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arXiv:cond-mat/9711212v1 [cond-mat.supr-con] 21 Nov 1997

Topological Multivortices Solutions of the Self-Dual Maxwell-Chern-Simons-Higgs System

Dongho Chae Department of Mathematics Seoul National University Seoul 151-742, Korea e-mail: dhchae@math.snu.ac.kr and Namkwon Kim Department of Mathematics POSTECH, Pohang 790-784, Korea e-mail: nkkim@math.snu.ac.kr

Abstract We study existence and various behaviors of topological multivortices solutions of the relativistic self-dual Maxwell-Chern-SimonsHiggs system. We ?rst prove existence of general topological solutions by applying variational methods to the newly discovered minimizing functional. Then, by an iteration method we prove existence of topological solutions satisfying some extra conditions, which we call admissible solutions. We establish asymptotic exponential decay estimates for these topological solutions. We also investigate the limiting behaviors of the admissible solutions as parameters in our system goes to some limits. For the Abelian Higgs limit we obtain strong convergence result, while for the Chern-Simons limit we only obtained that our admissible solutions are weakly approximating one of the Chern-Simons solutions.

1

Introduction

Since the pioneering works by Ginzburg and Landau on the superconductivity there are many studies on the Abelian Higgs system[10], [12](and references therein). In particular in [10] Ja?e and Taubes established the unique existence of general ?nite energy multi-vortices solutions for the Bogomol’nyi equations. (See also [12] for more constructive existence proof together with explicit numerical solutions.) More recently, motivated largely by the physics of high temperature superconductivity the self-dual Chern-Simons system(hereafter Chern-Simons system) was modeled in [2] and [3].(See [1] for a general survey.) The general existence theorem of topological solutions for the corresponding Bogomol’nyi equations was established in [11] by a variational method, and in [8] by an iteration argument. For the nontopological boundary condition we have only general existence result for the radial solutions for vortices in a single point[9]. We recall that in the Lagrangian of the Chern-Simons system there is no Maxwell term appearing in the Abelian Higgs system, while the former includes the Chern-Simons term which is not present in the later. Naive inclusion of both of the two terms in the Lagrangian makes the system non self-dual(i.e. there is no Bogomol’nyi type equations for the nontrivial global minimizer of the energy functional.) In [5], however, a self-dual system including both of the Maxwell and the Chern-Simons terms, so called (relativistic) self-dual Maxwell-ChernSimons-Higgs system was successfully modeled using the N = 2 supersymmetry argument[4], [6]. It was found that we need extra neutral scalar ?eld to make the system self-dual. In this paper we ?rst prove general existence theorem for topological multi-vortex solutions of the corresponding Bogomol’nyi equations of this system by a variational method. Then, using an iteration argument we constructively prove existence of a class of solutions enjoying some extra conditions. We call topological solutions satisfying these extra conditions the admissible topological solutions. We prove asymptotic exponential decay estimates for the various terms in our Lagrangian for the general topological solutions. One of the most interesting facts for the admissible topological solutions is that these solutions are really ”interpolated” between the Abelian Higgs solution and the Chern-Simons solutions in the the following sense: for ?xed electric charge, when the Chern-Simons coupling constant goes

2

to zero, our solution converges to the solution of the Abelian Higgs system. The convergence in this case is very strong. On the other hand, when both the Chern-Simons coupling constant and the electric charge goes to in?nity with some constraints between the two constants, we proved that our solution is ”weakly approximately” satisfying the Bogomol’nyi equations for Chern-Simons system. In the existence proof for the admissible topological solutions, although we used iteration method in R2 directly, we could start iteration in a bounded domain with suitable boundary condition to obtain a solution in that domain, and then enlarge this domain to the whole of R2 as is done in [8], and [12] in a much simpler case than ours. In this way it would be possible to obtain an explicit numerical solution. The organization of this paper is following. In the section 1 we introduce the action functional for the self-dual Maxwell-Chern-Simons-Higgs system, and deduce a system of second order elliptic partial di?erential equations which is a reduced version of the Bogomol’nyi system. In the section 2 by a variational method we prove a general existence of topological solutions. Then, we introduce the notion of admissible topological solution. In the section 3, 4 and 5 we prove existence of admissible topological solutions by an iteration method. In the section 5, in particular, we establish exponential decay estimates for our solutions. In the section 6 we prove strong convergence of the admissible topological solutions to the Abelian Higgs solution. The last section considers the Chern-Simons limit, and we prove that admissible topological solutions are weakly consistent to the Chern-Simons equation in this limit. (After ?nishing this work, we found that there was a study on the non relativistic version of our model by Spruck-Yang in [7].)

1

Preliminaries

The Lagrangian density for the (relativistic) self-dual Maxwell-ChernSimons-Higgs system in (2 + 1)-D Lagrangian density modeled by C. Lee et al[5] is κ 1 L = ? F ?ν F?ν + ??νλ F?ν Aλ + D? φD? φ? 4 4 1 1 + ?? N ? ? N ? q 2 N 2 |φ|2 ? (q|φ|2 + κN ? q)2 2 2

(1)

3

where φ is a complex scalar ?eld, N is a real scalar ?eld, A = (A0 , A1 , A2 ) is a vector ?eld, F?ν = ?? Aν ? ?ν A? , D? = ?? ? iqA? , ? = 0, 1, 2, ? ? ?0 = ?t , ?j = ?xj , j = 1, 2, q > 0 is the charge of electron, and κ > 0 is a coupling constant for Chern-Simons term. The action functional for this system is given by A= Ldx. (2)

R3

The static energy functional for the above system is E = 1 2 1 1 2 |D0 φ|2 + |Dj φ|2 + Fj0 + F12 + (?j N )2 2 2 2 1 +q 2 N 2 |φ|2 + (q|φ|2 + κN ? q)2 dx. 2

R2

(3)

with the Gauss law constraint (?? + 2q 2 |φ|2 )A0 = ?κF12 . (4)

This is the Euler-Lagrange equation with variation of the action taken with respect to A0 . Integrating by parts, using (4), we obtain from the energy functional E = 1 |(D1 ± iD2 )φ|2 + |D0 φ ? iqφN |2 + (Fj0 ± ?j N )2 2 R2 1 + |F12 ± (q|φ|2 + κN ? q)|2 dx ± q F12 dx. (5) 2 R2

This implies the lower bound for the energy E ≥q F12 dx ,

R2

which is saturated by the solutions of the equations(the Bogomol’nyi equations) for (φ, A, N ) A0 = ?N (6) (7) (8)

2 2 2 2 2

(D1 ± iD2 )φ = 0

F12 ± (q|φ| + κN ? q) = 0

?A0 = ±(κq(1 ? |φ| ) + κ A0 ) + 2q |φ| A0

(9)

4

as |x| → ∞. The former is called non-topological, and the latter is called topological. In this paper we are considering only topological boundary condition. We set φ = e 2 (u+iθ) ,

1

where the upper(lower) sign corresponds to positive(negative) values of R2 F12 dx, and (9) follows from the Gauss law combined with (8). If (φ, A, N ) is a solution that makes E ?nite, then either q φ → 0 and N = ?A0 → , κ or |φ|2 → 1 and N = ?A0 → 0

m

θ=

j=1

2nj arg(z ? zj ),

nj ∈ Z + ,

where z = (x, y) is the canonical coordinates in R2 , and each zj = (xj , yj ) is a zero of φ with winding number nj , which corresponds to the multiplicity of the j?th vortex. After similar reduction procedure similar to [10], we obtain the equations(we have chosen the upper sign.)

m

?u = 2q 2 (eu ? 1) ? 2qκA0 + 4π

j=1

nj δ(z ? zj )

(10) (11)

?A0 = κq(1 ? eu ) + (κ2 + 2q 2 eu )A0 with the boundary condition (14). We de?ne

m

f=

j=1

nj ln

|z ? zj |2 1 + |z ? zj |2

,

and we set u = v + f to remove the singular inhomogeneous term in (10). Then (10) and (11) become ?A0 = κq(1 ? e with the boundary condition

|z|→∞

?v = 2q 2 (ev+f ? 1) ? 2qκA0 + g,

v+f

(12) )A0 (13)

) + (κ + 2q e

2

2 v+f

lim v = 0,

m

|z|→∞

lim A0 = 0,

(14)

where g=

4nj (1 + |z ? zj |2 )2 j=1

5

2

Existence of a Variational Solution

?2 v ? (κ2 + 4q 2 ev+f )?v + 4q 4 ev+f (ev+f ? 1)

Solving (12) for A0 , and substituting this into (13), we obtain = 2q 2 |?(v + f )|2 ev+f ? 4q 2 gev+f + ?g ? κ2 g

(15)

If we formally set κ = 0 in this equation, then we have ?(?v ? 2q 2 (ev+f ? 1) ? g) = 0 which, if we ask v ∈ H 2 (R2 ), recovers the Abelian Higgs system studied in [10]. On the other hand, if we take the limit κ, q → ∞ with q 2 /κ = l ?xed number, then after formally dropping the lower order terms in o(q) and o(κ), we obtain ?v = 4l2 ev+f (ev+f ? 1) + g. This is the equation corresponding to the pure Chern-Simons system studied in [8], [11], etc. Later in this paper we provide rigorous justi?cations for these two limiting behaviors of the solutions. By a direct calculation we ?nd that the equation (15) is a variational equation of the following functional. F(v) = 1 |?v|2 ? (?g ? κ2 g)v + 2q 4 (ev+f ? 1)2 2 1 + κ2 |?v|2 + 2q 2 ev+f |?(v + f )|2 dx 2 (16)

The above functional is well-de?ned in H 2 (R2 ) since ef |?f |2 ∈ L1 (R2 ). We now prove existence of solution of (15) in H 2 (R2 ). Further regularity then v ∈ H 2 (R2 ) follows from the standard regularity results for the nonhomogeneous biharmonic equations. Theorem 1 The functional (16) is coercive, and weakly lower semicontinuous in H 2 (R2 ), and thus there is a global minimizer of the functional (16) in H 2 (R2 ). Proof: If a sequence {vk } converges to v weakly in H 2 (R2 ), vk → v strongly in L∞ (BR ) and in H 1 (BR ) for any ball BR = B(0, R) ? R2 by Rellich’s compactness theorem. Thus we observe that to prove the

6

lower semi-continuity of the functional (16), it is su?cient to prove the lower semi-continuity of R2 ev+f |?(v + f )|2 dx. We have lim inf

k→∞ R2

evk +f |?(vk + f )|2 dx ≥ lim inf

k→∞

BR

evk +f |?(vk + f )|2 dx

=

BR

ev+f |?(v + f )|2 dx.

Letting R → ∞, we obtain the desired weak lower semi-continuity. On the other hand, we note that the coercivity of F in H 2 (R2 ), is a simple corollary of the inequality: v

2 L2 (R2 )

≤ C(1 + ?v

2 L2 (R2 )

+ ev+f ? 1

2 L2 (R2 ) ),

(17)

since we have | (?g ? κ2 g)vdx| ≤ C +? v ?

2 L2 (R2 )

for any ? > 0, and by the Calderon-Zygmund inequality we have D2 v

L2 (R2 )

≤ C ?v

L2 (R2 ) .

For the proof of (17), we just recall the inequality (4.10) in [11], which immediately implies; v

2 L2 (R2 )

≤ C(1 + ?v

2 L2 (R2 )

+ ev+f ? 1

2 L2 (R2 ) ).

Now, for any η > 0 we have |?v|2 dx = ? v?vdx ≤ η v

2 L2 (R2 )

+ Cη ?v

2 L2 (R2 ) .

Taking η small enough, (17) follows. This completes the proof of the theorem.

Proposition 1 Let (v, A0 ) be any topological solution of (12)-(13), q and va be the ?nite energy solution of the Abelian Higgs system. Then the following conditions are equivalent. (ii) v ≤ ?f (i) A0 ≤ 0

7

q (iii) A0 ≥ κ (ev+f ? 1) q (iv) v ≤ va .

Proof: q (i)?(ii), (i)?(iv): We assume A0 ≤ 0. Let va be the solution of the q satis?es Abelian Higgs system, i.e. va

q ?va = 2q 2 (eva +f ? 1) + g.

q

(18)

q The existence and uniqueness of va ∈ H 2 (R2 ) ∩ C ∞ (R2 ) satisfying q va ≤ ?f is well-known[10]. From (12) with A0 ≤ 0 we have

?v ≥ 2q 2 (ev+f ? 1) + g. Thus,

q q ?(va ? v) ≤ 2q 2 (eva f ? ev+f ) = 2q 2 eλ+f (va ? v)

q

by the mean value theorem, where λ is between v and v κ,q . By the maximum principle we have

q v ≤ va ≤ ?f.

(ii)?(i): We assume v ≤ ?f . From (13) we have ?A ≥ (κ2 + 2q 2 ev+f )A. Thus (i) follows from the maximum principle. q (i)?(iii): We assume A0 ≤ 0. Set G = κ (1 ? ev+f ). Then, we compute. q q ?G = ? |?(v + f )|2 ev+f ? ?(v + f )ev+f κ κ q v+f 2 v+f [2q (e ? 1) ? 2qκA0 ] ≤ ? e κ = 2q 2 ev+f G + 2q 2 A0 Thus, we have ?(G + A0 ) ≤ (κ2 + 2q 2 ev+f )(G + A0 ) + 2q 2 A0 ≤ (κ2 + 2q 2 ev+f )(G + A0 ). Since G → 0 as |z| → ∞, we have G + A ≥ 0 by the maximum principle.

8

(iv)?(ii): This is obvious, and included in the above proof. (iii)?(i): Assuming (iii), we have from (13) ?A0 ≥ 2q 2 ev+f A0 . Thus, (i) follows again by the maximum principle. This completes the proof of the proposition.

De?nition 1 We call a topological solution (v, A0 ) satisfying any one of the four conditions in Proposition 1 by an admissible topological solution.

3

Iteration Scheme

In this section we construct an approximate multi-vortices solution sequence of our Bogomol’nyi equations by an iteration scheme. Later this approximate solution sequence will be found to converge to an admissible topological solution. Our iteration scheme is similar to that of [8], but is substantially extended in form.

q q De?nition 2 We set v 0 = va , A0 = 0, where va is the ?nite energy 0 solution of the Abelian Higgs system. De?ne (v i , Ai ) ∈ H 2 (R2 ) ∩ 0 C ∞ (R2 ), i ≥ 1 iteratively as follows: First de?ne v i from (v i?1 , Ai?1 ) by solving: 0

(? ? d)v i = 2q 2 (ev (? ? κ2 ? 2q 2 ev

i?1 +f

? 1) ? 2qκAi?1 + g ? dv i?1 , 0

i +f

(19)

and then de?ne Ai from (v i , Ai?1 ) by solving: 0 0

i +f

? d)Ai = κq(1 ? ev 0

) ? dAi?1 . 0

(20)

Here, d ≥ 2q 2 is a constant that will be ?xed later. Lemma 1 The scheme (19)-(20) is well-de?ned, and the iteration sequence (v i , Ai ) satis?es the monotonicity, i.e. 0 Ai ≤ Ai?1 ≤ · · · ≤ A0 = 0. 0 0 0 v i ≤ v i?1 ≤ · · · ≤ v 0 ≤ ?f

9

Proof: We proceed by an induction. For i = 1 we have from (19) (? ? d)v 1 = 2q 2 (ev On the other hand v 0 satis?es ?v 0 = 2q 2 (ev Thus we have From this we obtain v 1 = v 0 ∈ H 2 (R2 ) ∩ C ∞ (R2 ), and obviously v 1 ≤ v 0 ≤ ?f. Now from (20) for i = 1 we have (? ? κ2 ? 2q 2 ev

1 +f 0 +f 0 +f

? 1) + g ? dv 0

? 1) + g.

(? ? d)(v 1 ? v 0 ) = 0.

? d)A1 = κq(1 ? ev 0

1 +f

).

(21)

Using the mean value theorem, we obtain (1 ? ev

1 +f

R2

)2 dx ≤

R2

(v 1 + f )2 eλ+f dx ≤

R2

(v 1 + f )2 dx < ∞,

where v 1 < λ < ?f , and we used the fact f ∈ L2 (R2 ). We also have 1 0 ≤ ev +f < 1. Thus, by the standard result of the linear elliptic theory the equation (21) de?nes A1 ∈ H 2 (R2 ) ∩ C ∞ (R2 ). 0 1 Furthermore, since κq(1 ? ev +f ) ≥ 0, by the maximum principle applied to (21) we have A1 ≤ A0 = 0. 0 0 Thus Lemma 1 is true for i = 1. Suppose the lemma is true up to i?1. Clearly (19)-(20) de?ne (v i , Ai ) ∈ H 2 ∩C ∞ (R2 ) from (v i?1 , Ai?1 ). We 0 0 only need to observe that (ev

i +f

R2

? 1)2 dx =

R2

(v i + f )2 eλ+f dx < ∞ ≤1

and 0 ≤ ev

i +f

10

if v i?1 ∈ L2 (R2 ) and v i?1 ≤ ?f . We also have (? ? d)(v i ? v i?1 ) = 2q 2 (ev ≥ d(ev

i?1 +f

?d(v i?1 ? v i?2 )

i?1 +f

? ev

i?2 +f

) ? 2qκ(Ai?1 ? Ai?2 )

= d(ef +λ ? 1)(v i?1 ? v i?2 ),

? ev

i?2 +f

? (v i?1 ? v i?2 ))

where v i?1 + f ≤ λ ≤ v i?2 + f , and we used the mean value theorem. Since i ew ≤ 1 and v i?1 ? v i?2 ≤ 0 by induction hypothesis, we obtain (? ? d)(v i ? v i?1 ) ≥ 0. Applying maximum principle again, we have v i ≤ v i?1 . On the other hand, (? ? d)(Ai ? Ai?1 ) = ?κq(ev 0 0

i +f

+(κ2 + 2q 2 e

i +f

? ev

i?1 +f

) )Ai?1 ? d(Ai?1 ? Ai?2 ) 0 0 0

vi +f

+2q 2 (ev

≥ (κ2 + 2q 2 ev

i +f

? ev

i?1 +f

i?1 )(Ai ? A0 ) 0

)(Ai ? Ai?1 ), 0 0

where we used the assumption that our lemma holds up to i ? 1, and v i ≤ v i?1 . Therefore, by the maximum principle, Ai ≤ Ai?1 0 0 Lemma 1 is thus proved.

Lemma 2 The iteration sequence (v i , Ai ) satis?es the inequality 0 Ai ≥ 0 for each i = 0, 1, 2, · · ·. q vi +f (e ? 1). κ

11

Proof: We use induction again. Lemma 2 is true for i = 1. We set i q Gi = κ (1 ? ev +f ). Suppose Lemma 2 holds for i ? 1, then q i ?Gi = ? ? · (?(v i + f )ev +f ) κ q q i i = ? |?(v i + f )|2 ev +f ? ?(v i + f )ev +f κ κ q vi +f 2 vi?1 +f ≤ ? e (2q (e ? 1) ? 2qκAi?1 + d(v i ? v i?1 )) 0 κ i?1 q i i?1 q i ≤ 2q 2 ev +f Gi + (ev +f ? ev +f ? dev +f (v i ? v i?1 ) κ κ i i?1 q ≤ 2q 2 ev +f Gi ? dev +f (v i ? v i?1 ) κ by Ai ≤ 0, i ≥ 0. Therefore 0 (? ? d)(Gi + Ai ) ≤ (κ2 + 2q 2 ev 0 )(Ai + Gi ) 0 q vi +f i i?1 i ?dG ? dA0 ? de (v ? v i?1 ) κ i ≤ (κ2 + 2q 2 ev +f )(Ai + Gi ) 0 q vi +f i?1 i?1 i ?d(A0 + G + (e ? ev +f ), κ

i +f

where we used the mean value theorem in the last step,and used the fact v i ≤ v i?1 . Rewriting it, we have (? ? κ2 ? 2q 2 ev

i +f

? d)(Gi + Ai ) ≤ ?d(Ai?1 + Gi?1 ) ≤ 0 0 0

by the induction hypothesis. By maximum principle we have Ai + Gi ≥ 0. This completes the proof of Lemma 2. 0

4

In this section we will prove the following:

Monotonicity of F(v i)

Lemma 3 Let {v i } given as in De?nition 2 and F(v) is given in (16). We have F(v i ) ≤ F(v i?1 ) ≤ · · · ≤ F(v 0 ). (22) To prove this we ?rstly begin with:

12

Lemma 4 Let (v i , Ai ) be as in De?nition 2, then 0 |1 ? ev

i +f

R2

+

d 2π κ i A0 | + 2 |v i ? v i+1 | dx = 2 q 2q q

m

nj

j=1

for all i ≥ 0. From Lemma 1 and 2 we have 1 ? ev We only need to prove

R2

i +f

+

κ i A , v i ? v i+1 ≥ 0. q 0

1 ? ev

i +f

+

κ i d 1 A0 + 2 (v i ? v i+1 ) dx = 2 q 2q 2q

R2

g dx.

Fix R > 0. Integrating (19) over BR = {|z| < R}, we obtain

BR

(1 ? ev 1 2q 2

i +f

+

d κ i A0 ) + 2 (v i ? v i+1 ) dx q 2q (23)

=

BR

(g ? ?v i+1 ) dx

By divergence theorem ?v i =

BR ?BR

?v i dσ. ?r

We note that v i ∈ H 1 (R2 ). Thus

?BR

|?v i |dσ ≤ 2πR

?BR

|?v i |2 dσ

1/2

(24)

by H¨lder’s inequality. Let o H(r) =

?Br

|?v i |2 dσ, H(r)dr < +∞

then

R2

|?v i |2 dx =

∞ 0

Therefore there exists an increasing sequence of radii, {rk }∞ , such k=1 that o(rk ) lim rk = +∞, and H(rk ) < k→∞ rk

13

Otherwise, there exists ? > 0 and r > 0 such that H(r) > ? i |2 dx = ∞. Thus (24) implies but then R2 |?v

?Brk

? r

for r > r , ?

|?v i |dσ ≤ (2πrk H(rk ))1/2 ≤ (2πo(rk ))1/2 . |?v i |dσ = 0.

Therefore

k→∞ ?Br k

lim

Choose R = rk , and let k → ∞ in (23), then we have

k→∞ Br k

lim

1 ? ev

i +f

+

d 1 κ i A0 + 2 (v i ? v i+1 ) dx = 2 q 2q 2q

R2

g dx.

This, together with

m R2 ∞

gdx =

j=1

8πnj

0

m rdr = 4π nj (1 + r 2 )2 j=1

completes the proof of the lemma. As a corollary of Lemma 4, we can get the following uniform bound. Corollary 1 Let (v i , Ai ) be as in De?nition 1, and de?ne 0 S = sup

i≥1

R2

e

vi +f

|?(v + f )| dx

i

2

1 2

(25)

Then S ≤ (4π

1 m j=1 nj ) 2 .

Proof: Multiplying (19) by ev ev

i +f

i +f

and integrating by parts, we have d(v i?1 ? v i )ev

i +f i +f

R2

|?(v i + f )|2 dx =

R2

+2q 2 ev ≤

(1 ? ev

i?1 +f

+

κ i?1 A ) dx q 0

i?1 +f

R2

d(v i?1 ? v i ) + 2q 2 (1 ? ev

m

+

κ i?1 A ) dx q 0

≤ 4π

nj

j=1

14

by Lemma 4. We now prove our main lemma in this section. Proof of Lemma 3: From (16) we have F(v i?1 ) ? F(v i ) = 1 |?(v i ? v i?1 )|2 ? ?v i ?(v i ? v i?1 ) 2 κ2 +(?g ? κ2 g)(v i ? v i?1 ) + |?(v i ? v i?1 )|2 2

R2

?κ2 ?(v i ? v i?1 ) · ?v i + II + III dx, where II = ?2q 4 (ev III = ?2q 2 ev We also set I = ??v i ?(v i ? v i?1 ) + (?g ? κ2 g)(v i ? v i?1 ) ? κ2 ?(v i ? v i?1 ) · ?v i . Then F(v i?1 ) ? F(v i ) = 1 κ2 |?(v i ? v i?1 )|2 + |?(v i ? v i?1 )|2 2 2 (26)

i +f

? 1)2 ? (ev

i?1 +f

? 1)2 |?(v i?1 + f )|2 .

i +f

|?(v i + f )|2 ? ev

i?1 +f

R2

+I + II + III dx. We ?rstly estimate I. From (19) we have

i?1 A0 =

1 i?1 2q 2 (ev +f ? 1) + g ? d(v i?1 ? v i ? ?v i ). 2qκ

Putting this into (20) after substituting i with i ? 1 in (20), we have ?2 v i ? (d + κ2 + 2q 2 ev = (2q 2 ev

i?1 +f i?1 +f

)?v i + (dκ2 + 2q 2 dev

i?1 +f

)v i

?4q 2 gev

i?1 +f

? d)?v i?1 + 2q 2 |?(v i?1 + f )|2 ev + ?g ? κ2 g + d(κ2 + 2q 2 ev (ev

i?1 +f i?1 +f

i?1 +f

)v i?1 (27)

?4q 4 ev

i?1 +f

? 1) ? 2qκd(Ai?1 ? Ai?2 ). 0 0

15

Multiplying (27) by v i ? v i?1 , integrating by parts, we have

R2

?v i ?(v i ? v i?1 ) + κ2 ?(v i ? v i?1 ) · ?v i ? (?g ? κ2 g)(v i ? v i?1 ) dx =?

R2

d|?(v i ? v i?1 )|2 + (dκ2 + 2q 2 dev

i?1 +f

)(v i ? v i?1 )2

+2qκd(Ai?1 ? Ai?2 )(v i ? v i?1 ) ? IV ? V dx, 0 0 where we set IV V = ?4q 4 ev = 2q 2 ev

i?1 +f

(ev

i?1 +f

i?1 +f

? 1)(v i ? v i?1 )

i?1 +f

(v i ? v i?1 )?(v i + v i?1 )

i?1 +f

+|?(v i?1 + f )|2 ev

(v i ? v i?1 ) + 2gev

(v i ? v i?1 ) .

i?1 Recalling the de?nition of I and observing (A0 ?Ai?2 )(v i ?v i?1 ) ≥ 0, 0 we get

I + IV + V ≥ d|?(v i ? v i?1 )|2 + (dκ2 + 2q 2 dev To calculate I + II + III, we observe II ? IV = 4q 4 (v i ? v i?1 ) ev

i?1 +f

i?1 +f

)(v i ? v i?1 )2 . (28)

(ev

i?1 +f

? 1) ? eλ+f (eλ+f ? 1)

= 4q 4 (v i ? v i?1 )(v i?1 ? λ)eη+f (2eη+f ? 1), where we used mean value theorem repeatedly with v i ≤ η ≤ λ ≤ v i?1 . Thus i?1 II ? IV ≥ ?4q 4 (v i ? v i?1 )2 ev +f . (29) Now we have III = ?2q 2 [(ev +e

i +f

vi?1 +f

= ?2q 2 [eλ+f (v i ? v i?1 )|?(v i + f )|2 +ev

i?1 +f

(|?(v i + f )|2 ? |?(v i?1 + f )|2 )]

? ev

i?1 +f

)|?(v i + f )|2

= V I + V II, V = 2q 2 ev

i?1 +f

?(v i ? v i?1 ) · ?(v i + v i?1 + 2f )] (v i ? v i?1 )[?(v i + v i?1 + 2f )

+|?(v i?1 + f )|2 ] = V III + IX,

16

where v i?1 ≥ λ ≥ v i by mean value theorem. We now calculate III ? V = (V II ? V III) ? IX + V I. By integration by parts we obtain

R2

[ V II ? V III ]dx

R2

= 2q 2 =

R2

ev

i?1 +f

(v i ? v i?1 )?(v i?1 + f ) · ?(v i + v i?1 + 2f ) dx

[X]dx,

i?1 +f

X ? IX = 2q 2 ev = XI.

(v i ? v i?1 )?(v i + f ) · ?(v i?1 + f )

Since V I ≤ 0, we have V I + XI ≥ ?2q 2 (v i ? v i?1 ) ev +(ev

i +f i +f

?(v i + f ) · ?(v i ? v i?1 )

? ev

i?1 +f

)?(v i + f ) · ?(v i?1 + f )

i +f

≥ ?2q 2 |v i ? v i?1 | ev +|ev +|ev

i +f i +f

|?(v i + f )||?(v i ? v i?1 )|

? ev ? ev

i?1 +f i?1 +f

||?(v i?1 + f )|2 ||?(v i ? v i?1 )||?(v i?1 + f )|

i +f

≥ ?2q 2 |v i ? v i?1 | (ev +ev

i?1 +f

|?(v i + f )|

|?(v i ? v i?1 )||?(v i?1 + f )|)

+eλ+f |v i ? v i?1 ||?(v i?1 + f )|2 by the mean value theorem where we used the fact |ev +f ? ev +f | ≤ i?1 ev +f in the last step. We use H¨lder’s inequality and interpolation o inequality to obtain

R2

i i?1

[ V I + XI ]dx ≥ ?2q 2 + ev

ev

i +f

?(v i + f )

L2 (R2 ) L2 (R2 ) L2 (R2 )

i?1 +f

× (v i ? v i?1 )

?(v i?1 + f )

L∞ (R2 )

?(v i ? v i?1 )

R2

?2q 2 v i ? v i?1

2 L∞ (R2 )

e

vi +f

|?(v i + f )|2 dx

17

≥ ?4q 2 CS v i ? v i?1 ?2q CS v ? × ?(v i ? v i?1 )

2 2 i

1 2

L2 (R2 )

?(v i ? v i?1 ) ?(v i ? v i?1 )

1 2

1 2

L2 (R2 )

L2 (R2 ) i?1 v L2 (R2 )

L2 (R2 ) ,

where C is an absolute constant and we set S = sup

i≥1

R2

e

vi +f

|?(v + f )| dx

i

2

as in Corollary 1. Applying Young’s inequality, we have

R2

[ III ? V ]dx ≥ ?Cq 4 (S 2 + S 4 ) v i ? v i?1

2 L2 (R2 )

?Cq 2 S ?(v i ? v i?1 ) 2 2 (R2 ) L 1 ? ?(v i ? v i?1 ) 2 2 (R2 ) L 4 Combining with (28), (29) and (30), (26) becomes F(v i?1 ) ? F(v i ) ≥

(30)

1 ?(v i ? v i?1 ) 2 2 (R2 ) + (dκ2 ? C) v i ? v i?1 L 4 +(d ? C) ?(v i ? v i?1 ) 2 2 (R2 ) , L

2 L2 (R2 )

where C is an absolute constant depending on q and m nj . Taking j=1 d large enough, and using the Calderon-Zygmund inequality, we have ?nally F(v i?1 ) ? F(v i ) ≥ C v i ? v i?1 H 2 (R2 ) ,

which is a stronger form of (22). This completes the proof of Lemma 3.

Corollary 2 Let v be any admissible topological solution of (12)-(13), q and va be the ?nite energy solution of the Abelian Higgs system. Then, we have q F(v) ≤ F(va ).

q Proof: Just substitute v i = v, v i?1 = va in the proof of Lemma 3, and instead of Lemma 4 we use

R2

|1 ? ev+f +

1 κ A0 | dx = 2 q 2q

R2

g=

2π q2

m

nj ,

j=1

18

which follows immediately from integration of (12) and Proposition 1.

5 Existence of Admissible Solutions and Asymptotic Decay

Based on the previous estimates for the iteration sequence {v i }, in this section, we prove the existence of admissible topological solutions of our Bogomol’nyi equations (12)-(13). We also establish asymptotic exponential decay estimates of these solutions as |z| → ∞. As a corollary of these decay estimates we prove that the action (2) and, hence the energy functional (3) are ?nite. Firstly we prove Theorem 2 Given zj ∈ R2 , nj ∈ Z+ with j = 1, · · · , m, there exists a smooth solution (φ, A, N ) to (6)-(9) such that φ = 0 at each z = zj with corresponding winding numbers nj , and satisfying 0 ≤ 1 ? |φ|2 ≤ for all q, κ > 0 Proof: By (22) the monotone decreasing sequence {v i } satis?es F(v i ) ≤ F(v 0 ) < ∞ This implies by (17) and (16) that vi Thus sup v i

i≥0 H 2 (R2 ) H 2 (R2 )

κ κ N = ? A0 q q

(31)

?i = 1, 2, · · · .

< CF(v i ) ≤ CF(v 0 ) ?i = 1, 2, · · · . < ∞.

(32)

On the other hand, from (19) Ai = 0 1 i 2q 2 (ev +f ? 1) + g + d(v i+1 ? v i ) ? ?v i+1 2qκ < ∞ by (20).

which belongs to L2 (R2 ) uniformly by (32). Thus, supi≥0 Ai L2 (R2 ) < ∞, and supi≥0 ?Ai 0 0

L2 (R2 )

19

Combining this with the Calderon-Zygmund inequality and the standard interpolation inequality, we obtain sup Ai 0

i≥0 H 2 (R2 )

<∞

1 both weakly in H 2 (R2 ) and strongly both in Hloc (R2 ) and in L∞ (R2 ) loc by Rellich’s compactness theorem. The limits v, A0 ∈ H 2 (R2 ) satis?es (12)-(13) in the weak sense, and by repeatedly using the standard linear elliptic regularity result we have v, A0 ∈ C ∞ (R2 ). Moreover, by construction we have q (33) v ≤ va ≤ ?f,

Thus there exists v, A0 ∈ H 2 (R2 ) and a subsequence (v i , Ai ) such 0 that v i → v and Ai → A0 0

and by Proposition 1 q v+f (e ? 1) ≤ A0 ≤ 0. κ We de?ne 1 N = ?A0 , φ = exp (v + f + iθ) 2 (34)

1 where θ = m 2nj arg(z ? zj ). For for α = 2 (A1 ? iA2 ) and ?z = j=1 1 2 (?1 ? i?2 ) we also de?ne

α = i?z lnφ. Explicit computation shows 1 A1 = (?2 v + ?2 b), 2 and A2 = where we set b=?

m j=1

(35)

1 (??1 v ? ?1 b), 2 nj ln(1 + |z ? zj |2 ).

(36)

Converting our reduction procedure from (12)-(13) to the Bogomol’nyi equations (6)-(9), we we ?nd that the ?elds A? , φ, N (? = 0, 1, 2)

20

satisfy the Bogomol’nyi equations (6)-(9). In particular (32) follows immediately from (34) and (35). We now establish asymptotic exponential decay estimates for admissible topological solutions of our Bogomol’nyi equations. Theorem 3 Let (φ, A0 , N ) be any admissible topological solution of the Bogomol’nyi equations (6)-(9). Suppose ? > 0 is given, then there exists r0 = r0 (?) > 0 and C = C? such that

1

0 ≤ 1 ? |φ|2 , |N |, |F12 | ≤ C? e?q(1?γ??) 2 |z| |D? φ|, |?A0 | ≤ C? e √ ?ρ2 +ρ ρ2 +8 and ρ = κ . if |z| > r0 . Here we set γ = 4 q

1 ?q(1?γ??) 2 |z|

(37) (38)

Remark: We note that γ was chosen(see the proof below) so that ρ2 = γ, thus 0 < γ < 1. ρ2 +2γ Proof of Theorem 3: From (10) and (11), ?u2 = 2|?u|2 + 2u?u ?A2 = 2|?A0 |2 + 2A0 ?A0 0 ≥ 4q 2 (eu ? 1)u ? 4qκA0 u,

≥ 2(κ2 + 2q 2 eu )A2 + 2κq(1 ? eu )A0 0

for |z| > supj {|zj |}. Let E = u2 + 2A2 , then we have 0 ?E ≥ 4q 2 eu ? 1 2 u + 2eu A2 + 4κ2 A2 + 4κqA0 (1 ? eu ? u) 0 0 u ≥ 4q 2 ξ(u)E + 4κ2 A2 ? 8κqA0 u (39) 0

et ?1 t }.

where we used the inequality t ≤ et ? 1, and set ξ(t) = min{et , Note that ξ(t) = et if t < 0 et ? 1 otherwise = t

21

Also, ξ > 0 and ξ(t) → 1 as t → 0. The last term in (39) is estimated by 8κq|A0 u| ≤ 4(κ2 + 2γq 2 )A2 + 0 4κ2 q 2 u2 κ2 + 2γq 2 4ρ2 q 2 u2 = 4κ2 A2 + 4γq 2 E, = 4(κ2 + 2γq 2 )A2 + 2 0 0 ρ + 2γ

since

ρ2 ρ2 +2γ

= γ. Thus, (39) becomes ?E ≥ 4(ξ(u) ? γ)q 2 E

1

(40)

Since u → 0 as |z| → ∞, given ? > 0, we can choose r0 so large that ξ(u) ≥ 1 ? ? on |z| > r0 . Thus, by comparing E with the function

on |z| > r0 , where C? was ?xed to compare E with β on {|z| = r0 }. (37) follows from the fact 0 ≤ 1 ? |φ|2 = 1 ? eu ≤ |u|, N = ?A0 , and |F12 | ≤ q(1 ? |φ|2 ) + κ|N |,

β(z) = C? e?2q(1?γ??) 2 |z| in |z| ≥ r0 , using the maximum principle, we deduce 1 |u|2 , |A0 |2 ≤ C? e?2q(1?γ??) 2 |z|

|D? φ|2 = |(?? ? iqA? )φ|2 1 u ⊥ e |(?? (u + iqθ) ? iq(?? u + ?? θ)|2 = 4 1 u ⊥ e |?? u ? iq?? u|2 , = 4 ⊥ where we used the notation (?? ) = (?0 , ??2 , ?1 ). Thus |D? φ|2 ≤ C|?u|2 . Therefore it is su?cient to have decay estimate for |?u|2 . A direct calculation gives ?|?u|2 = 2|?2 u|2 + 2?u · ??u κ ≥ 4q 2 ?u · ?(eu ? 1 ? A0 ) q 2 u 2 = 4q e |?u| ? 4qκ?u · ?A0 ≥ 2(κ2 + 2q 2 eu )|?A0 |2 ? (2qκeu ? 4q 2 eu A0 )?A0 · ?u

which follows from (8). Next, we estimate the asymptotic decay of |Di φ|2 . We observe

?|?A0 |2 = 2|?2 A0 |2 + 2?A0 · ??A0

22

for |z| > supj {|zj |}. We set J = |?u|2 + 2|?A0 |2 , then we have ?J ≥ 4q 2 eu J + 4κ2 |?A0 |2 ? (8κq + 4q 2 eu |A0 |)|?u||?A0 | 1 ≥ 4q 2 eu (1 ? |A0 |)J + 4κ2 |?A0 |2 ? 8κq|?u||?A0 |, (41) 2

where we used |?u||?A0 | ≤ 1/2J in the second inequality. (41) is the same form as (39), observing in case of (41) we have 1 eu (1 ? |A0 |) → 1 2 as |z| → ∞.

Given ? > 0, we apply Young’s inequality to the term |?u||?A0 | similarly to the previous case, and get ?J ≥ 4q 2 (1 ? γ ? ?)J when |z| > r0 for r0 large enough. The above equation is the same as (40). Thus J satis?es the estimate (38). This completes the proof of Theorem 5. Now we complete proof of our existence theorem by proving that the solutions constructed in Theorem 2 make our action in (2) ?nite. This follows if we prove that any admissible topological solution makes the action ?nite. Corollary 3 Let (A, φ, N ) be the solution of the Bogomol’nyi equations (6)- (9) constructed Theorem 2, then we have A = A(A, φ, N ) < ∞. Proof: Since N = ?A0 ∈ H 1 (R2 ) and |φ| ≤ 1,

R2

(?? N )2 dx < ∞,

R2

N 2 |φ|2 dx < ∞.

From (4) and (10) we have F12 = q|φ|2 ? κN ? q ∈ L2 (R2 ). Clearly F0i = ??i A0 ∈ L2 (R2 ), Therefore

R2

i = 1, 2.

|F?ν |2 dx < ∞.

23

We now consider the Chern-Simons term. Firstly we have |F12 A0 |dx ≤ |F12 | dx

2

1 2

R2

R2

R2

|A0 | dx

2

1 2

< ∞.

Thus it su?ces to prove F01 A2 = ??1 A0 A2 , F02 A1 = ?2 A0 A1 ∈ L1 (R2 ). (42)

Since A1 , A2 ∈ Lp (R2 ) for all p ∈ (2, ∞] from (35)- (36), and ?A0 ∈ Lq (R2 ), for q ∈ [1, ∞] from (38), (42) follows immediately by the H¨lder inequality. Therefore we also have that o ??νλ F?ν Aλ ∈ L1 (R2 ). Finally from (38) we have |D? φ|2 ∈ L1 (R2 ). This completes the proof of the corollary.

6

Abelian Higgs Limit

In this section we prove that, for q ?xed, the sequence of admissible q topological solutions, (v κ,q , Aκ,q ) converges to (va , 0), as κ goes to zero, 0 q where va is the ?nite energy solution of the Abelian Higgs system. Firstly we establish: Lemma 5 Let (v κ,q , Aκ,q ) be any admissible topological solution of 0 (12) and (13). Then, for each ?xed q ∈ (0, ∞), we have sup

0<κ<1

v κ,q

H 2 (R2 )

< ∞, sup

0<κ<1

Aκ,q 0

H 2 (R2 ) .

<∞

(43)

Thus, by the Sobolev embedding we have sup

0<κ<1

v κ,q

L∞ (R2 )

< ∞, sup

0<κ<1

Aκ,q 0

L∞ (R2 )

<∞

(44)

24

Proof: Let κ ∈ (0, 1). From Corollary 2 and (17) we have v κ,q

H 2 (R2 )

≤ (1 + κ2 )C1 + C2 F(v κ,q )

q ≤ (1 + κ2 )C1 + C2 F(va ) ≤ C(1 + κ2 ),

where C1 , C2 and C are constants independent of κ. Thus the ?rst inequality of (43) follows. Now, taking L2 (R2 ) inner product (13) with Aκ,q , we have after integration by part 0

R2

|?Aκ,q |2 + (κ2 + 2q 2 ev 0 ≤ ≤ κ2 2 κ2 2

κ,q +f

)|Aκ,q |2 dx = 0

R2

R2

κq(1 ? ev )2 dx

κ,q +f

)|Aκ,q | dx 0

q2 2 R2 q2 |Aκ,q |2 dx + 0 2 R2 |Aκ,q |2 dx + 0

(1 ? ev

κ,q +f

R2

|v κ,q + f |2 dx.

Thus, by Young’s inequality and the ?rst inequality in (43) we obtain

R2

|?Aκ,q |2 dx + 0

R2

(κ2 + 4q 2 ev

κ,q +f

)|Aκ,q |2 dx ≤ C, 0

(45)

where C is independent of κ. From (13) and (45) we have

R2

|?Aκ,q |2 dx ≤ 2κ2 q 2 0 ≤ 2κ2 q 2 ≤ 2q 2

R2

R2

(1 ? ev

κ,q +f

)2 dx + 2

R2

(κ2 + 2q 2 ev

R2

κ,q +f

)2 |Aκ,q |2 dx 0 )|Aκ,q |2 dx 0

|v κ,q + f |2 e2(λ+f ) dx + 2(κ2 + 2q 2 )

R2

(κ2 + 4q 2 ev

κ,q +f

R2

|v κ,q + f |2 dx + 2(1 + q 2 )

(κ2 + 4q 2 ev

κ,q +f

)|Aκ,q |2 dx ≤ C 0

for a constant C independent of κ, where λ ∈ (v κ,q +f, 0), and we used the mean value theorem. Thus, by the Calderon-Zygmund inequality

R2

|D 2 Aκ,q |2 dx ≤ C. 0

(46)

By Sobolev’s embedding for a bounded domain, for any ball BR = {|z| < R} ? R2 , we have. Aκ,q 0

L∞ (BR )

≤ C,

where C is independent of κ. We take R > max1≤j≤m {|zj |+1}. Then,

R2

|Aκ,q |2 dx = 0

BR

|Aκ,q |2 dx + 0

R2 ?BR

|Aκ,q |2 dx 0

25

≤ πR2 Aκ,q 0

2 L∞ (BR )

+ ev

κ,q +f

L∞ (R2 ?BR )

R2 ?BR

ev

κ,q +f

|Aκ,q |2 dx 0 (47)

≤ C1 (R) + C2 ef

L∞ (R2 ?BR )

R2

ev

κ,q +f

|Aκ,q |2 dx ≤ C(R), 0

where C1 (R), C2 and C(R) are independent of κ. Combining (45)(47), we obtain the second inequality in (43). This completes the proof of Lemma 5.

Theorem 4 Let v κ,q , Aκ,q be the admissible topological solutions of 0 q (12) and (13), and va a ?nite energy solution of the Abelian Higgs system. Let q be ?xed. For all k ∈ Z+ we have

q v κ,q → va ,

and

Aκ,q → 0 0

in

H k (R2 ).

as κ → 0. Proof: We have by mean value theorem

q ?(v κ,q ? va ) = 2q 2 (ev

κ,q +f

? eva +f ) ? 2κqAκ,q 0 (48)

q

q q where λ ∈ (v κ,q , va ). Multiplying (48) by v κ,q ? va , we have after integration by parts q q |?(v κ,q ?va )|2 +2q 2 eλ+f (v κ,q ?va )2 dx = 2κq q Aκ,q (v κ,q ?va ) dx 0

q = 2q 2 eλ+f (v κ,q ? va ) ? 2κqAκ,q 0

R2

R2

≤ 2κq Aκ,q 0 λ

L2 (R2 )

q v κ,q ? va

L2 (R2 )

≤ κC ≤C

where C is independent of κ by Lemma 5. Since

L∞ (R2 )

≤ v κ,q

L∞ (R2 )

q + va

L∞ (R2 )

independently of κ < 1, we have from the above estimate

R2 q q |?(v κ,q ? va )|2 + ef |v κ,q ? va |2 dx → 0

as κ → 0. Let ?δ = ∪m {|z ? zj | < δ}. Now, j=1

?δ q q |v κ,q ? va |2 dx ≤ πmδ2 v κ,q ? va 2 L∞ (R2 )

≤ Cδ2 .

26

where C is independent of κ by Lemma 5. Thus, for any given ? > 0, we can choose δ independently of κ so that

?δ q |v κ,q ? va |2 dx ≤

? . 2

For such δ we have

R2 q |v κ,q ? va |2 dx = ?δ q |v κ,q ? va |2 dx + R2 ??δ q |v κ,q ? va |2 dx

? + sup {e|f | } 2 R2 ??δ ? ? ≤ + =? 2 2 for su?ciently small κ, i.e. ≤

R2

R2

q ef |v κ,q ? va |2 dx

q |v κ,q ? va |2 dx → 0

as κ → 0. Combining the above results, we obtain Now we prove the convergence for Aκ,q . Multiplying (13) by Aκ,q and 0 0 integrating, we estimate

R2 q v κ,q → va in H 1 (R2 ) as

κ → 0.

|?Aκ,q |2 +(κ2 +2q 2 ev 0

L2 (R2 )

κ,q +f

)|Aκ,q |2 dx ≤ κq 0

L2 (R2 )

R2

(1?ev

κ,q +f

)|Aκ,q | dx 0 ≤ Cκ.

where we used Lemma 5 in the ?rst and third step and use the fact 1 ? et ≤ t for t ≤ 0 in the second step. Using the fact |v κ,q | < C uniformly in κ < 1, we obtain from this

R2

≤ κq Aκ,q 0

1 ? ev

κ,q +f

≤ Cκ v κ,q + f

L2 (R2 )

|?Aκ,q |2 + ef |Aκ,q |2 dx → 0 0 0

as κ → 0. Since |Aκ,q | < C uniformly in κ < 1, by Lemma 5 we can 0 deduce Aκ,q → 0 in H 1 (R2 ) similarly to the case of v κ,q . From these 0 results together with uniform bounds v κ,q , Aκ,q ≤ C, applying the 0 standard elliptic regularity to (48) and (13) repeatedly, we obtain

q (v κ,q , Aκ,q ) → (va , 0) in 0

[H k (R2 )]2 ,

?k ≥ 1

27

7

Chern-Simons Limit

In this section we study the behaviors of v κ,q , Aκ,q as κ, q → ∞ with 0 l = q 2 /κ kept ?xed for the admissible topological solutions. Although we could not obtain the strong convergence to a solution of the ChernSimons equation, instead, we will prove that the sequence {v κ,q } is ”weakly approximating” the Chern-Simons equation: ?v = 4l2 ev+f (ev+f ? 1) + g. We denote l = q 2 /κ the ?xed number, and ακ,q = qAκ,q throughout 0 this section. Theorem 5 Let {(v κ,q , Aκ,q )} be a sequence of admissible topological 0 ∞ solutions of (12)-(13). For any ψ ∈ C0 (R2 ) we have

κ,q→∞ R2

lim

?v κ,q ? 4l2 ev

κ,q +f

(ev

κ,q +f

? 1) ? g ψdx = 0.

(49)

For proof of this theorem we ?rstly establish the following lemma which is interesting in itself. Lemma 6 Let {(v κ,q , Aκ,q )} be given as in Theorem 5. For any ?xed 0 p ∈ [1, ∞) we have

κ,q→∞

lim

ακ,q ? l(ev

κ,q +f

? 1)

Lp (R2 )

= 0.

Proof: Firstly we have from v κ,q + f ≤ 0 ev

κ,q +f

?1

L∞ (R2 )

≤ 1.

Also, from 0 ≥ ακ,q ≥ l(ev ακ,q From (12) we have

R2

κ,q +f

? 1),

κ,q +f

L∞ (R2 )

≤ l ev

?1

L∞ (R2 )

≤ l.

|ακ,q ? l(ev

κ,q +f

? 1)|dx = =

R2

ακ,q ? l(ev gdx.

κ,q +f

? 1) dx

1 q

R2

Thus

κ,q→∞

lim

ακ,q ? l(ev

κ,q +f

? 1)

L1 (R2 )

= 0.

28

By a standard interpolation inequality ακ,q ? l(ev ≤

κ,q +f

? 1)

Lp (R2 )

ακ,q ? l(ev

1 1? p

κ,q +f

? 1)

1 p

L1 (R2 )

ακ,q ? l(ev

1 p

κ,q +f

? 1)

1? 1 p L∞ (R2 )

≤ (2l)

ακ,q ? l(ev

κ,q +f

? 1)

L1 (R2 )

→0

as κ, q → ∞ with q 2 /κ = l ?xed. Proof of Theorem 5: From (13) added by (14)×q/κ we obtain ?(v κ,q + 2l κ,q κ,q α ) = g + 4lev +f ακ,q . q

κ,q +f κ,q +f

∞ Multiplying ψ ∈ C0 (R2 ), and integrating by parts, we obtain R2

?v κ,q ψ = 4l2

R2

ev

(ev

? 1) + g ψdx

2l κ,q κ,q ακ,q ?ψdx + 4l ev +f [ακ,q ? l(ev +f ? 1)]ψdx. 2 q R2 R Now we have 2l 2l2 ακ,q ?ψdx ≤ lim |?ψ|dx = 0. lim κ,q→∞ q R2 κ,q→∞ q R2 + and by Lemma 6

κ,q→∞ R2

lim

ev

κ,q +f

[ακ,q ? l(ev

κ,q +f

? 1)]dx ? 1)|dx = 0.

≤ ψ

lim L∞ (R2 ) κ,q→∞

R2

|ακ,q ? l(ev

κ,q +f

Thus, Theorem 5 follows. Remark: If we could have uniform L1 (R2 ) estimate of ?v κ,q , then we could prove existence of subsequence {v κ,q } and its Lq (R2 ) (1 ≤ loc q < 2)-limit v such that v is a smooth solution of the Chern-Simons equation. Acknowledgements The authors would like to deeply thank to Professor Choonkyu Lee for introducing the problems issued in this paper for them, and many helpful discussions. This research is supported partially by KOSEF(K94073, K95070), BSRI(N94121), GARC-KOSEF and SNU(95-03-1038).

29

References

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30

赞助商链接

- Remarks on 2+1 Self-dual Chern-Simons Gravity
- Self-dual Chern-Simons solitons in noncommutative space
- Non-Abelian, Self-Dual Chern-Simons Vortices Coupled to Gravity
- Quasi-hole solutions in finite noncommutative Maxwell-Chern-Simons theory
- Vacuum Mass Spectra for SU(N) Self-Dual Chern-Simons-Higgs Systems
- The existence of non-topological multivortex solutions in the relativistic self-dual Chern-
- Vortex Condensates in the Relativistic Self-Dual Maxwell-Chern-Simons-higgs System
- Non-topological condensates for the self-dual Chern-Simons-Higgs model
- Vortex Dynamics in Self-Dual Chern-Simons Higgs Systems
- Quantum equivalence between the self-dual and the Maxwell-Chern-Simons models nonlinearly c
- Sen Equivalence of the Self-Dual Model and Maxwell{Chern{Simons theory on Arbitrary Manifol
- BATALIN-FRADKIN-TYUTIN EMBEDDING OF A SELF-DUAL MODEL AND THE MAXWELL-CHERN-SIMONS THEORY
- Topological quantization of self-dual Chern-Simons vortices on Riemann Surfaces
- Solitons of the Self-dual Chern-Simons Theory on a Cylinder
- Self-dual Vortices in the Generalized Abelian Higgs Model with Independent Chern-Simons Int

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