# The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-
Sekerka Flow in Two Dimensions

Vesa Julin  Matematiikan ja Tilastotieteen Laitos, Jyväskylän Yliopisto,
Finland<EMAIL_ADDRESS>,  Massimiliano Morini  Dipartimento di Scienze
Matematiche Fisiche e Informatiche, Università di Parma, Italy
<EMAIL_ADDRESS>,  Marcello Ponsiglione  Dip. di Matematica,
Univ. Roma-I ”La Sapienza”, Roma, Italy<EMAIL_ADDRESS>and
Emanuele Spadaro  Dip. di Matematica, Univ. Roma-I ”La Sapienza” Roma, Italy
<EMAIL_ADDRESS>
###### Abstract.

We provide the first general result for the asymptotics of the area preserving
mean curvature flow in two dimensions showing that flat flow solutions,
starting from any bounded set of finite perimeter, converge with exponential
rate to a finite union of equally sized disjoint disks. A similar result is
established also for the periodic two-phase Mullins-Sekerka flow.

##  1\. Introduction

In this paper we address the long-time behaviour of two physically relevant
area preserving nonlocal geometric flows in the plane: the area-preserving
mean curvature and the Mullins-Sekerka flow.

We start by recalling that a smooth flow of sets
$(E_{t})_{t\in[0,T)}\subset\mathbb{R}^{2}$, for some $T>0$, is a solution to
the area preserving mean curvature flow if it satisfies

(1.1) $V_{t}=-\kappa_{E_{t}}+\overline{\kappa}_{E_{t}}\quad\text{on }\partial
E_{t}\,,$

where $V_{t}$ denotes the normal velocity, $\kappa_{E_{t}}$ the curvature and
$\overline{\kappa}_{E_{t}}:=-\kern-8.96268pt\int_{\partial
E_{t}}\kappa_{E_{t}}\,d\mathcal{H}^{1}$ the integral average of the curvature
of the evolving boundary $\partial E_{t}$. Such a geometric flow has been
proposed in the physical literature as a model for coarsening phenomena. For
example, one can consider systems that, after a first relaxation time, can be
described by two subdomains of nearly pure phases far from equilibrium,
evolving in a way to decrease the total interfacial length between the phases
while keeping their area constant (for the physical background see [7, 34, 42,
43]). An important feature of the flow is that it can be regarded as a
gradient flow of the perimeter with respect to a suitable (formal)
$L^{2}$-type Riemannian structure.

The second geometric evolution we consider, the two-phase Mullins-Sekerka flow
in the flat torus $\mathbb{T}^{2}$, is governed by the law

(1.2) $\begin{cases}V_{t}=[\partial_{\nu_{t}}u_{t}]&\text{on $\partial
E_{t}$,}\\\ -\Delta u_{t}=0&\text{in $\mathbb{T}^{2}\setminus\partial
E_{t}$,}\\\ u_{t}=\kappa_{E_{t}}&\text{on $\partial E_{t}$},\end{cases}$

where $\nu_{t}$ denotes the external normal to $\partial E_{t}$,
$[\partial_{\nu_{t}}u_{t}]$ denotes the jump of the normal derivative of
$u_{t}$ at $\partial E_{t}$, i.e.,
$[\partial_{\nu_{t}}u_{t}]:=\partial_{\nu_{t}}u^{+}_{t}-\partial_{\nu_{t}}u^{-}_{t}$,
with $u^{+}_{t}$ and $u^{-}_{t}$ denoting the restrictions of $u_{t}$ to
$\mathbb{T}^{2}\setminus E_{t}$ and $E_{t}$ respectively, and $\kappa_{E_{t}}$
is as before the curvature of the evolving boundary. Let us notice that the
choice of the flat torus $\mathbb{T}^{2}$ instead of a bounded domain $\Omega$
is made to avoid in the first place boundary effects. The Mullins-Sekerka flow
is a nonlocal generalization of the mean curvature flow arising from physics.
It can be seen as a quasistatic variant of the Stefan problem and it was
originally proposed as an isotropic model for solidification and liquefaction
phenomena when the specific heat is negligible, see [36]. Moreover, it arises
as a singular limit of the Cahn-Hilliard equation, see [3, 38]. Common
features with (1.1) are the area preserving character and the gradient flow
structure (this time with respect to a suitable $H^{-1}$-Riemannian
structure).

It is well-known that, in general, smooth solutions of (1.1) may develop
singularities in finite time, such as disappearance and coalescence of
components, pinch-offs and curvature blow-up, even in two dimensions (see for
instance [17, 30, 31]). The same can be expected for the flow (1.2). The
possible singular behaviour of (1.1) and (1.2) is even wilder than that of the
unconstrained mean curvature flow, due to their nonlocal character and the
subsequent lack of a comparison principle. Thus, for a well defined global-in-
time evolution one has to introduce suitable notion of weak solution which is
capable of handling singularities, changes in topology and, possibly, rough
initial data. This is a well-established feature of curvature flows, and for
several geometric motions, different definitions of weak solutions have been
introduced in the literature.

Due to the lack of a comparison principle and based on the underlying gradient
flow structure, a natural choice for (1.1) and (1.2) is the minimizing
movement approach proposed for the mean curvature flow indepentently by
Almgren, Taylor and Wang [4] and by Luckhaus and Sturzenhecker [28], and
adapted to the volume constrained case in [35]. Note that Luckhaus and
Sturzenhecker [28] introduce a similar variational scheme for (1.2) as well,
see also [39] where the same scheme is further analyzed. We recall that the
minimizing movement method is based on implicit time-discretization and
recursive minimization of suitable incremental problems. The limiting time-
continuous evolutions constructed in this way are usually referred to as _flat
flows_. We refer to Sections 3 and 4 for the precise definition of flat flow
solution of (1.1) and (1.2), respectively.

Once a global-in-time weak solution has been constructed, it is a natural
problem to investigate its asymptotics. The focus of the paper is the long-
time behaviour of flat flows in two dimensions. Previous results on the long-
time convergence of volume preserving flows are mostly confined to the case of
smooth solutions starting from specific classes of initial regular sets, see
for instance [18, 24, 37] for the volume preserving mean curvature flow and
[1, 9, 19, 23] for the Mullins-Sekerka flow. For less general initial data,
the long time behaviour of the volume preserving mean curvature flow starting
from convex and star-shaped sets (see [6, 26]) has been characterized only up
to (possibly diverging in the case of [6]) translations. Finally, concerning
flat flow solutions the most general result is due to [25] where the
asymptotic convergence to finitely many disjoint balls is proven in two and
three dimensions for arbitrary bounded initial sets of finite perimeter, but
only up to (possibly diverging) translations and without a convergence rate.

In our main result we are able to rule out translations and we provide in two
dimensions the first full convergence result for the asymptotics of the area
preserving mean curvature and the Mullins-Sekerka flow. We show that every
flat flow solutions of (1.1) starting from _any_ set of finite perimeter
asymptotically converge, with _exponential rate_ , to a finite disjoint union
of (possibly tangent) equally sized discs. Under the additional assumption
that the perimeter of the initial set is smaller than $2$, we establish a
similar result also for (1.2)). Note that such an additional condition is
assumed for simplicity to rule out lamellae as possible limiting sets (see
Sections 4 for further details). We refer to the next section for the precise
statements.

Let us finally mention that the analysis of this paper extends in two
dimensions the results proven in [33] (see also [13] for related results in
the flat torus) for the discrete minimizing movements of the volume preserving
mean curvature flow to the time-continuous limiting evolutions.

###  1.1. Statement of the main results

In the previous work [33] three of the authors prove that in all dimensions
the discrete approximate volume preserving mean curvature flow converges
exponentially fast to a disjoint union of balls with equal size. This is the
optimal convergence result but it leaves open the question of the convergence
of the limiting flat flow. On the other hand, in [25] the first author and
Niinikoski prove that the limiting flat flow converges in low dimensions
$\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ to a disjoint union of balls, up to
possible translations of the components. Again this result does not prove the
full convergence nor does it provide any rate of convergence. In both papers
it was observed that a key technical issue is to prove a quantitative version
of the Alexandrov theorem, which in the classical form states that the only
compact smooth hypersurfaces with constant mean curvature are union of
spheres. In this paper we develop this idea further and observe that we may
prove a geometric inequality, very much related to the quantitative Alexandrov
theorem, which implies the full convergence of the flow and also gives the
exponential rate of convergence.

There has been a lot of recent research on generalizations and quantifications
of the Alexandrov theorem. We refer to [10] for an overview of this
challenging problem, and mention the works [14, 15, 16] on the
characterization of critical sets of the isoperimetric problem and [11, 12,
27] on quantification of the Alexandrov theorem.

We state our quantitative version in a form that is suitable for the study of
the equation (1.1). We denote the length of the boundary or more generally the
perimeter of a set $E$ by $P(E)$ and by $|E|$ its area. We also denote by
$P_{d}=2\sqrt{\pi md}$ the perimeter of the union of $d$ disks with total area
$m$. Our first result reads as follows.

######  Theorem 1.1.

Let $m,M>0$ and let $E\subset\mathbb{R}^{2}$ be a bounded open set of class
$C^{2}$, with $|E|=m$ and $P(E)\leq M$. Then there exists a constant
$C(m,M)>0$ such that

$\min_{d\in\mathbb{N}}|P(E)-P_{d}|\leq
C\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial E)}^{2}.$

Moreover, if $\delta_{0}>0$ and $d\in\mathbb{N}$, are such that $P_{d}\leq
P(E)\leq P_{d+1}-\delta_{0}$, then it holds

(1.3) $P(E)-P_{d}\leq
C_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial E)}^{2},$

with $C_{0}=C_{0}(m,M,\delta_{0})$.

The novelty of the above result is that on the right-hand-side we have
quadratic dependence on the curvature which is clearly optimal. One may
compare this result to the quantification of the Willmore energy [40] or to
the optimal quantitative isoperimetric inequality [22], which both have
similar scaling. The inequalities in Theorem 1.1 are geometric and do not
measure how close the set $E$ is to the union of disks. In the planar case the
closeness of $E$ to the union of disks is proven in [21] (see also Proposition
2.1 in Section 2). The above result is proven in the planar case but it could
be true also in higher dimensions.

As we already mentioned, the motivation for the geometric inequality in
Theorem 1.1 is the proof of the asymptotic convergence of the area-preserving
mean curvature flow equation (1.1).

######  Theorem 1.2.

Let $\\{E(t)\\}_{t\geq 0}$ be an area-preserving flat flow for (1.1) starting
from a bounded set of finite perimeter $E(0)\subset\mathbb{R}^{2}$. Then,
there exist $d\in\mathbb{N}$ disjoint open disks in the plane
$D_{r}(x_{1}),\ldots,D_{r}(x_{d})$, with $\pi r^{2}d=|E(0)|$, and there exists
a constant $C>1$ such that, setting
$E_{\infty}=\bigcup_{i=1}^{d}D_{r}(x_{i})$, it holds

(1.4) $\sup_{x\in E(t)\Delta E_{\infty}}\textup{dist}(x,\partial
E_{\infty})+|P(E(t))-P(E_{\infty})|\leq Ce^{-\frac{t}{C}}$

for all $t\geq 0$.

The above theorem gives the full characterization and quantitative speed of
convergence of the equation (1.1). We expect the result to be sharp, in the
sense that the flow may, indeed, converge to a union of tangent disks. In [21,
Theorem 1.4] the authors consider the case when the initial set is a union of
two ellipses and show that the equation (1.1) is well defined and smooth for
all times and converges to two tangent disks. In particular, we may not
improve the Hausdroff convergence in Theorem 1.2 to $C^{1}$-convergence of the
sets. The exponential convergence rate is optimal but we note that the flow
may in fact converge to the limiting disks also in finite time. This is the
case when we consider as an initial set a union of two disks $D_{1},D_{2}$,
which are far apart and $D_{2}$ is much smaller than $D_{1}$. Then along the
flow the larger disk grows and the smaller one shrinks until it vanishes
completely and the flow reaches its equilibrium state in finite time. The same
phenomenon occurs when $D_{2}$ is only slightly smaller than $D_{1}$ but the
time to reach the equilibrium state tends to infinity when the size of $D_{2}$
gets closer to the size of $D_{1}$. This shows that we cannot bound the
constant $C$ by a universal constant, but it may depend on the initial set in
a rather complicated way.

We note that our method can be also used to study asymptotic behavior of other
geometric flows, and to emphasize this we also address the asymptotics of the
two-phase Mullins-Sekerka flow (1.2). To avoid boundary effects we consider
periodic conditions and set the problem in the flat torus $\mathbb{T}^{2}$
and, as a further simplification, we consider initial configurations with
perimeter smaller than that of the single lammella (alternatively, we can
think that the size of the torus is big enough compared to the perimeter of
the initial set).

The main result is the following. We denote the perimeter of a set $E$ in the
flat torus by $P_{\mathbb{T}^{2}}(E)$.

######  Theorem 1.3.

Let $\\{E(t)\\}_{t\geq 0}$ be a flat flow solution to the Mullins-Sekerka flow
(1.2) in the flat torus $\mathbb{T}^{2}$ starting from a set of finite
perimeter $E(0)\subset\mathbb{T}^{2}$, with $P_{\mathbb{T}^{2}}(E)<2$. Then,
there exist $d\in\mathbb{N}$ disjoint open disks
$D_{r}(x_{1}),\ldots,D_{r}(x_{d})$, with $\pi r^{2}d=|E(0)|$, and there exists
a constant $C>1$ such that it holds

$|E(t)\Delta E_{\infty}|+|P(E(t))-P(E_{\infty})|\leq Ce^{-\frac{t}{C}}$

for all $t\geq 0$, where $E_{\infty}$ either coincides with
$\bigcup_{i=1}^{d}D_{r}(x_{i})$ or with its complement in $\mathbb{T}^{2}$.

The proof of Theorem 1.3 is similar to that of the previous theorem. We use
Theorem 1.1 and a result by Schätzle [41] to obtain a functional inequality
(see Corollary 4.3), which is in the spirit of the quantitative Alexandrov
theorem, stated now in terms of the potential $u_{t}$.

We remark that one could also consider the one-phase model for the Mullins-
Sekerka as in [8] in the whole $\mathbb{R}^{2}$ and expect the above
convergence to hold also in this case. We also expect the convergence of the
sets in Theorem 1.3 to hold with respect to Hausdorff distance but we do not
prove it here.

###  1.2. Structure of the paper

Section 2 is purely geometric and in Proposition 2.1 we prove our quantitative
version of the Alexandrov theorem which then implies Theorem 1.1 as a
corollary. In Section 3 we first introduce the incremental minimization
problem for the minimizing movements scheme, and recall some basic results
related to its minimizers. Then we recall the construction of the flat flow
and give the proof of Theorem 1.2 at the end of the section. In Section 4 we
introduce the incremental minimization problem and the flat flow for the
Mullins-Sekerka equation. We then state and prove in Proposition 4.2 a crucial
functional inequality which is related to Proposition 2.1. The section
concludes with the proof of Theorem 1.3.

##  2\. A sharp quantitative Alexandrov theorem in two-dimensions

Let us first recall that for measurable sets $E\subset\mathbb{R}^{2}$, the
perimeter is defined by

$P(E):=\sup\Big{\\{}\int_{E}\mathrm{div}X\,dx:X\in
C_{c}^{1}(\mathbb{R}^{2},\mathbb{R}^{2}),\,\|X\|_{L^{\infty}}\leq 1\Big{\\}}.$

If $P(E)<\infty$ we say that $E$ is a set of finite perimeter. We also recall
that if $E$ is regular enough, say a domain with Lipschitz boundary, then
$P(E)=\mathcal{H}^{1}(\partial E)$. For the general properties of sets of
finite perimeter we refer to the monographs [5, 29].

In the following we fix the prescribed area $m>0$ of a set $E$ and a constant
$M>0$ representing an upper bound for the perimeter of $E$. For
$d\in\mathbb{N}$ we denote by $P_{d}$ the perimeter of any union of $d$
disjoint disks with equal areas $m/d$, i.e.,

$P_{d}:=2\sqrt{\pi md}\,.$

For a set of $E\subset\mathbb{R}^{2}$ of class $C^{2}$ we denote by
$\kappa_{E}$ its curvature (with the sign defined so that $\kappa_{E}$ is
positive for convex sets) and we set

$\overline{\kappa}_{E}:=-\kern-10.66977pt\int_{\partial
E}\kappa_{E}\,d\mathcal{H}^{1}=\frac{1}{\mathcal{H}^{1}(\partial
E)}\int_{\partial E}\kappa_{E}\,d\mathcal{H}^{1}.$

In [21] it is proven that if $E\subset\mathbb{R}^{2}$ is a set of class
$C^{2}$ with area $|E|=m$ and
$\|\kappa_{E}-\overline{\kappa}_{E}\|^{2}_{L^{1}(\partial
E)}\leq\varepsilon_{0}$, for $\varepsilon_{0}$ small enough, then $E$ is
$C^{1}$-diffeomorphic to a disjoint union of disks $D_{1},\dots,D_{d}$ and it
holds

$|P(E)-P_{d}|\leq C\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{1}(\partial
E)}\,.$

Our first result improves the above inequality by showing that a similar
estimate holds with quadratic right-hand side, which is the optimal scaling of
the quantitative Alexandrov theorem. We also consider $L^{2}$-norms as this is
more natural in our variational framework. We state this in the following
proposition.

######  Proposition 2.1.

Let $m,M>0$. There exist $\varepsilon_{0}=\varepsilon_{0}(m,M)\in(0,1)$ and
$C_{0}=C_{0}(m,M)>1$ with the following property: Let $E\subset\mathbb{R}^{2}$
be a bounded open set of class $C^{2}$, with $|E|=m$ and $P(E)\leq M$, such
that $\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)}\leq\varepsilon_{0}$. Then $E$ is diffeomorphic to a union of $d$ disjoint
disks $D_{1}$, …, $D_{d}$, with equal areas $m/{d}$ and
$\textup{dist}(D_{i},D_{j})>0$ for $i\neq j$, and

(2.1) $|P(E)-P_{d}|\leq
C_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|^{2}_{L^{2}(\partial E)}\,.$

Moreover, $d$ is bounded from above by a constant depending only on $m,\,M$.

Finally, for $\varepsilon_{0}$ sufficiently small, the boundary of every
connected component of the set $E$ can be parametrized as a normal graph over
one of the discs $D_{i}$ with $C^{1,\frac{1}{2}}$ norm of the parametrization
vanishing as $\varepsilon_{0}\to 0$.

###### Proof.

Let $E$ be as in the statement and let $E_{1}$,…, $E_{d}$ be the collection of
its connected components. For each component $E_{i}$ we denote by $\Gamma_{i}$
the outer component of $\partial E_{i}$ and by $\hat{E}_{i}$ the bounded
region enclosed by $\Gamma_{i}$, i.e., the set obtained by filling the “holes”
of $E_{i}$.

We split the proof into several steps. Notice that in what follows
$m_{0}\in(0,1)$ and $M_{0}>1$ will denote “universal” constants, i.e,
constants depending only on $m,M$, which may change from line to line.

Step 1. We claim that

(2.2) $|\hat{E}_{\bar{k}}|\geq m_{0}\quad\text{for some
}{\bar{k}}\in\\{1,\dots,d\\}.$

Indeed, by translating the components if necessary we may assume that
$\textup{dist}(\hat{E}_{i},\hat{E}_{j})>\sqrt{2}$. Setting
$Q:=(0,1)\times(0,1)$, we may use [33, Lemma 2.1] to infer that there exist
$z\in\mathbb{Z}^{2}$ such that

$|E\cap(z+Q)|\geq c\min\Big{\\{}\frac{m^{2}}{M^{2}},1\Big{\\}}\,,$

with $c>0$ a universal constant. Since $z+Q$ can only intersect one component
$\hat{E}_{i}$, the claim follows.

Step 2. We claim that

(2.3) $|\overline{\kappa}_{E}|\leq M_{0}\,.$

To this aim, note that by the Isoperimetric Inequality and by (2.2), we have

(2.4) $\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})\geq 2\sqrt{\pi m_{0}}\,.$

Now,

$\int_{\partial\hat{E}_{\bar{k}}}\Big{|}\kappa_{E}-\frac{2\pi}{\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})}\Big{|}^{2}\,d\mathcal{H}^{1}\leq\int_{\partial
E}|\kappa_{E}-\overline{\kappa}_{E}|^{2}\,d\mathcal{H}^{1}\leq\varepsilon_{0}^{2}\,,$

where we used the simply connectedness of $\hat{E}_{\bar{k}}$ and Gauss-Bonnet
Theorem to get
$\overline{\kappa}_{\hat{E}_{\bar{k}}}=2\pi/\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})$.
In turn,

$\begin{split}\frac{1}{\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})}&\Big{|}2\pi-\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})\overline{\kappa}_{E}\Big{|}^{2}=\int_{\partial\hat{E}_{\bar{k}}}\Big{|}\frac{2\pi}{\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})}-\overline{\kappa}_{E}\Big{|}^{2}\,d\mathcal{H}^{1}\\\
&\leq
2\int_{\partial\hat{E}_{\bar{k}}}\Big{|}\kappa_{E}-\frac{2\pi}{\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})}\Big{|}^{2}\,d\mathcal{H}^{1}+2\int_{\partial
E}\Big{|}\kappa_{E}-\overline{\kappa}_{E}\Big{|}^{2}\,d\mathcal{H}^{1}\leq
4\varepsilon_{0}^{2}\leq 4\,.\end{split}$

Hence
$\Big{|}2\pi-\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})\overline{\kappa}_{E}\Big{|}\leq
2\sqrt{\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})}$, so that, using also
(2.4),

$2\sqrt{\pi m_{0}}|\overline{\kappa}_{E}|\leq
2\pi+|2\pi-\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})\overline{\kappa}_{E}|\leq
2\pi+2\sqrt{\mathcal{H}^{1}(\partial\hat{E}_{\bar{k}})}\leq
2\pi(1+\sqrt{M})\,,$

and the claim follows.

Step 3. We claim that

(2.5) $\mathcal{H}^{1}(\Gamma)\geq m_{0}\qquad\text{for any component
}\Gamma\text{ of }\partial E\,.$

Indeed, using again Gauss-Bonnet Theorem,

$\begin{split}M|\overline{\kappa}_{E}|^{2}+1&\geq\mathcal{H}^{1}(\Gamma)|\overline{\kappa}_{E}|^{2}+\varepsilon_{0}^{2}\geq\mathcal{H}^{1}(\Gamma)|\overline{\kappa}_{E}|^{2}+\int_{\Gamma}|\kappa_{E}-\overline{\kappa}_{E}|^{2}\,d\mathcal{H}^{1}\\\
&\geq\frac{1}{2}\int_{\Gamma}|\kappa_{E}|^{2}\,d\mathcal{H}^{1}\geq\frac{1}{2\mathcal{H}^{1}(\Gamma)}\Big{(}\int_{\Gamma}\kappa_{E}\Big{)}^{2}=\frac{1}{2\mathcal{H}^{1}(\Gamma)}4\pi^{2}\,,\end{split}$

and the claim follows taking into account (2.3).

Step 4. We claim that if $\varepsilon_{0}$ is sufficiently small, then $E$ has
$d\leq M_{0}$ connected components which are simply connected.

We argue by contradiction. Suppose there exists a connected component $E_{i}$
which is not simply connected. Then there exists a component
$\Gamma\subset\partial E$ contained in $\hat{E}_{i}$ such that
$\int_{\Gamma}\kappa_{E}\,d\mathcal{H}^{1}=-2\pi$. We observe that then it
holds

$-\kern-10.66977pt\int_{\Gamma}\kappa_{E}d\mathcal{H}^{1}=-\frac{2\pi}{\mathcal{H}^{1}(\Gamma)}\qquad\text{and}\qquad-\kern-10.66977pt\int_{\partial\hat{E}_{i}}\kappa_{E}d\mathcal{H}^{1}=\frac{2\pi}{\mathcal{H}^{1}(\partial\hat{E}_{i})}$

and therefore

$\int_{\Gamma}\Big{|}\kappa_{E}+\frac{2\pi}{\mathcal{H}^{1}(\Gamma)}\Big{|}^{2}d\mathcal{H}^{1}+\int_{\partial\hat{E}_{i}}\Big{|}\kappa_{E}-\frac{2\pi}{\mathcal{H}^{1}(\partial\hat{E}_{i})}\Big{|}^{2}d\mathcal{H}^{1}\leq
2\int_{\partial E}|\kappa_{E}-\overline{\kappa}_{E}|^{2}d\mathcal{H}^{1}\leq
2\varepsilon_{0}^{2}.$

We then infer that by (2.5)

$\displaystyle\frac{16\pi^{2}}{M^{2}}$
$\displaystyle\leq\left|\frac{2\pi}{\mathcal{H}^{1}(\Gamma)}+\frac{2\pi}{\mathcal{H}^{1}(\partial\hat{E}_{i})}\right|^{2}\leq
2\left|\overline{\kappa}_{E}+\frac{2\pi}{\mathcal{H}^{1}(\Gamma)}\right|^{2}+2\left|\overline{\kappa}_{E}-\frac{2\pi}{\mathcal{H}^{1}(\partial\hat{E}_{i})}\right|^{2}$
$\displaystyle\leq
2-\kern-10.66977pt\int_{\Gamma}\left|\kappa_{E}+\frac{2\pi}{\mathcal{H}^{1}(\Gamma)}\right|^{2}d\mathcal{H}^{1}+2-\kern-10.66977pt\int_{\partial\hat{E}_{i}}\left|\kappa_{E}-\frac{2\pi}{\mathcal{H}^{1}(\partial\hat{E}_{i})}\right|^{2}d\mathcal{H}^{1}\leq\frac{4\varepsilon_{0}^{2}}{m_{0}^{2}}.$

Therefore, for $\varepsilon_{0}$ sufficiently small we reach a contradiction.

Every component of $E$ is thus simply connected and by (2.5) their perimeter
is bounded from below. Therefore the number $d$ of the components is bounded
from above $d\leq M_{0}$. Note that in particular
$\overline{\kappa}_{E}=\frac{2\pi d}{\mathcal{H}^{1}(\partial E)}$.

Step 5. Let us show that if $\varepsilon_{0}$ is sufficiently small, then each
connected component $E_{i}$ is a nearly spherical set, parametrized over a
disks $D_{r_{i}}(x_{i})$ with $|D_{r_{i}}(x_{i})|=|E_{i}|$ and the
$C^{1,\frac{1}{2}}$ norm of the parametrization is infinitesimal with
$\varepsilon_{0}\to 0$.

We adapt the argument of [21, Lemma 3.2]. Let us fix a component $E_{i}$ and
denote its perimeter by $l_{i}$, i.e. $H^{1}(\partial E_{i})=l_{i}$. By Gauss-
Bonnet it holds $\overline{\kappa}_{E_{i}}=\frac{2\pi}{l_{i}}$. Since the
boundary $\partial E_{i}$ is connected we may parametrize it by a unit speed
curve $\gamma:[0,l_{i}]\to\mathbb{R}^{2}$, $\gamma(s)=(x(s),y(s))$ with
counterclockwise orientation. Define
$\theta(s):=\int_{0}^{s}\kappa_{E_{i}}(\gamma(\tau))\,d\tau$ so that
$\theta(0)=0$ and $\theta(l_{i})=2\pi$. Then, for every $0\leq s_{1}<s_{2}\leq
l_{i}$, it holds by Hölder’s inequality

(2.6)
$\begin{split}|\theta(s_{2})-s_{2}\overline{\kappa}_{E}-(\theta(s_{1})-s_{1}\overline{\kappa}_{E})|&\leq\int_{s_{1}}^{s_{2}}|\kappa_{E}-\overline{\kappa}_{E}|\\\
&\leq\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)}|s_{2}-s_{1}|^{\frac{1}{2}}\\\
&\leq\varepsilon_{0}|s_{2}-s_{1}|^{\frac{1}{2}}.\end{split}$

In particular, applying (2.6) to $s_{1}=0$ and $s_{2}=s\in[0,l_{i}]$ generic,
we get

(2.7) $\big{|}\theta(s)-s\overline{\kappa}_{E}\big{|}\leq
M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial E)}\leq
M_{0}\varepsilon_{0},$

and for $s_{2}=l_{i}$ it yields

(2.8) $\left|2\pi-l_{i}\overline{\kappa}_{E}\right|\leq
M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial E)}\leq
M_{0}\varepsilon_{0}.$

By possibly rotating the set $E_{i}$ we have

$x^{\prime}(s)=-\sin\theta(s)\qquad\text{and}\qquad
y^{\prime}(s)=\cos\theta(s)\quad\text{ for all }s\in(0,l_{i}).$

We obtain by (2.7) and (2.8) that

(2.9) $\Big{|}x^{\prime}(s)+\sin\big{(}\frac{2\pi
s}{l_{i}}\big{)}\Big{|}+\Big{|}y^{\prime}(s)-\cos\big{(}\frac{2\pi
s}{l_{i}}\big{)}\Big{|}\leq
M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial E)}\leq
M_{0}\varepsilon_{0}$

for all $s\in[0,l_{i}]$. Integrating (2.9) we deduce that there are numbers
$a$ and $b$ such that

$\displaystyle\Big{|}x(s)-a-\frac{l_{i}}{2\pi}\cos\big{(}\frac{2\pi
s}{l_{i}}\big{)}\Big{|}$
$\displaystyle+\Big{|}y(s)-b-\frac{l_{i}}{2\pi}\sin\big{(}\frac{2\pi
s}{l_{i}}\big{)}\Big{|}$ (2.10) $\displaystyle\leq
M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial E)}\leq
M_{0}\varepsilon_{0}$

for all $s\in[0,l_{i}]$. We set $x_{i}=(a,b)$ and note that from (2) we infer
that

(2.11)
$D_{\frac{l_{i}}{2\pi}-M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)}}(x_{i})\subset E_{i}\subset
D_{\frac{l_{i}}{2\pi}+M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)}}(x_{i}).$

In particular, if $r_{i}$ is chosen in such a way that $|E_{i}|=\pi
r_{i}^{2}=|D_{r_{i}}(x_{i})|$, then (2.11) yields

(2.12)
$\frac{l_{i}}{2\pi}-M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)}\leq
r_{i}\leq\frac{l_{i}}{2\pi}+M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)},$

and

$\displaystyle\Big{|}x(s)-a-r_{i}\cos\big{(}\frac{2\pi
s}{l_{i}}\big{)}\Big{|}$
$\displaystyle+\Big{|}y(s)-b-r_{i}\sin\big{(}\frac{2\pi
s}{l_{i}}\big{)}\Big{|}$ (2.13) $\displaystyle\leq
M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial E)}\leq
M_{0}\varepsilon_{0}\qquad\forall\;s\in[0,l_{i}].$

By (2.9) and (2) the boundary of the component $E_{i}$ is parametrized by a
small perturbation of the boundary of the disc $\partial D_{r_{i}}(x_{i})$
given by $c:[0,l_{i}]\to\mathbb{R}^{2}$ with $c(s)=r_{i}(\cos(\frac{2\pi
s}{l_{i}}),\sin(\frac{2\pi s}{l_{i}}))$:

$\displaystyle\gamma(s)=c(s)+\sigma(s)$ (2.14)
$\displaystyle\|\sigma\|_{L^{\infty}}+\|\sigma^{\prime}\|_{L^{\infty}}\leq
M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial E)}\leq
M_{0}\varepsilon_{0}.$

Now it is a simple consequence of (2.6), (2.9) and (2) to verify that
$\partial E_{i}$ is as nearly spherical sets over $D_{r_{i}}(x_{i})$, with
$|D_{r_{i}}(x_{i})|=|E_{i}|$, by functions $f_{i}\in C^{1,1/2}(\partial
D_{r_{i}}(x_{i}))$ with $C^{1,1/2}$ norm convering to zero as
$\varepsilon_{0}\to 0$.

Step 6. Quantitative Alexandrov Theorem.

We use the quantitative Alexandrov theorem proven in [33] to infer that, if
$f_{i}$ is the parametrization of the component $E_{i}$, then

$\|f_{i}\|_{H^{1}(\partial D_{r_{i}}(x_{i}))}^{2}\leq
C\|\kappa_{E_{i}}-\overline{\kappa}_{E_{i}}\|_{L^{2}(\partial E_{i})}^{2}.$

Recall that $r_{i}$ is such that $|E_{i}|=|D_{r_{i}}(x_{i})|$. By the area
formula, see e.g. [33, (1.3)], and a simple linearization we infer that

$0\leq P(E_{i})-P(D_{r_{i}}(x_{i}))\leq C\|f_{i}\|_{H^{1}(\partial
D_{r_{i}}(x_{i}))}^{2}.$

Summing over the connected components yields

(2.15) $\begin{split}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)}^{2}&\geq\sum_{i=1}^{d}\|\kappa_{E_{i}}-\overline{\kappa}_{E_{i}}\|_{L^{2}(\partial
E_{i})}^{2}\geq c\sum_{i=1}^{d}\|f_{i}\|_{H^{1}(\partial
D_{r_{i}}(x_{i}))}^{2}\\\ &\geq
c\,\big{|}\sum_{i=1}^{d}P(E_{i})-P(D_{r_{i}}(x_{i}))\big{|}.\end{split}$

Step 7. Conclusion.

Let $r>0$ be such that the disk $D_{r}(x_{i})$ has area $|D_{r}(x_{i})|=m/d$ ,
where $m=|E|$. In other words $\sum_{i=1}^{d}P(D_{r}(x_{i}))=2\pi\,rd=P_{d}$.
Recall that the disks $D_{r_{i}}(x_{i})$ are defined such that
$|D_{r_{i}}(x_{i})|=|E_{i}|$ for every component $E_{i}$ and thus

(2.16) $\sum_{i}^{d}r_{i}^{2}=dr^{2}.$

Recall also that by the previous estimates it holds $m_{0}\leq r_{i},r,d\leq
M_{0}$. By (2.11) and (2.12) we infer that

(2.17) $D_{r_{i}-M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)}}(x_{i})\subset E_{i}\subset
D_{r_{i}+M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)}}(x_{i}).$

and therefore

(2.18) $|r-r_{i}|\leq
M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial E)}.$

Thus, by simple algebra, by (2.16) and by (2.12), if $d>1$ we deduce

$\begin{split}\big{|}P_{d}-\sum_{i=1}^{d}P(D_{r_{i}}(x_{i}))\big{|}&=2\pi\big{|}d\,r-\sum_{i=1}^{d}r_{i}\big{|}=2\pi\left|\sqrt{d}\left(\sum_{i=1}^{d}r_{i}^{2}\right)^{\frac{1}{2}}-\sum_{i=1}^{d}r_{i}\right|\\\
&\leq
M_{0}\left(d\sum_{i=1}^{d}r_{i}^{2}-\left(\sum_{i=1}^{d}r_{i}\right)^{2}\right)\\\
&=M_{0}\sum_{1\leq i<j\leq d}(r_{i}-r_{j})^{2}\\\ &\leq
C\sum_{i=1}^{d}(r_{i}-r)^{2}\stackrel{{\scriptstyle\eqref{eq:step7-2}}}{{\leq}}M_{0}\|\kappa_{E}-\overline{\kappa}_{E}\|_{L^{2}(\partial
E)}^{2}.\end{split}$

Hence, the inequality (2.1) then follows by combining the above estimate with
(2.15). Finally, by the very same argument of step 5 and by (2.18) we deduce
that the connected components $E_{i}$ can be parametrized as nearly spherical
sets over the discs $D_{r}(x_{i})$. ∎

Proposition 2.1 immediately implies the sharp geometric inequality in the
plane stated in Theorem 1.1.

###### Proof of Theorem 1.1.

Let $\varepsilon_{0}>0$ be from Proposition 2.1. If
$\|\kappa_{E}-\bar{\kappa}_{E}\|_{L^{2}(\partial E)}\leq\varepsilon_{0}$ then
the inequality holds by Proposition 2.1. If
$\|\kappa_{E}-\bar{\kappa}_{E}\|_{L^{2}(\partial E)}\geq\varepsilon_{0}$, then
the inequality holds trivially as

$|P(E)-P_{d}|\leq
3M\leq\frac{3M}{\varepsilon_{0}^{2}}\|\kappa_{E}-\bar{\kappa}_{E}\|_{L^{2}(\partial
E)}^{2}.$

The inequality (1.3) follows similarly. ∎

##  3\. The asymptotics of the area preserving curvature flow in the plane

Let us first introduce the setting for the construction of the flat flows. We
use the notation from [33] and refer to [35, 33] for more detailed
introduction. We denote the signed distance function by $d_{E}$ and define it
as

$d_{E}(x)=\textup{dist}(x,E)-\textup{dist}(x,\mathbb{R}^{2}\setminus E).$

Then clearly $|d_{E}(x)|=\textup{dist}(x,\partial E)$.

We fix the volume $m>0$ and the time step $h>0$, and given a bounded set $E$
we consider the minimization problem

(3.1)
$\min\Big{\\{}P(F)+\frac{1}{h}\int_{F}d_{E}\,dx+\frac{1}{\sqrt{h}}\big{|}|F|-m\big{|}\Big{\\}}$

and note that the minimizer exists but might not be unique. We define the
dissipation of a set $F$ with respect to a set $E$ as

(3.2) $\mathcal{D}(F,E):=\int_{F\Delta E}\textup{dist}(x,\partial E)\,dx$

and observe that we may write the minimization problem (3.1) as

$\min\Big{\\{}P(F)+\frac{1}{h}\mathcal{D}(F,E)+\frac{1}{\sqrt{h}}\big{|}|F|-m\big{|}\Big{\\}}.$

Let us then recall the construction of the flat flow for the volume preserving
mean curvature flow (1.1) from [35]. Let $E(0)\subset\mathbb{R}^{2}$ be a
bounded set of finite perimeter which coincides with its Lebesgue
representative. We fix a minimizer of (3.1), with $E=E(0)$, denote it by
$E_{1}^{(h)}$ and consider its Lebesgue representative. We construct the
discrete-in-time evolution $\\{E_{k}^{(h)}\\}_{k\in\mathbb{N}}$ by recursion
such that assuming that $E_{k}^{(h)}$ is defined we set $E_{k+1}^{(h)}$ to be
a minimizer of (3.1) with $E=E_{k}^{(h)}$. By [35, Lemma 3.1] it holds for all
$k=0,1,\dots$

(3.3)
$P(E_{k+1}^{(h)})+\frac{1}{\sqrt{h}}\big{|}|E_{k+1}^{(h)}|-m\big{|}+\frac{1}{h}\mathcal{D}(E_{k+1}^{(h)},E_{k}^{(h)})\leq
P(E_{k}^{(h)})+\frac{1}{\sqrt{h}}\big{|}|E_{k}^{(h)}|-m\big{|}.$

Also the set $E_{k+1}^{(h)}$ is $C^{2,\alpha}$-regular and satisfies the
Euler-Lagrange equation

(3.4)
$\frac{d_{E_{k}^{(h)}}}{h}=-\kappa_{E_{k+1}^{(h)}}+\lambda_{k+1}^{(h)}\qquad\text{on
}\,\partial E_{k+1}^{(h)},$

in the classical sense, where $\lambda_{k+1}^{(h)}$ is the Lagrange multiplier
due to the volume penalization. Finally we define the approximative flat flow
$\\{E^{(h)}(t)\\}_{t\geq 0}$ by setting

$E^{(h)}(t)=E_{k}^{(h)}\qquad\text{for }\,t\in[kh,(k+1)h).$

We define a flat flow solution of (1.1) to be any family of sets
$\\{E(t)\\}_{t\geq 0}$ which is a cluster point of $\\{E^{(h)}(t)\\}_{t\geq
0}$, i.e.,

$E^{(h_{n})}(t)\to E(t)\quad\text{as }\,h_{n}\to 0\quad\text{in
}\,L^{1}\quad\text{for almost every }\,t>0.$

By [35, Theorem 2.2] there exists a flat flow starting from $E(0)$ such that
$P(E(t))\leq P(E(0))$ and $|E(t)|=m$ for every $t\geq 0$.

We are interested in the long time behavior of the flow. To this aim we need
two technical lemmas. The first lemma is algebraic.

######  Lemma 3.1.

Let $K\in\mathbb{N}$ and $\\{a_{k}\\}_{k\in\\{1,\ldots,K\\}}$ be a sequence of
non-negative numbers and let $\mathcal{I}\subset\\{1,\ldots,K\\}$. Assume that
there exists $c>1$ such that

$\sum_{k=i}^{K}a_{k}\leq ca_{i}$

for every $i\in\\{1,\ldots,K\\}\setminus\mathcal{I}$. Then,

$\sum_{k=i+1}^{K}a_{k}\leq\Bigl{(}1-\frac{1}{c}\Bigr{)}^{i-|\mathcal{I}|}S$

for every $i\in\\{1,\ldots,K\\}$, where $S:=\sum_{k=1}^{K}a_{k}$ and
$|\mathcal{I}|$ denotes the cardinality of $\mathcal{I}$.

###### Proof.

Set $F(i):=\sum_{k=i}^{K}a_{k}$ and note that by assumption $F(i)\leq
c(F(i)-F(i+1))$ for every $i\in\\{1,\ldots,K\\}\setminus\mathcal{I}$. Hence,
we have

$F(i+1)\leq\begin{cases}\Bigl{(}1-\frac{1}{c}\Bigr{)}F(i)&\textup{if
}i\not\in\mathcal{I},\\\ F(i)&\textup{if }i\in\mathcal{I}.\end{cases}$

By iterating the previous estimate (note that at least $K-|\mathcal{I}|$ times
the first instance must hold), we conclude. ∎

The second lemma is in the spirit of Ekeland variational principle.

######  Lemma 3.2.

Let $d\in\mathbb{N}$ and $D_{r}(x_{1}),\dots,D_{r}(x_{d})$ be disjoint disks
and denote $F=\bigcup_{i=1}^{d}D_{r}(x_{i})$. Then there is a constant $C$,
which depends only on $d$ and $r$, such that for every set of finite perimeter
$E\subset\mathbb{R}^{2}$ it holds

$P(F)\leq P(E)+C|E\Delta F|^{\frac{1}{3}}.$

###### Proof.

Let us fix a set $E$ and let $\rho\leq r/10$ be a positive number which choice
will be clear later. We begin by constructing a set $F_{\rho}$ of class
$C^{1,1}$, which contains the union of disks $F\subset F_{\rho}$, satisfies
interior and exterior ball condition with radius $\rho$ and

(3.5) $P(F)\leq P(F_{\rho})+C\sqrt{\rho},\quad\text{and }\quad|F\Delta
F_{\rho}|\leq C\rho^{\frac{3}{2}}.$

Let $x_{1},\dots,x_{d}$ be the centerpoints of the disks. If it holds
$|x_{i}-x_{j}|>2r+\rho$ for every $i\neq j$ we simply choose $F_{\rho}=F$. If
$|x_{i}-x_{j}|\leq 2r+\rho$ for some $i\neq j$ we connect the disks
$D_{r}(x_{i})$ and $D_{r}(x_{j})$ with a thin neck around the midpoint
$(x_{i}+x_{j})/2$ as follows. We first enlarge the disks by $\rho$ and
consider the union $\tilde{F}_{\rho}^{ij}:=D_{r+\rho}(x_{i})\cup
D_{r+\rho}(x_{j})$, which overlap around the midpoint $(x_{i}+x_{j})/2$ . We
then decrease the union back by $\rho$ and define

$F_{\rho}^{i,j}=\\{x\in\mathbb{R}^{2}:\textup{dist}(x,\mathbb{R}^{2}\setminus\tilde{F}_{\rho}^{ij})>\rho\\}.$

Since $|x_{i}-x_{j}|\leq 2r+\rho$, the set $F_{\rho}^{i,j}$ is connected and
contains the disks $D_{r}(x_{i})$ and $D_{r}(x_{j})$. The part of the boundary
of $F_{\rho}^{i,j}$, which is not contained in
$\bar{D}_{r}(x_{i})\cup\bar{D}_{r}(x_{j})$, consists of two arcs, see Figure
1. In particular, the set $F_{\rho}^{i,j}$ satisfies interior and exterior
ball condition with radius $\rho$. We repeat the same construction for all
disks $D_{r}(x_{i})$ and $D_{r}(x_{j})$ which are close to each other in the
sense that $|x_{i}-x_{j}|\leq 2r+\rho$, and obtain $F_{\rho}$ which satisfies
(3.5).

$x_{j}$$x_{i}$ Figure 1. If two disks are close to each other, we connect them
with a neck given by two arcs

The rest of the proof follows from standard calibration argument (see e.g. [2,
Proof of Theorem 4.3]) and we only give the sketch of the argument. We
construct a vector field $X\in C^{1,1}(\mathbb{R}^{2},\mathbb{R}^{2})$ such
that

$X(x)=\nabla d_{F_{\rho}}(x)\zeta(x)$

where $0\leq\zeta\leq 1$ is a smooth cut-off function such that $\zeta(x)=1$
for $|d_{F_{\rho}}(x)|\leq\rho/4$, $\zeta(x)=0$ for
$|d_{F_{\rho}}(x)|\geq\rho/2$ and $|\nabla\zeta|\leq C/\rho$. In particular,
it holds $|X|\leq 1$ in $\mathbb{R}^{2}$ and $X=\nu_{F_{\rho}}$ on $\partial
F_{\rho}$. Moreover, since $F_{\rho}$ satisfies interior and exterior ball
condition with radius $\rho$ it holds $|\Delta d_{F_{\rho}}(x)|\leq C/\rho$
for $|d_{F_{\rho}}(x)|\leq\rho/2$. Therefore by the divergence theorem

$P(F_{\rho})-P(E)\leq\int_{F_{\rho}\Delta
E}|\mathrm{div}(X)|\,dx\leq\frac{C}{\rho}|F_{\rho}\Delta E|.$

We combine the above inequality with (3.5) and deduce

$P(F)\leq P(E)+\frac{C}{\rho}|E\Delta F|+C\sqrt{\rho}.$

Choosing $\rho=\min\\{|E\Delta F|^{\frac{2}{3}},r/10\\}$ yields the claim. ∎

We may now give the proof of the convergence of the area-preserving mean
curvature flow.

###### Proof of Theorem 1.2.

Let $\\{E(t)\\}_{t\geq 0}$ be an area-preserving flat flow and let
$\\{E^{(h_{n})}(t)\\}_{t\geq 0}$ be an approximate flow converging to $E(t)$.
Set

$f_{n}(t)=P(E^{(h_{n})}(t))+\frac{1}{\sqrt{h_{n}}}\big{|}|E^{(h_{n})}(t)|-m\big{|}.$

By (3.3) the $f_{n}$’s are monotone non-increasing functions which are bounded
by $P(E(0))$. Therefore, by Helly’s selection theorem, up to passing to a
further subsequence (not relabeled), the functions $f_{n}$’s converge
pointwise to some non-increasing function
$f_{\infty}:[0,+\infty)\to\mathbb{R}$. Set
$F_{\infty}=\lim_{t\to+\infty}f_{\infty}(t)$. In what follows we also set

$v^{(h_{n})}_{t}=\frac{d_{E^{(h_{n})}_{k}}}{h_{n}}\,,\qquad\text{where
}\,k=\left\lfloor\frac{t}{h_{n}}\right\rfloor-1$

the approximate velocity of the approximate flow at time $t$. Moreover, $C$
will denote a positive constant, which may change from line to line and might
depend on the flat flow itself (but not on $h_{n}$ nor on the discrete step of
the minimizing movements).

We divide the proof in two cases.

Case 1: There exists $d\in\mathbb{N}\setminus\\{0\\}$ such that either
$P_{d}<F_{\infty}<P_{d+1}$ or $F_{\infty}=P_{d}$ and $f_{\infty}(t)>P_{d}$ for
every $t\in[0,+\infty)$.

In this case, there exists $\bar{t}>0$ such that, for every $T>\bar{t}$ there
exist $\bar{n}\in\mathbb{N}\setminus\\{0\\}$ such that

(3.6) $\displaystyle P_{d}\leq f_{n}(t)<P_{d+1}\qquad\text{and}\qquad
P_{d+1}-f_{n}(t)\geq\frac{P_{d+1}-F_{\infty}}{2}=:\delta_{0}$

for every $n\geq\bar{n}$ and $t\in[\bar{t},T]$. Set
$\mathcal{I}^{(h_{n})}=\left\\{i\in\big{\\{}\lfloor\frac{\bar{t}}{h_{n}}\rfloor,\ldots,\lfloor\frac{T}{h_{n}}\rfloor\big{\\}}:|E^{(h_{n})}_{i}|\neq
m\right\\}$. By [35, Cor. 3.10] there exists a constant $C_{T}>0$ such that

(3.7) $|\mathcal{I}^{(h_{n})}|\leq C_{T}$

for $n$ sufficiently large. For every $i\not\in\mathcal{I}^{(h_{n})}$ we have
by iterating (3.3) and using (3.6)

$\frac{1}{h_{n}}\sum_{k=i+1}^{\lfloor\frac{T}{h_{n}}\rfloor}\mathcal{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\leq
P(E_{i}^{(h_{n})})-P(E^{(h_{n})}_{\lfloor\frac{T}{h_{n}}\rfloor})\leq
P(E_{i}^{(h_{n})})-P_{d}.$

Then by (1.3) and by the Euler-Lagrange equation (3.4)

(3.8)
$\begin{split}\frac{1}{h_{n}}\sum_{k=i+1}^{\lfloor\frac{T}{h_{n}}\rfloor}\mathcal{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})&\leq
P(E_{i}^{(h_{n})})-P_{d}\leq
C_{0}\|\kappa_{E_{i}^{(h_{n})}}-\overline{\kappa}_{E_{i}^{(h_{n})}}\|_{L^{2}(\partial
E_{i}^{(h_{n})})}^{2}\\\ &\leq
C_{0}\|\kappa_{E_{i}^{(h_{n})}}-\lambda_{i}^{(h_{n})}\|_{L^{2}(\partial
E_{i}^{(h_{n})})}^{2}=\frac{C_{0}}{h_{n}^{2}}\int_{\partial
E_{i}^{(h_{n})}}d_{E_{i-1}^{(h_{n})}}^{2}\,d\mathcal{H}^{1}.\end{split}$

In [35] it is proven (formula after (3.25)) that

(3.9) $\int_{\partial
E_{i}^{(h_{n})}}d_{E_{i-1}^{(h_{n})}}^{2}\,d\mathcal{H}^{1}\leq
C\mathcal{D}(E^{(h_{n})}_{i},E^{(h_{n})}_{i-1}).$

Therefore from (3.8) we conclude

$\sum_{k=i+1}^{\lfloor\frac{T}{h_{n}}\rfloor}\mathcal{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\leq\frac{C_{0}^{\prime}}{h_{n}}\mathcal{D}(E^{(h_{n})}_{i},E^{(h_{n})}_{i-1}).$

Setting
$a^{(h_{n})}_{k}=h_{n}^{-1}\mathcal{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})$ we
have that for every
$i\in\big{\\{}\lfloor\frac{\bar{t}}{h_{n}}\rfloor,\ldots,\lfloor\frac{T}{h_{n}}\rfloor\big{\\}}\setminus\mathcal{I}^{(h_{n})}$
it holds

$\sum_{k=i}^{\lfloor\frac{T}{h_{n}}\rfloor}a_{k}^{(h_{n})}\leq\frac{C^{\prime}_{0}+h_{n}}{h_{n}}a_{i}^{(h_{n})}\leq\frac{2C^{\prime}_{0}}{h_{n}}a_{i}^{(h_{n})}.$

Moreover it holds by (3.3) $\sum_{k=1}^{\infty}a_{k}\leq P(E(0))\leq M$. By
Lemma 3.1 we infer that

$\sum_{k=i+1}^{\lfloor\frac{T}{h_{n}}\rfloor}a_{k}^{(h_{n})}\leq
M\left(1-\frac{h_{n}}{2C^{\prime}_{0}}\right)^{i-C_{T}-\frac{\bar{t}}{h_{n}}}\qquad\text{for
all
}\,i=\lfloor\frac{\bar{t}}{h_{n}}\rfloor,\ldots,\lfloor\frac{T}{h_{n}}\rfloor.$

In other words for every $t\in[\bar{t},T]$ we have

(3.10)
$\sum_{k=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{T}{h_{n}}\rfloor}h_{n}^{-1}\mathcal{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\leq
M\left(1-\frac{h_{n}}{2C^{\prime}_{0}}\right)^{\lfloor\frac{t}{h_{n}}\rfloor-
C_{T}-\frac{\bar{t}}{h_{n}}}\leq Ce^{-\frac{t}{2C^{\prime}_{0}}}$

for $h_{n}\leq h_{0}(T)$.

By [35, Proposition 3.4] it holds

$|E^{(h_{n})}_{i}\Delta E^{(h_{n})}_{i-1}|\leq C\ell
P(E^{(h_{n})}_{i})+\frac{C}{\ell}\int_{E^{(h_{n})}_{i}\Delta
E^{(h_{n})}_{i-1}}|d_{E^{(h_{n})}_{i-1}}|\,dx$

for all $\ell\leq\frac{1}{C}\sqrt{h_{n}}$. Therefore, by the inequality above
and by (3.10) we infer that for every $\bar{t}\leq t<s\leq T$ we have

$\begin{split}|E^{(h_{n})}(t)\Delta
E^{(h_{n})}(s)|&=\sum_{i=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{s}{h_{n}}\rfloor}|E^{(h_{n})}_{i}\Delta
E^{(h_{n})}_{i-1}|\\\ &\leq
C\sum_{i=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{s}{h_{n}}\rfloor}\left(\ell
P(E^{(h_{n})}_{i})+\frac{1}{\ell}\int_{E^{(h_{n})}_{i}\Delta
E^{(h_{n})}_{i-1}}|d_{E^{(h_{n})}_{i-1}}|\,dx\right)\\\ &\leq
CP(E(0))\ell\frac{s-t}{h_{n}}+\frac{C}{\ell}\sum_{i=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{s}{h_{n}}\rfloor}\mathcal{D}(E^{(h_{n})}_{i},E^{(h_{n})}_{i-1})\\\
&\leq
CM\ell\frac{s-t}{h_{n}}M+\frac{C\,h_{n}}{\ell}e^{-\frac{t}{2C^{\prime}_{0}}},\end{split}$

for all $\ell\leq\frac{1}{C}\sqrt{h_{n}}$ and $h_{n}\leq h_{0}$. In
particular, choosing $\ell=\frac{h_{n}}{e^{\alpha t}}$ with
$\alpha=\frac{1}{4C^{\prime}_{0}}$ and $s\leq t+1$, we have

$|E^{(h_{n})}(t)\Delta E^{(h_{n})}(s)|\leq CMe^{-\frac{t}{4C^{\prime}_{0}}}.$

Passing to the limit as $h_{n}\to 0$, we get

(3.11) $|E(t)\Delta E(s)|\leq CMe^{-\frac{t}{4C^{\prime}_{0}}}\qquad\text{for
all}\,\,\bar{t}\leq t\leq s\leq t+1.$

Hence, we deduce that $E(t)$ converges exponentially fast to a set of finite
perimeter $E_{\infty}$ in $L^{1}$ and $|E_{\infty}|=m$.

We now show that the limiting set $E_{\infty}$ is the union of disjoint open
disks with the same radius. Denote by $S_{\infty}$ the countable set of
discontinuity points of $f_{\infty}$ and note that for any
$t\in(0,+\infty)\setminus S_{\infty}$ and any sequence $t_{n}\to t$ we have
$f_{n}(t_{n})\to f_{\infty}(t)$.

Fix $t\geq\bar{t}$, $0<\alpha<\frac{1}{2C^{\prime}_{0}}$, and an open set
$A(t)$ such that $S_{\infty}\cap[t,t+e^{-\alpha t}]\subset
A(t)\subset[t,t+e^{-\alpha t}]$ and $|A(t)|\leq e^{-\alpha^{\prime}t}$, with
$\alpha^{\prime}>\alpha$. By (3.9) and (3.10) we have

(3.12) $\begin{split}&\int_{[t,t+e^{-\alpha t}]\setminus
A(t)}\Big{(}\int_{\partial
E^{(h_{n})}(s)}(v^{(h_{n})}_{s})^{2}d\mathcal{H}^{1}\Big{)}ds\leq\frac{1}{h_{n}}\sum_{i=\lfloor\frac{t}{h_{n}}\rfloor}^{\lfloor\frac{t+e^{-\alpha
t}}{h_{n}}\rfloor}\int_{\partial
E_{i}^{(h_{n})}}d_{E_{i-1}^{(h_{n})}}^{2}\,d\mathcal{H}^{1}\\\ &\leq
C\sum_{i=\lfloor\frac{t}{h_{n}}\rfloor}^{\lfloor\frac{t+e^{-\alpha
t}}{h_{n}}\rfloor}\frac{1}{h_{n}}\mathcal{D}(E^{(h_{n})}_{i},E^{(h_{n})}_{i-1})\leq
Ce^{-\frac{t}{2C^{\prime}_{0}}}\,,\end{split}$

for $n$ sufficiently large. By possibly increasing $\bar{t}$ we have
$|[t,t+e^{-\alpha t}]\setminus A(t)|>\frac{1}{2}e^{-\alpha t}$ for
$t\geq\bar{t}$. Moreover by (3.7) it holds

$|\\{s\in[t,t+e^{-\alpha t}]:\,|E^{(h_{n})}(s)|\neq m\\}|\to 0\,,\quad\text{as
}\,n\to\infty.$

Then by (3.12) and by the mean value theorem there exists
$s_{n}\in[t,t+e^{-\alpha t}]\setminus A(t)$ such that

(3.13)
$\|\kappa_{E^{(h_{n})}(s_{n})}-\overline{\kappa}_{E^{(h_{n})}(s_{n})}\|_{L^{2}(\partial
E^{(h_{n})}(s_{n}))}^{2}\leq\int_{\partial
E^{(h_{n})}(s_{n})}(v^{(h_{n})}_{s_{n}})^{2}d\mathcal{H}^{1}\leq
Ce^{-\big{(}\frac{1}{2C^{\prime}_{0}}-\alpha\big{)}t}\,,$

$|E^{(h_{n})}(s_{n})|=m$, and thus, in particular,
$f_{n}(s_{n})=P(E^{(h_{n})}(s_{n}))$. From Proposition 2.1 and (3.13), we
infer that, for $t\geq\tilde{t}$, where $\tilde{t}$ is sufficiently large,
$E^{(h_{n})}(s_{n})$ is diffeomorphic to a union of $d$ disjoint disks and

(3.14) $|P(E^{(h_{n})}(s_{n}))-P_{d}|\leq
Ce^{-\big{(}\frac{1}{2C^{\prime}_{0}}-\alpha\big{)}t}.$

In particular, passing to the limit in $h_{n}\to 0$ (up to a further not
relabelled subsequence, if needed), there exists $s_{t}\in[t,t+e^{-\alpha
t}]\setminus A(t)$ such that $s_{n}\to s_{t}$ and thus $E^{(h_{n})}(s_{n})\to
E(s_{t})$ in $L^{1}$ and $P(E^{(h_{n})}(s_{n}))=f_{n}(s_{n})\to
f_{\infty}(s_{t})$. In fact, by the uniform $C^{1,\frac{1}{2}}$-bounds
provided by (3.13) and Proposition 2.1 we deduce that
$P(E^{(h_{n})}(s_{n}))\to P(E(s_{t}))$ and thus
$f_{\infty}(s_{t})=P(E(s_{t}))$, and that $E(s_{t})$ is the union of $d$
nearly spherical sets parametrized over $d$ disjoint open disks
$D_{r}(x_{i}(t))$, $i=1,\dots d$ of volume $m/d$, with
$C^{1,\frac{1}{2}}$-norm of the parametrizations (exponentially) small. In
particular, setting $F(t):=\cup_{i=1}^{d}D_{r}(x_{i}(t))$, we have that
$\sup_{x\in E(s_{t})\Delta F(t)}\textup{dist}(x,\partial F(t))$ decays
exponentially to zero as $t\to+\infty$, $E_{\infty}$ is a union of $d$
disjoint open disks of volume $m/d$, and $F(t)\to E_{\infty}$ in the Hausdorff
sense exponentially fast.

Summarizing, and recalling also the first inequality in (3.6) and (3.14), we
have shown that for every $t$ sufficiently large, there exists
$s_{t}\in[t,t+e^{-\alpha t}]$ such that $E(s_{t})$ is the union of $d$
disjoint nearly spherical sets parametrized over the disjoint open disks of
$E_{\infty}$ and

(3.15) $P_{d}\leq f_{\infty}(s_{t})=P(E(s_{t}))\leq
P_{d}+Ce^{-(1/C-\alpha)t}\,,\qquad\sup_{x\in E(s_{t})\Delta
E_{\infty}}\textup{dist}(x,\partial E_{\infty})\leq Ce^{-\frac{t}{C}}\,,$

for a suitable constant $C>1$.

From the first inequality in (3.15) and by the monotonicity of $f_{\infty}$ we
obtain for all $s$ sufficiently large that by choosing $t$ such that
$s=t+e^{-\alpha t}$ it holds

$P(E(s))\leq f_{\infty}(s)\leq f_{\infty}(s_{t})\leq
P_{d}+Ce^{-(1/C-\alpha)(s-e^{-\alpha t})}\leq
P_{d}+Ce^{-\frac{(1/C-\alpha)s}{2}}\,.$

On the other hand, by Lemma 3.2 and (3.11) we obtain

$P_{d}\leq P(E(t))+C|E(t)\Delta E_{\infty}|^{\frac{1}{3}}\leq
P(E(t))+C^{\prime}e^{-\frac{t}{12C_{0}^{\prime}}}\,.$

Hence, we have the exponential convergence of the perimeters in (1.4).

The first part of the inequality in (1.4) follows from the second inequality
in (3.15) and from [25, Lemma 4.3].

Case 2: There exist $d\in\mathbb{N}\setminus\\{0\\}$ and $\bar{t}>0$ such that
$F_{\infty}=P_{d}=f_{\infty}(t)$ for every $t\geq\bar{t}$.

In this case, using the monotonicity of the functions $f_{n}$’s, we deduce
that for every $T>\bar{t}$ the functions $f_{n}$ converge uniformly to
$f_{\infty}\equiv F_{\infty}$ in $[\bar{t},T]$. In particular, using that

$\frac{1}{h_{n}}\mathcal{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\leq
f_{n}((k-1)h_{n})-f_{n}(kh_{n}),$

we deduce that for every $t\in[\bar{t}+h_{n},T]$ we have

$\sum_{k=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{T}{h_{n}}\rfloor}h_{n}^{-1}\mathcal{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\leq
f_{n}\Big{(}\lfloor\frac{t}{h_{n}}\rfloor
h_{n}\Big{)}-f_{n}\Big{(}\lfloor\frac{T}{h_{n}}\rfloor h_{n}\Big{)}=:b_{n}\to
F_{\infty}-F_{\infty}=0\qquad\textup{as }h_{n}\to 0.$

Arguing as above, for every $\bar{t}+h_{n}\leq t<s\leq T$, we get

$\displaystyle|E^{(h_{n})}(t)\Delta E^{(h_{n})}(s)|\leq
C\ell\frac{s-t}{h_{n}}P(E(0))+\frac{C}{\ell}\sum_{i=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{s}{h_{n}}\rfloor}\mathcal{D}(E^{(h_{n})}_{i},E^{(h_{n})}_{i-1}),$

for all $\ell\leq\frac{1}{C}\sqrt{h_{n}}$ and, choosing
$\ell=\sqrt{b_{n}}h_{n}$, we conclude that

$\displaystyle|E^{(h_{n})}(t)\Delta E^{(h_{n})}(s)|\leq
C\sqrt{b_{n}}(s-t)P(E(0))+C\sqrt{b_{n}}\to 0,$

that is $E(t)=E(s)$ for every $\bar{t}<t<s<T$.

The final part of the proof consists in showing that the limiting set
$E_{\infty}$ is the union of disjoint open disks with the same radius. We have

$\int_{t}^{T}\int_{\partial
E^{(h_{n})}(t)}(v_{t}^{(h_{n})})^{2}d\mathcal{H}^{1}=\frac{1}{h_{n}}\sum_{i=\lfloor\frac{t}{h_{n}}\rfloor}^{\lfloor\frac{T}{h_{n}}\rfloor}\int_{\partial
E_{i}^{(h_{n})}}d_{E_{i-1}^{(h_{n})}}^{2}=\sum_{i=\lfloor\frac{t}{h_{n}}\rfloor}^{\lfloor\frac{T}{h_{n}}\rfloor}\frac{1}{h_{n}}\mathcal{D}(E^{(h_{n})}_{i},E^{(h_{n})}_{i-1})=o(1).$

By the mean value theorem, for ever $T$ sufficiently large there exists
$t_{n}\in[T,T+1]$ such that

$\|\kappa_{E^{(h_{n})}(t_{n})}-\overline{\kappa}_{E^{(h_{n})}(t_{n})}\|_{L^{2}(\partial
E^{(h_{n})}(t_{n}))}^{2}\leq\int_{\partial
E^{(h_{n})}(t_{n})}(v_{t_{n}}^{(h_{n})})^{2}d\mathcal{H}^{1}=o(1).$

As before, by Proposition 2.1 the sets $E^{(h_{n})}(t_{n})$ are nearly
spherical and converges to union of $d$ disjoint open balls. From here the
conclusion follows. ∎

##  4\. The asymptotics of the 2D Mullins-Sekerka flow

Let us first construct a flat flow solution for the Mullins-Sekerka flow in
the 2-dimensional flat torus. The construction in the case of bounded domain
is due to Luckhaus and Stürzenhecker [28] and the same construction can be
applied to the periodic setting with obvious changes. We denote the perimeter
of a set $E$ in the flat torus $\mathbb{T}^{2}$ by $P_{\mathbb{T}^{2}}(E)$ and
recall that it is defined as

$P_{\mathbb{T}^{2}}(E):=\sup\Big{\\{}\int_{E}\mathrm{div}X\,dx:X\in
C^{1}(\mathbb{T}^{2},\mathbb{R}^{2}),\,\|X\|_{L^{\infty}}\leq 1\Big{\\}}.$

Here $X\in C^{1}(\mathbb{T}^{2},\mathbb{R}^{2})$ means that the
$\mathbb{Z}^{2}$-periodic extension of $X$ to $\mathbb{R}^{2}$ is continuously
differentiable. For a given set of finite perimeter $E\subset\mathbb{T}^{2}$,
with $|E|=m$, we consider the minimization problem

(4.1)
$\min\Big{\\{}P_{\mathbb{T}^{2}}(F)+\frac{h}{2}\int_{\mathbb{T}^{2}}|\nabla
U_{F,E}|^{2}\,dx:\quad\text{with }\,|F|=|E|=m\Big{\\}},$

where the function $U_{F,E}\in H^{1}(\mathbb{T}^{2})$ is the solution of

(4.2) $-\Delta U_{F,E}=\frac{1}{h}\left(\chi_{F}-\chi_{E}\right)$

with zero average. As proven in [28, 39] there exists a minimizer for (4.1),
but it might not be unique. Concerning the regularity of the minimizers we may
argue as in [2, Theorem 2.8.] (see also [33, Proposition 2.2]) to deduce that
the minimizing set $F$ is $C^{3,\alpha}$-regular. Let us briefly sketch the
argument. First, we may replace the volume constraint in (4.1) by volume
penalization as in [2, 20] and conclude that the minimizer is a
$\Lambda$-minimizer of the perimeter. This implies that the minimizer is
$C^{1,\alpha}$-regular and satisfies the associated Euler-Lagrange equation

$U_{F,E}=-\kappa_{F}+\lambda\qquad\text{on }\,\partial F$

in a weak sense, where $\lambda$ is the Lagrange multiplier. Since $U_{F,E}$
is the solution of (4.2), by standard elliptic regularity it holds $U_{F,E}\in
C^{1,\alpha}(\mathbb{T}^{2})$. Then by the Euler-Lagrange equation we deduce
that $F$ is in fact $C^{3,\alpha}$-regular and the Euler-Lagrange equation
holds in the classical sense.

Let us denote

(4.3) $\mathfrak{D}(F,E):=\int_{\mathbb{T}^{2}}|\nabla U_{F,E}|^{2}\,dx$

where $U_{F,E}$ is defined in (4.2). We define the $H^{-1}$-norm by duality as

$\|f\|_{H^{-1}(\mathbb{T}^{2})}:=\sup\Big{\\{}\int_{\mathbb{T}^{2}}\varphi\,f\,dx:\|\nabla\varphi\|_{L^{2}(\mathbb{T}^{2})}\leq
1\Big{\\}}.$

Then by integrating (4.2) by parts yields

(4.4) $\|\chi_{F}-\chi_{E}\|_{H^{-1}(\mathbb{T}^{2})}^{2}\leq h^{2}\,\|\nabla
U_{F,E}\|_{L^{2}(\mathbb{T}^{2})}^{2}=h^{2}\,\mathfrak{D}(F,E).$

We fix the time step $h>0$ and our initial set $E(0)\subset\mathbb{T}^{2}$ and
let $E_{1}^{(h)}$ be a minimizer of (4.1) with $E(0)=E$. We construct the
discrete-in-time evolution $(E_{k}^{(h)})_{k\in\mathbb{N}}$ as before by
induction such that, assuming that $E_{k}^{(h)}$ is defined, we set
$E_{k+1}^{(h)}$ to be a minimizer of (4.1) with $E=E_{k}^{(h)}$ and denote the
associated potential for short by $U_{k+1}^{(h)}$, which is the solution of

(4.5) $-\Delta
U_{k+1}^{(h)}=\frac{1}{h}\left(\chi_{E_{k+1}^{(h)}}-\chi_{E_{k}^{(h)}}\right)$

with zero average. The Euler-Lagrange equation now reads as

(4.6) $U_{k+1}^{(h)}=-\kappa_{E_{k+1}^{(h)}}+\lambda_{k+1}^{(h)}\qquad\text{on
}\,\partial E_{k+1}^{(h)}.$

By a direct energy comparison (formula (3.6) in [39]) we obtain

(4.7)
$P_{\mathbb{T}^{2}}(E_{k+1}^{(h)})+\frac{h}{2}\mathfrak{D}(E_{k+1}^{(h)},E_{k}^{(h)})\leq
P_{\mathbb{T}^{2}}(E_{k}^{(h)}),$

where $\mathfrak{D}(E_{k+1}^{(h)},E_{k}^{(h)})$ is defined in (4.3).

As before we define the approximative flat flow $\\{E^{(h)}(t)\\}_{t\geq 0}$
by setting

$E^{(h)}(t)=E_{k}^{(h)}\qquad\text{for }\,t\in[kh,(k+1)h)$

and we call a _flat flow solution_ of (1.2) any cluster point
$\\{E(t)\\}_{t\geq 0}$ of $\\{E^{(h)}(t)\\}_{t\geq 0}$, as $h\to 0$; i.e.,

$E^{(h_{n})}(t)\to E(t)\quad\text{ in $L^{1}$ for almost every }\,t>0\text{
and for some }h_{n}\to 0.$

Arguing exactly as in [39, Proposition 3.1] we may conclude that there exists
a flat flow starting from $E(0)$ such that $P_{\mathbb{T}^{2}}(E(t))\leq
P_{\mathbb{T}^{2}}(E(0))$, $|E(t)|=|E(0)|$ for every $t\geq 0$ and
$\\{E(t)\\}_{t\geq 0}$ satisfies the equation (1.2) in a weak sense.

To proceed, we need the analogue of Proposition 2.1 for the Mullins-Sekerka
flow. To this aim we first prove the following lemma, which is similar to [41,
Lemma 2.1].

######  Lemma 4.1.

Let $E\subset\mathbb{T}^{2}$ be a set of class $C^{3}$, with
$|E|\leq\frac{1}{2}$ and $P_{\mathbb{T}^{2}}(E)<2$, and let $u_{E}\in
C^{1}(\mathbb{T}^{2})$ be a function with zero average such that $\|\nabla
u_{E}\|_{L^{2}(\mathbb{T}^{2})}\leq M$ and

(4.8) $\kappa_{E}=-u_{E}+\lambda\quad\text{on }\,\partial E\quad\text{ for
some }\lambda\in\mathbb{R}.$

Then it holds

(4.9) $\sup_{x\in\mathbb{T}^{2},\rho>0}\frac{\mathcal{H}^{1}(\partial E\cap
D_{\rho}(x))}{\rho}\leq K,$

where the constant $K>0$ depends only on $|E|$ and $M$.

###### Proof.

We note that by (4.8) for every $X\in C^{1}(\mathbb{T}^{2};\mathbb{R}^{2})$ it
holds

(4.10) $\int_{\partial
E}\mathrm{div}_{\tau}X\,d\mathcal{H}^{1}=\int_{E}\mathrm{div}\big{(}(-u_{E}+\lambda)X\big{)}\,dx.$

Therefore the statement follows from [41, Lemma 2.1] once we bound the
Lagrange multiplier $\lambda\in\mathbb{R}$. To this aim, and for future
purpose, we show that there is $\delta>0$ such that every component $E_{i}$ of
$E$ is contained in a cube $Q_{1-\delta}(x_{i}):=(1-\delta)^{2}+\\{x_{i}\\}$
for some $x_{i}$.

Let us first show that every component $\Gamma_{i}$ of the boundary $\partial
E$ divides the torus $\mathbb{T}^{2}$ in two components and thus it is the
boundary of a set. Indeed, if this is not the case then necessarily
$\mathcal{H}^{1}(\Gamma_{i})\geq 1$. Since $\Gamma_{i}$ is not a boundary of a
set then $\partial E$ must have another component, say $\Gamma_{j}$, such that
$\mathcal{H}^{1}(\Gamma_{j})\geq 1$. But this implies
$P_{\mathbb{T}^{2}}(E)=\mathcal{H}^{1}(\partial E)\geq 2$, which contradicts
the assumption $P_{\mathbb{T}^{2}}(E)<2$.

Let us next show that $\Gamma_{i}$ is contained in a cube
$Q_{1-\delta}(x_{i})$ for some $x_{i}$. Let
$\pi_{1}:\mathbb{T}^{2}\to\mathbb{T}$ be the projection onto the $x_{1}$-axis
i.e., $\pi_{1}(x_{1},x_{2})=x_{1}$. Then we deduce from
$\mathcal{H}^{1}(\Gamma_{i})<2$ and from the fact that $\Gamma_{i}$ is the
boundary of a set that $\mathcal{H}^{1}(\pi_{1}(\Gamma_{i}))<1$. Similarly it
holds $\mathcal{H}^{1}(\pi_{2}(\Gamma_{i}))<1$, where $\pi_{2}$ is the
projection onto the $x_{2}$-axis. This implies that $\Gamma_{i}\subset
Q_{1-\delta}(x_{i})$ for some $\delta>0$ and $x_{i}$. Let us from now on
denote the set enclosed by $\Gamma_{i}$ which is inside the cube
$Q_{1-\delta}(x_{i})$ by $F_{i}$.

Let $\Gamma_{1},\dots,\Gamma_{n}$ be the components of the boundary $\partial
E$ which enclose the sets $F_{1},\dots,F_{n}$. Let us show that

(4.11) $E\subset\bigcup_{i=1}^{n}F_{i}.$

Since $F_{i}\subset Q_{1-\delta}(x_{i})$ we have by the Isoperimetric
Inequality $2\sqrt{\pi|F_{i}|}\leq\mathcal{H}^{1}(\Gamma_{i})$. Therefore by
the assumption on the perimeter, $P_{\mathbb{T}^{2}}(E)<2$, we have

$4\pi\big{|}\bigcup_{i=1}^{n}F_{i}\big{|}\leq
4\pi\sum_{i=1}^{n}|F_{i}|\leq\sum_{i=1}^{n}\mathcal{H}^{1}(\Gamma_{i})^{2}\leq\Big{(}\sum_{i=1}^{n}\mathcal{H}^{1}(\Gamma_{i})\Big{)}^{2}\leq
P_{\mathbb{T}^{2}}(E)^{2}<4.$

Therefore $\big{|}\bigcup_{i=1}^{n}F_{i}\big{|}<\frac{1}{\pi}<\frac{1}{3}$.
Since, $|E|\leq\frac{1}{2}$ then necessarily $E\subset\bigcup_{i=1}^{n}F_{i}$.

We conclude from (4.11) that a component $E_{j}$ of $E$ is contained in
$F_{i}$ for some $i$. Therefore since $F_{i}\subset Q_{1-\delta}(x_{i})$, then
also $E_{j}\subset Q_{1-\delta}(x_{i})$.

We may finally bound the Lagrange multiplier in (4.10) by a standard argument.
Indeed, let $E_{j}$ be a component of $E$. Since $E_{j}\subset
Q_{1-\delta}(x_{j})$ we may define $X\in
C_{0}^{1}\big{(}Q_{1-\delta/3}(x_{j})\big{)}$ such that $X(x)=x$ in $E_{j}$
and $X(x)=0$ in $E\setminus E_{j}$. We apply (4.10) with this choice of $X$
and have

$\begin{split}P_{\mathbb{T}^{2}}(E_{j})=\int_{\partial
E_{j}}\mathrm{div}_{\tau}x\,d\mathcal{H}^{1}&=\int_{E_{j}}\mathrm{div}\big{(}(-u_{E}+\lambda)x\big{)}\,dx\\\
&=-\int_{E_{j}}\mathrm{div}(u_{E}\,x)\,dx+2\lambda|E_{j}|.\end{split}$

We have $\big{|}\int_{E_{j}}\mathrm{div}(u_{E}\,x)\,dx\big{|}\leq
C\|u_{E}\|_{H^{1}(E_{j})}$. By repeating the argument for every component we
obtain by the Poincaré inequality

$|\lambda||E|\leq P_{\mathbb{T}^{2}}(E)+C\|u_{E}\|_{H^{1}(E)}\leq
P_{\mathbb{T}^{2}}(E)+C\|u_{E}\|_{H^{1}(\mathbb{T}^{2})}\leq
P_{\mathbb{T}^{2}}(E)+C\|\nabla u_{E}\|_{L^{2}(\mathbb{T}^{2})}.$

This yields the required bound on the Lagrange multiplier. ∎

We also recall the result by Meyers-Ziemer [32, Theorem 4.7] which implies
that if $E$ satisfies (4.9) then for every $\varphi\in C^{1}(\mathbb{T}^{2})$
it holds

(4.12) $\big{|}\int_{\partial E}\varphi\,d\mathcal{H}^{1}\big{|}\leq
C\|\varphi\|_{W^{1,1}(\mathbb{T}^{2})}\,,$

with $C$ depending on $K$ (and thus on $|E|$ and $M$).

We are now ready to state and prove the analogue of Proposition 2.1, which is
suited for the Mullins-Sekerka flow.

######  Proposition 4.2.

Let $E\subset\mathbb{T}^{2}$ be a set of class $C^{3}$, with
$|E|=m\leq\frac{1}{2}$ and $P_{\mathbb{T}^{2}}(E)<2$, and let $u_{E}\in
C^{1}(\mathbb{T}^{2})$ be a function with zero average such that

$\kappa_{E}=-u_{E}+\lambda\quad\text{on }\,\partial E$

for some $\lambda\in\mathbb{R}$. Then, there exist
$\varepsilon_{0}=\varepsilon_{0}(m)\in(0,1)$ and $C_{0}=C_{0}(m)>1$ such that
if

$\|\nabla u_{E}\|_{L^{2}(\mathbb{T}^{2})}\leq\varepsilon_{0}$

then $E$ is diffeomorphic to a union of $d$ disjoint disks $D_{1}$, …, $D_{d}$
with equal areas $m/{d}$ and $\textup{dist}(D_{i},D_{j})>0$ for $i\neq j$.
Moreover,

$|P_{\mathbb{T}^{2}}(E)-P_{d}|\leq C_{0}\|\nabla
u_{E}\|_{L^{2}(\mathbb{T}^{2})}^{2}$

and for $\varepsilon_{0}$ sufficiently small the boundary of every connected
component of the set $E$ can be parametrized as a normal graph over a one of
the disc $D_{i}$ with $C^{1,\frac{1}{2}}$ norm of the parametrization
vanishing as $\varepsilon_{0}\to 0$.

We note that we need the assumption $P_{\mathbb{T}^{2}}(E)<2$ to exclude the
case when $E$ is a strip or a union of strips.

###### Proof.

We recall that the argument in the proof of Lemma 4.1 implies that every
component $E_{i}$ of $E$ is contained in a cube $Q_{1-\delta}(x_{i})$ for some
$x_{i}$. By Lemma 4.1 we can apply (4.12) with $\varphi=u_{E}^{2}$ and obtain

$\int_{\partial E}u_{E}^{2}\,d\mathcal{H}^{1}\leq
C\|u_{E}^{2}\|_{W^{1,1}(\mathbb{T}^{2})}\leq
C\|u_{E}\|_{H^{1}(\mathbb{T}^{2})}^{2}\leq C\|\nabla
u_{E}\|_{L^{2}(\mathbb{T}^{2})}^{2},$

where the last inequality follows from Poincaré inequality. Since $u_{E}$
satisfies (4.8) we deduce by the assumption $\|\nabla
u_{E}\|_{L^{2}(\mathbb{T}^{2})}\leq\varepsilon_{0}$ that

$\int_{\partial
E}|\kappa_{E}-\overline{\kappa}_{E}|^{2}\,d\mathcal{H}^{1}\leq\int_{\partial
E}|\kappa_{E}-\lambda|^{2}\,d\mathcal{H}^{1}=\int_{\partial
E}u_{E}^{2}\,d\mathcal{H}^{1}\leq C\|\nabla
u_{E}\|_{L^{2}(\mathbb{T}^{2})}^{2}\leq C\varepsilon_{0}^{2}.$

Hence, the claim follows from Proposition 2.1. ∎

Proposition 4.2 immediately implies the following corollary.

######  Corollary 4.3.

Let $E\subset\mathbb{T}^{2}$ be a set of class $C^{3}$, with
$|E|=m\leq\frac{1}{2}$ and $P_{\mathbb{T}^{2}}(E)<2$ and let $u_{E}\in
C^{1}(\mathbb{T}^{2})$ be a function with zero average such that
$\kappa_{E}=-u_{E}+\lambda$ on $\partial E$ for some $\lambda\in\mathbb{R}$.
If $\delta_{0}>0$ and $d\in\mathbb{N}$, are such that $P_{d}\leq P(E)\leq
P_{d+1}-\delta_{0}$, then it holds

$P(E)-P_{d}\leq C_{0}\|\nabla u_{E}\|_{L^{2}(\mathbb{T}^{2})}^{2}$

for $C_{0}=C_{0}(m,\delta_{0})$.

We also need the following lemma which is essentially a restatement of [28,
Lemma 3.1]. The proof can also be found in [8, Lemma 2], but we recall it for
the reader’s convenience.

######  Lemma 4.4.

Let $\varphi\in\text{BV}(\mathbb{T}^{2})$. There is a constants $C>1$ and
$\rho_{0}>0$ such for all $\rho\leq\rho_{0}$ it holds

$\|\varphi\|_{L^{1}(\mathbb{T}^{2})}\leq
C\rho\,\|\varphi\|_{BV(\mathbb{T}^{2})}+C\rho^{-1}\|\varphi\|_{H^{-1}(\mathbb{T}^{2})}.$

###### Proof.

Let us fix $\rho>0$ and let $\eta_{\rho}(x)=\rho^{-2}\eta(\tfrac{x}{\rho})$ be
the standard mollifier. Then we write

$\|\varphi\|_{L^{1}(\mathbb{T}^{2})}\leq\int_{\mathbb{T}^{2}}|\varphi-\varphi*\eta_{\rho}|\,dx+\int_{\mathbb{T}^{2}}|\varphi*\eta_{\rho}|\,dx.$

Let us first bound the second term on the RHS. Since
$\|\eta_{\rho}\|_{H^{1}(\mathbb{T}^{2})}\leq C/\rho$ we obtain by the
definition of the $H^{-1}$-norm

$\begin{split}\int_{\mathbb{T}^{2}}|\varphi*\eta_{\rho}|\,dx=\int_{\mathbb{T}^{2}}\big{|}\int_{\mathbb{T}^{2}}\varphi(y)\eta_{\rho}(y-x)\,dy\big{|}dx\leq\|\varphi\|_{H^{-1}(\mathbb{T}^{2})}\|\eta_{\rho}\|_{H^{1}(\mathbb{T}^{2})}\leq
C\rho^{-1}\,\|\varphi\|_{H^{-1}(\mathbb{T}^{2})}.\end{split}$

We bound the first term by change of variables

$\begin{split}\int_{\mathbb{T}^{2}}|\varphi-\varphi*\eta_{\rho}|\,dx&=\int_{\mathbb{T}^{2}}\big{|}\int_{\mathbb{T}^{2}}\big{(}\varphi(x)-\varphi(x+\rho
y)\big{)}\eta(y)\,dy\big{|}\,dx\\\
&=\int_{\mathbb{T}^{2}}\big{|}\int_{\mathbb{T}^{2}}\int_{0}^{\rho}-\frac{\partial}{\partial\tau}\varphi(x+\tau
y)\eta(y)\,d\tau dy\big{|}\,dx\\\ &\leq
C\rho\,\|\varphi\|_{BV(\mathbb{T}^{2})}\,.\end{split}$

∎

We are ready to prove the convergence of the Mullisk-Sekerka flow in the flat
torus $\mathbb{T}^{2}$.

###### Proof of Theorem 1.3.

The proof is similar to the proof of Theorem 1.2 but we highlight the main
differences. Let $\\{E(t)\\}_{t\geq 0}$ be a flat flow for the Mullins-Sekerka
and let $\\{E^{(h_{n})}(t)\\}_{t\geq 0}$ be an approximate flow converging to
$E(t)$. Since $\\{\mathbb{T}^{2}\setminus E(t)\\}_{t\geq 0}$ is a flat flow
starting from $\mathbb{T}^{2}\setminus E(0)$, by replacing $E(0)$ with its
complement in $\mathbb{T}^{2}$ if needed, we may assume without loss of
generality that $|E(0)|\leq\frac{1}{2}$. We will show that in this case the
limiting set is a finite union of disjoint open discs with equal radii.

Arguing as before we deduce that by (4.7) the functions

$f_{n}(t)=P_{\mathbb{T}^{2}}(E^{(h_{n})}(t))$

are monotone non-increasing with $f_{n}(t)<2$ and (possibly up to a further
unrelabelled subsequence) converge pointwise to a non-increasing function
$f_{\infty}:[0,+\infty)\to\mathbb{R}$. Set
$F_{\infty}=\lim_{t\to+\infty}f_{\infty}(t)$. Again we divide the proof in two
cases.

Case 1: There exists $d\in\mathbb{N}\setminus\\{0\\}$ such that either
$P_{d}<F_{\infty}<P_{d+1}$, or $F_{\infty}=P_{d}$ and $f_{\infty}(t)>P_{d}$
for every $t\in[0,+\infty)$. In this case, there exists $\bar{t}\geq 1$ such
that, for every $T>\bar{t}$ there exist $\bar{n}\in\mathbb{N}\setminus\\{0\\}$
such that

(4.13) $\displaystyle P_{d}\leq f_{n}(t)<P_{d+1}\qquad\text{and}\qquad
P_{d+1}-f_{n}(t)\geq\frac{P_{d+1}-F_{\infty}}{2}=:\delta_{0}$

for every $n\geq\bar{n}$ and $t\in[\bar{t},T]$. By summing (4.7) and using
(4.13) we obtain for every
$i\in\big{\\{}\lfloor\frac{\bar{t}}{h_{n}}\rfloor,\ldots,\lfloor\frac{T}{h_{n}}\rfloor\big{\\}}$
that

(4.14)
$\frac{h_{n}}{2}\sum_{k=i+1}^{\lfloor\frac{T}{h_{n}}\rfloor}\mathfrak{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\leq
P_{\mathbb{T}^{2}}(E_{i}^{(h_{n})})-P_{\mathbb{T}^{2}}\big{(}E^{(h_{n})}_{\lfloor\frac{T}{h_{n}}\rfloor}\big{)}\leq
P_{\mathbb{T}^{2}}(E_{i}^{(h_{n})})-P_{d},$

where $\mathfrak{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})$ is defined in (4.3).
Then by (4.14) and by Corollary 4.3 it holds

$\frac{h_{n}}{2}\sum_{k=i+1}^{\lfloor\frac{T}{h_{n}}\rfloor}\mathfrak{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\leq
P_{\mathbb{T}^{2}}(E_{i}^{(h_{n})})-P_{d}\leq C_{0}\|\nabla
U_{i}^{(h)}\|_{L^{2}(\mathbb{T}^{2})}^{2}=C_{0}\mathfrak{D}(E^{(h_{n})}_{i},E^{(h_{n})}_{i-1}).$

Therefore we conclude

$\sum_{k=i+1}^{\lfloor\frac{T}{h_{n}}\rfloor}\mathfrak{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\leq\frac{2C_{0}}{h_{n}}\mathfrak{D}(E^{(h_{n})}_{i},E^{(h_{n})}_{i-1}).$

Setting
$a^{(h_{n})}_{k}=h_{n}\,\mathfrak{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})$ we
have that for every
$i\in\big{\\{}\lfloor\frac{\bar{t}}{h_{n}}\rfloor,\ldots,\lfloor\frac{T}{h_{n}}\rfloor\big{\\}}$
it holds

$\sum_{k=i}^{\lfloor\frac{T}{h_{n}}\rfloor}a_{k}^{(h_{n})}\leq\frac{2C_{0}+h_{n}}{h_{n}}a_{i}^{(h_{n})}\leq\frac{3C_{0}}{h_{n}}a_{i}^{(h_{n})}$

and by applying (4.14) with $i=\lfloor\frac{\bar{t}}{h_{n}}\rfloor$ yields

$\sum_{k=\lfloor\frac{\bar{t}}{h_{n}}\rfloor+1}^{\lfloor\frac{T}{h_{n}}\rfloor}a_{k}^{(h_{n})}\leq
P_{\mathbb{T}^{2}}\Big{(}E_{\lfloor\frac{\bar{t}}{h_{n}}\rfloor}^{(h_{n})}\Big{)}\leq
P_{\mathbb{T}^{2}}(E(0))<2.$

Therefore Lemma 3.1, with $\mathcal{I}$ being the empty set this time, implies

$\sum_{k=i+1}^{\lfloor\frac{T}{h_{n}}\rfloor}a_{k}^{(h_{n})}\leq
2\left(1-\frac{h_{n}}{3C_{0}}\right)^{i-\frac{\bar{t}}{h_{n}}}\qquad\text{for
all
}\,i=\lfloor\frac{\bar{t}}{h_{n}}\rfloor,\ldots,\lfloor\frac{T}{h_{n}}\rfloor.$

In other words for every $t\in[\bar{t},T]$ we have

$\sum_{k=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{T}{h_{n}}\rfloor}h_{n}\,\mathfrak{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\leq
2\left(1-\frac{h_{n}}{3C_{0}}\right)^{\lfloor\frac{t}{h_{n}}\rfloor-\frac{\bar{t}}{h_{n}}}\leq
Ce^{-\frac{t}{3C_{0}}}$

for $h_{n}\leq h_{0}(T)$. Then, by (4.4) and by the above inequality we have
that for $\bar{t}\leq t<s\leq T$ with $s\leq t+1$ it holds

(4.15)
$\begin{split}\|\chi_{E^{(h_{n})}(s)}&-\chi_{E^{(h_{n})}(t)}\|_{H^{-1}(\mathbb{T}^{2})}\leq\sum_{k=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{s}{h_{n}}\rfloor}\|\chi_{E^{(h_{n})}_{k}}-\chi_{E^{(h_{n})}_{k-1}}\|_{H^{-1}(\mathbb{T}^{2})}\\\
&\leq\frac{\sqrt{s-t}}{\sqrt{h_{n}}}\Big{(}\sum_{k=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{T}{h_{n}}\rfloor}\|\chi_{E^{(h_{n})}_{k}}-\chi_{E^{(h_{n})}_{k-1}}\|_{H^{-1}(\mathbb{T}^{2})}^{2}\Big{)}^{\frac{1}{2}}\\\
&\leq\frac{1}{\sqrt{h_{n}}}\Big{(}\sum_{k=\lfloor\frac{t}{h_{n}}\rfloor+1}^{\lfloor\frac{T}{h_{n}}\rfloor}h_{n}^{2}\,\mathfrak{D}(E^{(h_{n})}_{k},E^{(h_{n})}_{k-1})\Big{)}^{\frac{1}{2}}\\\
&\leq C\,e^{-\frac{t}{6C_{0}}},\end{split}$

when $h_{n}\leq h_{0}(T)$.

Recall for all $t>0$ it holds
$\|\chi_{E^{(h_{n})}(t)}\|_{\text{BV}(\mathbb{T}^{2})}\leq\|\chi_{E(0)}\|_{\text{BV}(\mathbb{T}^{2})}\leq
3$. We use Lemma 4.4 and (4.15) to deduce

$\begin{split}\|\chi_{E^{(h_{n})}(s)}&-\chi_{E^{(h_{n})}(t)}\|_{L^{1}(\mathbb{T}^{2})}\\\
&\leq
C\varepsilon\,\|\chi_{E^{(h_{n})}(s)}-\chi_{E^{(h_{n})}(t)}\|_{\text{BV}(\mathbb{T}^{2})}+C\varepsilon^{-1}\|\chi_{E^{(h_{n})}(s)}-\chi_{E^{(h_{n})}(t)}\|_{H^{-1}(\mathbb{T}^{2})}\\\
&\leq C\varepsilon+C\varepsilon^{-1}e^{-\frac{t}{6C_{0}}}.\end{split}$

Choosing $\varepsilon=e^{-\frac{t}{12C_{0}}}$ yields

$\|\chi_{E^{(h_{n})}(s)}-\chi_{E^{(h_{n})}(t)}\|_{L^{1}(\mathbb{T}^{2})}\leq
Ce^{-\frac{t}{12C_{0}}}.$

Letting $h_{n}\to 0$ we obtain for the limit flow

$|E(s)\Delta E(t)|\leq C\,e^{-\frac{t}{12C_{0}}}.$

From here we conclude that $E(t)$ converges to a set of finite perimeter
$E_{\infty}$ exponentially fast.

We may characterize the limit set $E_{\infty}$ as a disjoint union of open
disks $D_{r}(x_{1}),\dots,D_{r}(x_{d})$ thanks to Proposition 4.2 by arguing
as in the proof of Theorem 1.2. Similarly, we obtain the convergence of the
perimeters. We leave the details for the reader.

Also the argument for the Case 2, when there exist
$d\in\mathbb{N}\setminus\\{0\\}$ and $\bar{t}>0$ such that
$F_{\infty}=P_{d}=f_{\infty}(t)$ for every $t\geq\bar{t}$, follows by the same
argument as in the proof of Theorem 1.2. ∎

## Acknowledgments

V. J. was supported by the Academy of Finland grant 314227. E. S. has been
supported by the ERC-STG grant 759229 HiCoS. M. M. was supported by the
University of Parma FIL grant ‘Regularity, Nonlinear Potential Theory and
related topics”.

## References

  * [1] E. Acerbi, N. Fusco, V. Julin, M. Morini, _Nonlinear stability results for the modified Mullins-Sekerka and the surface diffusion flow._ J. Differential Geom. 113 (2019), 1–53.
  * [2] E. Acerbi, N. Fusco, M. Morini. _Minimality via second variation for a nonlocal isoperimetric problem._ Comm. Math. Phys. 322 (2013), 515–557.
  * [3] N.D. Alikakos, P.W. Bates & X. Chen, _Convergence of the Cahn-Hilliard equation to the Hele-Shaw model._ Arch. Rational Mech. Anal. 128 (1994), 165–205.
  * [4] F. Almgren, J. Taylor, L. Wang. _Curvature-driven flows:a variational approach_. SIAM J. Optim. 31 (1993), 387–438.
  * [5] L. Ambrosio, N. Fusco, D. Pallara _Functions of bounded variation and free discontinuity problems._ Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.
  * [6] G. Bellettini , V. Caselles , A. Chambolle , M. Novaga, _The volume preserving crystalline mean curvatureflow of convex sets in $\mathbb{R}^{N}$._ J. Math. Pure Appl. 92 (2009), 499–527.
  * [7] W. Carter, A. Roosen, J. Cahn, and J. Taylor. _Shape evolution by surface diffusion and surface attachment limited kinetics on completely faceted surfaces._ Acta Metallurgica et Materialia 43 (1995), 4309–4323.
  * [8] A. Chambolle, T. Laux. _Mullins-Sekerka as the Wasserstein flow of the perimeter._ Proc. Amer. Math. Soc. 149 (2021), 2943–2956.
  * [9] X. Chen, _The Hele-Shaw problem and area-preserving curve-shortening motions._ Arch. Rational Mech. Anal. 123 (1993), 117–151.
  * [10] G. Ciraolo, _Quantitative estimates for almost constant mean curvature hypersurfaces_. Boll. Unione Mat. Ital. 14 (2021), 137–150.
  * [11] G. Ciraolo, F. Maggi, _On the shape of compact hypersurfaces with almost-constant mean curvature_. Comm. Pure Appl. Math. 70 (2017), 665–716.
  * [12] G. Ciraolo, L. Vezzoni. _A sharp quantitative version of Alexandrov’s theorem via the method of moving planes_. J. Eur. Math. Soc. (JEMS) 20 (2018), 261–299.
  * [13] D. De Gennaro, A. Kubin, _Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus._ Preprint 2021.
  * [14] M. Delgadino, F. Maggi, _Alexandrov’s theorem revisited_. Anal. PDE 12 (2019), 1613–1642.
  * [15] M. Delgadino, F. Maggi, C. Mihaila, R. Neumayer, _Bubbling with $L^{2}$-almost constant mean curvature and an Alexandrov-type theorem for crystals_. Arch. Ration. Mech. Anal. 230 (2018), 1131–1177.
  * [16] A. De Rosa, S. Kolasinski, M. Santilli _Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets_. Arch. Ration. Mech. Anal. 238 (2020), 1157–1198.
  * [17] J. Escher, K. Ito, _Some dynamic properties of volume preserving curvature driven flows._ Math. Ann. 333 (2005), 213–230.
  * [18] J. Escher, G. Simonett. _The volume preserving mean curvature flow near spheres_. Proc. Am. Math. Soc. 126 (1998), 2789–2796.
  * [19] J. Escher; G. Simonett, _A center manifold analysis for the Mullins-Sekerka model._ J. Differential Equations 143 (1998), 267–292.
  * [20] L. Esposito, N. Fusco. _A remark on a free interface problem with volume constraint_. J. Convex Anal. 18 (2011), 417–426.
  * [21] N. Fusco, V. Julin, M. Morini. _Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane_ , to appear in J. Geom. Anal.
  * [22] N. Fusco, F. Maggi, A. Pratelli, _The sharp quantitative isoperimetric inequality_. Ann. of Math. 168, 941–980 (2008).
  * [23] H. Garcke, M. Rauchecker, _Stability analysis for stationary solutions of the Mullins-Sekerka flow with boundary contact._ Preprint 2019.
  * [24] G. Huisken. _The volume preserving mean curvature flow._ J. Rein. Angew. Math 382 (1987), 35–48.
  * [25] V. Julin, J. Niinikoski, _Quantitative Alexandrov Theorem and asymptotic behavior of the volume preserving mean curvature flow_. Preprint 2020.
  * [26] I. Kim, D. Kwon. _Volume preserving mean curvature flow for star-shaped sets._ Comm. Partial Diffenretial Equations 45 (2020), 414–455.
  * [27] B. Krummel, F. Maggi, _Isoperimetry with upper mean curvature bounds and sharp stability estimates._ Calc. Var. Partial Differ. Equ. 56 (2017), Article n. 53.
  * [28] S. Luckhaus, T. Stürzenhecker, _Implicit time discretization for the mean curvature flow equation_. Calc. Var. PDEs, 3 (1995), 253–271.
  * [29] F. Maggi, _Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory._ Cambridge Studies in Advanced Mathematics, 135. Cambridge University Press, Cambridge, 2012.
  * [30] U. F. Mayer, _A singular example for the average mean curvature flow._ Experimental Mathematics 10 (2001), 103–107.
  * [31] U. F. Mayer, G. Simonett, _Self-intersections for the surface diffusion and the volume-preserving mean curvature flow._ Differential and Integral Equations 13 (2000), 1189–1199.
  * [32] N. Meyers, W. P. Ziemer, _Integral inequalities of Poincaré and Wirtinger type for BV-functions_. Amer. J. of Math. 99 (1977) 1345–1360.
  * [33] M. Morini, M. Ponsiglione, E. Spadaro, _Long time behaviour of discrete volume preserving mean curvature flows._ Preprint 2020. https://arxiv.org/abs/2004.04799
  * [34] L. Mugnai, C. Seis, _On the coarsening rates for attachment-limited kinetics._ SIAM J. Math. Anal. 45 (2013), 324–344.
  * [35] L. Mugnai, C. Seis, E. Spadaro, _Global solutions to the volume-preserving mean-curvature flow._ Calc. Var. and PDEs 55(2016), Article n. 18.
  * [36] W.W Mullins, R.F. Sekerka, _Morphological stability of a particle growing by diffusion or heat flow,_ Fundamental contributions to the continuum theory of evolving phase interfaces in solids, 75–81, Springer, Berlin, 1999.
  * [37] J. Niinikoski, _Volume preserving mean curvature flows near strictly stable sets in flat torus._ J. Differential Equations 276 (2021), 149–186.
  * [38] R.L. Pego, _Front migration in the nonlinear Cahn-Hilliard equation._ Proc. Roy. Soc. London Ser. A 422 (1989), 261–278.
  * [39] M. Röger, _Existence of weak solutions for the Mullins-Sekerka flow_. Siam J. Math. Anal. 37 (2005), 291–301.
  * [40] M. Röger, R. Schätzle, _Control of the isoperimetric deficit by the Willmore deficit_. Analysis (Munich) 32 (2012), 1–7.
  * [41] R. Schätzle, _Hypersurfaces with mean curvature given by an ambient Sobolev function_ , J. Differential Geom. 58 (2001), 371–420.
  * [42] L. A. Tarshis, J. L. Walker and M. F. X. Gigliotti, _Solidification._ Annual Review of Materials Science 2(1972), 181–216.
  * [43] C. Wagner, _Theorie der Alterung von Niederschlägen durch Umlösen(Ostwald-Reifung)._ Zeitschrift für Elektrochemie, Berichte der Bunsengesellschaft für physikalische Chemie 65 (1961), 581–591.