On the existence of canonical multi-phase Brakke flows
-
Salvatore Stuvard
and
Yoshihiro Tonegawa
Abstract
This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. As a consequence of the results recently established in [J. Fischer, S. Hensel, T. Laux and T. M. Simon, The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions, preprint 2020, https://arxiv.org/abs/2003.05478], under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows.
1 Introduction
Arising as the gradient flow of the area functional, the mean curvature flow (henceforth abbreviated as MCF) is arguably the most fundamental geometric flow involving extrinsic curvatures. The unknown of MCF is a one-parameter family
The aim of the present paper is to establish the global-in-time
existence of a canonical multi-phase Brakke flow evolving from an arbitrary rectifiable initial datum.
The attribute multi-phase here refers to the fact that the evolving surfaces are, in fact, boundaries of finitely many, but at least two, labelled open subsets
of
In the present paper, we work with an arbitrary (but finite) number of grains. The solution we construct consists of two objects: the flow of the evolving grains and a Brakke flow, intertwined as follows. The Brakke flow – a measure-theoretic generalization of MCF, particularly suited to describe the evolution of surfaces through singularities (see Definition 2.1) – is essentially supported on the topological boundary of the grains, and it keeps track of multiplicities. Additionally, the mean curvature of the Brakke flow determines the distributional velocity at which the reduced boundary of each grain moves. As a result of the latter property, the change of volume of each grain between two instants of time can be recovered by integrating the mean curvature over the reduced boundary, a property certainly expected for a smooth MCF but quite non-trivial in a context where singularities and multiplicities occur. The attribute canonical refers to this very precise interplay between the Brakke flow and the evolution of the grains. Note that Brakke flows are non-unique in general due to the nature of the formulation, but the existence of the grains prevents redundant non-uniqueness such as sudden vanishing, for example. In the absence of higher multiplicities of the Brakke flow, we show that the collection of the grains (or, more precisely, the collection of their indicator functions) constitutes a BV solution of the MCF (see Definition 2.4). In this case, the grain boundaries are, in fact, a smooth MCF almost everywhere in space and time by Brakke’s partial regularity theory [6, 20, 38].
In certain instances, the additional
BV characterization may lead to the uniqueness of the canonical Brakke flow.
For instance, in ambient dimension
In more precise and technical terms, the highlight of the main results of the present paper may be stated as follows (see the complete statements in Section 2.3).
Theorem A.
Let
and satisfying the following properties, writing
For each
(1.1)If the Brakke flow is locally a unit density flow, then, locally, we have
for a.e. t.
In the above statement,
When integrated, formula (1.1) provides, as a byproduct, the change of the
We emphasize the following point in particular: formulae (1.1) and (1.2)
hold true even if there is a “higher
multiplicity portion” of
For general Brakke flows, there is no clear pathway leading from the characterization of Brakke flow, which
consists of a variational inequality dictating an upper bound on the rate of change of the mass of the evolving surfaces,
to formula (1.1), even under
the unit density assumption. In fact, as mentioned already,
by the partial regularity theorem for unit density
Brakke flows [6, 20, 38], in this latter case it is known that
Next, we discuss closely related works, particularly on the aspect of existence of generalized MCF. For two-phase MCF, the level-set method [8, 12] provides a general existence and uniqueness result even past the time after singularities appear. On the other hand, the level-set may develop a non-trivial interior, a phenomenon called “fattening”, due to the singular behavior of the MCF. Also, the uniqueness of the level-set solutions depends essentially on the maximum principle and it cannot handle general multi-phase MCF of more than two phases.
For the general multi-phase problem,
it is natural to consider an initial datum
Luckhaus and Sturzenhecker [25] introduced the formulation of BV solution of the two-phase MCF, which can be extended naturally to the multi-phase MCF. Their existence result is conditional, in the sense that a BV solution is shown to exist under the assumption that the time-discrete approximate solutions converge to their limit without loss of surface energy. Laux and Otto [23] proved that a sequence arising from the thresholding scheme of Merriman, Bence and Osher converges conditionally to a BV solution using the interpretation in terms of minimizing movements due to Esedoḡlu and Otto [10], again under an assumption similar to [25] (see also [24] for a similar convergence result of the parabolic Allen–Cahn system). The BV solution of [25] was partly motivated by the minimizing movements scheme of Almgren, Taylor and Wang [2], and the multi-phase version has been studied recently by Bellettini and Kholmatov [4].
On the side of Brakke flows, Ilmanen [18]
proved the existence of a rectifiable Brakke flow arising as a limit
of solutions to a parabolic Allen–Cahn equation, and the second-named author proved the
integrality of the Brakke flow [37]. These results are for two-phase
MCF, but the relation to the BV solution as formulated in [25]
remained obscure. In a different but
related problem – the Allen–Cahn action functional –, Mugnai and Röger
[29] introduced a notion of
Finally, we mention again that a global-in-time existence result of a multi-phase Brakke flow which is equipped with moving grain boundaries was given by Kim and the second-named author in [21], reworking the pioneering paper by Brakke [6] within a different formulation. The grains in [21] move continuously with respect to the Lebesgue measure, but the problem concerning the validity of an exact identity involving the volume change was not addressed in there. Previous works by the authors of the present paper (see [35, 34]), in which certain Brakke flows are constructed with an approximation scheme analogous to that introduced in [21], could be reworked so that the additional conclusions concerning the interplay between the flow of the grains and the Brakke flow can be drawn in those contexts as well: in particular, it is possible to have the Brakke flow with prescribed boundary constructed in [35] satisfy formulae (1.1) and (1.2) (see Section 7.2).
2 Notation and main results
2.1 Basic notation
We shall use the same notation adopted in [21, Section 2]. In particular, the ambient space we will be working in is the Euclidean space
A positive Radon measure μ on
The upper and lower k-dimensional densities of a Radon measure μ at
respectively. If
Given an open set
If
We say that
For a set
A subset
where
A (positive) measure μ on
that is,
When this happens, the plane π is unique, and it is denoted by
A k-dimensional varifold in
2.2 Three weak notions of MCF
As anticipated in Section 1, in the last few decades several alternative notions of weak solution to the MCF have been proposed. In this subsection, we briefly define and comment upon the three of interest in the present paper: Brakke flows,
Definition 2.1 (Brakke flow).
Let
For a.e.
For a.e.
The generalized mean curvature
For all
(2.1)
The inequality in (2.1) is typically referred to as Brakke’s inequality. It is not difficult to show that if
In particular, if
Definition 2.2.
A Brakke flow is said to be a unit density flow
in
The following definition of
Definition 2.3 (
L
2
flow).
Let
The generalized mean curvature
and
There exist a vector field
For every
(2.3)
Any function
The definition can be easily motivated by considering (2.2) once again: the latter, indeed, implies (2.3) if both h and v are in
Finally, the following definition of
Definition 2.4 (
BV
flow).
Suppose
For a.e.
(2.4)There exist scalar functions
(2.5)and with the property that
(2.6)for a.e.
Setting
The functions
(2.7)for all vector fields
The following inequality holds for a.e.
(2.8)
Equation (2.6) characterizes
and thus (2.7)
formally characterizes
(v) corresponds precisely to the energy dissipation inequality for the hypersurface measures of the MCF.
2.3 Main results
Definition 2.5.
We will denote by Ω a function in
for all
The function Ω is introduced as a weight function to treat unbounded MCF, which may have infinite n-dimensional Hausdorff measure in
Assumption 2.6.
Suppose that
Moreover, assume that there are
By (2.11) and
For a given
The following Theorems 2.7–2.12 all hold under Assumption
2.11. First, we claim the existence of a Brakke flow starting with
Theorem 2.7.
There exists an n-dimensional Brakke flow
If
for all
If
For any
Remark 2.8.
If
Definition 2.9.
With
The following Theorem 2.10 is satisfied in general for Brakke flows, and thus it holds, in particular, for the flow produced in Theorem 2.7.
Theorem 2.10.
The Radon measure μ and the Brakke flow
The following theorem shows that, in addition to the Brakke flow of Theorem 2.7,
there are evolving domains
Theorem 2.11.
For each
the following assertions hold:
For all
For every
so that ( 2.6 ) reads as follows: for every
(2.12)for any
If
If
Notice that in (vi) we speak about generalized
Theorem 2.12.
Suppose that, for
Furthermore, (2.7) holds true with
If we assume the following additional conditions on
Theorem 2.13.
Suppose that
Then there exists
If we set “the maximal existence time” of unit density Brakke flow as
under the assumption of Theorem 2.13, either
is also a BV solution of MCF.
Finally, the validity of identity (2.12) allows to employ a simple argument to deduce a lower bound on the
extinction time for
Before stating the result, let us notice that if
Theorem 2.14.
If
Observe that if
2.4 Outline of the proofs and plan of the paper
In what follows, we assume to have fixed the following:
The strategy towards the proof of the existence of the Brakke flow
The open partition
The only difference between the scheme employed in [21] and the one devised in the present paper is in the choice of the Lipschitz deformations in the first step. While in [21] one only requires that the change of volumes of the grains due to Lipschitz deformation is small (for a certain smallness scale), in the present paper we shall require that the change of volume is small compared to the reduction in surface measure. We define such new class of volume-controlled Lipschitz deformations in Section 3.1, and then we claim that the construction of [21] can be carried over using volume-controlled deformations, as outlined in Section 3.2: by the results of [21], this completes the proof of Theorem 2.7 (i)–(iv) (with some additional remarks), Theorem 2.10 (i) and Theorem 2.11 (i)–(v). There are technical details for verifying that the modifications to the volume-controlled deformations do not pose any difficulties, and we defer it to Section A.
In Section 4, the main
result is Theorem 2.11 (vi), namely identity (2.12). For that, we first prove that
any Brakke flow is
In Section 5, we prove Theorem 2.10 (ii)–(iv), which, in particular, implies that the support
Section 6 contains the proofs of Theorem 2.11 (vii) and (viii).
They are improved versions of the integrality theorem [21, Theorem 8.6] in that the behaviors of
approximating flows and their grains are tracked more in detail.
A characterization in [22, Section 4] of limiting behavior within a length scale of
Finally, Section 7 contains some final remarks on the notion of generalized
2.5 Further notation
We collect here some further notation and results which will be extensively used throughout the paper. We begin with the notion of open partitions of
Definition 2.15.
Let
It holds
The set
The set of all Ω-finite open partitions of N elements is denoted by
Note that it is allowed for some
We define
and with a slight abuse of notation we use the same symbol to denote the varifold
namely the unit density varifold induced from the countably n-rectifiable set
Definition 2.16.
Given
It holds
The following lemma is [21, Lemma 4.4].
Lemma 2.17.
Let
Next, we define a class of test functions with good properties. For
Note that
Finally, we give the notion of smoothed mean curvature vector of a varifold. Let
Then, for every
where
We shall call
In the above formula,
for all
by means of the identity
Analogously,
for all
We state the following lemma concerning the smoothed mean curvature vector to be used in the subsequent sections. For the proof, the reader can consult [21, Lemma 5.1].
Lemma 2.18.
For every
3 Existence of multi-phase Brakke flow
3.1 Volume-controlled Lipschitz deformations
In this subsection, we introduce the modified class of Lipschitz deformations used in the present paper to gain improved control on the volume change of partitions. For
Definition 3.1.
For
It holds
The difference between the above class
and thus the right-hand side of (ii) is non-negative.
In particular, the identity map
Definition 3.2.
For
and for a compact set
Even with this modification, the claims in [21, Section 4] remain essentially the same.
Lemma 3.3.
For compact sets
and
Lemma 3.4.
Suppose that
We remark that the corresponding statement in [21, Lemma 4.11] contains the additional assumption
Proof.
By (3.2),
for any compact set C.
Let
We define a map
where we used (ii) for
By letting
The following lemma is similar to [21, Lemma 4.12] with a minor change. The proof is identical.
Lemma 3.5.
Suppose that
Then we have
3.2 The constructive scheme
In this subsection, we provide the detailed construction of the sequence of piecewise constant-in-time approximating flows of open partitions leading to the existence result of a multi-phase Brakke flow. We let the weight function Ω be as in Definition 2.5, and we consider an initial rectifiable set
In (3.5), the epoch length is
First, one chooses
and sets
Thus, in particular,
Next, one defines the map
where
and thus
Notice that the scheme just defined differs from that adopted in [21] only in step (1) of the algorithm, where volume-controlled Lipschitz deformations are used. In spite of such modification, we claim that the proof from [21] can be essentially carried over to this framework, and thus leading to the following theorem.
Theorem 3.6.
There is a constant
a subsequence
reals
integers
a family
a family
such that the approximating flow of open partitions
Furthermore, the following assertions hold:
There exists a subset
The flow of grains
The proofs of the claims contained in the statement of Theorem 3.6 are analogous to the corresponding ones outlined in [21], modulo a few technical modifications in consequence of the use of volume-controlled Lipschitz deformations. More precisely, the following assertions hold:
The conclusion (i) is [21, Lemma 9.1].
The conclusion (ii) is [21, Theorem 3.2, Proposition 3.4], except that Theorem 2.7 (ii) follows from the argument in [35, Proposition 6.10] and Theorem 2.7 (iv) follows from Brakke’s inequality and Theorem 2.7 (iii) with
as
We will not repeat the proofs of the above results, but we will detail the aforementioned changes due to volume-controlled Lipschitz deformations in Section A. For the time being, we will assume the validity of Theorem 3.6, and we will proceed with the derivation of all remaining results claimed in Section 2.
4 BV flow: Proof of Theorem 2.11 (vi)
The main result of this section is the proof of Theorem 2.11 (vi), which we isolate as the following theorem.
Theorem 4.1.
The one-parameter family
for every
Remark 4.2.
Introducing the notation
This implies, in particular, that
where
The proof of Theorem 4.1 will be obtained in three main steps, which are the content of Sections
4.1–4.3, respectively. First, we check
that Brakke flow implies
4.1 Characterization as
L
2
flow
In this subsection, we first prove Theorem 2.7 (iv), which we isolate as the following theorem.
Theorem 4.3.
Let
Proof.
We verify the requirements of Definition 2.3. Conditions (i) and (ii) are satisfied by
so that
for every
In particular, the assignment
defines a positive linear functional on
defined according to the following prescriptions:
Let now
so that the linearity and monotonicity of L yield
Notice that for such a ϕ the space integration can be restricted to U (as the – continuous – derivatives of ϕ are necessarily zero on
We next proceed to estimate
We see then that
The following is a simple corollary of Theorem 4.3 and [29, Proposition 3.3].
Corollary 4.4.
Let
at μ-a.e.
4.2 Existence of measure-theoretic velocities
We have the following proposition.
Proposition 4.5.
Let
Then, for every i, there exists a function
for every
Proposition 4.5 readily implies the following corollary.
Corollary 4.6.
For every
In particular, since
Proof of Proposition 4.5.
We shall divide the proof into several steps. At first, we will drop the dependence of the test function on time, which will be introduced again at a later step. The idea of the proof is to obtain an evolution identity for the approximating flow
Step one: preliminary reductions.
By density, it is evidently sufficient to show that identity (4.5) holds when
and we can further decompose each summand on the right-hand side of (4.6) as
where
Here,
where
Step two: evaluation of
so that, using Definition 3.1 (ii) and (3.1), it holds
We then proceed with
Using that g is a diffeomorphism and changing variable in the second integral in
where
together with (2.15), we have that if
Notice that, in order to apply estimate (2.15), we are using that
which is bounded by a quantity depending only on T, the initial datum and
where we have used that
that
where
as soon as ε is small enough to absorb the constants depending on
and the divergence theorem, we have
Recall that, in the right-hand side of (4.15),
where
Since
we can write
Recall now that the ε-smoothed mean curvature vector
In particular, using [21, (5.74) and Lemma 4.17], it is easy to estimate the second summand on the right-hand side of (4.16) by
For the first summand on the right-hand side of (4.16), instead,
by Lemma 2.18. Thus, we can finally estimate
where
by
for all ε sufficiently small. We have thus proved that
where
satisfy estimates (4.8), (4.14), (4.17) and (4.18).
Step three: sum of contributions and limit
we obtain from (4.6) that
with
Moreover, since
where
so that the end-point
due to (3.10), whereas
To this end, we first observe that, since
and using that, by definition,
Therefore, for every
where the last line follows from (4.23).
In particular, using (3.6) and (3.7), for any fixed
are (signed) Radon measures in
in the sense of measures. Estimate (4.24) also implies that
Hence, we can now calculate the limit in (4.22) (along the aforementioned subsequence, not relabeled), which gives
We have then concluded the existence of
for all
Step four: the case of time-dependent ϕ. We now consider the general case of a test function
The proof is analogous, with a few minor modifications which take the dependence on t into account. First, formula (4.6) becomes
whereas formula (4.7) becomes
where
By the fundamental theorem of calculus and Fubini’s theorem, we can immediately calculate
The summands
Introducing back the subscripts
where
Summing over
where
where we have used (2.15). In particular, when
and similarly for others, and they all vanish as
where
Step five: support of
Notice that
In the second case, we have that, necessarily,
which then gives
Since ϕ is arbitrary, we deduce then that
4.3 Boundaries move by their mean curvature
In this subsection, we complete the proof of Theorem 4.1, by achieving the representation for the measure-theoretic time derivative
Lemma 4.7.
For every
The tangent
Relation ( 4.4 ) holds.
The proof of Lemma 4.7 can be deduced with relatively little effort from arguments already contained in [29], but we include it for the reader’s convenience. Before proceeding, we will need some consequences of the celebrated monotonicity formula by Huisken for MCF. Since said consequences will be needed also later on in the paper, we record them here. First, let us set some notation. For
as well as the truncated kernel
where
Lemma 4.8.
There exists
As a consequence, one has the following lemma [21, Lemma 10.4], which provides a uniform upper bound on mass density ratios.
Lemma 4.9.
For every
Proof of Lemma 4.7.
Fix
where
On the other hand, it is not difficult to see that
Indeed, first notice that, as an immediate corollary of Lemma 4.9, one has
Let then A be a set with
Then, by [33, Theorem 3.2], one has
so that the conclusion
Combining (4.32) with (4.33), we immediately have that
Similarly, by taking the Radon–Nikodým derivative of
Since
If
This shows that
for every
Next, to prove (iii) and (iv), we are going to use the coarea formula for sets of finite perimeter, slicing
Then using, e.g., [26, Theorem 18.11] together with the fact that
for a.e.
for
and for every
so that
where in the last identity we have used the coarea formula [3, (2.72)] and where
which readily implies that
is then obtained by repeating the argument in (i) at fixed t.
Finally, (v) is always true at points satisfying (i)–(iv): indeed, if
This completes the proof. ∎
Proof of Theorem 4.1.
By virtue of Proposition 4.5, we only need to show the validity of equation (4.2). Fix
Let
and thus
where we have used that
Due to (4.41), at points
so that the chain of identities in (4.42) can be continued as
again by the coarea formula. This completes the proof. ∎
5 Ilmanen’s density lower bound
This section contains a proof of the following fact (Theorem 5.3 below),
from which we may conclude the proofs of Theorem 2.10 (ii)–(iv): if
Using Lemma 4.8 and Lemma 4.9, we prove next the following lemma, which is a variant of Brakke’s clearing out lemma (cf. [6, Section 6.3], [18, Section 6.1] and [21, Lemma 10.6]).
Recalling the notation set in (4.28) and (4.29), we define, for
Lemma 5.1.
For any
with the following property: if
then
Proof.
For
Consider first the term
By the change of variable
We can then conclude
We let then
Choosing
Remark 5.2.
Observe that, since
Lemma 5.1 immediately implies the following: if
then
Theorem 5.3.
For
Then there exists
Proof.
For every
so that
Let
Furthermore, since
Hence, for γ small enough (depending on the ratio
In particular,
and thus
which would then imply
We have then concluded the following dichotomy:
Therefore, if
intersects
so that a countable union of such sets covers
For every center
are a covering of
where we have used (5.6). Letting
by monotone convergence. Hence, by taking countable unions,
with the obvious meaning of the symbols and taking into account that
The following is the immediate corollary of Theorem 5.3, which proves Theorem 2.10 (ii).
Corollary 5.4.
There exists
In particular, by recalling from Theorem 2.10 (i) and Theorem 2.11 (iii)
that
is
Since
6 Two-sidedness at unit density point
In this section, we prove Theorem 2.11 (vii) and (viii), which are
restated in Propositions 6.3 and 6.4, respectively. To do so, we
need to analyze the behavior of approximating flows. The first Lemma
6.1 shows that, for a.e. t, only
the reduced boundaries of approximating grains contribute to the limit
measure and that the measures coming from “interior boundaries”
Lemma 6.1.
For a.e.
Proof.
We fix t such that
which holds for a.e.
for arbitrary
As in [22, Section 4], let
where
It holds
For all k,
where
We may use Lemma 3.5 with
for all sufficiently large
Since
By the Besicovitch covering theorem, we have a set of mutually disjoint balls
such that
Because of the lower bound of the measure
combined with (6.7), we would prove (6.6), ending the proof. Assume for a contradiction that (6.8) were not true and we had a
subsequence (denoted by the same index)
Consider a rescaling
On the other hand, (6.9) implies that
Since
for each k, (6.10)–(6.12) show
Since
The next Lemma 6.2 shows that
Lemma 6.2.
For
as varifolds,
for all
The claims (6.14) and (6.15) also hold with
Proof.
Without loss of generality, assume that
which holds for a.e.
By Corollary 5.4, for a.e.
where
In the following, we use the same argument in the proof of [21, Theorem 8.6].
Let
for all
Following the argument in [21, p. 113], one can prove that
Thus, for
where
Note that the choice with (6.19) is possible due to
(6.1). We then proceed verbatim as in [21, pp. 114–120] with
These claims are, respectively,
[21, (8.154), (8.156) (stated differently) and (8.159)].
The measurability of
Note that
as varifolds. We also have
Since
Since
Then (6.22) and (6.27) show (6.23). Finally, we define
Since
Proposition 6.3.
For
Proof.
We prove (6.29) for a.e. t such that the conclusion of Lemma 6.2 holds.
With such t fixed, we drop t and suppose that (6.29) were not true for a contradiction.
We may apply the result of Lemma 6.2 and find a point
Without loss of generality, we may assume that
For simplicity, write the union of the measure-theoretic boundaries as
We also define the measure-theoretic interior and exterior of
and
We note from the definition that
We next use some results in the proof of [11, Theorem 5.23]
on the property of measure-theoretic interior and exterior. For each
and
Here
for all
and with
Let
Then, by the Fubini theorem and (6.35), we may choose a sequence
we have
On cylinders
and similarly if
Note that
In particular,
which converges to 0 as
this implies that
Proposition 6.4.
Assume
Proof.
The proof proceeds similarly to the proof of Proposition 6.3, except that we need to localize the argument to
each layers. Fix a bounded open set
On the other hand, for
we have
We use Lemma 6.2, and in the proof of Lemma 6.2 we may additionally assume that the chosen subsequence satisfies
if
if
Set
Given any
Then, for
We cover
where
so that
We now proceed similarly to the previous Proposition 6.3, and choose
At each point
we may choose
and
or the inequalities replacing the role of
since a is a Lebesgue point of
With these properties, we can make sure that, with respect to
By using (6.37), for
Finally, we comment on the proofs of Theorem 2.12 and 2.13. If
the left-hand side of (2.7)
is equal to
such that (recall (4.28))
for all
is non-increasing on
Since
7 Final remarks
7.1 On generalized BV solutions
As explained in Definition 2.4, a BV flow is classically defined as consisting of two objects:
N families of sets of finite perimeter
and check that (2.7)
implies that the generalized mean curvature
may not correspond, in general, to the generalized
mean curvature vector of
Then
7.2 MCF with fixed boundary conditions
In [35], given a strictly convex bounded
domain
7.3 A lower bound estimate for extinction time
Proof of Theorem 2.14.
Suppose that
for all
Next, set
Combining (7.1) and (7.2) then gives the inequality
for a.e.
Since
that is, (2.14). ∎
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 18H03670
Award Identifier / Grant number: 19H00639
Award Identifier / Grant number: 17H01092
Funding statement: The first author is partially supported by the GNAMPA – Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni of INdAM. The second author is partially supported by JSPS 18H03670, 19H00639, 17H01092.
A The existence theorem of Kim and Tonegawa revisited
Here we point out places which require a change to
A.1 Construction of approximate flows
For the construction of discrete approximate sequence in [21, Section 6],
we simply replace
by Definition 3.1 (ii) and (3.1), we have for each
The change in [21, (6.9)] is also reflected in the estimate [21, (6.4)] and we have
A.2 Proofs of rectifiability and integrality
For the proofs of rectifiability and integrality of
(i) In [21, Proposition 7.2], it is proved
that there exists an
(ii) In [21, Theorem 7.3], assumption (iv) should be replaced by the volume-controlled counterpart. In the proof, on [21, pp. 94–95], a Lipschitz map f is defined with the properties stated in the bottom of [21, p. 94] using [21, Proposition 7.2]. Using [21, (7.40)], which gives
one can proceed in [21, (7.43)] as
Thus, for all sufficiently large l, Definition 3.1 (ii) is satisfied.
The use of [21, Lemma 4.12] is also justified,
and we have
(iii) Throughout [21, Section 8], the only
crucial point that needs to be checked is in [21, Lemma 8.1], which involves the actual construction of
a measure-reducing Lipschitz deformation. It proves roughly that when
and, with
The map
As one can see in [21, (8.67)], the reduction of measure is
and we also have from [21, (8.8)] that
We need to see the difference with the weight Ω,
and since
In the last line, we used (A.2) and (A.3). Note that the first term of the last line is a negative term
of order
for all sufficiently large j, and we have
A.3 Volume change of grains
The motivation of having
A.4 Changes in the later paper by Kim and Tonegawa
The results from [22] are used in the present paper
and the modifications are needed there as well. On the other hand,
similarly, one can check that the
change to the volume-controlled counterpart does
not cause any difficulties. The part which
is relevant to the change is [22, Section 4]. In the proof of [22, Lemma 4.2], a Lipschitz retraction map
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Articles in the same Issue
- Frontmatter
- On the Hölder regularity of all extrema in Hilbert’s 19th Problem
- A partially overdetermined problem in domains with partial umbilical boundary in space forms
- On the existence of canonical multi-phase Brakke flows
- Concavity properties for solutions to p-Laplace equations with concave nonlinearities
- On the Almgren minimality of the product of a paired calibrated set and a calibrated manifold of codimension 1
- The Neumann and Dirichlet problems for the total variation flow in metric measure spaces
- No Lavrentiev gap for some double phase integrals
- Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity
- Properties of the free boundaries for the obstacle problem of the porous medium equations
- A generalized fractional Pohozaev identity and applications
Articles in the same Issue
- Frontmatter
- On the Hölder regularity of all extrema in Hilbert’s 19th Problem
- A partially overdetermined problem in domains with partial umbilical boundary in space forms
- On the existence of canonical multi-phase Brakke flows
- Concavity properties for solutions to p-Laplace equations with concave nonlinearities
- On the Almgren minimality of the product of a paired calibrated set and a calibrated manifold of codimension 1
- The Neumann and Dirichlet problems for the total variation flow in metric measure spaces
- No Lavrentiev gap for some double phase integrals
- Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity
- Properties of the free boundaries for the obstacle problem of the porous medium equations
- A generalized fractional Pohozaev identity and applications
