2$ Hölder-continuity in time from the embedding

$\operatorname{W}^{1,2}\left([0,T];\operatorname{L}^{2}\left(\operatorname{\mathbb{T}};\mathbb{R}^{N}\right)\right)\hookrightarrow\operatorname{C}^{1/2}\left([0,T];\operatorname{L}^{2}\left(\operatorname{\mathbb{T}};\mathbb{R}^{N}\right)\right),$
(20)

which follows from a generalized version of the fundamental theorem of
calculus and Hölder’s inequality.

With Definition 3.1, we are able to state our existence result for a solution
of the Allen–Cahn equation [LS18, Lemma 2.3] which uses De Giorgi’s minimizing
movements scheme. Note moreover that one would expect even more regularity for
a solution of the Allen–Cahn equation. In fact, it has been shown by De
Mottoni and Schatzmann in [DS95] that in the scalar case, a solution u is
smooth for positive times, at least for bounded initial data. Their Ansatz is
a variation of parameters and the regularity then follows from the smoothing
effect of the heat kernel.

######  Theorem 3.3.

Let $u^{0}\colon\operatorname{\mathbb{T}}\to\mathbb{R}^{N}$ be such that
$\operatorname{E}_{\varepsilon}(u^{0})<\infty$. Then there exists a weak
solution $u_{\varepsilon}$ to the Allen–Cahn equation (10) in the sense of
Definition 3.1 with initial data $u^{0}$. Furthermore the solution satisfies
the energy dissipation inequality

$\operatorname{E}_{\varepsilon}(u_{\varepsilon}(t))+\int_{0}^{t}\int\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}\differential{x}\differential{s}\leq\operatorname{E}_{\varepsilon}(u^{0})$
(21)

for every $t\in[0,T]$. We additionally have
$\partial_{i,j}^{2}u_{\varepsilon},\nabla
W(u_{\varepsilon})\in\operatorname{L}^{2}([0,T]\times\operatorname{\mathbb{T}};\mathbb{R}^{N})$
for all $1\leq i,j\leq d$. In particular we can test the weak form (18) with
$\partial_{i,j}^{2}u_{\varepsilon}$.

###  3.2 Convergence to an evolving partition

We now want to study of solutions to the Allen-Cahn equation (10) as
$\varepsilon$ tends to zero. Let us first fix some notation. For a function
$u=\sum_{i=1}^{P}\mathds{1}_{\Omega_{i}}\alpha_{i}$ with sets of finite
perimeter $\Omega_{i}$ and the corresponding interfaces
$\Sigma_{ij}\coloneqq\partial_{\ast}\Omega_{i}\cap\partial_{\ast}\Omega_{j}$,
we define for a given continuous function
$\varphi\in\operatorname{C}(\operatorname{\mathbb{T}})$ the energy measures by

$\displaystyle\operatorname{E}_{\varepsilon}(u_{\varepsilon};\varphi)$
$\displaystyle\coloneqq\int\varphi\left(\frac{1}{\varepsilon}W(u_{\varepsilon})+\frac{\varepsilon}{2}\absolutevalue{\nabla
u_{\varepsilon}}^{2}\right)\differential{x}$ (22) and
$\displaystyle\operatorname{E}(u;\varphi)$
$\displaystyle\coloneqq\sum_{i<j}\sigma_{ij}\int_{\Sigma_{ij}}\varphi\differential{\operatorname{\mathcal{H}}^{d-1}}.$
(23)

The latter is motivated through the perimeter functional introduced in Section
2.2.

One of the many difficulties in the vectorial case is that there is no easy
choice of a primitive for $\sqrt{2W(u)}$ compared to the scalar case. This is
crucial for the classic Modica–Mortola trick to obtain
$\operatorname{BV}$-compactness. As a suitable replacement, we consider the
_geodesic distance_ defined as

$\displaystyle\operatorname{\mathrm{d}_{\mathnormal{W}}}_{d}istance(u,v)$
$\displaystyle\coloneqq{}$
$\displaystyle\inf\left\\{\int_{0}^{1}\sqrt{2W(\gamma)}\absolutevalue{\dot{\gamma}}\differential{t}\,\colon\,\gamma\in\mathrm{C}^{1}\left([0,1];\mathbb{R}^{N}\right)\text{
with }\gamma(0)=u,\,\gamma(1)=v\right\\}$

motivated by Baldo in [Bal90]. This indeed defines a metric on
$\mathbb{R}^{N}$: If we have
$\operatorname{\mathrm{d}_{\mathnormal{W}}}_{d}istance(v,w)=0$, then by the
continuity of $W$ and since it only has a discrete set of zeros, we may deduce
that $v=w$. Symmetry can be seen by reversing a given path between two points
and the triangle inequality follows from concatenation of two paths.

The _surface tensions_ generated by $W$ are defined as

$\sigma_{ij}\coloneqq\operatorname{\mathrm{d}_{\mathnormal{W}}}_{d}istance(\alpha_{i},\alpha_{j})$
(24)

and as a consequence of
$\operatorname{\mathrm{d}_{\mathnormal{W}}}_{d}istance$ being a metric satisfy
the triangle inequality

$\sigma_{ik}\leq\sigma_{ij}+\sigma_{jk}.$

Furthermore $\sigma_{ij}$ is zero if and only if $i$ is equal to $j$ and by
symmetry, $\sigma_{ij}$ is always the same as $\sigma_{ji}$.

Our replacement for the primitive of $\sqrt{2W}$ is given for $1\leq i\leq P$
by the geodesic distance function

$\phi_{i}(u)\coloneqq\operatorname{\mathrm{d}_{\mathnormal{W}}}_{d}istance(\alpha_{i},u).$

Our first obstacle is the regularity of the functions
$\psi_{\varepsilon,i}\coloneqq\phi_{i}\circ u_{\varepsilon}$. A priori we only
know that $\phi_{i}$ is locally Lipschitz continuous on $\mathbb{R}^{N}$ and
thus differentiable almost everywhere. If $N=1$, this would already suffice to
deduce that $\psi_{\varepsilon,i}$ is weakly differentiable. In higher
dimensions however, $u$ could for example move along a hypersurface where
$\phi_{i}$ could in theory be nowhere differentiable since the hypersurface is
a Lebesgue nullset. This can be salvaged through the chain rule for
distributional derivatives by Ambrosio and Dal Maso [AM90, Cor. 3.2]. Moreover
we have the crucial inequality for its gradient given by

$\absolutevalue{\nabla\psi_{i}(u)}\leq\sqrt{2W(u)}.$ (25)

We then arrive at the following convergence result, see [LS18, Prop. 2.7].

######  Proposition 3.4.

Let $u_{\varepsilon}^{0}$ be well prepared initial data in the sense that

$\lim_{\varepsilon\to
0}\operatorname{E}_{\varepsilon}(u_{\varepsilon}^{0})=\operatorname{E}(u^{0})\eqqcolon\operatorname{E}_{0}<\infty.$

Then there exists for any sequence $\varepsilon\to 0$ some non-relabelled
subsequence such that the solutions $u_{\varepsilon}$ of the Allen–Cahn
equation (10) with initial conditions $u_{\varepsilon}^{0}$ converge in
$\operatorname{L}^{1}\left((0,T)\times\operatorname{\mathbb{T}};\mathbb{R}^{N}\right)$
to some $u=\sum_{i=1}^{P}\chi_{i}\alpha_{i}$ with a partition

$\chi\in\operatorname{BV}\left((0,T)\times\operatorname{\mathbb{T}};\\{0,1\\}^{P}\right).$

Furthermore we have

$\operatorname{E}(u(t))\leq\liminf_{\varepsilon\to
0}\operatorname{E}_{\varepsilon}(u_{\varepsilon}(t))\leq\operatorname{E}_{0}$
(26)

for almost every time $0\leq t\leq T$ and $u$ attains the initial data $u^{0}$
continuously in
$\operatorname{L}^{2}\left(\operatorname{\mathbb{T}};\mathbb{R}^{d}\right)$.
Moreover for all $1\leq i\leq P$, the compositions
$\psi_{\varepsilon,i}\coloneqq\phi_{i}\circ u_{\varepsilon}$ are uniformly
bounded in $\operatorname{BV}((0,T)\times\operatorname{\mathbb{T}})$ and
converge to $\psi_{i}\coloneqq\phi_{i}\circ u$ in
$\operatorname{L}^{1}\left((0,T)\times\operatorname{\mathbb{T}}\right)$.

##  4 De Giorgi’s mean curvature flow

In this chapter, we build on the results of [LS18], but introduce a different
solution concept for mean curvature flow, namely a De Giorgi type
$\operatorname{BV}$-solution to mean curvature flow. A similar solution
concept has been introduced in [LO16] and [LL21, Def. 1], but in the context
of convergence of the thresholding scheme to mean curvature flow. Moreover, we
will also compare it to the solution concept [HL21, Def. 1], which permits
oriented varifolds to be solutions to mean curvature flow. This provides a
more general notion of solution and is of use when the assumption of energy
convergence falls away. See also the later discussion in Section 4.4.

###  4.1 Conditional convergence to De Giorgi’s mean curvature flow

In this section, we shall state our solution concept and prove convergence to
the aforementioned.

######  Definition 4.1 (De Giorgi type $\operatorname{BV}$-solution to
multiphase mean curvature flow).

Fix some finite time horizon $T<\infty$, a $(P\times P)$-matrix of surface
tensions $\sigma$ and initial data
$\chi^{0}\colon\operatorname{\mathbb{T}}\to\\{0,1\\}^{P}$ with
$\operatorname{E}_{0}\coloneqq\operatorname{E}(\chi^{0})<\infty$ and
$\sum_{i=1}^{P}\chi_{i}^{0}=1$ almost everywhere. We say that

$\chi\in\operatorname{C}\left([0,T];\operatorname{L}^{2}\left(\operatorname{\mathbb{T}};\\{0,1\\}^{P}\right)\right)\cap\operatorname{BV}\left((0,T)\times\operatorname{\mathbb{T}};\\{0,1\\}^{P}\right)$

with $\operatorname*{ess\,sup}_{0\leq t\leq T}\operatorname{E}(\chi)<\infty$
and $\sum_{i=1}^{P}\chi_{i}=\sum_{i=1}^{P}\mathds{1}_{\Omega_{i}}=1$ almost
everywhere is a _De Giorgi type_ $\operatorname{BV}$_-solution to multiphase
mean curvature flow with initial data_ $\chi^{0}$ _and surface tensions_
$\sigma$ if the following holds.

  1. 1.

For all $1\leq i\leq P$, there exists a normal velocity
$V_{i}\in\operatorname{L}^{2}(\absolutevalue{\nabla\chi_{i}}\differential{t})$
such that

$\partial_{t}\chi_{i}=V_{i}\absolutevalue{\nabla\chi_{i}}\differential{t}$

holds in the distributional sense on $(0,T)\times\operatorname{\mathbb{T}}$.

  2. 2.

There exist a mean curvature vector
$H\in\operatorname{L}^{2}(\operatorname{E}(u;\cdot)\differential{t};\mathbb{R}^{d})$
which satisfies

$\displaystyle\sum_{1\leq i<j\leq
P}\sigma_{ij}\int_{0}^{T}\int_{\Sigma_{ij}}\left\langle
H\,,\,\xi\right\rangle\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}$
$\displaystyle={}$ $\displaystyle-\sum_{1\leq i<j\leq
P}\sigma_{ij}\int_{0}^{T}\int_{\Sigma_{ij}}\left\langle\mathrm{D}\xi\,,\,\mathrm{Id}-\nu_{i}\otimes\nu_{i}\right\rangle\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}$
(27)

for all test vector fields
$\xi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left((0,T)\times\operatorname{\mathbb{T}};\mathbb{R}^{d}\right)$,
where $\nu_{i}\coloneqq\nabla\chi_{i}/\absolutevalue{\nabla\chi_{i}}$ are the
inner unit normals and
$\Sigma_{ij}\coloneqq\partial_{\ast}\Omega_{i}\cap\partial_{\ast}\Omega_{j}$
is the $(i,j)$-th interface.

  3. 3.

The partition $\chi$ satisfies a De Giorgi type optimal energy dissipation
inequality in the sense that for almost every time $0<T^{\prime}<T$, we have

$\operatorname{E}(\chi(T^{\prime}))+\frac{1}{2}\sum_{1\leq i<j\leq
P}\sigma_{ij}\int_{0}^{T^{\prime}}\int_{\Sigma_{ij}}V_{i}^{2}+\absolutevalue{H}^{2}\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}\leq\operatorname{E}_{0}.$
(28)

  4. 4.

The initial data is attained in
$\operatorname{C}\left([0,T];\operatorname{L}^{2}(\operatorname{\mathbb{T}})\right)$.

###  4.2 Localization estimates

Indeed, we obtain the existence of such solutions through a phase-field
approximation with the Allen-Cahn equation as long as we assume the
convergence of the integrated energies given by

$\lim_{\varepsilon\to
0}\int_{0}^{T}\operatorname{E}_{\varepsilon}(u_{\varepsilon})\differential{t}=\int_{0}^{T}\operatorname{E}(u)\differential{t}.$
(29)

A consequence of this is the _equipartition of energies_ , already observed by
Ilmanen in [Ilm93] for the scalar case under weaker assumptions using a
comparison principle. The proof for the multiphase case can be found in [LS18,
Lemma 2.11].

######  Lemma 4.2.

In the situation of Proposition 3.4 and under the energy convergence
assumption (29), we have for any continuous function
$\varphi\in\operatorname{C}(\operatorname{\mathbb{T}})$ that

$\displaystyle\operatorname{E}(u;\varphi)=\lim_{\varepsilon\to
0}\operatorname{E}_{\varepsilon}(u_{\varepsilon};\varphi)$
$\displaystyle=\lim_{\varepsilon\to
0}\int\varphi\sqrt{2W(u_{\varepsilon})}\absolutevalue{\nabla
u_{\varepsilon}}\differential{x}$ $\displaystyle=\lim_{\varepsilon\to
0}\int\varphi\varepsilon\absolutevalue{\nabla
u_{\varepsilon}}^{2}\differential{x}$ $\displaystyle=\lim_{\varepsilon\to
0}\int\varphi\frac{1}{\varepsilon}2W(u_{\varepsilon})\differential{x}$

for almost every time $0\leq t\leq T$.

In order to prove convergence, we want to reduce the multiphase case to the
two-phase case. The central idea here is to cover the flat torus with a
suitable collection of balls. Then we argue that, up to a small error, we can
choose for each ball a pair of majority phases $(i,j)$ such that the partition
looks like a two-phase mean curvature flow on the ball, see Figure 3.

$\Omega_{1}$$\Omega_{2}$$\Omega_{3}$$\Omega_{4}$ Figure 3: Localization of
multiphase mean curvature flow. For the green balls, we choose the majority
phase (2,3), for the blue ones (1,3) and for the yellow balls (3,4). The red
balls however give us an error.

To formulate this rigorously let $r>0$ and define the covering
$\mathcal{B}_{r}$ of the flat torus by

$\mathcal{B}_{r}\coloneqq\left\\{B_{r}(c)\,\colon\,c\in\mathcal{L}_{r}\right\\},$

where the set of centers $\mathcal{L}_{r}$ is given by
$\mathcal{L}_{r}\coloneqq\operatorname{\mathbb{T}}\cap(r/\sqrt{d})\mathbb{Z}^{d}$.
Moreover let $\rho_{B}$ be a smooth cutoff for the ball $B$ with support in
the ball with the same center, but double the radius.

Additionally to choosing a majority phase, we can also argue that along the
chosen majority phase, we have a local flatness. Thus we may approximate the
inner unit normal up to an arbitrarily small error by a constant unit vector.
This is captured by the following lemma found in [LO16]. The proof is based on
De Giorgi’s structure theorem for sets of finite perimeter.

######  Lemma 4.3.

For every $\delta>0$ and every partition
$\chi\colon\operatorname{\mathbb{T}}\to\\{0,1\\}^{P}$ such that
$\chi_{i}\in\operatorname{BV}(\operatorname{\mathbb{T}})$ holds for all $1\leq
i\leq P$, there exist some $r_{0}>0$ such that for all $0<r<r_{0}$, we find
for every ball $B\in\mathcal{B}_{r}$ some unit vector $\nu_{B}$ such that

$\displaystyle\sum_{B\in\mathcal{B}_{r}}\min_{i\neq
j}\int\rho_{B}\absolutevalue{\nu_{i}-\nu_{B}}^{2}\absolutevalue{\nabla\chi_{i}}+\int\rho_{B}\absolutevalue{\nu_{j}+\nu_{B}}^{2}\absolutevalue{\nabla\chi_{j}}+\sum_{k\notin\\{i,j\\}}\int\rho_{B}\absolutevalue{\nabla\chi_{k}}$
$\displaystyle\lesssim{}$ $\displaystyle\delta\operatorname{E}(\chi).$

Note that summing $\absolutevalue{\nabla\chi_{k}}$ over all $k\notin\\{i,j\\}$
corresponds to summing over all interfaces which are not the $(i,j)$-th
interface, see also [Bal90]. Moreover the second summand is redundant. In fact
the last summand provides us with a localization on the $(i,j)$-th interface,
on which we have $\nu_{i}=-\nu_{j}$. However it is convenient to keep it so
that we do not have to repeat this argument.

######  Remark 4.4.

This localization estimate also implies the smallness of

$\sum_{B\in\mathcal{B}_{r}}\min_{i}\absolutevalue{\operatorname{E}(\chi;\rho_{B})-\int\rho_{B}\absolutevalue{\nabla\psi_{i}}}=\sum_{B\in\mathcal{B}_{r}}\operatorname{E}(\chi;\rho_{B})-\max_{i}\int\rho_{B}\absolutevalue{\nabla\psi_{i}}$
(30)

in the same sense as in the above Lemma 4.3. Notice that the equality (30)
follows from

$\absolutevalue{\nabla\psi_{i}}=\sum_{1\leq k<l\leq
P}\absolutevalue{\sigma_{ik}-\sigma_{il}}\operatorname{\mathcal{H}}^{d-1}\llcorner_{\Sigma_{kl}}\leq\operatorname{E}(\chi;\cdot),$
(31)

where we remind that $\psi_{i}\coloneqq\phi_{i}\circ u$. The smallness follows
since we have for every $i\neq j$ that

$\displaystyle\operatorname{E}(\chi;\rho_{B})-\int\rho_{B}\absolutevalue{\nabla\psi_{i}}$
$\displaystyle={}$ $\displaystyle\sum_{1\leq k<l\leq
P}\sigma_{kl}\int_{\Sigma_{kl}}\rho_{B}\differential{\operatorname{\mathcal{H}}^{d-1}}-\sum_{1\leq
k<l\leq
P}\absolutevalue{\sigma_{ik}-\sigma_{il}}\int_{\Sigma_{kl}}\rho_{B}\differential{\operatorname{\mathcal{H}}^{d-1}}$
$\displaystyle={}$ $\displaystyle\sum_{\begin{subarray}{c}1\leq k<l\leq P\\\
(k,l)\neq(i,j)\end{subarray}}\left(\sigma_{kl}-\absolutevalue{\sigma_{ik}-\sigma_{il}}\right)\int_{\Sigma_{kl}}\rho_{B}\differential{\operatorname{\mathcal{H}}^{d-1}}$
$\displaystyle\lesssim{}$
$\displaystyle\sum_{k\notin\\{i,j\\}}\int\rho_{B}\absolutevalue{\nabla\chi_{k}}.$

Thus we can estimate the error by the last summand of the error in Lemma 4.3.

###  4.3 Conditional convergence of Allen–Cahn equations to De Giorgi’s
multiphase mean curvature flow

We are now in the position to prove the desired conditional convergence
result.

######  Theorem 4.5.

Let $W\colon\mathbb{R}^{N}\to[0,\infty)$ be a smooth multiwell potential
satisfying the assumptions (12)-(15). Let $T<\infty$ be an arbitrary finite
time horizon. Let
$u_{\varepsilon}^{0}\colon\operatorname{\mathbb{T}}\to\mathbb{R}^{N}$ be a
sequence of initial data approximating a partition
$\chi^{0}\in\operatorname{BV}\left(\operatorname{\mathbb{T}};\\{0,1\\}^{P}\right)$
in the sense that $u_{\varepsilon}^{0}\to u^{0}=\sum_{1\leq i\leq
P}\chi_{i}^{0}\alpha_{i}$ holds pointwise almost everywhere and

$\operatorname{E}_{0}\coloneqq\operatorname{E}(\chi^{0})=\lim_{\varepsilon\to
0}\operatorname{E}_{\varepsilon}(u_{\varepsilon}^{0})<\infty.$

Then we have for some subsequence of solutions $u_{\varepsilon}$ to the
Allen–Cahn equation (10) with initial datum $u_{\varepsilon}^{0}$ that there
exists a time-dependent partition $\chi$ with

$\chi\in\operatorname{BV}\left((0,T)\times\operatorname{\mathbb{T}};\\{0,1\\}^{P}\right)$

and
$\chi\in\operatorname{C}\left([0,T];\operatorname{L}^{2}\left(\operatorname{\mathbb{T}};\\{0,1\\}^{P}\right)\right)$
such that $u_{\varepsilon}$ converges to $u\coloneqq\sum_{1\leq i\leq
P}\chi_{i}\alpha_{i}$ in
$\operatorname{L}^{1}\left((0,T)\times\operatorname{\mathbb{T}};\mathbb{R}^{N}\right)$.
Moreover $u$ attains the initial data $u^{0}$ continuously in
$\operatorname{L}^{2}\left(\operatorname{\mathbb{T}};\mathbb{R}^{N}\right)$.

If we additionally assume that the time-integrated energies converge (29),
then $\chi$ is a De Giorgi type $\operatorname{BV}$-solution to multiphase
mean curvature flow in the sense of Definition 4.1.

###### Proof.

Proposition 3.4 already provides us with all claims except the last. The idea
of the proof is that we already have an optimal energy dissipation inequality
for the Allen–Cahn equation given by (21). If we additionally use the
Allen–Cahn equation once, we arrive at the De Giorgi type optimal energy
dissipation inequality given by

$\operatorname{E}_{\varepsilon}(u_{\varepsilon}(T^{\prime}))+\frac{1}{2}\int_{0}^{T^{\prime}}\int\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}+\frac{1}{\varepsilon}\absolutevalue{\varepsilon\Delta
u_{\varepsilon}-\frac{1}{\varepsilon}\nabla
W(u_{\varepsilon})}^{2}\differential{x}\differential{t}\leq\operatorname{E}_{\varepsilon}(u_{\varepsilon}^{0}).$

Our hope is that as $\varepsilon$ tends to zero, we can pass to the optimal
energy dissipation inequality (28) for $\chi$ through lower semicontinuity.
Since we assume the convergence of the initial energies and energy convergence
for almost every time, the only terms we have to care about are the velocity
and curvature term. The desired lower semicontinuity of the velocity term
reads

$\liminf_{\varepsilon\to
0}\frac{1}{2}\int_{0}^{T^{\prime}}\int\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}\differential{x}\differential{t}\geq\frac{1}{2}\sum_{i<j}\sigma_{ij}\int_{0}^{T^{\prime}}\int_{\Sigma_{ij}}V_{i}^{2}\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}$
(32)

and the lower semicontinuity of the curvature term is given by

$\displaystyle\liminf_{\varepsilon\to
0}\frac{1}{2}\int_{0}^{T^{\prime}}\int\frac{1}{\varepsilon}\absolutevalue{\varepsilon\Delta
u_{\varepsilon}-\frac{1}{\varepsilon}\nabla
W(u_{\varepsilon})}^{2}\differential{x}\differential{t}$ $\displaystyle\geq{}$
$\displaystyle\frac{1}{2}\sum_{i<j}\sigma_{ij}\int_{0}^{T^{\prime}}\int_{\Sigma_{ij}}\absolutevalue{H}^{2}\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}.$
(33)

The existence of square integrable distributional normal velocities has been
proven in [LS18, Prop. 2.10]. Moreover we have to show the existence of the
mean curvature vector $H$. We could cheat in this step and simply use that by
the main result [LS18, Thm. 1.2], we already know that the tangential
divergence applied to $\xi$ is given by the velocity. In other words, that we
already have $V_{i}\nu_{i}=-H$ on $\Sigma_{ij}$. But we want to present a more
direct approach.

Consider the linear functional

$L(\xi)\coloneqq-\sum_{1\leq i<j\leq
P}\sigma_{ij}\int_{0}^{T}\int_{\Sigma_{ij}}\left\langle\mathrm{D}\xi\,,\,\mathrm{Id}-\nu_{i}\otimes\nu_{i}\right\rangle\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}$

defined on test vector fields $\xi$. Then $L$ is bounded with respect to the
$\operatorname{L}^{2}$-norm on $(0,T)\times\operatorname{\mathbb{T}}$ equipped
with the measure $\operatorname{E}(u;\cdot)\differential{t}$ since by the
convergence of the curvature term observed in [LS18, Prop. 3.1], we have

$\displaystyle\absolutevalue{L(\xi)}$ $\displaystyle={}$
$\displaystyle\lim_{\varepsilon\to
0}\absolutevalue{-\int_{0}^{T}\int\left\langle\varepsilon\Delta
u_{\varepsilon}-\frac{1}{\varepsilon}\nabla
W(u_{\varepsilon})\,,\,\mathrm{D}u_{\varepsilon}\xi\right\rangle\differential{x}\differential{t}}$
$\displaystyle\leq{}$ $\displaystyle\liminf_{\varepsilon\to
0}\left(\int_{0}^{T}\int\frac{1}{\varepsilon}\absolutevalue{\varepsilon\Delta
u_{\varepsilon}-\frac{1}{\varepsilon}\nabla
W(u_{\varepsilon})}^{2}\differential{x}\differential{t}\right)^{1/2}\left(\int_{0}^{T}\int\varepsilon\absolutevalue{\mathrm{D}u_{\varepsilon}\xi}^{2}\differential{x}\differential{t}\right)^{1/2}$
$\displaystyle\leq{}$ $\displaystyle\liminf_{\varepsilon\to
0}\left(\int_{0}^{T}\int\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}\differential{x}\differential{t}\right)^{1/2}\left(\int_{0}^{T}\int\varepsilon\absolutevalue{\nabla
u_{\varepsilon}}^{2}\absolutevalue{\xi}^{2}\differential{x}\differential{t}\right)^{1/2}.$

The first factor stays uniformly bounded due to the energy dissipation
inequality (21), and by the equipartition of energies (Lemma 4.2), the second
factor converges to the $\operatorname{L}^{2}$-norm of $\xi$ with respect to
the energy measure, proving our claim. Therefore we can extend the functional
to the square integrable functions with respect to the energy measure. By
Riesz representation theorem we obtain the existence of the desired mean
curvature vector $H$.

Let us now consider the lower semicontinuity of the curvature term, which in
particular yields a sharp version of the previous estimate. Let again $\xi$ be
some test vector field. Then for all $\varepsilon>0$ and some fixed time, we
have by Young’s inequality

$\displaystyle\liminf_{\varepsilon\to
0}\frac{1}{2}\int\frac{1}{\varepsilon}\absolutevalue{\varepsilon\Delta
u_{\varepsilon}-\frac{1}{\varepsilon}\nabla
W(u_{\varepsilon})}^{2}\differential{x}$ $\displaystyle\geq{}$
$\displaystyle\liminf_{\varepsilon\to 0}\int\left\langle\varepsilon\Delta
u_{\varepsilon}-\frac{1}{\varepsilon}\nabla
W(u_{\varepsilon})\,,\,\mathrm{D}u_{\varepsilon}\xi\right\rangle\differential{x}-\frac{1}{2}\int\varepsilon\absolutevalue{\mathrm{D}u_{\varepsilon}\xi}^{2}\differential{x}$
$\displaystyle\geq{}$ $\displaystyle\liminf_{\varepsilon\to
0}\int\left\langle\varepsilon\Delta
u_{\varepsilon}-\frac{1}{\varepsilon}\nabla
W(u_{\varepsilon})\,,\,\mathrm{D}u_{\varepsilon}\xi\right\rangle\differential{x}-\limsup_{\varepsilon\to
0}\frac{1}{2}\int\varepsilon\absolutevalue{\mathrm{D}u_{\varepsilon}}^{2}\absolutevalue{\xi}^{2}$
$\displaystyle\geq{}$ $\displaystyle-\operatorname{E}\left(\chi;\left\langle
H\,,\,\xi\right\rangle\right)-\frac{1}{2}\operatorname{E}\left(\chi;\absolutevalue{\xi}^{2}\right).$

Here the last inequality is due to the above mentioned convergence of the
curvature term and the equipartition of energies Lemma 4.2. Since this
inequality holds for any test vector field, we may take a sequence of test
vector fields $\xi_{n}$ satisfying

$\lim_{n\to\infty}\norm{\xi_{n}+H}_{\operatorname{L}^{2}\left(\operatorname{\mathbb{T}},\operatorname{E}(\chi;\cdot);\mathbb{R}^{d}\right)}=0.$

This then yields the desired inequality (4.3) by applying Fatou’s lemma.

In principle the proof is now already done, since the main result by Laux and
Simon already gives us that for almost every time $t$, we have
$V_{i}\nu_{i}=-H$ on $\Sigma_{i,j}$ $\operatorname{\mathcal{H}}^{d-1}$-almost
everywhere. We instead want to present another proof which directly proves the
lower semicontinuity of the velocity term and may leave more room for future
generalizations.

Let us first consider the two-phase case $N=1$ and $P=2$. By a similar duality
argument as for the lower semicontinuity of the curvature term, we compute
that for every test function $\varphi$, we have

$\displaystyle\liminf_{\varepsilon\to
0}\frac{1}{2}\int_{0}^{T}\int\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}\differential{x}\differential{t}$
$\displaystyle\geq{}$ $\displaystyle\liminf_{\varepsilon\to
0}\int_{0}^{T}\int\partial_{t}u_{\varepsilon}\phi^{\prime}(u_{\varepsilon})\varphi\differential{x}\differential{t}-\frac{1}{2}\int_{0}^{T}\int\frac{1}{\varepsilon}\left(\phi^{\prime}(u_{\varepsilon})\varphi\right)^{2}\differential{x}\differential{t}$
$\displaystyle={}$ $\displaystyle\liminf_{\varepsilon\to
0}\int_{0}^{T}\int\partial_{t}\psi_{\varepsilon}\varphi\differential{x}\differential{t}-\frac{1}{2}\int_{0}^{T}\int\frac{1}{\varepsilon}2W(u_{\varepsilon})\varphi^{2}\differential{x}\differential{t}$
$\displaystyle={}$ $\displaystyle\sigma\int_{0}^{T}\int_{\Sigma}\varphi
V\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}-\frac{1}{2}\sigma\int_{0}^{T}\int_{\Sigma}\varphi^{2}\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}.$

Here the last equality uses the weak convergence of $\partial_{t}(\phi\circ
u_{\varepsilon})=\partial_{t}\psi_{\varepsilon}$ to $\partial_{t}(\phi\circ
u)=\partial_{t}\psi=\sigma V\absolutevalue{\nabla\chi}\differential{t}$ given
by Proposition 3.4 for the first summand and the equipartition of energies
(Lemma 4.2) for the second summand. Since the inequality holds for any test
function $\varphi$, we may plug in a sequence of test functions $\varphi_{n}$
satisfying

$\lim_{n\to\infty}\norm{\varphi_{n}-V}_{\operatorname{L}^{2}\left((0,T)\times\operatorname{\mathbb{T}},\operatorname{\mathcal{H}}^{d-1}\llcorner_{\Sigma}\differential{t}\right)}=0$

and thereby obtain the desired inequality (32).

For the multiphase case, we do not find an immediate generalization of this
proof, but rather have to work with a localization argument in order to obtain
a reduction to the two-phase case.

For a localization in time, let $\delta>0$ and
$0=T_{0}<T_{1}<\dotsc<T_{K}=T^{\prime}$ be a partition of $[0,T^{\prime}]$.
Let $(g_{k})_{k=1,\dotsc,K}$ be a partition of unity with respect to the
intervals $\left((T_{k-1}-\delta,T_{k}+\delta)\right)_{1\leq k\leq K}$, let
$r>0$ and $\eta_{B}$ a partition of unity in space as in Lemma 4.3. Moreover
we fix some $R>0$. Then we estimate by Young’s inequality

$\displaystyle\liminf_{\varepsilon\to
0}\frac{1}{2}\int_{0}^{T}\int\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}\differential{x}\differential{t}$
$\displaystyle={}$ $\displaystyle\liminf_{\varepsilon\to
0}\sum_{k=1}^{K}\sum_{B\in\mathcal{B}_{r}}\frac{1}{2}\int_{0}^{T}\int
g_{k}\eta_{B}\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}\differential{x}\differential{t}$
$\displaystyle\geq{}$ $\displaystyle\liminf_{\varepsilon\to
0}\sum_{k=1}^{K}\sum_{B\in\mathcal{B}_{r}}\max_{1\leq i\leq
P}\sup_{\begin{subarray}{c}\varphi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left((0,T)\times\operatorname{\mathbb{T}}\right)\\\
\absolutevalue{\varphi}\leq R\end{subarray}}\int_{0}^{T}\int
g_{k}\eta_{B}\left\langle\nabla\phi_{i}(u_{\varepsilon})\,,\,\partial_{t}u_{\varepsilon}\right\rangle\varphi\differential{x}\differential{t}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-\frac{1}{2}\int_{0}^{T}\int
g_{k}\eta_{B}\frac{1}{\varepsilon}\absolutevalue{\nabla\phi_{i}(u_{\varepsilon})}^{2}\varphi^{2}\differential{x}\differential{t}.$

We identify via the chain rule that
$\left\langle\nabla\phi_{i}(u_{\varepsilon})\,,\,\partial_{t}u_{\varepsilon}\right\rangle=\partial_{t}\psi_{\varepsilon,i}$
and moreover remember $\absolutevalue{\nabla\phi_{i}}\leq\sqrt{2W}$ from
inequality (25). We pull the limit inferior inside the double sum and the
suprema and thus obtain that this term can be estimated from below by

$\displaystyle\sum_{k=1}^{K}\sum_{B\in\mathcal{B}_{r}}\max_{1\leq i\leq
P}\sup_{\begin{subarray}{c}\varphi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left((0,T)\times\operatorname{\mathbb{T}}\right)\\\
\absolutevalue{\varphi}\leq R\end{subarray}}\liminf_{\varepsilon\to
0}\int_{0}^{T}\int g_{k}\eta_{B}\varphi\partial_{t}\psi_{\varepsilon,i}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\;\;-\limsup_{\varepsilon\to
0}\frac{1}{2}\int_{0}^{T}\int
g_{k}\eta_{B}\frac{2}{\varepsilon}W(u_{\varepsilon})\varphi^{2}\differential{x}\differential{t}$
$\displaystyle={}$
$\displaystyle\sum_{k=1}^{K}\sum_{B\in\mathcal{B}_{r}}\max_{1\leq i\leq
P}\sup_{\begin{subarray}{c}\varphi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left((0,T)\times\operatorname{\mathbb{T}}\right)\\\
\absolutevalue{\varphi}\leq R\end{subarray}}\int_{0}^{T}\int
g_{k}\eta_{B}\varphi\partial_{t}\psi_{i}-\frac{1}{2}\int_{0}^{T}\operatorname{E}\left(\chi;g_{k}\eta_{B}\varphi^{2}\right)\differential{t}.$

For the equality we used the equipartition of energies Lemma 4.2 and the
$\operatorname{L}^{1}$-convergence of $\psi_{\varepsilon,i}$ to $\psi_{i}$
given by Proposition 3.4. By adding zero, we get

$\displaystyle\int_{0}^{T}\int
g_{k}\eta_{B}\varphi\partial_{t}\psi_{i}\differential{x}\differential{t}-\frac{1}{2}\int_{0}^{T}\operatorname{E}\left(\chi;g_{k}\eta_{B}\varphi^{2}\right)\differential{t}$
$\displaystyle={}$
$\displaystyle\sum_{j=1}^{P}\left(\sigma_{ij}\int_{0}^{T}\int
g_{k}\eta_{B}\varphi
V_{j}\absolutevalue{\nabla\chi_{j}}\differential{t}-\frac{1}{2}\sigma_{ij}\int_{0}^{T}\int
g_{k}\eta_{B}\varphi^{2}\absolutevalue{\nabla\chi_{j}}\differential{t}\right)$
$\displaystyle-\frac{1}{2}\int_{0}^{T}\left(\operatorname{E}\left(\chi;g_{k}\eta_{B}\varphi^{2}\right)-\int
g_{k}\eta_{B}\varphi^{2}\absolutevalue{\nabla\psi_{i}}\right)\differential{t}.$

Since no derivative has fallen on $g_{k}$, we may send $\delta$ to zero and
obtain by the dominated convergence theorem that we can replace $g_{k}$ by
$\mathds{1}_{(T_{k-1},T_{k})}$. We thus end up with a good summand consisting
of

$\sum_{k=1}^{K}\sum_{B\in\mathcal{B}_{r}}\max_{1\leq i\leq
P}\sup_{\begin{subarray}{c}\varphi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left((0,T)\times\operatorname{\mathbb{T}}\right)\\\
\absolutevalue{\varphi}\leq
R\end{subarray}}\sum_{j=1}^{P}\sigma_{ij}\int_{T_{k-1}}^{T_{k}}\int\eta_{B}\varphi\left(V_{j}-\frac{1}{2}\varphi\right)\absolutevalue{\nabla\chi_{j}}\differential{t}$
(34)

and an error summand given by

$\displaystyle\sum_{k=1}^{K}\sum_{B\in\mathcal{B}_{r}}\max_{1\leq i\leq
P}\sup_{\begin{subarray}{c}\varphi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left((0,T)\times\operatorname{\mathbb{T}}\right)\\\
\absolutevalue{\varphi}\leq
R\end{subarray}}-\frac{1}{2}\int_{T_{k-1}}^{T_{k}}\left(\operatorname{E}\left(\chi;\eta_{B}\varphi^{2}\right)-\int\eta_{B}\varphi^{2}\absolutevalue{\nabla\psi_{i}}\right)\differential{t}$
$\displaystyle\geq{}$
$\displaystyle\sum_{k=1}^{K}\sum_{B\in\mathcal{B}_{r}}\max_{1\leq i\leq
P}-\frac{R^{2}}{2}\int_{T_{k-1}}^{T_{k}}\left(\operatorname{E}\left(\chi;\eta_{B}\right)-\int\eta_{B}\absolutevalue{\nabla\psi_{i}}\right)\differential{t},$
(35)

where we used
$\absolutevalue{\nabla\psi_{i}}\leq\operatorname{E}(\chi,\cdot)$. We choose a
majority phase $(i,j)$ and estimate the summands of (34) from below using

$\displaystyle\max_{1\leq i\leq
P}\sup_{\begin{subarray}{c}\varphi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left((0,T)\times\operatorname{\mathbb{T}}\right)\\\
\absolutevalue{\varphi}\leq
R\end{subarray}}\sum_{j=1}^{P}\sigma_{ij}\int_{T_{k-1}}^{T_{k}}\int\eta_{B}\varphi\left(V_{j}-\frac{1}{2}\varphi\right)\absolutevalue{\nabla\chi_{j}}\differential{t}$
$\displaystyle\geq{}$
$\displaystyle\max_{i<j}\sup_{\begin{subarray}{c}\varphi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left((0,T)\times\operatorname{\mathbb{T}}\right)\\\
\absolutevalue{\varphi}\leq
R\end{subarray}}\sigma_{ij}\int_{T_{k-1}}^{T_{k}}\int\eta_{B}\varphi\left(V_{j}-\frac{1}{2}\varphi\right)\absolutevalue{\nabla\chi_{j}}\differential{t}$
$\displaystyle\qquad\qquad\qquad\qquad-C\sum_{l\notin\\{i,j\\}}\int_{T_{k-1}}^{T_{k}}\int\eta_{B}\left(R\absolutevalue{V_{l}}+R^{2}\right)\absolutevalue{\nabla\chi_{l}}\differential{t}.$
(36)

Concerning the error term, we have by Remark 4.4 that

$\frac{R^{2}}{2}\int_{T_{k-1}}^{T_{k}}\operatorname{E}\left(\chi;\eta_{B}\right)-\int\eta_{B}\absolutevalue{\nabla\psi_{i}}\differential{t}\lesssim
R^{2}\sum_{l\notin\\{i,j\\}}\int_{T_{k-1}}^{T_{k}}\int\eta_{B}\absolutevalue{\nabla\chi_{l}}\differential{t},$

which enables us to absorb the error (35) into the error term (36). By
applying Young’s inequality, we get that for every parameter $\alpha\in(0,1)$,
we have

$\int_{T_{k-1}}^{T_{k}}\int\eta_{B}R\absolutevalue{V_{l}}\absolutevalue{\nabla\chi_{l}}\differential{t}\lesssim\alpha\int_{T_{k-1}}^{T_{k}}\int\eta_{B}V_{l}^{2}\absolutevalue{\nabla\chi_{l}}\differential{t}+\frac{R^{2}}{\alpha}\int_{T_{k-1}}^{T_{k}}\int\eta_{B}\absolutevalue{\nabla\chi_{l}}\differential{t}.$

Collecting our estimates, we end up with an error term which can be estimated
from below up to a constant by

$\displaystyle-\alpha\sum_{l=1}^{P}\int_{0}^{T}\int
V_{l}^{2}\absolutevalue{\nabla\chi_{l}}\differential{t}-\frac{R^{2}}{\alpha}\sum_{k=1}^{K}\sum_{B\in\mathcal{B}_{r}}\max_{i<j}\sum_{l\notin\\{i,j\\}}\int_{T_{k-1}}^{T_{k}}\int\eta_{B}\absolutevalue{\nabla\chi_{l}}\differential{t}.$

Moreover, we can choose for fixed $k,B$ and tuple $(i,j)$ a sequence of test
functions $\varphi_{n}$ with $\absolutevalue{\varphi_{n}}\leq R$ which
converge to $V_{j}\mathds{1}_{\absolutevalue{V_{j}\leq R}}$ in the sense that

$\lim_{n\to\infty}\norm{\varphi_{n}-V_{j}\mathds{1}_{\absolutevalue{V_{j}}\leq
R}}_{\operatorname{L}^{2}\left((T_{k-1},T_{k})\times\operatorname{\mathbb{T}},\absolutevalue{\nabla\chi_{j}}\differential{t}\right)}=0.$

Combining these three arguments, we arrive at the estimate

$\displaystyle\liminf_{\varepsilon\to
0}\frac{1}{2}\int_{0}^{T}\int\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}\differential{x}\differential{t}$
$\displaystyle\geq{}$ $\displaystyle-C\alpha\sum_{1\leq l\leq
P}\int_{0}^{T}\int V_{l}^{2}\absolutevalue{\nabla\chi_{l}}\differential{t}$
$\displaystyle+\sum_{k=1}^{K}\sum_{B\in\mathcal{B}_{r}}\max_{i<j}\int_{T_{k-1}}^{T_{k}}\frac{\sigma_{ij}}{2}\int\eta_{B}\absolutevalue{V_{j}\mathds{1}_{\absolutevalue{V_{j}}\leq
R}}^{2}\absolutevalue{\nabla\chi_{j}}-C\frac{R^{2}}{\alpha}\sum_{l\notin\\{i,j\\}}\int\eta_{B}\absolutevalue{\nabla\chi_{l}}\differential{t}$

for every parameter $\alpha\in(0,1)$. We choose partitions whose width tends
to zero and which are contained in each other. By the monotone convergence
theorem and using Lebesgue points, we can pull the maximum inside the time
integral to obtain

$\displaystyle\liminf_{\varepsilon\to
0}\frac{1}{2}\int_{0}^{T}\int\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}\differential{x}\differential{t}$
$\displaystyle\geq{}$ $\displaystyle-\alpha C\sum_{1\leq l\leq
P}\int_{0}^{T}\int V_{l}^{2}\absolutevalue{\nabla\chi_{l}}\differential{t}$
$\displaystyle+\int_{0}^{T}\sum_{B\in\mathcal{B}_{r}}\max_{i<j}\frac{\sigma_{ij}}{2}\int\eta_{B}\absolutevalue{V_{j}\mathds{1}_{\absolutevalue{V_{j}}\leq
R}}^{2}\absolutevalue{\nabla\chi_{j}}-C\frac{R^{2}}{\alpha}\sum_{l\notin\\{i,j\\}}\int\eta_{B}\absolutevalue{\nabla\chi_{l}}\differential{t}$
$\displaystyle\geq{}$ $\displaystyle\frac{1}{2}\sum_{1\leq i<j\leq
P}\sigma_{ij}\int_{0}^{T}\int_{\Sigma_{ij}}V_{i}^{2}\mathds{1}_{\absolutevalue{V_{i}}\leq
R}\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}$
$\displaystyle-C\left(\alpha\sum_{1\leq l\leq P}\int_{0}^{T}\int
V_{l}^{2}\absolutevalue{\nabla\chi_{l}}\differential{t}+\frac{R^{2}}{\alpha}\int_{0}^{T}\sum_{B\in\mathcal{B}_{r}}\min_{i\neq
j}\sum_{k\notin\\{i,j\\}}\int\eta_{B}\absolutevalue{\nabla\chi_{k}}\differential{t}\right).$

We first send $r\to 0$ and then $\alpha\to 0$ . By the localization result
Lemma 4.3 we thus obtain that for all $R>0$

$\liminf_{\varepsilon\to
0}\frac{1}{2}\int_{0}^{T}\int\varepsilon\absolutevalue{\partial_{t}u_{\varepsilon}}^{2}\differential{x}\differential{t}\geq\frac{1}{2}\sum_{1\leq
i<j\leq
P}\sigma_{ij}\int_{0}^{T}\int_{\Sigma_{ij}}V_{i}^{2}\mathds{1}_{\absolutevalue{V_{i}}\leq
R}\differential{\operatorname{\mathcal{H}}^{d-1}}\differential{t}.$

Therefore the lower semicontinuity of the velocity term now follows from the
monotone convergence theorem. ∎

###  4.4 De Giorgi type varifold solutions for mean curvature flow

Up until this point, we have always made the crucial assumption of energy
convergence (29) for our proofs. However this is usually a very strong
assumption. One thing which could go wrong is for example illustrated in
Figure 4. There we see that two approximate interfaces of the first and second
phase collapse as $\varepsilon$ tends to zero. This results in a loss of
energy since the measure theoretic boundary of a set with finite perimeter
does not see such lines.

$\approx\alpha_{1}$$\approx\alpha_{2}$$\approx\alpha_{1}$ (a)
$1>\varepsilon>0$

$\approx\alpha_{1}$$\approx\alpha_{2}$$\approx\alpha_{1}$ (b)
$1\gg\varepsilon>0$

$=\alpha_{1}$$=\alpha_{1}$ (c) $\varepsilon=0$

Figure 4: Profile of the solution $u_{\varepsilon}$ as $\varepsilon$ tends to
zero.

We now introduce the solution concept by Hensel and Laux in [HL21] which
tackles this issue. For the two-phase case, the definition is as follows.

######  Definition 4.6 (De Giorgi type varifold solution for two-phase mean
curvature flow).

Let $T<\infty$ be an arbitrary finite time horizon and let
$\mu=\operatorname{\mathcal{L}}^{1}\otimes(\mu_{t})_{t\in(0,T)}$ be a family
of oriented varifolds
$\mu_{t}\in\mathcal{M}\left(\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}\right)$
for $t\in(0,T)$ such that
$t\mapsto\int_{\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}}\eta(t,x,p)\differential\mu_{t}(x,p)$
is measurable for all
$\eta\in\operatorname{L}^{1}\left((0,T);\operatorname{C}\left(\operatorname{\mathbb{T}},\mathbb{S}^{d-1}\right)\right)$.
Consider also a family $A=(A_{t})_{t\in(0,T)}$ of subsets of
$\operatorname{\mathbb{T}}$ with finite perimeter such that the associated
indicator function $\chi(x,t)\coloneqq\chi_{A_{t}}(x)$ is in the space
$\operatorname{L}^{\infty}\left((0,T);\operatorname{BV}(\operatorname{\mathbb{T}};\\{0,1\\})\right)$.
Let $\sigma>0$ be a surface tension constant.

Given an initial energy
$\omega^{0}\in\mathcal{M}\left(\operatorname{\mathbb{T}}\right)$ and initial
data
$\chi^{0}\in\operatorname{BV}\left(\operatorname{\mathbb{T}};\\{0,1\\}\right)$,
we call the pair $(\mu,\chi)$ a _De Giorgi type varifold solution to two-phase
mean curvature flow with initial data_ $(\omega^{0},\chi^{0})$ if the
following holds.

  1. 1.

(Existence of a normal speed) Writing
$\mu_{t}=\omega_{t}\otimes(\lambda_{t,x})_{x\in\operatorname{\mathbb{T}}}$ for
the disintegration of $\mu_{t}$, we require the existence of some
$V\in\operatorname{L}^{2}\left((0,T)\times\operatorname{\mathbb{T}},\omega_{t}\right)$
encoding a normal velocity in the sense of

$\displaystyle\sigma\int\chi(T^{\prime},x)\varphi(T^{\prime},x)-\chi^{0}(x)\varphi(0,x)\differential{x}$
$\displaystyle={}$
$\displaystyle\sigma\int_{0}^{T^{\prime}}\int\chi\partial_{t}\varphi\differential{x}\differential{t}+\int_{0}^{T^{\prime}}\int
V\varphi\differential{\omega_{t}}\differential{t}$ (37)

for almost every $T^{\prime}\in(0,T)$ and all
$\varphi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left([0,T)\times\operatorname{\mathbb{T}}\right)$.

  2. 2.

(Existence of a generalized mean curvature vector) We require the existence of
some
$H\in\operatorname{L}^{2}\left((0,T)\times\operatorname{\mathbb{T}},\omega_{t};\mathbb{R}^{d}\right)$
encoding a generalized mean curvature vector by

$\int_{0}^{T}\int\left\langle
H\,,\,\xi\right\rangle\differential{\omega_{t}}\differential{t}=-\int_{0}^{T}\int_{\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}}\left\langle\xi\,,\,\mathrm{Id}-p\otimes
p\right\rangle\differential{\mu_{t}(x,p)}\differential{t}$ (38)

for all
$\xi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left([0,T)\times\operatorname{\mathbb{T}};\mathbb{R}^{d}\right)$.

  3. 3.

(De Giorgi type optimal energy dissipation inequality) A sharp energy
dissipation inequality holds in form of

$\omega_{T^{\prime}}(\operatorname{\mathbb{T}})+\frac{1}{2}\int_{0}^{T^{\prime}}\int
V^{2}+\absolutevalue{H}^{2}\differential{\omega_{t}}\differential{t}\leq\omega^{0}(\operatorname{\mathbb{T}})$
(39)

for almost every $T^{\prime}\in(0,T)$.

  4. 4.

(Compatibility) For almost every $t\in(0,T)$ and all
$\xi\in\operatorname{C}^{\infty}\left(\operatorname{\mathbb{T}};\mathbb{R}^{d}\right)$,
it holds that

$\sigma\int\left\langle\xi\,,\,\nabla\chi(t,\cdot)\right\rangle=\int_{\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}}\left\langle\xi\,,\,p\right\rangle\differential{\mu_{t}(x,p)}.$
(40)

We firstly want to discuss this definition. If we have a De Giorgi type
$\operatorname{BV}$-solution $\chi$ to two-phase mean curvature flow in the
sense of Definition 4.1, we can think of the oriented varifold $\mu$ as the
measure

$\mu=\sigma\operatorname{\mathcal{L}}^{1}|_{(0,T)}\otimes(\absolutevalue{\nabla\chi(t)})_{t\in(0,T)}\otimes(\delta_{\nu(t,x)})_{t\in(0,T),x\in\operatorname{\mathbb{T}}}$
(41)

and therefore the measure $\omega_{t}$ becomes the energy measure
$\operatorname{E}(\chi(t),\cdot)$. The advantage of the varifold formulation
is that our new energy measure $\omega_{t}$ is not restricted to only seeing
the measure theoretic boundary of $\chi$, but can actually capture phenomena
as described in Figure 4. For example in such a scenario we would expect the
measure $\mu_{t}$ to be defined by

$\mu_{t}\coloneqq
2\sigma\operatorname{\mathcal{H}}^{1}|_{l}\otimes\left(\frac{1}{2}\delta_{e_{1}}+\frac{1}{2}\delta_{-e_{1}}\right)_{x\in\operatorname{\mathbb{T}}},$
(42)

where $l$ is the red line to which the phase of $\alpha_{2}$ shrank down as
$\varepsilon$ approached zero and $e_{1}=(1,0)^{\top}$. The factor 2 comes
from the fact that we obtain energy from both of the collapsing interfaces.

The equation (37) for the normal speed is simply motivated through the
fundamental theorem of calculus. Assuming that everything is nice and smooth,
we can compute that

$\displaystyle\sigma\int\chi(T,x)\varphi(T,x)-\chi^{0}(x)\varphi(0,x)\differential{x}$
$\displaystyle={}$
$\displaystyle\sigma\int_{0}^{T}\int\partial_{t}\left(\chi(t,x)\varphi(t,x)\right)\differential{x}\differential{t}$
$\displaystyle={}$
$\displaystyle\sigma\int_{0}^{T}\int\partial_{t}\chi(t,x)\varphi(t,x)+\chi(t,x)\partial_{t}\varphi(t,x)\differential{x}\differential{t}$
$\displaystyle={}$ $\displaystyle\int_{0}^{T}\int
V\varphi\differential{\omega_{t}}\differential{t}+\sigma\int_{0}^{T}\int\chi(t,x)\partial_{t}\varphi(t,x)\differential{x}\differential{t},$

where for the last equality, we used
$\partial_{t}\chi=V\absolutevalue{\nabla\chi}\differential{t}$ and
$\omega_{t}=\sigma\absolutevalue{\nabla\chi(t)}$.

The equation (38) for the generalized mean curvature is straightforward. In
fact if we assume that the varifold is given through equation (41), then we
have that the right hand side of equation (38) reads for a fixed time $t$

$\int_{\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}}\left\langle\xi\,,\,\mathrm{Id}-p\otimes
p\right\rangle\differential{\omega_{t}}=\sigma\int\left\langle\xi\,,\,\mathrm{Id}-\nu\otimes\nu\right\rangle\absolutevalue{\nabla\chi}.$

This is exactly the distributional formulation for the mean curvature vector.

De Giorgis inequality (39) is self-explanatory since $\omega_{t}$ is the
energy measure. The compatibility condition (40) is necessary to couple the
evolving set $A_{t}$ to the varifold. Notice that in our above example (42),
this condition is still satisfied. Even though the energy measure $\omega_{t}$
sees the red strip and the measure theoretic boundary of the corresponding
indicator function does not, the term on the right hand side of (40) is zero
since

$\int_{\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}}\left\langle\xi\,,\,p\right\rangle\differential{\mu_{t}}=\sigma\int_{l}\left\langle\xi\,,\,e_{1}-e_{1}\right\rangle\differential{\operatorname{\mathcal{H}}^{1}}=0.$

For the multiphase case, we propose the following solution concept, which
generalizes [HL21, Def. 2] to the case of arbitrary surface tensions.

######  Definition 4.7 (De Giorgi type varifold solution to multiphase mean
curvature flow).

Let $T<\infty$ be an arbitrary finite time horizon and let
$P\in\mathbb{N}_{\geq 2}$ be the number of phases. For each pair of phases
$(i,j)\in\\{1,\dotsc,P\\}^{2}$, let
$\mu_{ij}=\operatorname{\mathcal{L}}^{1}|_{(0,T)}\otimes(\mu_{t,ij})_{t\in(0,T)}$
be a family of oriented varifolds
$\mu_{t,ij}\in\mathcal{M}\left(\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}\right)$
for $t\in(0,T)$ such that the map
$t\mapsto\int_{\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}}\eta(t,x,p)\differential{\mu_{t,ij}(x,p)}$
is measurable for all
$\eta\in\operatorname{L}^{1}\left((0,T);\operatorname{C}\left(\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}\right)\right)$.
Define the evolving oriented varifolds
$\mu_{i}=\operatorname{\mathcal{L}}^{1}|_{(0,T)}\otimes(\mu_{t,i})_{t\in(0,T)}$
for $i\in\\{1,\dotsc,P\\}$ and
$\mu=\operatorname{\mathcal{L}}^{1}|_{(0,T)}\otimes(\mu_{t})_{t\in(0,T)}$ by

$\mu_{t,i}\coloneqq 2\mu_{t,ii}+\sum_{j=1,j\neq
i}^{P}\mu_{t,ij}\quad\text{and}\quad\mu_{t}\coloneqq\frac{1}{2}\sum_{i=1}^{P}\mu_{t,i}.$
(43)

The disintegration of $\mu_{t,ij}$ is expressed in form of
$\mu_{t,ij}=\omega_{t,ij}\otimes\left(\lambda_{t,x,ij}\right)_{x\in\operatorname{\mathbb{T}}}$
with expected value
$\langle\lambda_{t,x,ij}\rangle\coloneqq\int_{\mathbb{S}^{d-1}}p\differential{\lambda_{t,x,ij}(p)}$.
Analogous expressions are introduced for the disintegrations of $\mu_{t,i}$
and $\mu_{t}$.

Furthermore consider a tuple $A=\left(A_{1},\dotsc,A_{P}\right)$ such that for
each phase $1\leq i\leq P$, we have a family $A_{i}=(A_{i}(t))_{t\in(0,T)}$ of
subsets of $\operatorname{\mathbb{T}}$ with finite perimeter. We also require
$\left(A_{1}(t),\dotsc,A_{P}(t)\right)$ to be a partition of
$\operatorname{\mathbb{T}}$ for all $t\in(0,T)$ and that for each $1\leq i\leq
P$, the associated indicator function satisfies
$\chi_{i}\in\operatorname{L}^{\infty}\left((0,T);\operatorname{BV}\left(\operatorname{\mathbb{T}};\\{0,1\\}\right)\right)$.
We shortly write $\chi=\left(\chi_{1},\dotsc,\chi_{P}\right)$.

Given initial data $(\omega^{0},(\chi_{i}^{0})_{1\leq i\leq P})$ of the above
form and a $(P\times P)$-matrix of surface tensions $\sigma$ such that
$\sigma_{ij}>0$ for $i\neq j$, we call the pair $\left(\mu,\chi\right)$ a _De
Giorgi type varifold solution to multiphase mean curvature flow with initial
data_ $(\omega^{0},\chi^{0})$ _and surface tensions_ $\sigma$ if the following
requirements hold true.

  1. 1.

(Existence of normal speeds) For each phase $1\leq i\leq P$, there exists a
normal speed
$V_{i}\in\operatorname{L}^{2}\left((0,T)\times\operatorname{\mathbb{T}},\omega_{i}\right)$
in the sense that

$\displaystyle\int\chi_{i}(T^{\prime},x)\varphi(T^{\prime},x)-\chi_{i}^{0}(x)\varphi(0,x)\differential{x}$
$\displaystyle={}$
$\displaystyle\int_{0}^{T^{\prime}}\int\chi_{i}\partial_{t}\varphi\differential{x}\differential{t}+\sum_{j=1,j\neq
i}^{P}\frac{1}{\sigma_{ij}}\int_{0}^{T^{\prime}}\int
V_{i}\varphi\differential{\omega_{t,ij}}\differential{t}$ (44)

for almost every $T^{\prime}\in(0,T)$ and all
$\varphi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left([0,T)\times\operatorname{\mathbb{T}}\right)$.

  2. 2.

(Existence of a generalized mean curvature vector) There exists a generalized
mean curvature vector
$H\in\operatorname{L}^{2}\left((0,T)\times\operatorname{\mathbb{T}},\omega;\mathbb{R}^{d}\right)$
in the sense that

$\int_{0}^{T}\int\left\langle
H\,,\,\xi\right\rangle\differential{\omega_{t}}\differential{t}=-\int_{0}^{T}\int_{\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}}\left\langle\mathrm{D}\xi\,,\,\mathrm{Id}-p\otimes
p\right\rangle\differential{\mu_{t}(x,p)}\differential{t}$ (45)

holds for all
$\xi\in\operatorname{C}_{\mathrm{c}}^{\infty}\left([0,T)\times\operatorname{\mathbb{T}};\mathbb{R}^{d}\right)$.

  3. 3.

(De Giorgi type optimal energy dissipation inequality) A sharp energy
dissipation inequality holds in form of

$\omega_{T^{\prime}}(\operatorname{\mathbb{T}})+\frac{1}{2}\sum_{i=1}^{P}\int_{0}^{T^{\prime}}\int
V_{i}^{2}\frac{1}{2}\differential{\omega_{t,i}}\differential{t}+\frac{1}{2}\int_{0}^{T^{\prime}}\int\absolutevalue{H}^{2}\differential{\omega_{t}}\differential{t}\leq\omega^{0}(\operatorname{\mathbb{T}})$
(46)

for almost every $T^{\prime}\in(0,T)$.

  4. 4.

(Compatibility conditions) For all $1\leq i,j\leq P$, we require

$\omega_{t,ij}=\omega_{t,ji}$ (47)

for almost every $t\in(0,T)$ and

$\displaystyle\langle\lambda_{t,x,ij}\rangle$
$\displaystyle=-\langle\lambda_{t,x,ji}\rangle,$ (48) $\displaystyle
V_{i}(t,x)$ $\displaystyle=-V_{j}(t,x),$ (49)
$\displaystyle\absolutevalue{\langle\lambda_{t,x,ij}\rangle}^{2}H(t,x)$
$\displaystyle=\left\langle
H(t,x)\,,\,\langle\lambda_{t,x,ij}\rangle\right\rangle\langle\lambda_{t,x}^{ij}\rangle$
(50)

for almost every $t\in(0,T)$ and $\omega_{t,ij}$ almost every
$x\in\operatorname{\mathbb{T}}$. Finally for all $1\leq i\leq P$, we have

$\int\left\langle\xi\,,\,\nabla\chi_{i}(t,\cdot)\right\rangle=\sum_{j=1,j\neq
i}^{P}\frac{1}{\sigma_{ij}}\int_{\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}}\left\langle\xi\,,\,p\right\rangle\differential{\mu_{t,ij}}$
(51)

for almost every $t\in(0,T)$ and every
$\xi\in\operatorname{C}^{\infty}\left(\operatorname{\mathbb{T}};\mathbb{R}^{d}\right)$.

As before, we first want to motivate this definition. If we have a De Giorgi
type $\operatorname{BV}$-solution to multiphase mean curvature flow in the
sense of Definition 4.1, we can think of the varifold $\mu_{t,ij}$ for $i\neq
j$ as

$\mu_{t,ij}=\sigma_{ij}\operatorname{\mathcal{H}}^{d-1}\llcorner_{\Sigma_{ij}}\otimes(\delta_{\nu_{i}})_{x\in\operatorname{\mathbb{T}}}$
(52)

and $\mu_{t,ii}=0$. But if the approximate interfaces collapse as in Figure 4,
then this will be captured by the varifolds $\mu_{t,ii}$, which will be
described by

$\mu_{t,11}=2\operatorname{\mathcal{H}}^{1}\llcorner_{l}\otimes\left(\frac{1}{2}\delta_{e_{1}}+\frac{1}{2}\delta_{-e_{1}}\right)_{x\in\operatorname{\mathbb{T}}}$
(53)

as in the two-phase case.

Considering the definition of $\mu_{t}$ by equality (43), the factor $1/2$ is
present since every interface $\mu_{t,ij}$, $i\neq j$, is counted twice. Thus
we need the factor 2 in front of $\mu_{t,ii}$ in the definition of $\mu_{t,i}$
since this interface is not counted twice.

The equation (44) is similar to the two-phase case. Note however that we only
consider the energy measures $\omega_{t,ij}$ for $i\neq j$ since we do not
expect $\omega_{t,ii}$ to be relevant for the motion of the $i$-th phase.
Equation (45) is as in the two-phase case. For the energy dissipation
inequality, we note again that we need the factor $1/2$ in front of
$\omega_{t,i}$ since each interface will be counted twice.

The first compatibility condition (47) follows simply from
$\sigma_{ij}=\sigma_{ji}$ and $\Sigma_{ij}=\Sigma_{ji}$ if we have a De Giorgi
type $\operatorname{BV}$-solution. But it should even hold true in situations
like Figure 4, since the equation (47) becomes trivial for $i=j$. In the same
situation, the second compatibility condition (48) states that even if
approximate interfaces collapse, we have $\nu_{i}=-\nu_{j}$ on $\Sigma_{ij}$
$\operatorname{\mathcal{H}}^{d-1}$-almost everywhere. Indeed, for $i=j$, the
equation states that $\langle\lambda_{t,x,ii}\rangle=0$, which is satisfied by
equation (53). Similarly we explain the anti-symmetry of the velocities (49).
The compatibility condition (50) encodes that the mean curvature vector should
always point in direction of the inner unit normal, which is not necessarily
the case for varifolds, as demonstrated in Example 4.9. Nevertheless this can
always be assured if we have a De Giorgi type $\operatorname{BV}$-solution,
see the proof of Theorem 4.8. For the last compatibility condition (51), we
again notice that it behaves similarly to its two-phase equivalent.

######  Theorem 4.8.

Every De Giorgi type $\operatorname{BV}$-solution to multiphase mean curvature
flow in the sense of Definition 4.1 is also a De Giorgi type varifold
solutions for multiphase mean curvature flow in the sense of Definition 4.7.
The varifold $\mu$ is given by

$\displaystyle\mu_{t,ij}$
$\displaystyle\coloneqq\sigma_{ij}\operatorname{\mathcal{H}}^{d-1}\llcorner_{\Sigma_{ij}(t)}\otimes(\delta_{\nu_{i}(t,x)})_{x\in\operatorname{\mathbb{T}}}$

for $i\neq j$, $\mu_{t,ii}=0$ and the initial energy is
$\omega^{0}=\operatorname{E}(\chi^{0},\cdot)$.

###### Proof.

We first note that

$\displaystyle\mu_{t,i}$ $\displaystyle=\sum_{j\neq
i}\sigma_{ij}\operatorname{\mathcal{H}}^{d-1}\llcorner_{\Sigma_{ij}}\otimes(\delta_{\nu_{i}})_{x\in\operatorname{\mathbb{T}}}$
and $\displaystyle\mu_{t}$ $\displaystyle=\sum_{1\leq i<j\leq
P}\sigma_{ij}\operatorname{\mathcal{H}}^{d-1}\llcorner_{\Sigma_{ij}}\otimes\left(\frac{1}{2}\delta_{\nu_{i}}+\frac{1}{2}\delta_{\nu_{j}}\right)_{x\in\operatorname{\mathbb{T}}}.$

Thus the energy measures are given by

$\displaystyle\omega_{t,i}$ $\displaystyle=\sum_{j\neq
i}\sigma_{ij}\operatorname{\mathcal{H}}^{d-1}\llcorner_{\Sigma_{ij}(t)}$ and
$\displaystyle\omega_{t}$ $\displaystyle=\sum_{1\leq i<j\leq
P}\sigma_{ij}\operatorname{\mathcal{H}}^{d-1}\llcorner_{\Sigma_{ij}(t)}=\operatorname{E}(\chi(t);\cdot).$

The requirement
$\chi_{i}\in\operatorname{L}^{\infty}\left((0,T);\operatorname{BV}(\operatorname{\mathbb{T}};\\{0,1\\})\right)$
is an immediate consequence of the energy dissipation inequality (28). The
existence of normal velocities is given, but we need to check that equation
(44) holds. If we assume that $\varphi$ is compactly supported in
$(0,T)\times\operatorname{\mathbb{T}}$, then the equation follows by
approximating $\chi_{i}$ with $\rho_{n}\ast\chi_{i}$, where $\rho_{n}$ is a
sequence of radial symmetric standard mollifiers. In general, we have the
problem that
$\partial_{t}\chi_{i}=V_{i}\absolutevalue{\nabla\chi_{i}}\differential{t}$
holds only tested against functions supported in
$(0,T)\times\operatorname{\mathbb{T}}$. Therefore we take a sequence
$\eta_{n}$ of non-decreasing smooth functions with compact support in $(0,T]$
which are equal to $1$ on $(1/n,T]$. Then we have

$\displaystyle\int\chi_{i}(T^{\prime},x)\varphi(T^{\prime},x)\differential{x}$
$\displaystyle={}$
$\displaystyle\lim_{n\to\infty}\int\chi_{i}(T^{\prime},x)\eta_{n}(T^{\prime})\varphi(T^{\prime},x)\differential{x}$
$\displaystyle={}$
$\displaystyle\lim_{n\to\infty}\int_{0}^{T^{\prime}}\int\eta_{n}\varphi
V_{i}\absolutevalue{\nabla\chi^{i}}\differential{t}+\int_{0}^{T^{\prime}}\int\chi_{i}\eta_{n}\partial_{t}\varphi\differential{x}\differential{t}+\int_{0}^{T^{\prime}}\int\chi_{i}\partial_{t}\eta_{n}\varphi\differential{x}\differential{t}$

for almost every $0<T^{\prime}<T$. Using the dominated convergence theorem, we
see that the first two summands converge to the right hand side of the
velocity equation (44). For the third summand, we compute that since
$\int_{0}^{T}\eta^{\prime}\differential{t}=1$, we have

$\displaystyle\absolutevalue{\int_{0}^{T^{\prime}}\partial_{t}\eta_{n}\int\chi_{i}\varphi\differential{x}\differential{t}-\int\chi^{0}_{i}(x)\varphi(0,x)\differential{x}}$
$\displaystyle={}$
$\displaystyle\absolutevalue{\int_{0}^{T^{\prime}}\partial_{t}\eta_{n}\int\chi_{i}\varphi-\chi^{0}_{i}\varphi(0,x)\differential{x}\differential{t}}$
$\displaystyle\lesssim{}$
$\displaystyle\int_{0}^{T^{\prime}}\partial_{t}\eta_{n}\left(\int\absolutevalue{\chi^{i}\varphi-\chi^{0}_{i}\varphi(0,x)}^{2}\differential{x}\right)^{1/2}\differential{t}$
$\displaystyle\leq{}$
$\displaystyle\sup_{t\in(0,\frac{1}{n})}\left(\int\absolutevalue{\chi_{i}\varphi-\chi_{i}^{0}\varphi(0,x)}^{2}\differential{x}\right)^{1/2}.$

This converges to zero since $\chi_{i}$ attains the initial data continuously
with respect to the $\operatorname{L}^{2}$-norm. Therefore the velocity
equation (44) follows.

The curvature equation (45) follows immediately from equation (27) since

$\int_{\operatorname{\mathbb{T}}\times\mathbb{S}^{d-1}}\left\langle\mathrm{D}\xi\,,\,\mathrm{Id}-p\otimes
p\right\rangle\differential{\mu_{t}}=\sum_{1\leq i<j\leq
P}\sigma_{ij}\int_{\Sigma_{ij}}\left\langle\mathrm{D}\xi\,,\,\mathrm{Id}-\nu_{i}\otimes\nu_{i}\right\rangle\differential{\operatorname{\mathcal{H}}^{d-1}}.$

We used here that on $\Sigma_{ij}$, we have
$\nu_{i}\otimes\nu_{i}=\nu_{j}\otimes\nu_{j}$ since $\nu_{i}=-\nu_{j}$. De
Giorgi’s optimal energy dissipation inequality is also identical.

The compatibility conditions (47)-(49) all hold restricted to $\Sigma_{ij}$
$\operatorname{\mathcal{H}}^{d-1}$-almost everywhere and are therefore
satisfied. For the compatibility condition (50), we have to prove that the
mean curvature vector $H$ points in the direction of $\nu_{i}$
$\operatorname{\mathcal{H}}^{d-1}$-almost everywhere since
$\langle\lambda_{t,x,ij}\rangle=\nu_{i}(t,x)$ holds for
$\operatorname{\mathcal{H}}^{d-1}$-almost every $x$ on $\Sigma_{ij}$. But this
has been proven by Brakke in [Bra78, Thm. 5.8]. Note that we apply it for a
fixed time t to the varifold $V=\mu_{t}$, which is strictly speaking no
integer varifold. But all results still hold since

$V=\sum_{1\leq i<j\leq P}\sigma_{ij}v(\Sigma_{ij})$

is a finite sum. Here $v(\Sigma_{ij})$ is the naturally associated varifold to
$\Sigma_{ij}$ as defined by Brakke. The last compatibility condition (51) is a
consequence of

$\absolutevalue{\nabla\chi_{i}}=\sum_{i\neq
j}\operatorname{\mathcal{H}}^{d-1}\llcorner_{\Sigma_{ij}},$

which holds since the sets $\Omega_{j}$ are a partition of the flat torus.
Thus we have collected all necessary claims and the proof is finished. ∎

######  Example 4.9.

We want to present an example of a varifold where the mean curvature does not
point in normal direction. Let $\operatorname{\mathbb{T}}=[0,\Lambda)^{2}$ be
the flat torus in two dimensions and let
$\rho\colon\mathbb{R}\to(-\infty,\infty)$ be some positive smooth
$\Lambda$-periodic function which is not constant. Then we consider the
varifold given by

$\mu=\rho(x_{1})\operatorname{\mathcal{H}}^{1}|_{[0,\Lambda)\times\\{0\\}}\otimes\delta_{e_{2}},$

where $e_{2}=(0,1)^{\top}$. We compute that for a given test vector field
$\xi$, we have

$\displaystyle\int\left\langle\mathrm{D}\xi\,,\,\mathrm{Id}-p\otimes
p\right\rangle\differential{\mu}$
$\displaystyle=\int_{0}^{\Lambda}\left\langle\mathrm{D}\xi(x_{1},0)\,,\,e_{1}\otimes
e_{1}\right\rangle\rho(x_{1})\differential{x_{1}}$
$\displaystyle=\int_{0}^{\Lambda}\partial_{x_{1}}\xi^{1}\rho(x_{1})\differential{x_{1}}$
$\displaystyle=-\int_{0}^{\Lambda}\xi^{1}\rho^{\prime}(x_{1})\differential{x_{1}}.$

If a mean curvature vector $H$ would exist, then this integral would have to
be equal to

$-\int_{0}^{\Lambda}\left\langle
H(x_{1},0)\,,\,\xi(x_{1},0)\right\rangle\rho(x_{1})\differential{x_{1}}.$

This implies that $H$ would be given on $[0,\Lambda)\times\\{0\\}$
$\operatorname{\mathcal{H}}^{1}$-almost everywhere by

$H(x)=\frac{\rho^{\prime}(x_{1})}{\rho(x_{1})}e_{1},$

which does not lie in the normal space of the varifold at points where
$\rho^{\prime}(x_{1})\neq 0$.

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