The Allen-Cahn Action functional in higher dimensions
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER
DIMENSIONS
LUCA MUGNAI AND MATTHIAS RÖGER
Abstract. The Allen–Cahn action functional is related to the probability
of rare events in the stochastically perturbed Allen–Cahn equation. Formal
calculations suggest a reduced action functional in the sharp interface limit.
We prove the corresponding lower bound in two and three space dimensions.
One difficulty is that diffuse interfaces may collapse in the limit. We therefore
consider the limit of diffuse surface area measures and introduce a general-
ized velocity and generalized reduced action functional in a class of evolving
measures.
1. Introduction
In this paper we study the (renormalized) Allen–Cahn action functional
Sε(u) :=
ε∂tu+
− ε∆u+
W ′(u)
dx dt. (1.1)
This functional arises in the analysis of the stochastically perturbed Allen–Cahn
equation [2, 21, 13, 30, 8, 10, 12] and is related to the probability of rare events
such as switching between deterministically stable states.
Compared to the purely deterministic setting, stochastic perturbations add new
features to the theory of phase separations, and the analysis of action functionals has
drawn attention [8, 13, 18, 19, 26]. Kohn et alii [18] considered the sharp-interface
limit ε → 0 of Sε and identified a reduced action functional that is more easily
accessible for a qualitative analysis. The sharp interface limit reveals a connection
between minimizers of Sε and mean curvature flow.
The reduced action functional in [18] is defined for phase indicator functions
u : (0, T ) × Ω → {−1, 1} with the additional properties that the measure of the
phase {u(t, ·) = 1} is continuous and the common boundary of the two phases
{u = 1} and {u = −1} is, apart from a countable set of singular times, given
as union of smoothly evolving hypersurfaces Σ := ∪t∈(0,T ){t} × Σt. The reduced
action functional is then defined as
S0(u) := c0
∣v(t, x)−H(t, x)
dHn−1(x)dt + 4S0nuc(u), (1.2)
S0nuc(u) := 2c0
Hn−1(Σi), (1.3)
Date: November 12, 2021.
2000 Mathematics Subject Classification. Primary 49J45; Secondary 35R60, 60F10, 53C44.
Key words and phrases. Allen-Cahn equation, stochastic partial differential equations, large
deviation theory, sharp interface limits, motion by mean curvature.
http://arxiv.org/abs/0704.1954v2
2 LUCA MUGNAI AND MATTHIAS RÖGER
where Σi denotes the i
th component of Σ at the time of creation, where v denotes
the normal velocity of the evolution (Σt)t∈(0,T ), where H(t, ·) denotes the mean
curvature vector of Σt, and where the constant c0 is determined by W ,
c0 :=
2W (s) ds. (1.4)
(See Section 9 for a more rigourous definition of S0).
Several arguments suggest that S0 describes the Gamma-limit of Sε:
• The upper bound necessary for the Gamma-convergence was formally
proved [18] by the construction of good ‘recovery sequences’.
• The lower bound was proved in [18] for sequences (uε)ε>0 such that the
associated ‘energy-measures’ have equipartitioned energy and single multi-
plicity as ε→ 0.
• In one space-dimension Reznikoff and Tonegawa [26] proved that Sε
Gamma-converges to an appropriate relaxation of the one-dimensional ver-
sion of S0.
The approach used in [18] is based on the evolution of the phases and is sensible
to cancellations of phase boundaries in the sharp interface limit. Therefore in [18]
a sharp lower bound is achieved only under a single-multiplicity assumption for
the limit of the diffuse interfaces. As a consequence, it could not be excluded that
creating multiple interfaces reduces the action.
In the present paper we prove a sharp lower-bound of the functional Sε in space
dimensions n = 2, 3 without any additional restrictions on the approximate se-
quences.
To circumvent problems with cancellations of interfaces we analyze the evolution
of the (diffuse) surface-area measures, which makes information available that is
lost in the limit of phase fields. With this aim we generalize the functional S0 to a
suitable class of evolving energy measures and introduce a generalized formulation
of velocity, similar to Brakke’s generalization of Mean Curvature Flow [5].
Let us informally describe our approach and main results. Comparing the two
functionals Sε and S0 the first and second term of the sum in the integrand (1.1)
describe a ‘diffuse velocity’ and ‘diffuse mean curvature’ respectively. We will make
this statement precise in (6.13) and (7.1). The mean curvature is given by the first
variation of the area functional, and a lower estimate for the square of the diffuse
mean curvature is available in a time-independent situation [28]. The velocity of
the evolution of the phase boundaries is determined by the time-derivative of the
surface-area measures and the nucleation term in the functional S0 in fact describes
a singular part of this time derivative.
Our first main result is a compactness result: the diffuse surface-area measures
converge to an evolution of measures with a square integrable generalized mean cur-
vature and a square integrable generalized velocity . In the class of such evolutions
of measures we provide a generalized formulation of the reduced action functional.
We prove a lower estimate that counts the propagation cost with the multiplicity of
the interface. This shows that it is more expensive to move phase boundaries with
higher multiplicity. Finally we prove two statements on the Gamma-convergence
(with respect to L1(ΩT )) of the action functional. The first result is for evolu-
tions in the domain of S0 that have nucleations only at the initial time. This is in
particular desirable since minimizers of S0 are supposed to be in this class. The
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 3
second result proves the Gamma convergence in L1(ΩT ) under an assumption on
the structure of the set of measures arising as sharp interface limits of sequences
with uniformly bounded action.
We give a precise statement of our main results in Section 4. In the remainder
of this introduction we describe some background and motivation.
1.1. Deterministic phase field models and sharp interface limits. Most
diffuse interface models are based on the Van der Waals–Cahn–Hilliard energy
Eε(u) :=
|∇u|2 + 1
W (u)
dx. (1.5)
The energy Eε favors a decomposition of Ω into two regions (phases) where u ≈ −1
and u ≈ 1, separated by a transition layer (diffuse interface) with a thickness of
order ε. Modica and Mortola [23, 22] proved that Eε Gamma-converges (with
respect to L1-convergence) to a constant multiple of the perimeter functional P ,
restricted to phase indicator functions,
Eε → c0P , P(u) :=
d|∇u| if u ∈ BV (Ω, {−1, 1}),
∞ otherwise.
P measures the surface-area of the phase boundary ∂∗{u = 1} ∩ Ω. In this sense
Eε describes a diffuse approximation of the surface-area functional.
Various tighter connections between the functionals Eε and P have been proved.
We mention here just two that are important for our analysis. The (accelerated)
L2-gradient flow of Eε is given by the Allen–Cahn equation
ε∂tu = ε∆u−
W ′(u) (1.6)
for phase fields in the time-space cylinder (0, T ) × Ω. It is proved in different
formulations [7, 9, 17] that (1.6) converges to the Mean Curvature Flow
H(t, ·) = v(t, ·) (1.7)
for the evolution of phase boundaries.
Another connection between the first variations of Eε and P is expressed in a
(modified) conjecture of De Giorgi [6]: Considering
Wε(u) :=
− ε∆u+ 1
W ′(u)
dx (1.8)
the sum Eε +Wε Gamma-converges up to the constant factor c0 to the sum of the
Perimeter functional and the Willmore functional W ,
Eε +Wε → c0P + c0W , W(u) =
H2 dHn−1, (1.9)
where Γ denotes the phase boundary ∂∗{u = 1} ∩ Ω. This statement was recently
proved by Röger and Schätzle [28] in space dimensions n = 2, 3 and is one essential
ingredient to obtain the lower bound for the action functional.
4 LUCA MUGNAI AND MATTHIAS RÖGER
1.2. Stochastic interpretation of the action functional. Phenomena such as
the nucleation of a new phase or the switching between two (local) energy minima
require an energy barrier crossing and are out of the scope of deterministic models
that are energy dissipative. If thermal fluctuations are taken into account such an
energy barrier crossing becomes possible. In [18] ‘thermally activated switching’
was considered for the stochastically perturbed Allen–Cahn equation
ε∂tu = ε∆u−
W ′(u) +
2γηλ (1.10)
Here γ > 0 is a parameter that represents the temperature of the system, η is a
time-space white noise, and ηλ is a spatial regularization with ηλ → η as λ → 0.
This regularization is necessary for n ≥ 2 since the white noise is too singular to
ensure well-posedness of (1.10) in higher space-dimensions.
Large deviation theory and (extensions of) results by Wentzell and Freidlin
[15, 14] yield an estimate on the probability distribution of solutions of stochas-
tic ODEs and PDEs in the small-noise limit. This estimate is expressed in terms
of a (deterministic) action functional. For instance, thermally activated switching
within a time T > 0 is described by the set of paths
u(0, ·) = −1, ‖u(t, ·)− 1‖L∞(Ω) ≤ δ for some t ≤ T
, (1.11)
where δ > 0 is a fixed constant. The probability of switching for solutions of (1.10)
then satisfies
γ lnProb(B) = − inf
S(λ)ε (u). (1.12)
Here S(λ)ε is the action functional associated to (1.10) and converges (formally) to
the action functional Sε as λ → 0 [18]. Large deviation theory not only estimates
the probability of rare events but also identifies the ‘most-likely switching path’ as
the minimizer u in (1.12).
We focus here on the sharp interface limit ε → 0 of the action functional Sε.
The small parameter ε > 0 corresponds to a specific diffusive scaling of the time-
and space domains. This choice was identified [8, 18] as particularly interesting,
exhibiting a competition between nucleation versus propagation to achieve the op-
timal switching. Depending on the value of |Ω|1/d/
T a cascade of more and more
complex spatial patterns is observed [8, 18, 19]. The interest in the sharp interface
limit is motivated by an interest in applications where the switching time is small
compared to the deterministic time-scale, see for instance [20].
1.3. Organization. We fix some notation and assumptions in the next section.
In Section 3 we introduce the concept of L2-flows and generalized velocity. Our
main results are stated in Section 4 and proved in the Sections 5-8. We discuss
some implications for the Gamma-convergence of the action functional in Section
9. Finally, in the Appendix we collect some definitions from Geometric Measure
Theory.
Acknowledgment. We wish to thank Maria Reznikoff, Yoshihiro Tonegawa, and
Stephan Luckhaus for several stimulating discussions. The first author thanks the
Eindhoven University of Technology for its hospitality during his stay in summer
2006.
The first author was partially supported by the Schwerpunktprogramm DFG SPP
1095 ‘Multiscale Problems’ and DFG Forschergruppe 718.
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 5
2. Notation and Assumptions
Throughout the paper we will adopt the following notation: Ω is an open
bounded subset of Rn with Lipschitz boundary; T > 0 is a real number and
ΩT := (0, T ) × Ω; x ∈ Ω and t ∈ (0, T ) denote the space- and time-variables
respectively; ∇ and ∆ denote the spatial gradient and Laplacian and ∇′ the full
gradient in R× Rn.
We choose W to be the standard quartic double-well potential
W (r) =
(1− r2)2.
For a family of measures (µt)t∈(0,T ) we denote by L1 ⊗ µt the product measure
defined by
L1 ⊗ µt
(η) :=
µt(η(t, ·)) dt
for any η ∈ C0c (ΩT ).
We next state our main assumptions.
Assumption 2.1. Let n = 2, 3 and let a sequence (uε)ε>0 of smooth functions be
given that satisfies for all ε > 0
Sε(uε) ≤ Λ1, (A1)
|∇uε|2 +
W (uε)
(0, x) dx ≤ Λ2, (A2)
where the constants Λ1,Λ2 are independent of ε > 0. Moreover we prescribe that
∇uε · νΩ = 0 on [0, T ]× ∂Ω. (A3)
Remark 2.2. It follows from (A3) that for any 0 ≤ t0 ≤ T
ε∂tuε +
− ε∆uε +
W ′(uε)
ε(∂tuε)
− ε∆uε +
W ′(uε)
|∇uε|2 +
W (uε)
(t0, x) dx − 2
|∇uε|2 +
W (uε)
(0, x) dx.
By the uniform bounds (A1), (A2) this implies that
ε(∂tuε)
− ε∆uε +
W ′(uε)
dxdt ≤ Λ3, (2.1)
0≤t≤T
|∇uε|2 +
W (uε)
(t, x) dx ≤ Λ4, (2.2)
where
Λ3 := Λ1 + 2Λ2, Λ4 :=
Λ1 + Λ2.
Remark 2.3. Our arguments would also work for any boundary conditions for
which ∂tu∇u · νΩ vanishes on ∂Ω, in particular for time-independent Dirichlet con-
ditions or periodic boundary conditions.
6 LUCA MUGNAI AND MATTHIAS RÖGER
We set
wε := −ε∆uε +
W ′(uε) (2.3)
and define for ε > 0, t ∈ (0, T ) a Radon measure µtε on Ω,
µtε :=
|∇uε|2(t, ·) +
W (uε(t, ·))
Ln, (2.4)
and for ε > 0 measures µε, αε on ΩT ,
µε :=
|∇uε|2 +
W (uε)
Ln+1, (2.5)
αε :=
ε1/2∂tuε + ε
−1/2wε
)2Ln+1. (2.6)
Eventually restricting ourselves to a subsequence ε→ 0 we may assume that
µε → µ as Radon-measures on ΩT , (2.7)
αε → α as Radon-measures on ΩT , (2.8)
for two Radon measures µ, α on ΩT , and that
α(ΩT ) = lim inf
αε(ΩT ). (2.9)
3. L2-flows
We will show that the uniform bound on the action implies the existence of
a square-integrable weak mean curvature and the existence of a square-integrable
generalized velocity. The formulation of weak mean curvature is standard in Geo-
metric Measure Theory [1, 31]. Our definition of L2-flow and generalized velocity
is similar to Brakke’s formulation of mean curvature flow [5].
Definition 3.1. Let (µt)t∈(0,T ) be any family of integer rectifiable Radon measures
such that µ := L1⊗µt defines a Radon measure on ΩT and such that µt has a weak
mean curvature H(t, ·) ∈ L2(µt) for almost all t ∈ (0, T ).
If there exists a positive constant C and a vector field v ∈ L2(µ,Rn) such that
v(t, x) ⊥ Txµt for µ-almost all (t, x) ∈ ΩT , (3.1)
∂tη +∇η · v
dµtdt
≤ C‖η‖C0(ΩT ) (3.2)
for all η ∈ C1c ((0, T ) × Ω), then we call the evolution (µt)t∈(0,T ) an L2-flow. A
function v ∈ L2(µ,Rn) satisfying (3.1), (3.2) is called a generalized velocity vector.
This definition is based on the observation that for a smooth evolution
(Mt)t∈(0,T ) with mean curvature H(t, ·) and normal velocity vector V (t, ·)
η(t, x) dHn−1(x) −
∂tη(t, x) dHn−1(x) −
∇η(t, x) · V (t, x) dHn−1(x)
H(t, x) · V (t, x)η(t, x) dHn−1(x).
Integrating this equality in time implies (3.2) for any evolution with square-
integrable velocity and mean curvature.
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 7
Remark 3.2. Choosing η(t, x) = ζ(t)ψ(x) with ζ ∈ C1c (0, T ), ψ ∈ C1(Ω), we
deduce from (3.2) that t 7→ µt(ψ) belongs to BV (0, T ). Choosing a countable dense
subset (ψi)i∈N ⊂ C0(Ω) this implies that there exists a countable set S ⊂ (0, T )
of singular times such that any good representative of t 7→ µt(ψ) is continuous in
(0, T ) \ S for all ψ ∈ C1(Ω).
Any generalized velocity is (in a set of good points) uniquely determined by the
evolution (µt)t∈(0,T ).
Proposition 3.3. Let (µt)t∈(0,T ) be an L
2-flow and set µ := L1⊗µt. Let v ∈ L2(µ)
be a generalized velocity field in the sense of Definition 3.1. Then
v(t0, x0)
∈ T(t0,x0)µ (3.3)
holds in µ-almost all points (t0, x0) ∈ ΩT where the tangential plane of µ exists.
The evolution (µt)t∈(0,T ) uniquely determines v in all points (t0, x0) ∈ ΩT where
both tangential planes T(t0,x0)µ and Tx0µ
t0 exist.
We postpone the proof to Section 8.
In the set of points where a tangential plane of µ exists, the generalized velocity
field v coincides with the normal velocity introduced in [4].
We turn now to the statement of a lower bound for sequences (uε)ε>0 satisfying
Assumption 2.1. As ε→ 0 we will obtain a phase indicator function u as the limit of
the sequence (uε)ε>0 and an L
2-flow (µt)t∈(0,T ) as the limit of the measures (µε)ε>0.
We will show that in Hn-almost all points of the phase boundary ∂∗{u = 1} ∩Ω a
tangential plane of µ exists. This implies the existence of a unique normal velocity
field of the phase boundary.
4. Lower bound for the action functional
In several steps we state a lower bound for the functionals Sε. We postpone all
proofs to Sections 5-8.
4.1. Lower estimate for the mean curvature. We start with an application of
the well-known results of Modica and Mortola [23, 22].
Proposition 4.1. There exists u ∈ BV (ΩT , {−1, 1})∩L∞(0, T ;BV (Ω)) such that
for a subsequence ε→ 0
uε → u in L1(ΩT ), (4.1)
uε(t, ·) → u(t, ·) in L1(Ω) for almost all t ∈ (0, T ). (4.2)
Moreover
d|∇′u| ≤ Λ3 + TΛ4,
d|∇u(t, ·)| ≤ Λ4 (4.3)
holds, where c0 was defined in (1.4).
The next proposition basically repeats the arguments in [19, Theorem 1.1].
Proposition 4.2. There exists a countable set S ⊂ (0, T ), a subsequence ε → 0
and Radon measures µt, t ∈ [0, T ] \ S, such that for all t ∈ [0, T ] \ S
µtε → µt as Radon measures on Ω, (4.4)
8 LUCA MUGNAI AND MATTHIAS RÖGER
such that
µ = L1 ⊗ µt, (4.5)
and such that for all ψ ∈ C1(Ω) the function
t 7→ µt(ψ) is of bounded variation in (0, T ) (4.6)
and has no jumps in (0, T ) \ S.
Exploiting the lower bound [28] for the diffuse approximation of the Willmore
functional (1.8) we obtain that the measures µt are up to a constant integer-
rectifiable with a weak mean curvature satisfying an appropriate lower estimate.
Theorem 4.3. For almost all t ∈ (0, T )
µt is an integral (n− 1)-varifold,
µt has weak mean curvature H(t, ·) ∈ L2(µt),
and the estimate
|H |2 dµ ≤ lim inf
w2ε dxdt (4.7)
holds.
4.2. Lower estimate for the generalized velocity.
Theorem 4.4. Let (µt)t∈(0,T ) be the limit measures obtained in Proposition 4.2.
Then there exists a generalized velocity v ∈ L2(µ,Rn) of (µt)t∈(0,T ). Moreover the
estimate
|v|2 dµ ≤ lim inf
ε(∂tuε)
2 dxdt (4.8)
is satisfied. In particular, ( 1
µt)t∈(0,T ) is an L
2-flow.
We obtain v as a limit of suitably defined approximate velocities, see Lemma 6.2.
On the phase boundary v coincides with the (standard) distributional velocity of the
bulk-phase {u(t, ·) = 1}. However, our definition extends the velocity also to ‘hid-
den boundaries’, which seems necessary in order to prove the Gamma-convergence
of the action functional; see the discussion in Section 9.
Proposition 4.5. Define the generalized normal velocity V in direction of the
inner normal of {u = 1} by
V (t, x) := v(t, x) ·
(t, x), for (t, x) ∈ ∂∗{u = 1}.
Then V ∈ L1(|∇u|) holds and V |∂∗{u=1} is the unique vector field that satisfies for
all η ∈ C1c (ΩT )
V (t, x)η(t, x) d|∇u(t, ·)|(x)dt = −
u∂tη dxdt. (4.9)
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 9
4.3. Lower estimate of the action functional. As our main result we obtain
the following lower estimate for Sε.
Theorem 4.6. Let Assumption 2.1 hold, and let µ, (µt)t∈[0,T ], and S be the mea-
sures and the countable set of singular times that we obtained in Proposition 4.2.
Define the nucleation cost Snuc(µ) by
Snuc(µ) :=
µt(ψ)− lim
µt(ψ)
+ sup
µt(ψ)− µ0(ψ)
+ sup
µT (ψ)− lim
µt(ψ)
(4.10)
where the sup is taken over all ψ ∈ C1(Ω) with 0 ≤ ψ ≤ 1. Then
lim inf
Sε(uε) ≥
|v −H |2 dµ+ 4Snuc(µ). (4.11)
In the previous definition of nucleation cost we have tacitly chosen good represen-
tatives of µt(ψ) (see [3]). With this choice the jump parts in (4.10) are well-defined.
Eventually let us remark that, in view of Theorem 4.3, we can conclude that
Snuc does indeed measure only (n− 1)-dimensional jumps.
Theorem 4.6 improves [18] in the higher-multiplicity case. We will discuss our
main results in Section 9.
4.4. Convergence of the Allen–Cahn equation to Mean curvature flow.
Let n = 2, 3 and consider solutions (uε)ε>0 of the Allen–Cahn equation (1.6) sat-
isfying (A2) and (A3). Then Sε(uε) = 0 and the results of Sections 4.1-4.3 apply:
There exists a subsequence ε → 0 such that the phase functions uε converge to a
phase indicator function u, such that the energy measures µtε converge an L
2-flow
(µt)t∈(0,T ), and such that µ-almost everywhere
H = v (4.12)
holds, where H(t, ·) denotes the weak mean curvature of µt and where v denotes the
generalized velocity of (µt)t∈(0,T ) in the sense of Definition 3.1. Moreover Snuc(µ) =
0, which shows that for any nonnegative ψ ∈ C1(Ω) the function t 7→ µt(ψ) cannot
jump upwards. From (1.6) and (5.3) below one obtains that for any ψ ∈ C1(Ω) and
all ζ ∈ C1c (0, T )
∂tζ µ
ε(ψ) dt = −
ψ(x)w2ε (t, x) +∇ψ(x) · ∇uεwε(t, x)
dxdt.
(4.13)
We will show that suitably defined ‘diffuse mean curvatures’ converge as ε→ 0, see
(7.1). Using this result we can pass to the limit in (4.13) and we obtain for any
nonnegative functions ψ ∈ C1(Ω), ζ ∈ C1c (0, T ) that
t(ψ) dt ≤ −
H2(t, x) +∇ψ(x) ·H(t, x)
dµt(x)dt,
which is an time-integrated version of Brakke’s inequality.
10 LUCA MUGNAI AND MATTHIAS RÖGER
5. Proofs of Propositions 4.1, 4.2 and Theorem 4.3
Proof of Proposition 4.1. By (2.1), (2.2) we obtain that
|∇′uε|2 +
W (uε)
dxdt ≤ Λ3 + TΛ4.
This implies by [22] the existence of a subsequence ε → 0 and of a function u ∈
BV (ΩT ; {−1, 1}) such that
uε → u in L1(ΩT )
d|∇′u| ≤ lim inf
|∇′uε|2 +
W (uε)
dxdt ≤
Λ3 + TΛ4
After possibly taking another subsequence, for almost all t ∈ (0, T )
uε(t, ·) → u(t, ·) in L1(Ω) (5.1)
holds. Using (2.2) and applying [22] for a fixed t ∈ (0, T ) with (5.1) we get that
d|∇u|(t, ·) ≤ lim inf
µtε(Ω) ≤ Λ4.
Before proving Proposition 4.2 we show that the time-derivative of the energy-
densities µtε is controlled.
Lemma 5.1. There exists C = C(Λ1,Λ3,Λ4) such that for all ψ ∈ C1(Ω)
|∂tµtε(ψ)| dt ≤ C‖ψ‖C1(Ω). (5.2)
Proof. Using (A3) we compute that
ε(ψ) =
ε∂tuε +
(t, x)ψ(x) dx −
ε(∂tuε)
(t, x)ψ(x) dx
ε∇ψ(x) · ∂tuε(t, x)∇uε(t, x) dx. (5.3)
By (2.1), (2.2) we estimate
ε∇ψ · ∂tuε∇uε dxdt
ε(∂tuε)
2 + ε|∇uε|2
≤ (Λ3 + TΛ4)‖∇ψ‖C0(Ω) (5.4)
and deduce from (A1), (2.1), (5.3) that
|∂tµtε(ψ)| dt ≤ (Λ1 + Λ3)‖ψ‖C0(Ω) + C(Λ3, TΛ4)‖∇ψ‖C0(Ω),
which proves (5.2). �
Proof of Proposition 4.2. By (2.7) µε → µ as Radon-measures on ΩT . Choose now
a countable family (ψi)i∈N ⊂ C1(Ω) which is dense in C0(Ω). By Lemma 5.1
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 11
and a diagonal-sequence argument there exists a subsequence ε→ 0 and functions
mi ∈ BV (0, T ), i ∈ N, such that for all i ∈ N
µtε(ψi) → mi(t) for almost-all t ∈ (0, T ), (5.5)
ε(ψi) → m′i as Radon measures on (0, T ). (5.6)
Let S denote the countable set of times t ∈ (0, T ) where for some i ∈ N the measure
m′i has an atomic part in t. We claim that (5.5) holds on (0, T ) \ S. To see this
we choose a point t ∈ (0, T ) \ S and a sequence of points (tj)j∈N in (0, T ) \ S, such
that tj ր t and (5.5) holds for all tj . We then obtain
m′i([tj , t]) = 0 for all i ∈ N, (5.7)
ε(ψi)([tj , t]) = m
i([tj , t]) for all i, j ∈ N. (5.8)
Moreover
|mi(t)− µtε(ψi)| ≤ |mi(t)−mi(tj)|+ |mi(tj)− µtjε (ψi)|+ |µtjε (ψi)− µtε(ψi)|
≤ |m′i([tj , t])|+ |mi(tj)− µtjε (ψi)|+ |∂tµtε(ψi)([tj , t])|
Taking first ε → 0 and then tj ր t we deduce by (5.7), (5.8) that (5.5) holds for
all i ∈ N and all t ∈ (0, T ) \ S.
Taking now an arbitrary t ∈ (0, T ) such that (5.5) holds, by (2.2) there exists a
subsequence ε→ 0 such that
µtε → µt as Radon-measures on Ω. (5.9)
We deduce that µt(ψi) = mi(t) and since (ψi)i∈N is dense in C
0(Ω) we can identify
any limits of (µtε)ε>0 and obtain (5.9) for the whole sequence selected in (5.5), (5.6)
and for all t ∈ (0, T ), for which (5.5) holds. Moreover for any ψ ∈ C0(Ω) the map
t 7→ µtε(ψ) has no jumps in (0, T ) \ S. After possibly taking another subsequence
we can also ensure that as ε→ 0
µ0ε → µ0, µTε → µT
as Radon measures on Ω. This proves (4.4).
By the Dominated Convergence Theorem we conclude that for any η ∈ C0(ΩT )
η dµ = lim
η dµε = lim
η(t, x) dµtε(x) dt =
η(t, x) dµt(x) dt,
which implies (4.5).
By (5.2), the L1(0, T )-compactness of sequences that are uniformly bounded in
BV (0, T ), the lower-semicontinuity of the BV -norm under L1-convergence, and
(4.4) we conclude that (4.6) holds. �
Proof of Theorem 4.3. For almost all t ∈ (0, T ) we obtain from Fatou’s Lemma and
(2.1), (2.2) that
lim inf
µtε(Ω) +
w2ε(t, x) dx
< ∞. (5.10)
12 LUCA MUGNAI AND MATTHIAS RÖGER
Let S ⊂ (0, T ) be as in Proposition 4.2 and fix a t ∈ (0, T ) \ S such that (5.10)
holds. Then we deduce from [28, Theorem 4.1, Theorem 5.1] and (4.4) that
µt is an integral (n− 1)-varifold,
µt ≥ c0
|∇u(t, ·)|,
and that µt has weak mean curvature H(t, ·) satisfying
|H(t, x)|2 dµt(x) ≤ lim inf
wε(t, x)
2 dx. (5.11)
By (5.11) and Fatou’s Lemma we obtain that
|H(t, x)|2 dµt(x) dt ≤
lim inf
wε(t, x)
≤ lim inf
w2ε dxdt,
which proves (4.7).
For later use we also associate general varifolds to µtε and consider their conver-
gence as ε→ 0. Let νε(t, ·) : Ω → Sn−11 (0) be an extension of∇uε(t, ·)/|∇uε(t, ·)| to
the set {∇uε(t, ·) = 0}. Define the projections Pε(t, x) := Id−νε(t, x)⊗νε(t, x) and
consider the general varifolds V tε and the integer rectifiable varifold c
t defined
V tε (f) :=
f(x, Pε(t, x)) dµ
ε(x), (5.12)
V t(f) :=
f(x, P (t, x)) dµt(x) (5.13)
for f ∈ C0c (Ω × Rn×n), where P (t, x) ∈ Rn×n denotes the projection onto the
tangential plane Txµ
t. Then we deduce from the proof of [28, Theorem 4.1] that
V tε → V t as ε→ 0 (5.14)
in the sense of varifolds. �
6. Proof of Theorem 4.4
6.1. Equipartition of energy. We start with a preliminary result, showing the
important equipartition of energy: the discrepancy measure
ξε :=
|∇uε|2 −
W (uε)
Ln+1 (6.1)
vanishes in the limit ε→ 0.
To prove this we combine results from [28] with a refined version of Lebesgue’s
dominated convergence Theorem [25], see also [27, Lemma 4.2].
Proposition 6.1. For a subsequence ε→ 0 we obtain that
|ξε| → 0 as Radon measures on ΩT . (6.2)
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 13
Proof. Let us define the measures
ξtε :=
|∇uε|2 −
W (uε)
(t, ·)Ln
on Ω. For ε > 0, k ∈ N, we define the sets
Bε,k := {t ∈ (0, T ) :
w2ε(t, x) dx > k}. (6.3)
We then obtain from (2.1) that
w2ε(t, x) dxdt ≥ |Bε,k|k. (6.4)
Next we define the (signed) Radon-measures ξtε,k by
ξtε,k :=
ξtε for t ∈ (0, T ) \ Bε,k,
0 for t ∈ Bε,k.
(6.5)
By [28, Proposition 4.9], we have
|ξtεj | → 0 (j → ∞) as Radon measures on Ω (6.6)
for any subsequence εj → 0 (j → ∞) such that
lim sup
w2εj (t, x) dx < ∞.
By (2.2), (6.5) we deduce that for any η ∈ C0(ΩT ,R+0 ), k ∈ N, and almost all
t ∈ (0, T )
|ξtε,k|(η(t, ·)) → 0 as ε→ 0 (6.7)
and that
|ξtε,k|(η(t, ·)) =
1−XBε,k(t)
|ξtε|
η(t, ·)
≤ Λ4‖η‖C0(ΩT ). (6.8)
By the Dominated Convergence Theorem, (6.7) and (6.8) imply that
|ξtε,k|(η(t, ·)) dt → 0 as ε→ 0. (6.9)
Further we obtain that
|ξtε|(η(t, ·)) dt ≤
|ξtε,k|(η(t, ·)) dt +
|ξtε|(η(t, ·)) dt
|ξtε,k|(η(t, ·)) dt +
µtε(η(t, ·)) dt. (6.10)
For k ∈ N fixed we deduce from (2.2), (6.4), (6.10) that
lim sup
|ξtε|(η(t, ·)) dt ≤ lim
|ξtε,k|(η(t, ·)) dt + ‖η‖C0(ΩT )Λ4
. (6.11)
By (6.9) and since k ∈ N was arbitrary this proves the Proposition. �
14 LUCA MUGNAI AND MATTHIAS RÖGER
6.2. Convergence of approximate velocities. In the next step in the proof of
Theorem 4.4 we define approximate velocity vectors and show their convergence as
ε→ 0.
Lemma 6.2. Define vε : ΩT → Rn by
vε :=
− ∂tuε
|∇uε|
|∇uε|
if |∇uε| 6= 0,
0 otherwise.
(6.12)
Then there exists a function v ∈ L2(µ,Rn) such that
(ε|∇uε|2 Ln+1, vε) → (µ, v) as ε→ 0 (6.13)
in the sense of measure function pair convergence (see the Appendix B) and such
that (4.8) is satisfied.
Proof. We define Radon measures
µ̃ε := ε|∇uε|2 Ln+1 = µε + ξε. (6.14)
From (2.7), (6.2) we deduce that
µ̃ε → µ as Radon measures on ΩT . (6.15)
Next we observe that (µ̃ε, vε) is a function-measure pair in the sense of [16] (see
also Definition B.1 in Appendix B) and that by (2.1)
|vε|2 dµ̃ε ≤
ε(∂tuε)
2 dxdt ≤ Λ3. (6.16)
By Theorem B.3 we therefore deduce that there exists a subsequence ε → 0 and a
function v ∈ L2(µ,Rn) such that (6.13) and (4.8) hold. �
Lemma 6.3. For µ-almost all (t, x) ∈ ΩT
v(t, x) ⊥ Txµt. (6.17)
Proof. We follow [24, Proposition 3.2]. Let νε : ΩT → Sn−11 (0) be an extension of
∇uε/|∇uε| to the set {∇uε = 0} and define projection-valued maps Pε : ΩT →
Pε := Id− νε ⊗ νε.
Consider next the general varifolds Ṽε, V defined by
Ṽε(f) :=
f(t, x, Pε(t, x)) dµ̃ε(t, x), (6.18)
V (f) :=
f(t, x, P (t, x)) dµt(x) (6.19)
for f ∈ C0c (ΩT × Rn×n), where P (t, x) ∈ Rn×n denotes the projection onto the
tangential plane Txµ
From (5.14), Proposition 6.1, and Lebesgue’s Dominated Convergence Theorem
we deduce that
Ṽε = V (6.20)
as Radon-measures on ΩT × Rn×n.
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 15
Next we define functions v̂ε on ΩT × Rn×n by
v̂ε(t, x, Y ) = vε(t, x) for all (t, x) ∈ ΩT , Y ∈ Rn×n.
We then observe that
ΩT×Rn×n
v̂2ε dVε =
v2ε dµ̃ε ≤ Λ3
and deduce from (6.20) and Theorem B.3 the existence of v̂ ∈ L2(V,Rn) such that
(Vε, v̂ε) converge to (V, v̂) as measure-function pairs on ΩT × Rn×n with values in
We consider now h ∈ C0c (Rn×n) such that h(Y ) = 1 for all projections Y . We
deduce that for any η ∈ C0c (ΩT ,Rn)
η · v dµ = lim
ΩT×Rn×n
η(t, x) · h(Y )v̂ε(t, x, Y ) dVε(t, x, Y )
η(t, x) · v̂(t, x, P (t, x)) dµ(t, x),
which shows that for µ-almost all (t, x) ∈ ΩT
v̂(t, x, P (t, x)) = v(t, x). (6.21)
Finally we observe that for h, η as above
η(t, x) · P (t, x)v(t, x) dµ(t, x)
ΩT×Rn×n
η(t, x)h(Y ) · Y v̂(t, x, Y ) dV (t, x, Y )
= lim
ΩT×Rn×n
η(t, x)h(Y ) · Y v̂ε(t, x, Y ) dVε(t, x, Y )
= lim
η(t, x) · Pε(t, x)vε(t, x) dµ̃ε(t, x) = 0
since Pεvε = 0. This shows that P (t, x)v(t, x) = 0 for µ-almost all (t, x) ∈ ΩT . �
Proof of Theorem 4.4. By (2.1) there exists a subsequence ε → 0 and a Radon
measure β on ΩT such that
ε(∂tuε)
Ln+1 → β, β(ΩT ) ≤ Λ3. (6.22)
Using (A3) we compute that for any η ∈ C1c ((0, T )× Ω)
η dαε =
ε(∂tuε)
dxdt − 2
∂tη dµε
ε∇η · ∂tuε∇uε dxdt .
(6.23)
As ε tends to zero the term on the left-hand side and the first two terms on the
right-hand-side converge by (2.7), (2.8) and (6.22). For the third term on the
16 LUCA MUGNAI AND MATTHIAS RÖGER
right-hand side of (6.23) we obtain from (6.13) that
∇η · ε∂tuε∇uε dxdt = − lim
∇η · vε ε|∇uε|2 dxdt
∇η · v dµ.
Therefore, taking ε→ 0 in (6.23) we deduce that
η dα =
η dβ − 2
∂tη dµ− 2
∇η · v dµ
holds for all η ∈ C1c ((0, T )× Ω). This yields that
∂tη +∇η · v dµ
≤ ‖η‖C0(ΩT )
α(ΩT ) + β(ΩT )
which shows together with (6.17) that v is a generalized velocity vector for
(µt)t∈(0,T ) in the sense of Definition 3.1. The estimate (4.8) was already proved in
Lemma 6.2. �
7. Proof of Theorem 4.6
We start with the convergence of a ‘diffuse mean curvature term’.
Lemma 7.1. Define
Hε :=
|∇uε|2
let µ̃ε = ε|∇uε|2 Ln+1, and let vε, v be as in (6.12), (6.13). Then
(µ̃ε, Hε) → (µ,H), (7.1)
(µ̃ε, vε −Hε) → (µ, v −H) (7.2)
as ε→ 0 in the sense of measure function pair convergence. In particular
η|v −H |2 dµ ≤ α(η) (7.3)
holds for all η ∈ C0(ΩT ,R+0 ).
Proof. We use similar arguments as in the proof of Proposition 6.1. For ε > 0,
k ∈ N, we define sets
Bε,k := {t ∈ (0, T ) :
wε(t, x)
2 dx > k}. (7.4)
We then obtain from (2.1) that
w2ε dxdt ≥ |Bε,k|k. (7.5)
Next we define functionals T tε,k ∈ C0c (Ω,Rn)∗ by
T tε,k(ψ) :=
ψ(x) · wε(t, x)∇uε(t, x) dx for t ∈ (0, T ) \ Bε,k,
ψ(x) ·H(t, x) dµt(x) for t ∈ Bε,k.
(7.6)
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 17
Considering the general (n− 1)-varifolds V tε , V t defined in (5.12), (5.13) we obtain
from [28, Proposition 4.10] and (5.14) that
ψ · wεj (t, x)∇uεj (t, x) dx = − lim
δV tεj (ψ)
= − δµt(ψ) =
ψ ·H(t, x) dµt(x)
(7.7)
for any subsequence εj → 0 (j → ∞) such that
lim sup
w2εj dxdt < ∞.
Therefore we deduce from (7.6), (7.7) that for all η ∈ C0c (ΩT ,Rn), k ∈ N, and
almost all t ∈ (0, T )
T tε,k(η(t, ·)) →
η(t, x) ·H(t, x) dµt(x) as ε→ 0 (7.8)
and that
∣T tε,k(η(t, ·))
1−XBε,k(t)
η(t, x) · wε(t, x)∇uε(t, x) dx
+ XBε,k(t)
η(t, x) ·H(t, x) dµt(x)
≤‖η‖C0(ΩT )
1−XBε,k(t)
wε(t, x)
)1/2(
|∇uε(t, x)|2 dx
|η(t, x)||H(t, x)| dµt(x)
≤‖η‖C0(ΩT )
|η(t, x)||H(t, x)| dµt(x), (7.9)
where the right-hand side is bounded in L1(0, T ), uniformly with respect to ε > 0.
By the Dominated Convergence Theorem, (7.8) and (7.9) imply that
T tε,k(η(t, ·)) dt →
η ·H dµ as ε→ 0. (7.10)
Further we obtain that
η · wε∇uε dxdt−
η ·H dµ
T tε,k(η(t, ·)) dt−
η ·H dµ
η(t, x) ·H(t, x) dµt(x)dt
η · wε∇uε dx dt
(7.11)
18 LUCA MUGNAI AND MATTHIAS RÖGER
The last term on the right-hand side we further estimate by
η(t, x) · wε(t, x)∇uε(t, x) dx dt
≤‖η‖C0(ΩT )
w2ε dxdt
|Bε,k|1/2
≤‖η‖C0(ΩT )Λ3
Λ4, (7.12)
where we have used (2.2) and (7.5). For the second term on the right-hand side of
(7.11) we obtain
η(t, x) ·H(t, x) dµt(x)dt
≤ |Bε,k|1/2‖η‖1/2C0(ΩT )
supp(η)
H2 dµ
‖η‖1/2
C0(ΩT )
Λ3, (7.13)
where we have used (4.7) and (2.1). Finally, for k ∈ N fixed, by (7.10) we deduce
T tε,k(η(t, ·)) dt −
η ·H dµ
= 0. (7.14)
Taking ε→ 0 in (7.11) we obtain by (7.12)-(7.14) that
η · wε∇uε dxdt−
η ·H dµ
≤ Λ3√
‖η‖C0(ΩT )
Λ3 (7.15)
for any k ∈ N, which proves (7.1). Using (6.13) this implies (7.2). Finally we fix
an arbitrary nonnegative η ∈ C0(ΩT ) and deduce that the measure-function pair
(µ̃ε,
η(vε − Hε)) converges to (µ,
η(v − H)). The estimate (7.3) then follows
from Theorem B.3. �
Let Π : [0, T ]× Ω → [0, T ] denote the projection onto the first component and
Π# the pushforward of measures by Π. For ψ ∈ C0(Ω) we consider the measures
αψ := Π#
on [0, T ], that means
αψ(ζ) :=
ζ(t)ψ(x) dα(t, x),
for ζ ∈ C0([0, T ]), and set
αΩ := Π#α.
We then can estimate the atomic part of αΩ in terms of the nucleation cost.
Lemma 7.2. Let Snuc(µ) be the nucleation cost defined in (4.10). Then
(αΩ)atomic[0, T ] ≥ 4Snuc(µ). (7.16)
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 19
Proof. Let η ∈ C1(ΩT ,R+0 ) be nonnegative. We compute that
ηdαε =
ε(∂tuε)
w2ε + 2∂tuεwε
η∂tuεwε dxdt
= − 4
∂tη dµε + 4
∇η · ε∂tuε∇uε dxdt
+ 4µTε (η(T, ·))− 4µ0ε(η(0, ·)). (7.17)
Passing to the limit ε→ 0 we obtain from (2.7), (4.4), (6.13) that
ηdα ≥ − 4
∂tη dµ− 4
∇η · v dµ+ 4µT (η(T, ·))− 4µ0(η(0, ·)). (7.18)
We now choose η(t, x) = ζ(t)ψ(x) where ζ ∈ C1([0, T ],R+0 ), ψ ∈ C1(Ω,R
0 ) in
(7.18) and deduce that
ζdαψ ≥ − 4
t(ψ) dt+ 4
∇ψ · v(t, x) dµt(x) dt
+ 4ζ(T )µT (ψ)− 4ζ(0)µ0(ψ). (7.19)
This shows that
αψ ≥ 4∂t(µt(ψ)) + 4
∇ψ(x) · v(t, x) dµt(x)
µT (ψ)− lim
µt(ψ)
δT + 4
µt(ψ)− µ0(ψ)
δ0. (7.20)
Evaluating the atomic parts we obtain that for any 0 < t0 < T
αψ({t0}) ≥ 4∂t(µt(ψ))({t0}),
which implies that
αΩ({t0}) ≥ 4 sup
t(ψ))({t0}). (7.21)
where the supremum is taken over all ψ ∈ C1(Ω) with 0 ≤ ψ ≤ 1.
Moreover we deduce from (7.20)
αΩ({0}) ≥ 4 sup
µt(ψ)− µ0(ψ)
, (7.22)
αΩ({T }) ≥ 4 sup
µT (ψ) − lim
µt(ψ)
, (7.23)
where the supremum is taken over ψ ∈ C(Ω) with 0 ≤ ψ ≤ 1. By (7.21)-(7.23) we
conclude that (7.16) holds. �
Proof of Theorem 4.6. By (7.3) we deduce that α ≥ |v −H |2µ. Since µ = L1 ⊗ µt
we deduce from the Radon-Nikodym Theorem that
(αΩ)ac[0, T ] ≥
|v −H |2 dµ, (7.24)
and from (7.16) that
(αΩ)atomic[0, T ] ≥ 4Snuc(µ), (7.25)
20 LUCA MUGNAI AND MATTHIAS RÖGER
where (αΩ)ac and (αΩ)atomic denote the absolutely continuous and atomic part
with respect to L1 of the measure αΩ. Adding the two estimates and recalling (2.9)
we obtain (4.11). �
8. Proofs of Proposition 3.3 and Proposition 4.5
Define for r > 0, (t0, x0) ∈ ΩT the cylinders
Qr(t0, x0) := (t0 − r, t0 + r) ×Bnr (x0).
Proof of Proposition 3.3. Define
Σn(µ) :=
(t, x) ∈ ΩT : the tangential plane of µ in (t, x) exists
(8.1)
and choose (t0, x0) ∈ Σn(µ) such that
v is approximately continuous with respect to µ in (t0, x0). (8.2)
Since v ∈ L2(µ) we deduce from [11, Theorem 2.9.13] that (8.2) holds µ-almost
everywhere. Let
P0 := T(t0,x0)µ, θ0 > 0 (8.3)
denote the tangential plane and multiplicity at (t0, x0) respectively, and define for
any ϕ ∈ C0c (Q1(0)) the scaled functions ϕ̺ ∈ C0c (Q̺(t0, x0)),
ϕ̺(t, x) := ̺
̺−1(t− t0), ̺−1(x− x0)
We then obtain from (8.3) that
ϕ̺ dµ → θ0
ϕdHn as ̺ց 0. (8.4)
From (3.2), the Hahn–Banach Theorem, and the Riesz Theorem we deduce that
ϑ ∈ C1c (ΩT )∗, ϑ(η) :=
∇′η ·
dµ (8.5)
can be extended to a (signed) Radon-measure on ΩT . Since by the Radon-Nikodym
Theorem Dµ|ϑ| exists and is finite µ-almost everywhere we may assume without
loss of generality that
Dµ|ϑ|(t0, x0) < ∞. (8.6)
We next fix η ∈ C1c (Q1(0)) and compute that
ϑ(̺η̺) =
dµ. (8.7)
From (8.2), (8.4) we deduce that the right-hand side converges in the limit ̺→ 0,
dµ = θ0
v(t0, x0)
∇′η dµ. (8.8)
For the left-hand side of (8.7) we deduce that
lim inf
|ϑ(̺η̺)| ≤ ‖η‖C0c (Q1(0)) lim inf̺ց0 ̺
−n+1|ϑ|(Q̺(t0, x0)) (8.9)
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 21
and observe that (8.6) implies
∞ > lim
|ϑ|(Q̺(t0, x0))
µ(Q̺(t0, x0))
≥ lim inf
̺−n|ϑ|(Q̺(t0, x0))
lim sup
̺−nµ(Q̺(t0, x0))
≥ c lim inf
̺−n|ϑ|(Q̺(t0, x0)), (8.10)
since by (8.4) for any ϕ ∈ C0c (Q2(0),R+0 ) with ϕ ≥ 1 on Q1(0)
lim sup
̺−nµ(Q̺(t0, x0)) ≤ lim sup
ϕ̺ dµ ≤ C(ϕ).
Therefore (8.7)-(8.10) yield
v(t0, x0)
∇′η dµ = 0. (8.11)
Now we observe that the integral over the projection of ∇′η onto P0 vanishes. This
shows that
∇′η dHn ∈ P⊥0 . (8.12)
Since η can be chosen such that the integral in (8.12) takes an arbitrary direction
normal to P0 we obtain from (8.11) that v(t0, x0) satisfies (3.3). If Tx0µ
t0 exists
T(t0,x0)µ =
{0} × Tx0µt0
⊕ span
v(x0)
and we obtain that v is uniquely determined. �
To prepare the proof of Proposition 4.5 we first show that µ is absolutely con-
tinuous with respect to Hn.
Proposition 8.1. For any D ⊂⊂ Ω there exists C(D) such that for all x0 ∈ D
and almost all t0 ∈ (0, T )
lim sup
Qr(t0, x0)
≤ C(D)Λ4 + lim inf
w2ε(t0, x) dx. (8.13)
In particular,
lim sup
µ(Bρ(t0, x0))
< ∞ for µ− almost every (t0, x0) (8.14)
and µ is absolutely continuous with respect to Hn,
µ << Hn. (8.15)
Proof. Let
r0 := min
dist(D, ∂Ω), |t0|, |T − t0|
Then we obtain for all r < r0, x0 ∈ D, from (6.2) and [28, Proposition 4.5] that
∫ t0+r
r1−nµt
Bnr (x0)
∫ t0+r
r1−n0 µ
Bnr0(x0)
4(n− 1)2
∫ t0+r
lim inf
w2ε(t, x) dx
(8.16)
22 LUCA MUGNAI AND MATTHIAS RÖGER
By Fatou’s Lemma and (2.1)
t 7→ lim inf
w2ε(t, x) dx is in L
1(0, T ) (8.17)
and by (2.2) we deduce for almost all t0 ∈ (0, T ) that
lim sup
∫ t0+r
r1−nµt
Bnr (x0)
≤ 2r1−n0 Λ4 +
2(n− 1)2
lim inf
w2ε(t0, x) dx.
Since r0 depends only on D,Ω the inequality (8.13) follows.
By (8.17) the right-hand side in (8.13) is finite for L1-almost all t0 ∈ (0, T ) and
θ∗n(µ, (t, x)) is bounded for almost all t ∈ (0, T ) and all x ∈ Ω. By (2.2) we deduce
that for any I ⊂ (0, T ) with |I| = 0
µ(I × Ω) ≤ Λ4|I| = 0
which implies (8.14).
To prove the final statement let B ⊂ ΩT be given with
Hn(B) = 0. (8.18)
Consider the family of sets (Dk)k∈N,
Dk := {z ∈ ΩT : θ∗n(µ, z) ≤ k}.
By (8.14), [31, Theorem 3.2], and (8.18) we obtain that for all k ∈ N
µ(B ∩Dk) ≤ 2nkHn(B ∩Dk) = 0. (8.19)
Moreover we have that
µ(B \
Dk) = 0 (8.20)
by (8.14). By (8.19), (8.20) we conclude that
µ(B) = 0,
which proves (8.15). �
To prove Proposition 4.5 we need that Hn-almost everywhere on ∂∗{u = 1} the
generalized tangent plane of µ exists. We first obtain the following relation between
the measures µ and |∇′u|.
Proposition 8.2. There exists a nonnegative function g ∈ L2(µ,R+0 ) such that
g µ ≥ c0
|∇′u|. (8.21)
In particular, |∇′u| is absolutely continuous with respect to µ,
|∇′u| << µ. (8.22)
Proof. Let
G(r) =
2W (s) ds. (8.23)
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 23
On the set {|∇uε| 6= 0} we have
|∇G(uε)| =
|∇G(uε)|
|∇′G(uε)|
|∇′G(uε)|
|∇G(uε)|
∂tG(uε)2 + |∇G(uε)|2
|∇′G(uε)|
1 + |v2ε |
|∇′G(uε)|. (8.24)
Letting µ̃ε as in (6.14) we get from (6.16), (2.2), and Theorem B.3 the existence of
a function g ∈ L2(µ) such that (up to a subsequence)
(µ̃ε,
1 + |vε|2) = (µ, g) (8.25)
as measure-function pairs on ΩT with values in R.
Let η ∈ C0c (ΩT ). Then
1 + |vε|2 |∇G(uε)| dxdt −
1 + |vε|2 dµ̃ε
1 + |vε|2
2W (uε)
ε|∇uε|
ε|∇uε| dxdt
η2(1 + |vε|2)ε|∇uε|2 dxdt
)1/2∥
2W (uε)
ε|∇uε|
L2(ΩT )
≤‖η‖L∞(2TΛ4 + Λ3)1/2(2|ξε|(ΩT ))1/2. (8.26)
Thanks to (8.25), (8.26) and (6.2) we conclude that
(|∇G(uε)| Ln+1,
1 + |vε|2) = (µ, g) (8.27)
as measure-function pairs on ΩT with values in R.
Again by (2.1) we have
{0=|∇uε|<W (uε)}
|∇′G(uε)| dxdt
{0=|∇uε|<W (uε)}
|∂tuε|
2W (uε) dxdt
ε(∂tuε)
2 dxdt
)1/2(
{0=|∇uε|<W (uε)}
W (uε)
2Λ3(|ξε|(ΩT ))1/2,
which vanishes by (6.2) as ε→ 0. This implies together with (8.24) and (8.27) that
η g dµ = lim
1 + |vε|2|∇G(uε)| dxdt
= lim
η|∇′G(uε)| dxdt ≥
η d|∇′u|,
where in the last line we used that
η d|∇′u| =
η d|∇′G(u)| ≤ lim inf
η|∇′G(uε)| dxdt.
24 LUCA MUGNAI AND MATTHIAS RÖGER
Considering now a set B ⊂ ∂∗{u = 1} with µ(B) = 0 we conclude that
|∇′u|(B) ≤ 2
g dµ = 0,
since g ∈ L2(µ). �
Proposition 8.3. In Hn-almost-all points in ∂∗{u = 1} the tangential-plane of µ
exists.
Proof. From the Radon-Nikodym Theorem we obtain that the derivative
f(z) := D|∇′u|µ(z) := lim
µ(Bn+1r (z))
|∇′u|(Bn+1r (z))
(8.28)
exists for |∇′u|-almost-all z ∈ ΩT and that f ∈ L1(|∇′u|). By (8.15) we deduce
µ⌊∂∗{u = 1} = f |∇′u|. (8.29)
Similarly we obtain that
= Dµ|∇′u|(z)
is finite for µ-almost all z ∈ ∂∗{u = 1}. By (8.22) this implies that
f > 0 |∇′u|-almost everywhere in ΩT . (8.30)
Since |∇′u| is rectifiable and f measurable with respect to |∇′u| we obtain from
(8.29), (8.30) and [31, Remark 11.5] that
µ⌊∂∗{u = 1} is rectifiable. (8.31)
Moreover Hn-almost-all z ∈ ∂∗{u = 1} satisfy that
µ(Bn+1r (z) \ ∂∗{u = 1})
µ(Bn+1r (z))
= 0, (8.32)
lim sup
µ(Bn+1r (z))
< ∞. (8.33)
In fact, (8.32) follows from [11, Theorem 2.9.11] and (8.22), and (8.33) from Propo-
sition 8.1 and (8.22). Let now z0 ∈ ∂∗{u = 1} satisfy (8.32), (8.33). For an
arbitrary η ∈ C0c (Bn+11 (0)) we then deduce that
lim sup
ΩT \∂∗{u=1}
r−1(z − z0)
r−n dµ(z)
≤‖η‖C0c (Bn+11 (0)) lim supr→0
Bn+1r (z0) \ ∂∗{u = 1}
Bn+1r (z0)
) lim sup
Bn+1r (z0)
by (8.32), (8.33). Therefore
r−1(z − z0)
r−n dµ(z) = lim
∂∗{u=1}
r−1(z − z0)
r−n dµ(z)
if the latter limit exists. By (8.31) we therefore conclude that in Hn-almost-all
points of ∂∗{u = 1} the tangent-plane of µ exists and coincides with the tangent
plane of µ⌊∂∗{u = 1}. �
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 25
Proof of Proposition 4.5. Since u ∈ BV (ΩT ) and u(t, ·) ∈ BV (Ω) for almost all
t ∈ (0, T ) we obtain that ∂tu,∇u are Radon measures on ΩT and that ∇u(t, ·)
is a Radon measure on Ω for almost all t ∈ (0, T ). Moreover we observe that
v ∈ L1(|∇u|) since
|v| d|∇u| ≤
|v| d|∇′u| ≤ 2
g|v| dµ ≤ 2
‖g‖L2(µ)‖v‖L2(µ) < ∞
by Theorem 4.4 and Proposition 8.2. From (3.3) and Proposition 8.3 we deduce
that for any η ∈ C1c (ΩT )
η d∂tu =
ηv d∇u =
d|∇u| =
ηV d|∇u(t, ·)| dt,
which proves (4.9). �
9. Conclusions
Theorem 4.6 suggests to define a generalized action functional S in the class of
L2-flows by
S(µ) := inf
|v −H |2 dµ+ 4Snuc(µ), (9.1)
where the infimum is taken over all generalized velocities v for the evolution
(µt)t∈(0,T ). In the class of n-rectifiable L
2-flows we have
S(µ) =
|v −H |2 dµ+ 4Snuc(µ), (9.2)
where v is the unique normal velocity of (µt)t∈(0,T ) (see Proposition 3.3).
In the present section we compare the functional S with the functional S0 defined
in [18] (see (1.2)) and discuss the implications of Theorem 4.6 on a full Gamma
convergence result for the action functional. For the ease of the exposition we focus
in this section on the switching scenario.
Assumption 9.1. Let a sequence (uε)ε>0 of smooth functions uε : ΩT → R be
given with uniformly bounded action (A1), zero Neumann boundary data (A3), and
assume for the initial- and final states that for all ε > 0
uε(0, ·) = −1, uε(T, ·) = 1 in Ω. (9.3)
Following [18] we define the reduced action functional on the set M ⊂
BV (ΩT , {−1, 1})∩ L∞(0, T, BV (Ω)) such that
• for every ψ ∈ C0c (Ω) the function
u(t, ·)ψ dx
is absolutely continuous on [0, T ];
• (∂∗{u(t, ·) = 1})t∈(0,T ) is up to countably many times given as a smooth
evolution of hypersurfaces.
26 LUCA MUGNAI AND MATTHIAS RÖGER
By Assumption 9.1 the functional S0nuc can be rewritten as
S0(u) := c0
∣v(t, x) −H(t, x)
dHn−1(x)dt + 4S0nuc(u), (9.4)
S0nuc(u) :=
|∇u(t, ·)|(ψ)− lim
|∇u(t, ·)|(ψ)
+ sup
|∇u(t, ·)|(ψ) (9.5)
where the sup is taken over all ψ ∈ C1(Ω) with 0 ≤ ψ ≤ 1.
In [18, Proposition 2.2] a (formal) proof of the limsup- estimate was given for a
subclass of ‘nice’ functions in M. Following the ideas of that proof, using the one-
dimensional construction [18, Proposition 3.1], and a density argument we expect
that the limsup-estimate can be extended to the whole set M. We do not give a
rigorous proof here but rather assume the limsup-estimate in the following.
Assumption 9.2. For all u ∈ M there exists a sequence (uε)ε>0 that satisfies
Assumption 9.1 such that
u = lim
uε, S0(u) ≥ lim sup
Sε(uε). (9.6)
The natural candidate for the Gamma-limit of Sε with respect to L1(ΩT ) is the
L1(ΩT )-lower semicontinuous envelope of S0,
S(u) := inf
lim inf
S0(uk) : (uk)k∈N ⊂ M, uk → u in L1(ΩT )
. (9.7)
9.1. Comparison of S and S0. If we associate with a function u ∈ M the measure
|∇u| on ΩT we can compare S0(u) and S( c02 |∇u|).
Proposition 9.3. Let u ∈ M and let µ = L1 ⊗ µt be an L2-flow of measures.
Assume that for almost all t ∈ (0, T )
µt ≥ c0
|∇u(t, ·)| (9.8)
and that the nucleation cost S0nuc(u) is not larger than the nucleation cost Snuc(µ).
S0(u) ≤ S(µ) (9.9)
holds. For µ = c0
|∇u| we obtain that
S0(u) = S(c0
|∇u|). (9.10)
Proof. The locality of the mean curvature [29] shows that the weak mean curvature
of µt and the (classical) mean curvature coincide on ∂{u(t, ·) = 1}. By Proposition
4.5 any generalized velocity v and the (classical) normal velocity V are equal on
the phase boundary. This shows that the integral part of S0(u) is not larger than
the integral part of S(µ), with equality if µt = c0
|∇u(t, ·)| for almost all t ∈ (0, T ).
This proves (9.9). For the measure c0
|∇u| we observe that the nucleation cost
Snuc( c02 µ) equals the nucleation cost S
nuc(u) and we obtain (9.10). �
If higher multiplicities occur for the measure µ, the nucleation costs of µ and
u may differ and the value of S0(u) might be larger than S(µ) as the following
example shows. Let Ω = (0, L), let {u = 1} be the shaded regions in Figure 1, and
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 27
x1 x2
Figure 1. The
phases {u = 1}
x1 x2
Figure 2. The
measure µ
let µ be the measure supported on the phase boundary and with double density on
a hidden boundary connecting the upper and lower part of the phase {u = 1}, see
Figure 2. At time t2 a new phase is nucleated but this time is not singular with
respect to the evolution (µt)t∈(0,T ). On the other hand, no propagation cost occurs
for the evolution (u(t, ·))t∈(t1,t0) whereas there is a propagation cost for (µt)t∈(t1,t2).
The difference in action is given by
S0(u)− S(µ) = 8c0 − 2c0
(x2 − x1)2
t2 − t1
where x1 is the annihilation point at time t1 and x2 the nucleation point at time
t2, see Figure 1. This shows that as soon as (x2 − x1) < 4
t2 − t1 we have
S(µ) < S0(u).
x1 x2
Figure 3. Phases {uk = 1}
1x x2=
Figure 4. The limit
28 LUCA MUGNAI AND MATTHIAS RÖGER
The same example with x2 = x1 shows that S0 is not lower-semicontinuous
and that a relaxation is necessary in order to obtain the Gamma-limit of Sε. In
fact consider a sequence (uk)k∈N with phases {uk = 1} given by the shaded region
in Figure 3. Assume that the neck connecting the upper and lower part of the
shaded region disappears with k → ∞ and that uk converges to the phase indicator
function u with phase {u = 1} indicated by the shaded regions in Figure 4. Then a
nucleation cost at time t2 appears for u. For the approximations uk however there
is no nucleation cost for t > 0 and the approximation can be made such that the
propagation cost in (t1, t2) is arbitrarily small, which shows that
S0(u) > lim inf
S0(uk).
The situation in higher space dimensions is even more involved than in the one-
dimensional examples discussed above. For instance one could create a circle with
double density (no new phase is created) at a time t1 and let this double-density
circle grow until a time t2 > t1 where the double-density circle splits and two circles
evolve in different directions, one of them shrinking and the other one growing. In
this way a new phase is created at time t2. In this example S counts the creation
of a double-density circle at time t1 and the cost of propagating the double-density
circle between the times t1, t2. In contrast S0 counts the nucleation cost of the new
phase at time t2, which is larger as the nucleation cost Snuc at times t1, but no
propagation cost between the times t1, t2.
The analysis in [18] suggests that minimizers of the action functional exhibit
nucleation and annihilation of phases only at the initial- and final time. This class
is therefore particularly interesting.
Theorem 9.4. Let (uε)ε>0 satisfy Assumption 9.1 and suppose that Assumption
9.2 holds. Suppose that uε → u in L1(ΩT ), u ∈ M, and that u exhibits nucleation
and annihilation of phases only at the final and initial time. Then
S(u) = S0(u) ≤ lim inf
Sε(uε) (9.11)
holds. In particular, Sε Gamma-converges to S0 for those evolutions in M that
have nucleations only at the initial time.
Proof. From the definition of the functional S we deduce that
S(u) ≤ S0(u) (9.12)
and that there exists a sequence (uk)k∈N ⊂ M such that
u = lim
uk, S(u) = lim
S0(uk).
Assumption 9.2 implies that for all k ∈ N there exists a sequence (uε,k)ε>0 such
uk = lim
uε,k, S0(uk) ≥ lim sup
Sε(uε,k). (9.13)
Therefore we can choose a diagonal-sequence (uε(k),k)k∈N such that
S(u) ≥ lim sup
Sε(k)(uε(k),k). (9.14)
By Proposition 4.1, 4.2 there exists a a subsequence k → ∞ such that
uε(k),k → u, µε(k),k → µ, µ ≥
|∇u|, (9.15)
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 29
where the last inequality follows from
η d|∇u(t, ·)| ≤ lim inf
η|∇G(uε)| dx ≤ lim inf
ηdµtε =
ηdµt,
with G as in (8.23). By Theorem 4.6 we further deduce that
lim inf
Sε(k)(uε(k),k) ≥ S(µ).
This implies by (9.14) that
S(u) ≥ S(µ). (9.16)
Since µ0 = 0 and µt ≥ c0
|∇u(t, ·)| the nucleation cost of µ at t = 0 is not lower
than the nucleation cost for u. Since by assumption there are no more nucleation
times we can apply Proposition 9.3 and obtain that S0(u) ≤ S(µ). By (9.12),
(9.16) we conclude that S0(u) = S(u) = S(µ).
Applying Proposition 4.1 and Theorem 4.6 to the sequence (uε)ε>0 we deduce
that there exists a subsequence ε→ 0 such that
µε → µ̃, µ̃ ≥
|∇u| (9.17)
and such that
lim inf
Sε(uε) ≥ S(µ̃).
Repeating the arguments above we deduce from Proposition 9.3 that S0(u) ≤ S(µ̃)
S0(u) ≤ lim inf
Sε(uε).
Combining the upper bound (9.6) with (9.11) proves the Gamma convergence of
Sε in u. �
9.2. Gamma convergence under an additional assumption. Using Theorem
4.6 we can prove the Gamma convergence of Sε under an additional assumption
on the structure of the set of those measures that arise as limit of sequences with
uniformly bounded action.
Assumption 9.5. Consider any sequence (uε)ε>0 with uε → u in L1(ΩT ) that
satisfies Assumption 9.1. Define the energy measures µε according to (2.5) and let
µ be any Radon measure such that for a subsequence ε→ 0
µ = lim
µε. (9.18)
Then we assume that there exists a sequence (uk)k∈N ⊂ M such that
u = lim
uk, S(µ) ≥ lim
S0(uk). (9.19)
For any u ∈ M that exhibits nucleation and annihilation only at initial and final
time the Assumption 9.5 is always satisfied: The proof of Theorem 9.4 and our
results in Section 4 show that for any limit µ as in (9.18) we can apply Proposition
9.3. Therefore S0(u) ≤ S(µ) and the constant sequence u satisfies (9.19). However,
a characterization of those u ∈ M such that Assumption 9.5 holds is open.
Theorem 9.6. Suppose that the Assumptions 9.1, 9.2, and 9.5 hold. Then
Sε → S as ε→ 0 (9.20)
in the sense of Gamma-convergence with respect to L1(ΩT ).
30 LUCA MUGNAI AND MATTHIAS RÖGER
Proof. We first prove the limsup-estimate for Sε,S. In fact, fix an arbitrary u ∈
L1(ΩT , {−1, 1}) with S(u) < ∞. We deduce that there exists a sequence (uk)k∈N
as in (9.7) such that
S(u) = lim
S0(uk). (9.21)
By (9.6) for all k ∈ N there exists a sequence (uε,k)ε>0 such that
uε,k = uk, S0(uk) ≥ lim sup
Sε(uε,k).
Choosing a suitable diagonal sequence uε(k),k we deduce that
S(u) ≥ lim
Sε(k)(uε(k),k), (9.22)
which proves the limsup-estimate.
We next prove the liminf -estimate. Consider an arbitrary sequence (uε)ε>0 that
satisfies the Assumption 9.1. By Theorem 4.6 there exists u ∈ BV (ΩT , {−1, 1})
and a measure µ on ΩT such that
uε → u in L1(ΩT ), µε → µ (9.23)
for a subsequence ε→ 0, and such that
lim inf
Sε(uε) ≥ S(µ). (9.24)
By Assumption 9.5 there exists a sequence (uk)k∈N ⊂ M such that (9.19) holds.
By (9.24) and the definition of S this yields that
lim inf
Sε(uε) ≥ S(µ) ≥ lim
S0(uk) ≥ S(u) (9.25)
and proves the liminf -estimate. �
Appendix A. Rectifiable measures and weak mean curvature
We briefly summarize some definitions from Geometric Measure Theory. We
always restrict ourselves to the hypersurface case, that is ‘tangential-plane’ and
‘rectifiability’ of a measure in Rd means ‘(d− 1)-dimensional tangential-plane’ and
‘(d− 1)-rectifiable’.
Definition A.1. Let µ be a Radon-measure in Rd, d ∈ N.
(1) We say that µ has a (generalized) tangent plane in z ∈ Rd if there exist a
number Θ > 0 and a (d− 1)-dimensional linear subspace T ⊂ Rd such that
r−d+1
y − z
dµ(y) = Θ
η dHd−1, for every η ∈ C0c (Rd).
(A.1)
We then set Tzµ := T and call Θ the multiplicity of µ in z.
(2) If for µ-almost all z ∈ Rd a tangential plane exists then we call µ rectifiable.
If in addition the multiplicity is integer-valued µ-almost everywhere we say
that µ is integer-rectifiable.
(3) The first variation δµ : C1c (R
d,Rd) of a rectifiable Radon-measure µ on Rd
is defined by
δµ(η) :=
divTzµ η dµ.
THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 31
If there exists a function H ∈ L1loc(µ) such that
δµ(η) = −
H · η dµ
we call H the weak mean-curvature vector of µ.
Appendix B. Measure-function pairs
We recall some basic facts about the notion of measure function pairs introduced
by Hutchinson in [16].
Definition B.1. Let E ⊂ Rd be an open subset. Let µ be a positive Radon-
measure on E. Suppose f : E → Rm is well defined µ-almost everywhere, and
f ∈ L1(µ,Rm). Then we say (µ, f) is a measure-function pair over E (with values
in Rm).
Next we define two notions of convergence for a sequence of measure-function
pairs on E with values in Rm.
Definition B.2. Suppose {(µk, fk)}k and (µ, f) are measure-function pairs over
E with values in Rm. Suppose
µk = µ,
as Radon-measures on E. Then we say (µk, fk) converges to (µ, f) in the weak
sense (in E) and write
(µk, fk) → (µ, f),
if µk⌊fk → µ⌊f in the sense of vector-valued measures, that means
fk · η dµk =
f · η dµ,
for all η ∈ C0c (E,Rm).
The following result is a slightly less general version of [16, Theorem 4.4.2],
however this is enough for our aims.
Theorem B.3. Let F : Rm → [0,+∞) be a continuous, convex function with
super-linear growth at infinity, that is:
|y|→∞
F (y)
= +∞.
Suppose {(µk, fk)}k are measure-function pairs over E ⊂ Rd with values in Rm.
Suppose µ is Radon-measure on E and µk → µ as k → ∞. Then the following are
true:
(1) if
F (fk) dµk < +∞, (B.1)
then some subsequence of {(µk, fk)} converges in the weak sense to some
measure function (µ, f) for some f .
(2) if (B.1) holds and (µk, fk) → (µ, f) then
lim inf
F (fk) dµk ≥
F (f) dµ. (B.2)
32 LUCA MUGNAI AND MATTHIAS RÖGER
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THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 33
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Luca Mugnai, Max Planck Institute for Mathematics in the Sciences, Inselstr. 22,
D-04103 Leipzig
Matthias Röger, Max Planck Institute for Mathematics in the Sciences, Inselstr.
22, D-04103 Leipzig
E-mail address: mugnai@mis.mpg.de, roeger@mis.mpg.de
	1. Introduction
	1.1. Deterministic phase field models and sharp interface limits
	1.2. Stochastic interpretation of the action functional
	1.3. Organization
	Acknowledgment
	2. Notation and Assumptions
	3. L2-flows
	4. Lower bound for the action functional
	4.1. Lower estimate for the mean curvature
	4.2. Lower estimate for the generalized velocity
	4.3. Lower estimate of the action functional
	4.4. Convergence of the Allen–Cahn equation to Mean curvature flow
	5. Proofs of Propositions ??, ?? and Theorem ??
	6. Proof of Theorem ??
	6.1. Equipartition of energy
	6.2. Convergence of approximate velocities
	7. Proof of Theorem ??
	8. Proofs of Proposition ?? and Proposition ??
	9. Conclusions
	9.1. Comparison of S and S0
	9.2. Gamma convergence under an additional assumption
	Appendix A. Rectifiable measures and weak mean curvature
	Appendix B. Measure-function pairs
	References
