Line vortices in the <span class="mathjax-formula">$U(1)$</span> - Higgs model

LINE VORTICES IN THE U(1) - HIGGS MODEL

TRISTAN RIVIERE

Abstract. For a given U(1)-bundle^Eover^M =^R3nfx1:::xng, where the^xiareⁿdistinct points of^R3, we minimise the U(1)-Higgs action and we make an asymptotic analysis of the minimizers when the coupling constant tends to innity. We prove that the curvature (= magnetic eld) converges to a limiting curvature that we give explicitely and which is singular along line vortices which connect the ^xi. This work is the three dimensional equivalent of previous works in dimension two (see 3]

and 4]). The results presented here were announced in 12].

1. Introduction

Consider

n

distinct points ^f

x

¹

:::x

^ng in ^R3 and

E

a

U

(1)-bundle over

M

^x =^R3nf

x

¹

:::x

^ng.

Recall that such a bundle is uniquely determined by its rst Chern class

c

¹(

E

)²

H

²(

M

^x

^Z) =^Z

:::

^Z(n summands), that is

n

integers

d

¹

:::d

ⁿ which correspond to the integrals of

c

¹(

E

) on small spheres surrounding the points

x

¹

:::x

ⁿ. In other words

E

is uniquely determined by a curvature

h

⁰²

H

²(

M

^x) on

E

and for such a curvature one has

21

Z

S

h

⁰ = ^X

i:xi2Int ^(S)

d

ⁱ

where

S

is any closed, oriented surface of ^R3 enclosing some of the points

x

ⁱ in its interior,

Int

(

S

).

We suppose that⁸

i d

ⁱ⁶= 0 and we make the following neutrality hypoth- esis

P

i=1:::n

d

ⁱ= 0

:

Under the previous hypothesis, we can write the sequence of points

f

x

ⁱ

:::x

^ng and the corresponding degrees ^f

d

¹

:::d

^ng in the form

f

P

¹

:::P

^k

N

¹

:::N

^kg where the

P

ⁱ( resp.

N

ⁱ) correspond to the

x

ⁱ having positive (resp. negative) integer

d

ⁱ and are repeated as many time as to the multiplicity ^j

d

^ij.

This paper deals with a variational problem suggested in 8] by J. Frohlich and M. Struwe. For any section

'

of the^C -line bundle associated to

E

and

Centre de Mathematiques et de Leurs Applications, CNRS, URA 1611, Ecole Normale Superieure de Cachan, 61 Avenue du President Wilson, 94235 Cachan Cedex, France.

Received by the journal October 3, 1995. Accepted for publication March 8, 1996.

78 TRISTAN RIVIERE

any connection ~

A

on

E

, consider the

U

(1)^;Higgs action

S

(

' A

^~) =

Z

M

x

jr

~

A

'

^j2+^j

d A

^~j2+

2(1^;j

'

^j2)²

where^rA~

'

is the covariant derivative of

'

relative to ~

A

,

d A

^~ is the curva- ture of ~

A

and

V

(^j

'

^j) = ²(1^;j

'

^j2)² is the Higgs potential depending on^j

'

^j, which, from a mathematical point of view, could be replaced, in the remain of the paper, by any \reasonable" potential on ^j

'

^j (see 7] part 3) which forces^j

'

^jto be close to 1. The aim is to minimize

S

(

' A

^~) and to describe the minimizers.

The words used above (section, connection, covariant derivative...) come from geometry but there are also parallel words coming from abelian gauge theory in physics, that we will mix with the previous ones, to designate the same objects. We will also say that

'

is a scalar (Higgs) eld, ~

A

is the gauge eld (or vector potential),

d A

^~ is the magnetic eld. We will also called ^j

'

^j the density of the scalar eld and (

i'

^rA

'

) the current where ( , ) is the usual scalar product on each ber^C and

i'

the ² rotation of

'

on the ori- ented bre.

As it is pointed out in 8] for^j

d

^ij6= 0

S

(

' A

^~) = +¹

in fact for any

i

²1

n

] and

r

suciently small

Z

@B

r (x

i )

d A

^~= 2

d

ⁱ

thus by Cauchy-Schwartz inequality

^j

d

^ij2 2

r

^{2 6}

Z

@B

r (x

i )

j

d A

^~j2

(1. 1)

integrating (1. 1) in

r

we get^Rj

d A

^~j2= +¹.

Thus we will adopt the renormalisation proposed in 8].

Consider the following closed 2-form on

M

^x

h

⁰ =^;1

2

P

i=1:::n

d

ⁱ

d

1

j

x

^;

x

^ij

:

(1. 2)

Where denote the Hodge operator.

h

⁰ is a curvature on

E

and it veries the following equations

8

>

>

>

<

>

>

>

:

dh

⁰ = 2

^Xn

i=1

d

ⁱ

^xi in ^D0(^R3)

d

h

⁰ = 0 in ^D0(^R3)

:

(1. 3) where

d

=

d

.

h

⁰will be called the reference curvature (or magnetic eld).

Consider a connection ~

A

on

E

and ~

h

=

d A

^~ it's curvature, we denote by h the dierence

h

= ~

h

^;

h

⁰

:

Formally since

dh

= 0 in ^R3nf

x

¹

:::x

^ng and

R

@

h:

= 0 for any regular bounded domain of ^R3, thus there exists a

Esaim : Cocv June 1996, Vol.1, pp. 77-167

global 1-form

A

on ^R such that

h

=

dA

and

Z

R 3

(~

h

^;

h

⁰)

:h

⁰=

Z

R 3

h:h

⁰ =

Z

R 3

dA

^{^}

h

⁰

= ^;Z

R 3

A

^{^}

d

h

⁰

= 0

:

(1. 4) (The previous armations and equalities could be justied rigourously by assuming some regularity on ~

h

in the neighborhood of the

x

ⁱ and innity.

For instance suppose ~

h

²

L

^p(^R3) for some 1

< p <

³² and

d

h

^{~ 2}

L

^q(^R3) for some

q >

³². This would imply

d

^~

h

= 2

^Pni=1

d

ⁱ

^xi in ^D0(^R3) and the existence of

A

²

W

^loc1s(^R3) for some

s >

3 such that

h

=

dA

in ^D0(^R3)).

In view of (1. 4) we deduce that, by considering ^RMx

j~

h

^;

h

^0j2 instead of ^RMxj~

h

^j2, on one hand we are substracting the \same innite quantity"

R

M

x

j

h

^0j2 independent of ~

A

and, on the other hand, ^RMx

j~

h

^;

h

^0j2 has no reason, any more, to be innite (take ~

h

=

h

⁰ !).

Thus we are tempted to replace the classical Higg's action by the following one

S

~(

' A

^~) =

Z

M

x

jr

~

A

'

^j2+^j

d A

^~;

h

^0j2+

2(1^;j

'

^j2)²

:

In view of formulating a well posed variational problem we introduce a nite covering

U

of^R3nf

x

¹

:::x

^ngin each open set of which we can trivialize the bundle. The transition functions are choosen to be regular and we introduce the following set of minimisation

V

=

8

<

:

(

'

A

^~)²

H

^loc1 (

U

^C)

H

^loc1 (

U

^R3)

s:t:

on

U

^\

U

'

=

g

'

and

i A

^~=

g

^;1

dg

+

i A

^~

S

~(

' A

^~)

<

+¹

9

=

We have the following theorem which was already established in 8] and whose proof is essentially based on the choice of a good gauge which makes the functional coercive.

Theorem 1.1.

min

('

~

A)2V

S

~(

' A

^~) is achieved

:

After having given a proof of theorem 1.1, in the remains of the paper, we will follow the approach of 2] and 4], that is, we will make the parameter

= ^1" tend to innity in view of forcing the density ^j

'

^jto be close to 1.

The problem is that we cannot expect, for topological reasons, to nd a section

'

of the previous non trivial bundle

E

taking it's values in the unit sphere of each ber. This occurs only out of lines which connect the points

x

ⁱ according to their multiplicity. The problem is similar to the one in di- mension 2, here in dimension 3

'

necessarily has to have a degree around those lines which is independent of the local trivialisation and this will makes the energy tend to innity. That is the main diculty that one encounters when one tries to understand the behaviour of the mininimizers as

tends to innity. The dierent quantities seem to diverge and one has to work

Esaim : Cocv June 1996, Vol.1, pp. 77-167

80 TRISTAN RIVIERE

harder to get a norm for which something will remain bounded and con- verge in some space.

The notation ~

S

(

' A

^~) is used in 8] where this variational problem was rst introduced but, since we make the parameter

tend to innity, to be con- sistent with 4], the energy denoted by ~

S

(

' A

^~) becomes

G

^"(

' A

^~) where

"

= ^p1

:

In fact, as we will see later, the scale

"

will be much more relevant than

. Before we state our main result, we need to introduce the notion of minimal connection already used in 6]. Let ^Sk be the set of all the permutations in ^f1

:::k

^g. The following quantity

L

is called minimal connection of the sequence (

x

ⁱ

d

ⁱ)^i=1:::n

L

= min

2S

k k

X

i=1

j

P

^i;

N

^(i)j

:

Consider any

²

S

^k which realizes the previous minimum (

is perhaps not unique), we will call also minimal connection the 1 dimensional current IL with support in^Ski=1

P

ⁱ

N

⁽ⁱ⁾] oriented by

P

^i;

N

⁽ⁱ⁾ and having the integer multiplicity

(

x

) = #

i s:t: x

²

P

ⁱ

N

⁽ⁱ⁾]

: L

is of course the mass of IL.

Our main result is the following:

Theorem 1.2. For any sequence

"

ⁿ tending to zero and for any sequence (

'

ⁿ

A

^~n) of minimizers of

G

^"n on

V

, one can extract a subsequence, still denoted (

'

ⁿ

A

^~n), such that, for any 1

< p <

3

=

2

h

ⁿ=

d A

^~n;

h

⁰ strongly converges in

H

^loc1 (^R3n

supp

IL)^\

W

^loc1p(^R3) where IL is a minimal connection associated to the sequence (

x

ⁱ

d

ⁱ)^i=1:::n

:

Moreover the limit

h

^? is the solution of

(^;

h

^?+

h

^?+

h

⁰) =^;2

IL

:

(1. 5) Remark 1.1. If, for instance, the minimal connection is unique, thus the complete sequence converges.

Remark 1.2. Contrary to the 2 dimensional case, we are not able to give a precise asymptotic expansion of the energy which should be of the form, when

d

ⁱ=1

G

^"n(

'

ⁿ

A

^~n) = 2

L

log 1

"

^{n +}

W

(

x

ⁱ

d

ⁱ) + 0(1)

:

Here

W

(

x

ⁱ

d

ⁱ) is a renormalised energy which represents the self interaction of the minimal connection on itself. But this has not the same importance as in the 2-dimensional problem considered in 3] and 4] since, here, we know where the singularities are going to be located and the renormalized energy

W

(

x

ⁱ

d

ⁱ) could be directly computed once we have xed the

x

ⁱ and

d

ⁱ.

Esaim : Cocv June 1996, Vol.1, pp. 77-167

The asymptotic expansion of the minimal energy when all the minimal con- nections contain a segment with integer multiplicity dierent from 1 is an interesting question see remarks 2.1 and 2.2.

Remark 1.3. The analysis developed in this paper could be adapted for solving non-gauge invariant problems in dimension 3, that is, variational problems for functionals of the form

E

^"(

u

) =

Z

R 3

jr

u

^j2+ 1

"

^2(1;j

u

^j2)²

:

For instance, take to be a convex domain of ^R3 and

g

^" a boundary con- dition which simulates the vorticity at the boundary. A particular choice of

g

^" could be a \

H

¹-approximation" of a map

g

^?

@

^!

S

¹ having some degree around dierent points of

@

i.e.

- ^j

g

^"j61 on

@

- ⁹

n

distinct points (^P

a

¹

:::a

ⁿ) of

@

and

n

integers (

d

¹

:::d

ⁿ) such that

i=1:::n

d

ⁱ= 0

j

g

^"j= 1 on

@

ⁿ(^Sni=1

B

^"(

a

ⁱ)^\

@

)

deg

(

g

^"

@B

^"(

a

ⁱ)^\

@

) =

d

ⁱ

- an upper bound for the Dirichlet energy of

g

^"

Z

@ jr

g

^"j26

2

^Xn

i=1

d

²ⁱ

!

log 1

"

⁺

K:

One can prove that, given

"

ⁿ tending to zero, from a sequence

u

^"n of mini- mizers of

E

^"n among

H

^g1"n(

^C) one can extract a subsequence which con- verges in

H

^loc1 ("ⁿ

Supp

IL) to an harmonic map

u

into

S

¹, where IL is a minimal connection between the (

a

ⁱ

d

ⁱ) and the degree of

u

arround the dierent segments of IL corresponds to the multiplicity of the minimal con- nection along its segments. Moreover, if the minimal connection is unique, the complete sequence converges. The proof of this theorem can be estab- lished by following the main ideas of the proof of theorem 2. For instance one can transpose word by word the

-compactness lemma (part 3) to the non-gauge invariant situation just omitting to write

hAA

⁰

h

⁰ ! The case where is not convex is more complicated since it may happen that the lines of singularities (i.e. the bad set^j

u

^"j61

=

2) tend to touch

@

as

"

^!0.

Remark 1.4. With some more work, one could establish a stronger con- vergence than

H

¹ away from the line singularities as in the two dimensional case (see 2]).

Remark 1.5. As in the two dimensional case, the description of the mini- mizers for

" >

0 is much more interesting and also more dicult than the one of the limiting situation (see in dimension 2 for

" >

0 11], 1],...). In dimension 3, also, we can ask similar question to the ones in dimension 2:

Are there several minimizers ? In the particular case of a dipole with mul- tiplicity 1, is the minimizer axially symmetric around the dipole ? We can ask also questions specic to the 3 dimension: For instance when the multi- plicity

d

of our dipole is no more 1 how does the zero set of ^j

'

^j look like ? We can expect to get

d

dierent lines connecting the two poles in a way that has to be described. The conguration of those lines seems to be given by a

Esaim : Cocv June 1996, Vol.1, pp. 77-167

82 TRISTAN RIVIERE

1-dimensional variational problem, similar to the zero dimensional problem in 2 dimensions given by the renormalised energy but depending strongly on

"

. What could be the resulting interaction energy between those lines ?

The paper is devoted to the proof of theorem 1.2. The organisation of the proof is similar to the one developed in 4], but the dimension 3 induces a lot of new technical diculties. It is for instance much more complicated to detect and to \catch" lines (which can be a-priori very bad) than a nite number of points.

One of the main dierences comes also from the stress-energy tensor (see part 2-5) which contains, in dimension 3, a new term preventing a direct local estimate of the density and its closeness to 1 (see 4] part 3). In dimension 2 this closeness ensured quite directly the convergence except on a nite number of small balls. Here, following ideas and classical methods of the regularity theory for non-linear elliptic problems, we deduce, from the conservation law of the stress-energy tensor, some kind of monotonicity formula which is the main tool for proving what we call an \

-compactness"

lemma, ( with reference to the classical

"

-regularity lemma of the above mentioned theory). This

-compactness lemma says that, if the energy on a ball is suciently small compared to the global bound (in log^1") established in 2-3, one can ensure that, in the ball of half radius, the density^j

'

^jis close to 1, this is necessary for being far away from the non-compactness locus (the bad part of^R3nf

x

ⁱ

:::x

^ng) . From the

-compactness Lemma 3.1 we deduce, in part 4, that, for any

<

1 the bad set is contained in ^c()" balls of radius

"

and that, in some sense, we can bound the vorticity around this union of balls (this is Lemma 4.1). Having located the bad set in something which looks like a line when

"

^!0 (^c()" ball or radius

"

), we nally establish a rst bound, independent of

"

, which is a bound of the

W

^1p norm of the magnetic eld for any 1

< p <

3

=

2 (see Lemma 5.1 ). This gives us a weak convergence, in

W

^1p, of the magnetic eld and Theorem 5.1 describes the limit. In part 6 we work out of the limiting bad set for establishing really the

H

¹ compactness, in fact we were not sure that, away from the limiting bad set, we could ensure the closeness to 1 for^j

'

^jwhich, combined with the weak

W

^1p convergence of the magnetic eld, leads directly to the compactness.

Here again the main ingredient is the

-compactness lemma. Finally, in part 7, we show that the limiting bad set which, for topological reasons, has to connect the dierent points

x

ⁱaccording to the multiplicities

d

ⁱ, generates an energy, at the level

"

, which is asymptotically greater than 2

log^1" times the connection it realizes between the (

x

ⁱ

d

ⁱ) minus a positive quantity as small as we want. This fact combined with the initial lower bound established in 2.3 forces the limiting bad set to be a minimal connection between the (

x

ⁱ

d

ⁱ). The diculty in this part is essentially technical, since, the limiting bad set, is a-priori only a 1-dimensional rectiable current whose support is

H

1-bounded.

In the appendix we prove some technical and basic Lemmas we use in the paper.

Esaim : Cocv June 1996, Vol.1, pp. 77-167

2. Preliminaries and Notations

2.1. Choice of a covering and a Coulomb Gauge

First of all we x a particular covering of^R3nf

x

¹

:::x

^ngon which we will be able to trivialize

E

and that will be used in the remains of the paper in particular in 2.3 for establishing an upper bound for the energy.

Suppose rst that the indexing of the (

N

¹

:::N

^k) is chosen such that the trivial permutation (

i

^7!

i

) realizes a minimal connection i.e.

k

X

i=1

j

P

^i;

N

^ij= min

2S

k k

X

i=1

j

P

^i;

N

^(i)j

:

Suppose now there exists

i

and

i

⁰such that

P

ⁱ

N

ⁱ]^\

P

ⁱ⁰

N

ⁱ⁰] contains more than 1 point.

1st case:

P

ⁱ

N

ⁱ] =

P

ⁱ⁰

N

ⁱ⁰]. We do not change the sequence (

P

ⁱ

:::P

^k

N

ⁱ

:::N

^k).

2nd case:

P

ⁱ =

P

ⁱ⁰ but

N

ⁱ⁶=

N

ⁱ⁰(or the opposite). Suppose

N

^{i 2}

P

ⁱ⁰

N

ⁱ⁰] thus we repeat one more time

N

ⁱin the negative terms and we add

N

ⁱ in the positive term thus

P

ⁱ

N

ⁱ]

P

ⁱ⁰

N

^I0] is replaced by

P

ⁱ

N

ⁱ]

P

ⁱ

N

ⁱ]

N

ⁱ

N

ⁱ⁰]

3nd case:

P

ⁱ

N

ⁱ]

P

ⁱ⁰

N

ⁱ⁰] with

P

^{i 6}=

P

ⁱ⁰ and

N

ⁱ⁶=

N

ⁱ⁰.

In the sequence (

P

ⁱ

:::P

^k

N

ⁱ

:::N

^k) we repeat

P

ⁱ and

N

ⁱ such that

P

ⁱ

N

ⁱ]

P

ⁱ⁰

N

ⁱ⁰] =

P

ⁱ⁰

P

ⁱ]

P

ⁱ

N

ⁱ]

P

ⁱ

N

ⁱ]

N

ⁱ

N

ⁱ⁰]

4th case:

P

ⁱ⁰

N

ⁱ]

P

ⁱ

N

ⁱ⁰] with

P

^{i0 6}=

P

ⁱ and

N

^{i0 6}=

N

ⁱ

In the sequence (

P

¹

:::P

^k

N

¹

:::N

^k) we repeat

P

ⁱ⁰ and

N

ⁱ such that

P

ⁱ

N

ⁱ]

P

ⁱ⁰

N

ⁱ⁰] =

P

ⁱ

P

ⁱ⁰]

P

ⁱ⁰

N

ⁱ⁰]

P

ⁱ⁰

N

ⁱ]

N

ⁱ

N

ⁱ⁰]

Thus in all the cases we can change the sequence (

P

ⁱ

N

ⁱ)^i=1:::k and the number

k

without changing neither the bundle nor the minimal connec- tion in such a way that for

i

⁶=

i

⁰

P

ⁱ

N

ⁱ]^\

P

ⁱ⁰

N

ⁱ⁰] = or a point or

P

ⁱ

N

ⁱ] =

P

ⁱ⁰

N

ⁱ⁰]

:

We proceed to a last reindexing by associating to a couple (

P

ⁱ

N

ⁱ) the positive integer

n

ⁱ which is the number of times the couple is re- peated in ((

P

ⁱ

N

ⁱ))^i=1:::k. The previous sequence is now represented by ((

P

ⁱ

N

ⁱ

n

ⁱ))^i=1:::k0 with

k

⁰⁶

k

.

k

⁰will be denoted by

k

.

Let

O

ⁱbe the middle point of the segment

P

ⁱ

N

ⁱ], and

C

ⁱ(

r

) be the union of the two half cones of vertices

P

ⁱ and

N

ⁱ and of base the disk perpendicular to

P

ⁱ

N

ⁱ], of centre

O

ⁱ and radius

r

.

We can choose

r

suciently small such that⁸

i

⁶=

i

⁰

C

ⁱ(

r

)^\

C

ⁱ⁰(

r

) =or

C

ⁱ(

r

) =

C

ⁱ⁰(

r

)

:

Let

U

ⁱ =

C

ⁱ(

r

) for

i

⁶

k

and

U

^{k +1} =^R3nki=1

C

^i;2r. (

U

ⁱ)^{i=1:::k +1} realizes a covering of ^R3nf

x

ⁱ

:::x

^ng on each open set

U

ⁱ of which we can make a

Esaim : Cocv June 1996, Vol.1, pp. 77-167

84 TRISTAN RIVIERE

trivialisation of E. By

d

ⁱ we denote the angular 1-form around the axis (

P

ⁱ

N

ⁱ) oriented by

P

^i;

N

ⁱ.

We will only consider trivialisations of

E

over the

U

ⁱ whose transition func- tions

g

^{ik +1}(

x

)²

S

¹ on

U

^i\

U

^{k +1}=

C

ⁱ(

r

)ⁿ

C

ⁱ(

r=

2) verify

8

x

²

C

ⁱ(

r

)ⁿ

C

ⁱ(

r=

2)

g

^{ik +1;1}

dg

^{ik +1}(

x

) =

in

ⁱ

d

ⁱ (2. 1) Suppose we have a trivialisation of

E

over the

U

ⁱ whose transition functions verify (2. 1) we get all the others (verifying (2.1)) simply by multiplying by

e

ⁱwhere

is a global function on all of^R3nf

x

¹

:::x

^ng. That is the change of trivialisation we will consider most of the time.

Consider the following 1-forms

A

^{0k +1}(

x

) = ^{12 P}

i=1:::k

n

^ijx;Nx;Nii j

; x;P

i

jx;P

i j

:

^jPPii^;Ni

;N

i

j

d

ⁱ in

U

^{k +1}

A

^0j(

x

) = ^{12 P}

i=1:::k

n

^ijx;Nx;Nii j

; x;P

i

jx;P

i j

:

^jPPii^;Ni

;N

i j

d

ⁱ

;

n

^j

d

^j in

U

^j

j

⁶

k

(2. 2) Clearly

A

^0l(

l

= 1

:::k

+ 1) is regular in

U

^l and we have

8

j

⁶

k A

^{0k +1}(

x

) =

A

^0j(

x

) +

n

^j

d

^j in

U

^{k +1\}

U

^j

:

(2. 3) Moreover one veries that

8

l

⁶

k

+ 1

dA

^0l(

x

) =

h

⁰(

x

) in

U

^l (2. 4) Thus, suppose we have chosen a trivialisation of

E

over the

U

ⁱwhich veries (2. 1), (2. 2) is the expression in this trivialisation of a connection whose curvature is

h

⁰.

Consider now a

H

¹-connection ~

A

on

E

that is ~

A

^{j 2}

H

^loc1 (

U

^j

^R3) for

j

⁶

k

+1 such that

A

~^{k +1}(

x

) = ~

A

^j(

x

) +

n

^j

d

^j in

U

^{k +1\}

U

^j

:

(2. 5) Let ~

h

=

d A

^~in ^R3nf

x

¹

:::x

^ng and suppose

h

= ~

h

^;

h

⁰²

L

²(^R3)

:

Denote by

A

the following global 1-form on ^R3

A

(

x

) =^; 1 4

d

h

¹

j

x

^j

=^; 1 4

3

X

k =1 Z

R 3

3

X

l=1

h

^{k l}(

y

)

x

^l;

y

^l

j

x

^;

y

^j3

dy dx

^k

:

(2. 6) Since

h

²

L

²(^R3),

dh

²

W

^;12(^R3) and since supp(

dh

) ^f

x

ⁱ

:::x

^ng we have

dh

0 in ^D0(^R3). Thus we have

dA

(

x

) =

dd

h

^;₄

^1j

x

^j

+

d

d

h

^;₄

^1j

x

^j

=

h

^;₄

^1j

x

^j

=

h

(

x

) in ^D0(^R3) (2. 7) This implies

d

A

^~;

A

^0;

A

= 0 in ^R3nf

x

¹

:::x

^ng

Esaim : Cocv June 1996, Vol.1, pp. 77-167

where ~

A

^;

A

is a global 1-form dened by

A

~^;

A

⁰(

x

) = ~

A

^l(

x

)^;

A

^0l(

x

) in

U

^l

:

Since

H

^dR1 (^R3nf

x

¹

:::x

^ng) = 0 there exists

²

W

^loc22(^R3nf

x

¹

:::x

^ngR) such that

A

~^;

A

^0;

A

(

x

) =

d

(

x

) in^R3nf

x

¹

:::x

^ng

:

Thus, if we make the global change of trivialisation by multiplying the ber over

x

by

e

^{;i (x)} the new expression of the connection ~

A

is given by

A

~^0l(

x

) = ~

A

^l;

d

(

x

) =

A

^0l(

x

) +

A

(

x

) in

U

^l

:

(2. 8) Consider now the set among which we are going to minimize

G

^",

V

=

8

<

:

(

'

^l

A

^~l)²

H

^loc1 (

U

^l

^C)

H

^loc1 (

U

^l

^R3) s.t.

8

j

⁶

k '

^{k +1}=

'

^j

e

^injj

A

^{~k +1}= ~

A

^j+

n

^j

d

^j in

U

^j\

U

^{k +1}

G

^"(

' A

^~)

<

+¹

9

=

(2. 9) where

e

^ij is the angular coordinate around the axis

P

^j

N

^j] in

C

^j(

r

)ⁿ

C

^j;r2 where we have xed some horizontal direction equal to 1.

Among

V

we say that (

' A

^~) is equivalent to (

'

⁰

A

^~0) if there exists

²

W

^loc22(^R3nf

x

¹

:::x

^ng

^R) such that

81⁶

l

⁶

k

+ 1 ~

A

^l = ~

A

^0l+

d

and

'

^l =

e

ⁱ

'

^0l in

U

^l (2. 10) Proposition 2.1. Let (

' A

^~) ²

V

, there exists (

'

⁰

A

^~0) ²

V

equivalent to (

' A

^~) such that

81⁶

l

⁶

k

+ 1 ~

A

^0l(

x

) =

A

^0l(

x

) +

A

(

x

) in

U

^l

where

A

(

x

) = ^;41

d

(~

h

^;

h

⁰) ^jxj1 : This is what we call the Coulomb gauge of ~

h

.

The proposition is proved just above, the name Coulomb gauge comes from the fact that

d

A

= 0 in ^D0(^R3)

:

(2. 11) Moreover one easily veries that

8

l

^2f1

:::k

^g

d

A

^0l = 0 in ^D0(

U

^l)

:

(2. 12) Thus

8

l

^2f1

:::k

^g

d

A

^~0l= 0 in ^D0(

U

^l)

:

(2. 13) 2.2. Existence of minimizers. Proof of theorem 1.1

In this part we prove the theorem 1 stated in the introduction. More precisely we prove the following,

Theorem 2.3. For any

" >

0 the minimum of

G

^"among

V

(dened by (2.

9)) is achieved by, at least, one class of couples (

'

^"

A

^~") (for the relation 2.

10). The (

'

^"

A

^~") verify

;

12 ^j

'

^"j2= 1

"

^2j

'

^"j2(1^;j

'

^"j2)^;jrA"~

'

^"j2 in ^D0(^R3nf

x

¹

:::x

^ng)

(2. 14)

Esaim : Cocv June 1996, Vol.1, pp. 77-167

86 TRISTAN RIVIERE

and

;

d

h

^"=

J

^" in ^D0(^R3)

(2. 15) where

J

^", the current, is the gauge invariant quantity equal to (

i'

^"

d

^A"~

'

^") in ^R3nf

x

¹

:::x

^ng. Moreover^j

'

^"j2

J

^"

h

^"are regular in ^R3nf

x

¹

:::x

^ng.

Proof of theorem 2.3

Let

"

be a xed positive real for the remain of the proof. Let (

'

^p

A

^~p) be a minimizing sequence of

G

^". We have

Z

R 3

j

h

^pj2=

Z

R 3

nfx

1 :::xng

j~

h

^p;

h

^0j2

< C

indep. of

p :

(2. 16) Thus by classical estimates for Calderon - Zygmund operators we have

Z

R 3

j

A

^pj6

1=6+

Z

R 3

jr

A

^pj2

1=2

6

C

Z

R 3

j

h

^pj2

1=2

6

C

(2. 17) where

A

^p is dened by (2. 6).

In the class of (

' A

^~) consider the Coulomb gauge given by Proposition II.1. For any

l

⁶

k

+ 1 and

K

compact set in

U

^l, we have

Z

U

l

\K jr

~

A

p

'

^pj26

C

thus

Z

U

l

\K

jr

'

^{plj2 6}

C

+ 2

Z

U

l

\K

j

A

~^plj2j

'

^pj2

6

C

+ 2

Z

U

l

\K

j

A

~^plj2+ + 2

Z

U

l

\K j

A

~^plj4

1=2 Z

U

l

\K

(1^;j

'

^pj2)²

1=2

(2. 18)

Since

A

^0l is regular on

U

^l,^RUl

\K

j

A

^0lj4 is bounded and from (2.15) we deduce

that Z

U

l

\K

j

A

^0l +

A

^pj46

C

indep. of

p :

(2. 19) Thus from (2. 18) and (2. 19) it follows that

'

^pl weakly converges in

W

^loc12(

U

^l

^C) (up to a subsequence) and this is also the case for ~

A

^pl =

A

^0l+

A

^p. By lower semi continuity of the

W

¹² it is clear that the limit (

'

^l"

A

^0l +

A

^p")^{l=1:::k +1} minimizes

G

^".

The fact that equations (2. 14) and (2. 15) are veried in^D0(^R3nf

x

¹

:::x

^ng) and that, moreover,^j

'

^"j2

J

^" and

h

^" are regular in^R3nf

x

¹

:::x

^ngis proved in 10]. Let us just prove that (2. 15) is veried in ^D0(^R3).

Let be a bounded domain of ^R3, since

h

^{" 2}

L

²(),

d

h

^{" 2}

W

^;12(), moreover, since ^RR3(1^;j

'

^"j2)²

<

+¹

^j

'

^{"j 2}

L

⁴() and this implies that

Esaim : Cocv June 1996, Vol.1, pp. 77-167

the function

J

^" on ^R dened by

J

^" (

i'

^"

d

^A"

'

^") in ^Rnf

x

¹

:::x

^ng is in

L

⁴⁼³()

W

^;12().

Thus

d

h

^"+

J

^{" 2}

W

^;12()

and

supp

(

d

h

^"+

J

^")^f

x

¹

:::x

^ng, so this implies (2. 15).

2.3. An upper bound for the energy In this section we prove the following lemma

Lemma 2.1. For any

>

0 we have min

V

G

^"62

(

L

+

)

Ln

¹

"

⁺

K

(

)

(2. 20) where

L

is the minimal connection between the (

x

ⁱ

d

ⁱ) and

K

only depends on

, not on

"

.

Remark 2.6. Contrary to the 2-dim. case, the upper bound for the energy requires the introduction of an external parameter

. This is not because the dimension 3 leeds to new technical diculties, this weaker upper-bound, Compare to the two dimensional one (see proposition 2.5 4]), is intrinsically linked to the dimension 3 for any minimal connection between the (

x

ⁱ

d

ⁱ) containing a segment with integer multiplicity dierent from 1.

Consider for instance the case of a dipole (

PN

) of charge

d >

1 ie ((

x

ⁱ

d

ⁱ)) = ((

Pd

)

(

N

^;

d

)). The minimal connection is the segment

PN

] with the integer multiplicity

d

, that is IL =

d

^j

PN

]^j] and

L

=

d

^j

P

^;

N

^j. The singular set tends to concentrate along this segment when

"

^7! 0. Suppose this conguration is adopted for any

" >

0. Then the energy would be greater than 2

d

^2j

P

^;

N

^jlog^1"! Thus at the stage

"

, what we call the bad set prefer to decompose

d

distinct tubes between

P

and

N

which are close to the segment

PN

] (but not too much so that the interaction energy between them is at most of orderlog log

"

).

Remark 2.7. If (

x

ⁱ

d

ⁱ) admits a minimal connection which contains no segment of integer multiplicity dierent from 1, then Lemma 2.1 holds for

= 0. It seems that this is a minimal connection that the singular set would prefer to adopt.

Proof of Lemma 2.1.

Let

>

0 and

" >

0. We construct (

'

^l

A

^~l) in

V

verifying (2. 20). First of all we construct^j

'

^j.

For 1⁶

i

⁶

k

consider

n

ⁱ the multiplicity of (

P

ⁱ

N

ⁱ) in the minimal connec- tion that we consider and for 1⁶

s

⁶

n

ⁱ introduce

L

^is to be the following segments.

If

n

ⁱ = 1, take

L

^is =

L

ⁱ¹ =

P

ⁱ

N

ⁱ]. and if

n

^{i 6}= 1 we identify the perpen- dicular plane to (

P

ⁱ

N

ⁱ) passing by

O

ⁱ, the center of

P

ⁱ

N

ⁱ], with ^C and denote by (

a

^is)^s=1:::ni the

n

ⁱ-th roots of 1 multiplied by

. (we suppose

<

^r4 in such a way that

a

^is2

C

ⁱ(^r2) and also for technical reasons which

Esaim : Cocv June 1996, Vol.1, pp. 77-167