Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 04, pp. 1–11.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

AN ASYMPTOTIC MONOTONICITY FORMULA FOR

MINIMIZERS OF ELLIPTIC SYSTEMS OF ALLEN-CAHN TYPE

AND THE LIOUVILLE PROPERTY

CHRISTOS SOURDIS

Abstract. We prove an asymptotic monotonicity formula for bounded, glob-
ally minimizing solutions (in the sense of Morse) to a class of semilinear elliptic

systems of the form ∆u = Wu(u), x ∈ Rn, n ≥ 2, with W : Rm → R, m ≥ 1,

nonnegative and vanishing at exactly one point (at least in the closure of the
image of the considered solution u). As an application, we can prove a Liouville

type theorem under various assumptions.

1. Introduction and statement of main results

We consider the semilinear elliptic equation

∆u = Wu(u) in Rn, n ≥ 2, (1.1)

where W : Rm → R, m ≥ 1, is sufficiently smooth and nonnegative (we use the
notation Wu = ∇uW ). This system has variational structure, as solutions (in a
smooth, bounded domain Ω ⊂ Rn) are critical points of the energy

E(v; Ω) =

∫
Ω

{1

2
|∇v|2 +W (v)

}
dx (1.2)

(subject to their own boundary conditions), where |∇v|2 =
∑n
i=1 |vxi |2. A solution

u ∈ C2(Rn;Rm) is called globally minimizing (in the sense of Morse) if

E(u; Ω) ≤ E(u+ ϕ; Ω) (1.3)

for every smooth, bounded domain Ω ⊂ Rn and for every ϕ ∈ W 1,2
0 (Ω;Rm) ∩

L∞(Ω;Rm) (see [7, 30] and the references therein).
If m ≥ 2, there are two main categories of such potentials W :

• Those that vanish only on a discrete set of points (usually finite); in this
case (1.1) is known as the vectorial Allen-Cahn equation and models multi-
phase transitions (see [7, 9, 16, 27] and some of the references that will
follow).
• Those that vanish on a continuum of points, as in the Ginzburg-Landau

system (see [14]) or the elliptic system modeling phase-separation in [13].

2010 Mathematics Subject Classification. 35J48, 35J20, 35J61.
Key words and phrases. Entire solutions; monotonicity formula; Allen-Cahn equation;

Liouville theorem; multi-phase transitions.
c©2021 Texas State University.

Submitted January 21, 2019. Published January 20, 2021.

1



2 C. SOURDIS EJDE-2021/04

This article is motivated from the first class. In this setting, an effective way to
construct entire, nontrivial solutions to (1.1) is to assume that W is symmetric with
respect to a finite reflection group and to look for equivariant solutions (one first
minimizes E(·;BR) in this class, under suitable boundary conditions on ∂BR, and
then lets R → ∞). Under proper assumptions, this roughly amounts to studying
bounded, globally minimizing solutions to (1.1) such that the closure of their image
contains exactly one global minimum of W . In the scalar case, that is m = 1,
this approach has been utilized, among others, in [17] and [22]. On the other
hand, recent progress has been made in the vector case in [4, 7, 11, 12, 29, 40]. In
our opinion, the main obstruction in the vector case is the lack of the maximum
principle. This short discussion motivates our main result which is the following.

Theorem 1.1. Assume that W ∈ C1(Rm;R), m ≥ 1, and that there exists a ∈ Rm
such that

W > 0 in Rm \ {a} and W (a) = 0. (1.4)

If u ∈ C2(Rn;Rm), n ≥ 2, is a bounded, globally minimizing solution to the elliptic
system (1.1), then

lim
R→∞

( 1

Rn−1

∫
BR

{1

2
|∇u|2 +W (u)

}
dx
)

= 0, (1.5)

where BR stands for the n-dimensional ball of radius R and center at the origin.

The above result may be interpreted as an asymptotic monotonicity formula (see
(2.19) below). We emphasize that there is no assumption for the behavior of W near
a. Our proof of Theorem 1.1 is based on an adaptation to this setting of the famous
“bad discs” construction of [14] from the study of vortices in the Ginzburg-Landau
model.

Under even more general assumptions on W , it is well known that every bounded
and globally minimizing solution to (1.1) satisfies

lim sup
R→∞

( 1

Rn−1

∫
BR

{1

2
|∇u|2 +W (u)

}
dx
)
<∞,

(see for example [7, Ch. 5], [19]). The above relation can be proven by comparing
the energy of u in BR to that of a suitable test function which agrees with u on ∂BR
and is equal to some zero of W in BR−1. This simple idea, which can actually be
traced back to the theory of minimal surfaces (see [20]), will also play an important
role in our analysis.

As an application of Theorem 1.1, we can prove the following Liouville type
theorem.

Theorem 1.2. Assume that W and u are as in Theorem 1.1. Then

u ≡ a,
provided that one of the following additional conditions is satisfied:

(a) m = 1 and W ∈ C1,1
loc (R;R); or m ≥ 1 and Modica’s gradient bound holds,

that is
1

2
|∇u|2 ≤W (u) in Rn. (1.6)

(b) n = 2 and there exists a small r0 > 0 such that the functions

r 7→W (a+ rν), ν ∈ Sm−1 are nondecreasing for r ∈ (0, r0]; (1.7)

or n = 2 and m = 1.



EJDE-2021/04 ASYMPTOTIC MONOTONICITY FORMULAS FOR MINIMIZERS 3

The above Liouville property was originally proven, for all n ≥ 2, by differ-
ent techniques in [30] (see also the earlier paper [29]), under the conditions that
W ∈ C2(Rm;R) and u satisfy the assumptions of Theorem 1.1, and that the func-
tions in (1.7) have a strictly positive second order derivative in (0, r0). In particular,
the approach of the latter references is based on a quantitative refinement of the
replacement lemmas in [4] and [28], combined with a rather involved iterative proce-
dure. If W additionally satisfies the stronger assumption that a is a non-degenerate
minimum, this theorem was recently reproven in [6] by extending to this setting
the density estimates of [19]. In the aforementioned references, the Liouville type
theorem was proven by an application of a basic pointwise estimate. However, it is
not difficult to convince oneself that going in the opposite direction is also possible,
i.e., the pointwise estimate follows from the Liouville property (see also [41] for this
viewpoint). We note that the pointwise estimate is the one that is directly appli-
cable in relation to the discussion preceding Theorem 1.1. This pointwise estimate
roughly says that if W (as in Theorem 1.1) is such that the Liouville type theorem
holds, then a globally minimizing solution, defined in a sufficiently large ball (with
the appropriate modifications in the definition) and bounded independently of the
size of the ball, has to be close to a in the ball of half the radius (with the same
center).

In light of the recent density estimates of [23], we expect that the assertions
of Theorems 1.1 and 1.2 should also remain valid under the complementary set of
assumptions that W ∈ C1 satisfies

c|u− a|p ≤W (u) ≤ C|u− a|p, u ∈ Rm, m ≥ 1,

for some constants c, C > 0, where

p ∈

{
(2,∞), n = 2,(
2, 2n

n−2

)
, n ≥ 3.

In the scalar case, under the assumptions of the first part of Case (a) above, this
Liouville property can also be proven by using radial barriers as in [41]. On the
other hand, in the ODE case (i.e. n = 1, m ≥ 1) the Liouville property is valid
solely under the assumptions of Theorem 1.1 on W and u (see [8]).

In our opinion, three are the main advantages of our approach. Firstly, we
can treat in a unified and coordinate way the various situations in Theorem 1.2.
Secondly, we find that our approach is considerably simpler than those in the afore-
mentioned references. Lastly, to the best of our knowledge, it provides the strongest
available result when n = 2 for any m ≥ 1, even for the extensively studied scalar
case. This may seem too restrictive at first, but keep in mind that the dimensions
n = 2, 3 are the ones with physical interest. In fact, the majority of papers on the
subject deal exclusively with these dimensions (see [1, 2, 15, 16, 31, 38] for n = 2,
and [34] for n = 3). If n = 2, we believe that our results strongly indicate that
the convexity of W near its global minima, that was assumed in some of the afore-
mentioned papers that deal with the existence of equivariant solutions to (1.1) (for
instance in [12]), can be relaxed to the monotonicity condition that is described
in (1.7). In this regard, we emphasize that systems of the form (1.1) where the
potentials have degenerate minima arise naturally in various physical models (see
for example [10]).



4 C. SOURDIS EJDE-2021/04

The proof of Theorem 1.2 is based on combining Theorem 1.1 with a variety of
results that are available in the literature. Next, we will provide the proofs of our
main results.

2. Proofs of the main results

Proof of Theorem 1.1. Throughout this proof, we denote the energy density of u
by

e(x) =
1

2
|∇u(x)|2 +W

(
u(x)

)
, x ∈ Rn. (2.1)

Firstly, we note that standard elliptic regularity theory and Sobolev imbeddings
[25, 33], in combination with the fact that u is bounded and W ∈ C1, yield

‖u‖C1,α(Rn;Rm) ≤ C1, (2.2)

for some constants α ∈ (0, 1) and C1 > 0 (in fact, it holds for any α ∈ (0, 1)
provided that C1 = C1(α) > 0).

Since u is a globally minimizing solution, by comparing its energy to that of a
suitable test function which agrees with u on ∂BR and is identically equal to a in
BR−1, thanks to (2.2), we find that∫

BR

e(x) dx ≤ C2R
n−1, R ≥ 1, (2.3)

for some C2 > 0 (see [7, Ch. 5], [19]).
Therefore, by (2.3), the coarea formula (see for instance [25, Ap. C]), the non-

negativity of W , and the mean value theorem, there exists

SR ∈ (R, 2R) (2.4)

such that ∫
∂BSR

e(x) dS(x) ≤ C3R
n−2, R ≥ 1, (2.5)

for some C3 > 0.
Let ε > 0 be any small number. We will show in the sequel that the subset of

∂BSR where e(x) is above ε is contained in at most O(Rn−2) many geodesic balls
of radius 1 as R →∞ (the so-called “bad balls”, see [14]). More precisely, we will
establish that there exist Nε,R ≥ 0 points {xR,1, · · · , xR,Nε,R} on ∂BSR such that

Nε,R ≤MεR
n−2, R� 1 (with Mε > 0 independent of R), (2.6)

and

e(x) ≤ ε if x ∈ ∂BSR \ ∪
Nε,R
i=1 UR(xR,i, 1), (2.7)

for R � 1, where UR(p, r) ⊂ ∂BSR stands for the geodesic ball with center at p
and radius r (for convenience, we have suppressed the explicit dependence of xR,i
on ε). We will prove the above properties by adapting some arguments from [14].

First, we will show the following clearing-out property, which is in the spirit of
[14, Thm. III.3] and is actually valid for any function u that satisfies (2.2). For
any ε ∈ (0, 1), there exists a µε < ε such that if∫

UR(y,2)

e(x)dS(x) < µε for some y ∈ ∂BSR ,



EJDE-2021/04 ASYMPTOTIC MONOTONICITY FORMULAS FOR MINIMIZERS 5

then e(x) ≤ ε for x ∈ UR(y, 1), with R ≥ 1. We will show this property by arguing
by contradiction. So, let us suppose that, no matter how small µ is, we have

e(z) ≥ ε for some z ∈ UR(y, 1). (2.8)

From (2.2), using again that W ∈ C1, there exists a C4 > 0 such that

‖e‖C0,α(Rn;R) ≤ C4.

It then follows that

e(x) ≥ ε− C4d
α, x ∈ UR(z, d),

for all d < min{1, ( ε
2C4

)1/α} (see also [43, Lem. 2.3]). Since e ≥ 0, we find that∫
UR(y,2)

e(x)dS(x) ≥
∫
UR(z,d)

e(x)dS(x)

≥ (ε− C4d
α)|UR(z, d)|

≥ ε

2
|UR(z, d)| = ε

2
|Sn−1|dn−1.

Hence, we can arrive at a contradiction by choosing

µε =
ε

2
|Sn−1|

(
min

{
1,
( ε

2C4

)1/α})n−1

. (2.9)

We consider a finite family of geodesic balls {UR(xi, 1)}i∈IR , IR ⊂ N, such that

UR
(
xi,

1

4

)
∩ UR

(
xk,

1

4

)
= ∅ if i 6= k, (2.10)

∪i∈IRUR(xi, 1) = ∂BSR , (2.11)

for all R ≥ 1 (having suppressed the obvious dependence of xi on R). This is indeed
possible by the Vitali covering theorem (see [24, Sec. 1.5] and keep in mind that
∂BSR becomes a metric space when equipped with the geodesic distance). We say
that the ball UR(xi, 1) is a good ball if∫

UR(xi,2)

e(x)dS(x) < µε,

and that UR(xi, 1) is a bad ball if∫
UR(xi,2)

e(x)dS(x) ≥ µε.

The collection of bad balls is indexed over

JR = {i ∈ IR : UR(xi, 1) is a bad ball}.

The main observation is that, by (2.10), there is a universal constant C5 > 0
(independent of both ε and R) such that∑

i∈IR

∫
UR(xi,2)

e(x)dS(x) ≤ C5

∫
∂BSR

e(x)dS(x),

owing to the fact that each point on ∂BSR is covered by at most C5 geodesic balls
UR(xi, 2) (see also [14, Ch. IV]). The latter property plainly follows by observing
that all such balls that contain the same point are certainly contained in a geodesic
ball of radius 10, and from the basic fact that any (n−1)-dimensional ball of radius



6 C. SOURDIS EJDE-2021/04

10 can contain only a certain number of disjoint balls of radius 1/4 (keep in mind
that ∂BSR is essentially a flat manifold for R� 1). Using (2.5), we then infer that

card JR ≤
C5C3

µε
Rn−2, R� 1. (2.12)

Now, let us consider an x ∈ ∂BSR \ ∪i∈JRUR(xi, 1). By (2.11), there exists some
k ∈ IR \ JR such that x ∈ UR(xk, 1) which is a good ball. It follows from the
definition of µε that

e(x) ≤ ε,
thereby completing the proof of (2.6) and (2.7).

In view of (1.4) and (2.7), we have

|∇u(x)|2 ≤ 2ε and |u(x)−a| ≤ σε if x ∈ ∂BSR \∪
Nε,R
i=1 UR(xR,i, 1), R� 1, (2.13)

where

σε → 0 as ε→ 0, (2.14)

(we point out that σε depends only on ε).
We consider the function vR ∈W 1,2 (BSR ;Rm)∩L∞(BSR ;Rm) which is defined

in terms of polar coordinates as

vR(r, θ) =

{
u(SR, θ) +

(
a− u(SR, θ)

)
(SR − r), r ∈ [SR − 1, SR], θ ∈ Sn−1,

a, r ∈ [0, SR − 1], θ ∈ Sn−1,

(having slightly abused notation, keep in mind that x = rθ). We note that vR
belongs in W 1,2 because it is the composition of a smooth function with a Lipschitz
continuous one (see [35, pg. 54] and keep in mind that we only use the polar
coordinates away from the origin). Clearly, we have

vR = u on ∂BSR . (2.15)

Let

AR = BSR \B(SR−1) and CR = ∪Nε,Ri=1

(
B̄10(xR,i) ∩ ĀR

)
,

where B10(xR,i) stands for the n-dimensional ball of radius 10 and center at xR,i.
If x = rθ ∈ AR \ CR, via (2.13), we obtain

|vR(x)− a| ≤ 2|u(SR, θ)− a| ≤ 2σε. (2.16)

Moreover for such x, using (2.4) and (2.13), we find that

|∇RnvR|2 = |u(SR, θ)− a|2 +
1

r2
|(1 + r − SR)∇Sn−1u(SR, θ)|2

≤ σ2
ε +

2

S2
R

|∇Sn−1u(SR, θ)|2

≤ σ2
ε + 2|∇Rnu(SRθ)|2

≤ σ2
ε + 4ε,

(2.17)

where we made repeated use of the identity

|∇Rnv|2 = |∂rv|2 +
1

R2
|∇Sn−1v|2 on ∂BR, R > 0;

see [44, Ch. 8]. It follows that∫
BSR

{1

2
|∇vR|2 +W (vR)} dx



EJDE-2021/04 ASYMPTOTIC MONOTONICITY FORMULAS FOR MINIMIZERS 7

=

∫
AR
{1

2
|∇vR|2 +W (vR)} dx

≤ C6Nε,R +

∫
AR\CR

{1

2
|∇vR|2 +W (vR)} dx (using (2.2) and part of (2.17))

≤ C6Nε,R +
(σ2

ε

2
+ 2ε+ C7σε

)
|AR \ CR| (using (2.16), (2.17))

≤ C6Nε,R + C8(σε + ε)Sn−1
R ,

where C6, C7, C8 > 0 are independent of both small ε and large R.
Since u is a globally minimizing solution, thanks to (2.15), we obtain∫

BSR

e(x) dx ≤ C6Nε,R + C8(σε + ε)Sn−1
R

≤ C6MεR
n−2 + 2n−1C8(σε + ε)Rn−1

(2.18)

for R� 1, were we used (2.4) and (2.6). Since ε > 0 is arbitrary, in light of (2.14),
we infer that (1.5) holds, as desired. �

Proof of Theorem 1.2. Case (a) If u satisfies (1.6), since W ≥ 0, it is known that
the following strong monotonicity formula holds

d

dR

( 1

Rn−1

∫
BR

{1

2
|∇u|2 +W (u)} dx

)
≥ 0, R > 0, (2.19)

(see [18, 37] for m = 1, and [3] for arbitrary m ≥ 1). Let us point out in passing
that u being a globally minimizing solution is not used for this. Hence, for any
positive r < R, we have

1

rn−1

∫
Br

{1

2
|∇u|2 +W (u)} dx ≤ 1

Rn−1

∫
BR

{1

2
|∇u|2 +W (u)} dx.

By Theorem 1.1, letting R→∞ in the above relation yields u ≡ a as desired.
To complete the proof in this case, we note that the gradient estimate (1.6)

was shown in [26] to hold for any bounded, entire solution when m = 1 and

W ∈ C1,1
loc (R;R) is nonnegative (see [18, 36] for earlier proofs which required higher

regularity on W ).

Case (b) Here we partly follow [42]. Since n = 2, by working as in (2.5), and using
the assertion of Theorem 1.1, we arrive at∫

∂BSR

W (u(x)) dS(x)→ 0, for some SR ∈ (R, 2R), as R→∞.

By using just the C1-bound in (2.2), and working as we did in order to exclude
(2.8), we deduce that

max
|x|=SR

|u(x)− a| → 0 as R→∞. (2.20)

Under the assumptions of the first part of Case (b), a recent variational maximum
principle from [5], as extended in [42] (to allow for non-strict monotonicity in (1.7)),
implies that

max
|x|≤SR

|u(x)− a| ≤ max
|x|=SR

|u(x)− a|.

In light of (2.20), by letting R → ∞ in the above relation, we can conclude that
the assertion of the theorem holds in the first scenario of (b).



8 C. SOURDIS EJDE-2021/04

We will establish the validity of the Liouville property in the second scenario in
(b) by borrowing some ideas from [45], while adopting a slightly more explanatory
viewpoint. To this end, we will argue by contradiction. Without loss of generality,
we may assume that there exists a sequence Rj →∞ and a δ > 0 such that

u(xj) = max
|x|≤SRj

u(x) ≥ a+ δ, j ≥ 1,

for some xj ∈ BSRj . In particular, there exists a d ∈ (0, δ) such that

W (a+ d) < W (u(xj)) , j ≥ 1.

By (2.20), we may further assume that

max
|x|=SRj

u(x) ≤ a+
d

2
, j ≥ 1. (2.21)

Let uj ∈ [a+ d, u(xj)) be such that

W (uj) = min
u∈[a+d,u(xj)]

W (u). (2.22)

We consider the competitor function

Vj(x) = min{u(x), uj}, x ∈ BSRj ,

which belongs in W 1,2
(
BSRj ;Rm

)
∩ L∞

(
BSRj ;Rm

)
(see for instance [21]) and,

thanks to (2.21), agrees with u on ∂BSRj . To conclude, we will show that

E
(
Vj ;BSRj

)
< E

(
u;BSRj

)
,

which contradicts the energy minimality character of u. To this aim, we set

Dj = {x ∈ BSRj : u(x) > uj}.

We observe that Dj is nonempty (since it contains xj) and strictly contained in
BSRj (from (2.21)). Then, to arrive at a contradiction we plainly note that

E
(
Vj ;BSRj \ Dj

)
= E

(
u;BSRj \ Dj

)
and E

(
Vj ;Dj

)
= E

(
uj ;Dj

)
< E

(
u;Dj

)
,

since (2.22) holds and there exists a connected component Ej of Dj , say the one
containing xj , where u is nonconstant (note that u = uj on ∂Dj). �

We refer to [32] for a class of systems (1.1) of Allen-Cahn type whose solutions
satisfy Modica’s gradient bound (1.6). To the best of our knowledge, there are no
counterexamples to Modica’s gradient bound for systems of Allen-Cahn type in the
case of minimizing solutions. In this regard, we refer the interested reader to [39].

Acknowledgments. This work has received funding from the Hellenic Foundation
for Research and Innovation (HFRI) and the General Secretariat for Research and
Technology (GSRT), under grant agreement No 1889.



EJDE-2021/04 ASYMPTOTIC MONOTONICITY FORMULAS FOR MINIMIZERS 9

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Christos Sourdis

Department of Mathematics, National and Kapodistrian University of Athens, Athens,
Greece

Email address: sourdis@uoc.gr


	1. Introduction and statement of main results
	2. Proofs of the main results
	Acknowledgments

	References