Interaction energy between vortices of vector fields on Riemannian surfaces

0.16 MB·

by  Radu Ignat

Interaction energy between vortices of vector fields on
Riemannian surfaces
7
Radu Ignat ∗ Robert L. Jerrard †
1
0
2 January 24, 2017

n
a
J

3 Abstract
2
WestudyavariationalGinzburg-Landautypemodeldependingonasmallparameterε>0
] for(tangent)vectorfieldsona2-dimensionalRiemanniansurface. Asε→0,thevectorfields
P
tend to be of unit length and will have singular points of a (non-zero) index, called vortices.
A
Ourmain result determinestheinteraction energy between thesevorticesas aΓ-limit (at the
. second order) as ε→0.
h
t
a
m 1 Introduction
[
Let (S,g) be a closed (i.e., compact, connected without boundary) 2-dimensional Riemannian
1
manifold of genus g. We will focus on (tangent) vector fields
v
6
u:S →TS, i.e., u(x)∈T S for every x∈S
4 x
5
6 whereTS =∪x∈STxS isthetangentbundleofS. Itiswellknownthattherearenosmoothvector
0 fields X(S) (or more generally, of Sobolev regularity X1,2(S)) of unit length |u| =1 on S (unless
g
. g=1). Infact, vectorfields ofunit lengthhavein generalsingularpoints witha (non-zero)index.
1
0 Ouraimistodeterminetheinteractionenergybetweenthesesingularpointsinavariationalmodel
7 of Ginzburg-Landau type depending on a small parameter ε > 0 where the penalty |u| = 1 in S
g
1 is relaxed.
:
v
i Model. For vector fields u:S →TS, we define the energy functional
X
r 1 1
a E (u)= e (u)vol , e (u):= |Du|2+ F(|u|2),
ε ε g ε 2 g 4ε2 g
ZS
where |Du|2 := |D u|2 +|D u|2 in S, vol is the volume 2-form on (S,g) and D denotes co-
g τ1 g τ2 g g v
variantdifferentiation (with respect to the Levi-Civita connection) of u in direction v and {τ ,τ }
1 2
is any local orthonormal basis of TS. The potential F : R → R is a continuous function with
+ +
F(1) = 0 and there exists some c > 0 such that F(s2) ≥ c(1−s)2 for every s ≥ 0; in particular,
1 is the unique zero of F. The parameter ε > 0 is small penalizing |u| 6= 1 in S; the goal is to
g
analyse the asymptotic behaviour of E in the framework of Γ-convergence (at first and second
ε
order) in the limit ε→0. This is a “toy” problem for some physical models arising for thin shells
∗Institut de Math´ematiques de Toulouse, Universit´e Paul Sabatier, 31062 Toulouse, France. Email:
[email protected]
†Department of Mathematics, University of Toronto, Toronto, Ontario, Canada. Email: rjer-
[email protected]
1
in micromagnetics or nematic liquid crystals (see e.g., [4, 5]).
Connection 1-form. On an open subset O ⊂ S, a moving frame is a pair of smooth, properly
oriented, orthonormal vector fields τ ∈ X(O), k = 1,2, i.e., (τ ,τ ) = δ , k,l = 1,2, and
k k l g kℓ
vol (τ ,τ ) = 1 in O, where (·,·) is the scalar product on TS. (We will use the same notation
g 1 2 g
(·,·) for the inner product associated to k-forms, k = 0,1,2.) Defining i : TS → TS such that i
g
is an isometry of T S to itself for every x∈S satisfying
x
i2w=−w, (iw,v) =−(w,iv) = vol (w,v),
g g g
theneverysmoothvectorfieldτ ∈X(O) ofunitlengthprovidesamovingframe{τ ,τ }:={τ,iτ}
1 2
on O. Moreover, if {τ ,τ } is any moving frame in O, then τ = iτ . 1 Given a moving frame
1 2 2 1
{τ ,τ } on an open subset O ⊂ S, the connection 1-form A associated to {τ ,τ } is defined for
1 2 1 2
every smooth vector field v ∈X(O):
A(v):=(D τ ,τ ) =−(D τ ,τ ) in O.
v 2 1 g v 1 2 g
In particular, D τ = −A(v)τ and D τ = A(v)τ in O. In complex notation, it yields for any
v 1 2 v 2 1
smooth complex-valued function φ on O:
D (φτ )=(dφ(v)−iA(v)φ)τ in O.
v 1 1
ThedefinitionofAdependsonthechoiceofthemovingframe. However,theexteriorderivativedA
of the connection 1-form is independent of the moving frame, in particular, the following identity
holds
dA=κ vol ,
g
where κ is the Gaussian curvature of S (see [7] Proposition 2, Chapter 5.3). We recall the Gauss-
Bonnet theorem that states
κvol =2πχ(S),
g
ZS
where χ(S) is the Euler characteristic, related to the genus g of S by χ(S)=2−2g.
Vortices. We will identify vortices of a vector field u with small geodesic balls centered at some
pointsaroundwhichuhasa(non-zero)index. Tobe moreprecise,weintroducethe Sobolevspace
X1,p(S) of vector fields u:S →TS such that |u| and |Du| belong to Lp(S) (with respect to the
g g
volume 2-form), p ≥ 1. Given u ∈ X1,p(S)∩Lq(S) such that 1 + 1 = 1, p,q ∈ [1,∞], we define
p q
the 1-form j(u) by 2
j(u)=(Du,iu) .
g
Inparticular,j(u)isawell-defined1-forminL1(S)ifu∈X1,1(S)with|u| =1almosteverywhere
g
in S; the same is true if u∈X1,p(S) for p≥ 4. To introduce the notion of index, we assume that
3
O is a simply connected open subset of S and u ∈X1,2(N) is a vector field in a neighborhood N
of ∂O such that |u| ≥ 1 a.e. in N; then the index (or winding number) of u along ∂O is defined
g 2
by
1 j(u)
deg(u;∂O):= + κ vol
2π |u|2 g
(cid:18)Z∂O g ZO (cid:19)
1IngeneralamovingframeexistsonlylocallyonS.
2Note thatif{τ1,τ2}isamovingframeonanopen setO⊂S,then the connection 1-formAassociated tothe
movingframeisgivenbyA=−j(τ1)onO. Inparticular,dj(u)=−kvolg inO foreverysmoothu∈X(O)ofunit
length.
2
(see [7] Chapter 6.1). In particular, if u is defined in O and has unit length on ∂O, then one has
ω(u)=2πdeg(u;∂O) where ω(u) is the vorticity associated to the vector field u:
ZO
ω(u):=dj(u)+κvol . (1)
g
SometimeswecanidentifytheindexofuatapointP ∈S withtheindexofualongacurvearound
P. Notethateverysmoothvectorfieldu∈X(O)(ormoregenerally,u∈X1,2(O))ofunitlengthin
Ohasdeg(u;∂O)=0;moreover,avortexwithnon-zeroindexwillcarryinfiniteenergyE asε→0.
ε
We will prove a Γ-convergence result (at the second order) of E as ε → 0. In particular, at
ε
the level of minimizers u of E , we show that u converges in X1,1(S) (for a subsequence) to a
ε ε ε
canonical harmonic vector field u∗ of unit length that is smooth 3 away from n = |χ(S)| distinct
singularpoints a ,...,a , eachsingularpoint a carryingthe same index d =signχ(S)so that 4
1 n k k
n
d =χ(S). (2)
k
k=1
X
The vorticity ω(u∗) detects the singular points {a }n of u∗:
k k=1
n
ω(u∗)=2π d δ in S, (3)
k ak
k=1
X
where δ is the Dirac measure (as a 2-form) at a . The expansion of the minimal energy E at
ak k ε
the second order is given by
1 1
E (u )=nπlog + lim |Du∗|2 vol +nπlogr +nγ +o(1), as ε→0,
ε ε ε r→0(cid:18)ZS\∪nk=1Br(ak) 2 g g (cid:19) F
where γ > 0 is a constant depending only on the potential F and B (a ) is the geodesic ball
F r k
centered at a of radius r. The second term in the above RHS is called the renormalized energy
k
between the vortices a ,...,a and governsthe optimal location of these singular points as in the
1 n
Euclidiancase(see the seminalbook[3]). Inparticular,if S is the unit sphereinR3 endowedwith
the standard metric g, then n=2 and a and a are two diametrically opposed points on S.
1 2
Outline of the note. The note is divided as follows. Section 2 is devoted to characterize canonical
harmonicvectorfields ofunit length. InSection 3, we determine the renormalizedenergy between
singularpoints ofcanonicalharmonic vectorfields. The mainΓ-convergenceresultis statedinthe
last section. The proofs of these results are part of our forthcoming article [9].
2 Canonical harmonic vector fields of unit length
We will say that a canonical harmonic vector field of unit length having the singular points
a ,...,a ∈ S of index d ,...,d ∈ Z for some n ≥ 1, is a vector field u∗ ∈ X1,1(S) such
1 n 1 n
that |u∗| =1 in S, (3) holds and
g
d∗j(u∗)=0 in S. (4)
3Inthecaseofasurface(S,g)withgenus1(i.e.,homeomorphicwiththeflattorus),thenn=0andu∗issmooth
inS.
4Infact,deg(u∗;γ)=dk foreveryclosedsimplecurveγ aroundak andlyingnearak.
3
Here, d∗ istheadjointoftheexteriorderivatived,i.e., d∗j(u∗)istheunique 0-formonS suchthat
d∗j(u∗),ζ) vol = j(u∗),dζ) vol for every smooth 0-form ζ.
g g g g
ZS ZS
(cid:0) (cid:0)
If u∗ satisfies (3), then the Gauss-Bonnet theorem combined with (1) imply that necessarily (2)
holds.
We will seethat condition(2) is alsosufficient. Indeed, if (2)holds, wewill constructsolutions
of (3) and (4), as follows: let ψ =ψ(a,d) be the unique 2-form on S solving:
n
−∆ψ =−κ vol +2π d δ in S, ψ =0, (5)
g k ak
k=1 ZS
X
with the sign convention that −∆=dd∗+d∗d. The idea is to find u∗ such that j(u∗)−d∗ψ is an
harmonic 1-form, i.e.,
Harm1(S)={integrable 1-forms η on S : dη =d∗η =0 as distributions}.
ThedimensionofthespaceHarm1(S)istwicethegenus(i.e.,2g)of(S,g)andwefixanorthonor-
mal basis η ,...,η of Harm1(S) such that
1 2g
(η ,η) vol =δ for k,l=1,...,2g.
k l g g kl
ZS
Therefore, it is expected that
2g
j(u∗)=d∗ψ+ Φ η in S (6)
k k
k=1
X
forsomeconstantvectorΦ=(Φ ,...,Φ )∈R2g. Theseconstantsarecalledflux integralsasthey
1 2g
can be recovered by
Φ = (j(u∗),η ) vol , for k =1,...,2g.
k k g g
ZS
Note that (6) combined with (5) automatically yield (3) and (4). One important point is to
characterize for which values of Φ the RHS of (6) arises as j(u∗) for some vector field u∗ of unit
length in S. For that condition, we need to recall the following theorem of Federer-Fleming [8]:
there exist 2g simple closed geodesics γ on S, ℓ = 1,...,2g, such that for any closed Lipschitz
ℓ
curve γ on S, one can find integers c ...,c such that
1 2g
2g
γ is homologous to c γ
ℓ ℓ
ℓ=1
X
i.e., there exists an integrable function f :S →Z such that
2g
ζ− c ζ = fdζ for all smooth 1-forms ζ.
ℓ
Zγ ℓ=1 Zγℓ ZS
X
Having chosenthe geodesic curves{γ }2g andthe harmonic1-forms {η }2g , we fix the notation
ℓ ℓ=1 k k=1
α := η , k,ℓ=1,...,2g. (7)
ℓk k
Zγℓ
4
Theorem 1 Let n ≥ 1 and d = (d ,...,d ) ∈ Zn satisfy (2). Then for every a = (a ,...,a ) ∈
1 n 1 n
Sn, there exists
ζ =ζ (a;d)∈R/2πZ, ℓ=1,...,2g
ℓ ℓ
such that if a vector field u∗ ∈X1,1(S) of unit length solves (3) and (4), then j(u∗) has the form
(6) for constants Φ ,...,Φ such that
1 2g
2g
α Φ +ζ (a,d)∈2πZ, ℓ=1,...,2g, (8)
ℓk k ℓ
k=1
X
where (α ) were defined in (7). Conversely, given any Φ ,...,Φ satisfying (8), there exists a
ℓk 1 2g
vector field u∗ ∈ X1,1(S) of unit length solving (3) and (4) and such that j(u∗) satisfies (6). In
addition, the following hold:
1) ζ (·;d) depends continuously on a∈Sn for every ℓ=1,...,2g. More generally, if 5
ℓ
nt n0
µt :=2π dtlδat →µ0 :=2π d0lδa0 in W−1,1 as t↓0,
l l
l=1 l=1
X X
{dt} areintegerswith (2)and nt |dt|isuniformlyboundedint,thenζ (at,dt)→ζ (a0,d0)
l l l=1 l ℓ ℓ
as t↓0.
P
2) any u∗ solving (3) and (4) belongs to X1,p(S) for all 1 ≤ p < 2, and is smooth away from
{a }n .
k k=1
3) If u∗,u˜∗ both satisfy (6) for the same (a,d) and the same {Φ }2g , then u˜∗ =eiβu∗ for some
k k=1
β ∈R.
The constants {ζ (a;d)}2g are determined as follows. For every ℓ = 1,...,2g, we let λ be
ℓ ℓ=1 ℓ
some smooth simple closed curve such that λ is homologous to γ (the geodesics fixed in (7)) so
ℓ ℓ
that {a }n is disjoint from λ ; for example, λ is either γ or, if γ intersects some a , a small
k k=1 ℓ ℓ ℓ ℓ k
perturbation thereof. We now define ζ (a,d) to be the element of R/2πZ such that
ℓ
ζ (a,d):= (d∗ψ+A) mod 2π, ℓ=1,...,2g, (9)
ℓ
Zλℓ
where ψ = ψ(a,d) is the 2-form given by (5) and A is the connection 1-form associated to any
moving frame defined ina neighborhoodofλ . The integralin (9) is independent, modulo 2πZ, of
ℓ
the choice of moving frame and of the curve λ homologous to γ . In examples in which it can be
ℓ ℓ
explicitly computed, in general ζ (a,d)6=0 mod 2π for ℓ=1,...,2g.
ℓ
3 Renormalized energy
For any n ≥ 1, we consider n distinct points a = (a ,...,a ) ∈ Sn. Let d = (d ,...,d ) ∈ Zn
1 n 1 n
satisfying (2), {ζ (a;d)}2g be given in Theorem 1 and Φ ∈ R2g be a constant vector inside the
ℓ ℓ=1
set:
2g
L(a,d):={Φ=(Φ ,...,Φ )∈R2g : α Φ +ζ (a,d)∈2πZ, ℓ=1,...,2g}.
1 2g ℓk k ℓ
k=1
X
5Ifµisa2-form(possiblymeasure-valued)thenwewriteforp,q∈[1,∞]with 1 + 1 =1:
p q
kµkW−1,p :=sup(cid:26)ZSfµ : f ∈W1,q(S;R),kfkW1,q :=kfkLq +kdfkLq ≤1(cid:27).
5
We define the renormalized energy between the vortices a of indices d by
n
1
W(a,d,Φ):= lim |Du∗|2 vol +πlogr d2 ,
r→0(cid:18)ZS\∪nk=1Br(ak) 2 g g Xk=1 k(cid:19)
where u∗ =u∗(a,d,Φ) is the unique (up to a multiplicative complex number) canonical harmonic
vector field given in Theorem 1 and B (a ) is the geodesic ball centered at a of radius r. Our
r k k
arguments show that the above limit indeed exists. As in the euclidian case (see [3]), we can
compute the renormalized energy by using the Green’s function. For that, let G(x,y) be the
unique function on S×S such that
vol
g
−∆ (G(·,y) vol )=δ − distributionally in S, G(x,y)vol (x)=0 for every y ∈S.
x g y g
Vol (S)
g ZS
with Vol (S):= vol . Then G may be represented in the form (see [2] Chapter 4.2):
g S g
R G(x,y)=G (x,y)+H(x,y), with H ∈C1(S×S),
0
where G is smooth away from the diagonal, with
0
1 1
G (x,y)=− log(dist(x,y)) if the geodesic distance dist(x,y)< (injectivity radius of S).
0
2π 2
The 2-form ψ =ψ(a,d) defined at (5) can be written as:
n
ψ =2π d G(·,a )vol +ψ vol in S,
k k g 0 g
k=1
X
where ψ ∈C∞(S) has zero average on S and solves
0
1 2πχ(S)
−∆ψ =−κ+κ¯, for κ¯ = κvol = . (10)
0 g
Vol(S) Vol(S)
ZS
In other words, the 2-form x 7→ ψ(x)+d logdist(x,a )vol is C1 in a neighborhood of a for
k k g k
every 1≤k ≤n. We have the following expression of the renormalized energy:
Proposition 2 Given n ≥ 1 distinct points a ,...,a ∈ S, integers d ,...,d with (2) and
1 n 1 n
Φ∈L(a,d), then
n 1 |dψ |2
W(a,d,Φ)=4π2 d d G(a ,a )+2π πd2H(a ,a )+d ψ (a ) + |Φ|2+ 0 vol ,
l k l k k k k k 0 k 2 2 g
l6=k k=1 ZS
X X(cid:2) (cid:3)
(11)
where ψ is defined in (10).
0
In the case of the unit sphere S in R3 endowed with the standard metric (in particular, ψ
0
vanishes in S), if n=2 and d =d =1, then the second term in the RHS of (11) is independent
1 2
of a (as x 7→H(x,x) is constant, see [14]); moreover, Φ = 0 and so, minimizing W is equivalent
k
by minimizing the Green’s function G(a ,a ) over the set of pairs (a ,a ) in S ×S, namely, the
1 2 1 2
minimizing pairs are diametrically opposed.
6
4 Γ-convergence
Given the potential F in Section 1, we compute the energy E of the radial profile of a vortex of
ε
index 1 inside a geodesic ball of radius R>0:
R f(r)2 1
I (R,ε):=inf π f′(r)2+ + F(f(r)2) rdr :f(0)=0,f(R)=1 .
F r2 2ε2
( Z0 (cid:20) (cid:21) )
Then I (R,ε) = I (λR,λε) = I (1, ε) =: I (ε) for every λ > 0, and the following limit exists
F F F R F R
(see [3]):
γ := lim(I (t)+πlogt).
F F
t→0
We state our main result:
Theorem 3 The following Γ-convergence result holds.
1) (Compactness) Let(u ) be afamily of vector fields on S satisfying E (u )≤Nπ|logε|+C
ε ε↓0 ε ε
for some integer N ≥0 and a constant C >0. Denoting by
Φ(u ):= (j(u ),η ) vol ,..., (j(u ),η ) vol ∈R2g,
ε ε 1 g g ε 2g g g
(cid:18)ZS ZS (cid:19)
then there exists a sequence ε↓0 such that
n
ω(u )−→2π d δ in W−1,1, Φ(u )→Φ as ε→0, (12)
ε k ak ε
k=1
X
where {a }n are distinct points in S and {d }n are nonzero integers satisfying (2) and
k k=1 k k=1
n n
|d |≤ N and Φ ∈ L(a,d). Moreover, if |d | = N, then n =N and |d | =1 for
k=1 k k=1 k k
every k =1,...,n (in particular, n=χ(S) modulo 2).
P P
2) (Γ-liminf inequality) Assume that the vector fields u ∈ X1,2(S) satisfy (12) for n distinct
ε
points {a }n ∈Sn and |d |=1, k =1,...n that satisfy (2) and Φ∈L(a,d). Then
k k=1 k
liminf[E (u )−nπ|logε|)] ≥ W(a,d,Φ)+nγ .
ε ε F
ε→0
3) (Γ-limsup inequality) For every n distinct points a ,...,a ∈ S and d ,...,d ∈ {±1}
1 n 1 n
satisfying (2) and every Φ∈L(a,d) there exists a sequence of vector fields u on S such that
ε
(12) holds and
E (u )−nπ|logε|−→W(a,d,Φ)+nγ as ε→0.
ε ε F
This theorem is the generalization of the Γ-convergence result for E in the euclidian case
ε
(see [6, 11, 13, 1]) and it is based on topological methods for energy concentration (vortex ball
construction, vorticity estimates etc.) as introduced in [10, 12].
Acknowledgements
R.I. acknowledges partial support by the ANR project ANR-14-CE25-0009-01. The research of
R.J.waspartiallysupportedbytheNationalScienceandEngineeringResearchCouncilofCanada
under operating grant 261955.
7
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