We prove that if a globally minimizing solution to the vectorial Allen-Cahn equation has finite potential energy, then it is a constant

Sourdis, Christos

English

ACMAC

ACMAC’s PrePrint Repository
A Liouville theorem for minimizers with ﬁnite potential energy for the
vectorial Allen-Cahn equation
Christos Sourdis
Original Citation:
Sourdis, Christos
(2014)
A Liouville theorem for minimizers with ﬁnite potential energy for the vectorial Allen-Cahn equation.
(Submitted)
This version is available at: http://preprints.acmac.uoc.gr/290/
Available in ACMAC’s PrePrint Repository: February 2014
ACMAC’s PrePrint Repository aim is to enable open access to the scholarly output of ACMAC.
http://preprints.acmac.uoc.gr/A LIOUVILLE THEOREM FOR MINIMIZERS WITH FINITE
POTENTIAL ENERGY FOR THE VECTORIAL ALLEN-CAHN
EQUATION
CHRISTOS SOURDIS
Abstract. We prove that if a globally minimizing solution to the vectorial Allen-Cahn
equation has ﬁnite potential energy, then it is a constant.
Consider the semilinear elliptic system
∆u = ∇W(u) in R
n,n ≥ 1, (0.1)
where W : Rm → R, m ≥ 1, is suﬃciently smooth and nonnegative. It has been recently
shown in [1] that each nonconstant solution to the system (0.1) satisﬁes:
￿
BR
￿
1
2
|∇u|
2 + W (u)
￿
dx ≥



cRn−2 if n ≥ 3,
clnR if n =2 ,
(0.2)
for all R>1, and some c>0, where BR stands for the n-dimensional ball of radius R,
centered at the origin.
On the other side, if additionally W vanishes at least at one point, it is easy to see that
￿
BR
￿
1
2
|∇u|
2 + W (u)
￿
dx ≤ CR
n−1,R > 0, (0.3)
for some C>0 (see [4]).
The system (0.1) with W ≥ 0 vanishing at a ﬁnite number of global minima (typically
nondegenerate), and coercive at inﬁnity, is used to model multi-phase transitions (see [4] and
the references therein). In this case, the system (0.1) is frequently referred to as the vecto-
rial Allen-Cahn equation. In [7], we showed the following theorem for globally minimizing
solutions (see [5, 7] for the precise deﬁnition).
Theorem 0.1. Assume that W ∈ C1(Rm;R), m ≥ 1, and that there exist ﬁnitely many
N ≥ 1 points ai ∈ Rm such that
W(u) > 0 in R
m \{ a1,···,a N}, (0.4)
and there exists small r0 > 0 such that the functions
r ￿→ W(ai+rν), where ν ∈ S
1, are strictly increasing for r ∈ (0,r 0),i =1 ,···,N. (0.5)
Moreover, we assume that
liminf
|u|→∞
W(u) > 0. (0.6)
If u ∈ C2(R2;Rm) is a bounded, nonconstant, and globally minimizing solution to the
elliptic system (0.1) with n = 2, there exist constants c0,R 0 > 0 such that
￿
BR
W (u(x))dx ≥ c0R for R ≥ R0.
12 CHRISTOS SOURDIS
In view of (0.3), the above result captures the optimal growth rate in the case n = 2.
The purpose of this note is to establish the following Liouville type theorem which holds
in any dimension. Similarly to [7], we combine ideas from the study of vortices in the
Ginzburg-Landau model [3] with variational maximum principles from the study of the
vector Allen-Cahn equation [2].
Theorem 0.2. Let W be as in Theorem 0.1. Suppose that u ∈ C2(Rn;Rm), n ≥ 2, is a
bounded and globally minimizing solution to the elliptic system (0.1) such that
￿
Rn
W (u(x))dx < ∞.
Then, we have that
u ≡ ai for some i ∈{ 1,···,N}.
Proof. It follows that there exists a constant C0 > 0 such that
￿
BR
W (u(x))dx ≤ C0,R > 0. (0.7)
Let
ε =
1
R
and uε(y)=u
￿y
ε
￿
,y ∈ B1.
Then, relation (0.7) becomes
￿
B1
W (uε(y))dy ≤ C1ε
n,ε > 0, (0.8)
for some C1 > 0. Note that, by standard elliptic regularity estimates [6], we have that
|uε| + ε|∇uε|≤C2 in R
n,ε > 0, (0.9)
for some C2 > 0.
Let d>0 be any small number. As in [3], by combining (0.8) and (0.9), we deduce that
the set where W(uε) is above d>0 is included in a uniformly bounded number of balls of
radius ε, as ε → 0. Certainly, there exists rε ∈ (1
4, 3
4) such that
W (uε(y)) ≤ d if |y| = rε.
Since d>0 is arbitrary, we are led to ˜ rε ∈ (1
4, 3
4) such that
max
|y|=˜ rε
W (uε(y)) → 0 as ε → 0.
In terms of u and R, we have
max
|x|=sR
W (u(x)) → 0 as R →∞ , for some sR ∈
￿
1
4
R,
3
4
R
￿
.
In view of (0.6), the above relation implies that there exist ij ∈{ 1,···,N} such that
max
|x|=sR
￿
￿u(x) − aij
￿
￿ → 0 as R →∞ .
By virtue of (0.5), as in [7], exploiting the fact that u is a globally minimizing solution, we
can apply a recent variational maximum principle from [2] to deduce that
max
|x|≤sR
￿
￿u(x) − aij
￿
￿ ≤ max
|x|=sR
￿
￿u(x) − aij
￿ ￿ for R ￿ 1.LOWER BOUNDS FOR THE VECTORIAL ALLEN-CAHN ENERGY 3
The above two relations imply the existence of an i0 ∈{ 1,···,N} such that
max
|x|≤sR
|u(x) − ai0|→0 as R →∞ .
Since sR →∞as R →∞ , we conclude that u ≡ ai0. ￿
References
[1] N. D. Alikakos, Some basic facts on the system ∆u−Wu(u) = 0, Proc. Amer. Math. Soc. 139 (2011),
153-162.
[2] N.D. Alikakos, and G. Fusco, A maximum principle for systems with variational structure and an
application to standing waves, arXiv:1311.1022
[3] F. B´ ethuel, H. Brezis, and F. H´ elein, Ginzburg-Landau vortices, PNLDE 13, Birkh¨ auser Boston,
1994.
[4] G. Fusco, Equivariant entire solutions to the elliptic system ∆u − Wu(u)=0for general G-invariant
potentials, Calc. Var. DOI 10.1007/s00526-013-0607-7
[5] G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy, Comm. Pure.
Appl. Analysis 13 (2014), 1045–1060.
[6] D. Gilbarg, and N. S. Trudinger, Elliptic partial diﬀerential equations of second order, second ed.,
Springer-Verlag, New York, 1983.
[7] C. Sourdis. Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn
equation, Arxiv 2014.
Department of Mathematics and Applied Mathematics, University of Crete.
E-mail address: csourdis@tem.uoc.gr

A maximum principle for systems with variational structure and an application to standing waves,

Elliptic partial diﬀerential equations of second order,

Equivariant entire solutions to the elliptic system ∆u

On some elementary properties of vector minimizers of the Allen-Cahn energy,

Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn equation,

Some basic facts on the system ∆u−Wu(u)

A Liouville theorem for minimizers with finite potential energy for the vectorial Allen-Cahn equation

A Liouville theorem for minimizers with finite potential energy for the vectorial Allen-Cahn equation

Abstract

Similar works

Full text

Available Versions

ACMAC