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SIAM J. MATH. ANAL. Vol. 37, No. 1, pp. 1–16
ASYMPTOTICS FOR SELFDUAL VORTICES ON THE TORUS AND ON THE PLANE: A GLUING TECHNIQUE∗ MARTA MACR`I† , MARGHERITA NOLASCO‡ ,
AND TONIA RICCIARDI†
Abstract. We consider multivortex solutions for the selfdual Abelian Higgs model, as the ratio of the vortex core size to the separation distance between vortex points (the scaling parameter) tends to zero. To this end, we use a gluing technique (a shadowing lemma) for solutions to the corresponding semilinear elliptic equation on the plane, allowing any number (finite or countable) of arbitrarily prescribed singular sources. Our approach is particularly convenient and natural for the study of the asymptotics. In particular, in the physically relevant cases where the vortex points are either finite or periodically arranged in the plane, we prove that a frequently used factorization ansatz for multivortex solutions is rigorously satisfied, up to an error which is exponentially small. Key words. selfdual Abelian Higgs model, elliptic equation, shadowing lemma AMS subject classifications. 35J60, 58E15, 81T13 DOI. 10.1137/040619843
1. Introduction. We consider the energy density for the static, two-dimensional selfdual Abelian Higgs model in the following form: Eδ (A, φ) = δ 2 |dA|2 + |Dφ|2 +
2 1 2 |φ| − 1 , 2 4δ
where A = A1 dx1 + A2 dx2 , A1 (x), A2 (x) ∈ R is a gauge potential (a connection over a principal U (1) bundle), φ, φ(x) ∈ C is a Higgs matter field (a section over an associated complex line bundle), D = d − iA is the covariant derivative, and δ > 0 is the scaling parameter. Eδ is a rescaling of E1 = Eδ |δ=1 , which coincides with the two-dimensional Ginzburg–Landau energy density in the so-called Bogomol’nyi limit. Such a limit describes the borderline between type I and type II superconductors; see, e.g., Jaffe and Taubes [12]. By the selfdual structure, solutions to the Euler–Lagrange equations of Eδ may be obtained from solutions to the first order system: (1.1)
(D1 ± iD2 )φ = 0,
(1.2)
F12 = ∂1 A2 − ∂2 A1 = ±
1 (1 − |φ|2 ). 2δ 2
The vortex-type critical points for the energy associated with Eδ , namely, the solutions of (1.1)–(1.2), have received considerable attention in recent years, in view of both their physical and geometrical interest; see, e.g., Garc´ia-Prada [8], Hong, Jost, and Struwe [10], Stuart [17], Taubes [19], Wang and Yang [20], and the references therein. In particular, Hong, Jost, and Struwe [10] consider (1.1)–(1.2) on a compact ∗ Received by the editors November 30, 2004; accepted for publication (in revised form) April 13, 2005; published electronically August 17, 2005. This research was supported in part by the M.I.U.R. Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari.” http://www.siam.org/journals/sima/37-1/61984.html † Dipartimento di Matematica e Applicazioni, Universit` a di Napoli Federico II, Via Cintia, 80126 Napoli, Italy ([email protected], [email protected]). The last author’s research was supported in part by Regione Campania L.R. 5/02. ‡ Dipartimento di Matematica Pura ed Applicata, Universit` a di L’Aquila, Via Vetoio, Coppito, 67010 L’Aquila, Italy ([email protected]).
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M. MACR`I, M. NOLASCO, AND T. RICCIARDI
Riemannian surface and perform a detailed analysis of the asymptotics as δ → 0+ . Indeed, the small δ > 0 regime, corresponding to the limit of small vortex core size with respect to the separation distance between vortices, is an appropriate approximation for the analysis near the vortex points of solutions of (1.1)–(1.2) with δ = 1. This type of asymptotics is also relevant in the context of Ginzburg–Landau vortices; see, e.g., Aftalion, Sandier, and Serfaty [2], Alama and Bronsard [3], Andr´e, Bauman, and Phillips [4], Bethuel, Brezis, and H´elein [6], Lin [13], Rubinstein and Sternberg [16], just to mention a few. It has also been widely investigated in the context of other selfdual gauge theories; see the monographs of Tarantello [18] and Yang [22]. The fundamental results concerning finite energy solutions of (1.1)–(1.2) on R2 were obtained by Taubes [12, 19]. In particular, Taubes showed that such solutions are completely determined by solutions to the singular elliptic problem s −Δu = δ −2 (1 − eu ) − 4π j=1 mj δpj on R2 , (1.3) u(x) → 0 as |x| → +∞. Here s ∈ N, and for j = 1, 2, . . . , s, pj ∈ R2 are the vortex points, mj ∈ N is the multiplicity of pj , and δpj is the Dirac measure at pj . By variational methods, Taubes proved that, for any δ > 0, there exists a unique solution of (1.3) such that the configuration (A, φ) defined in complex notation by s φ(z) = exp{ 12 u(z) ± i j=1 mj arg(z − pj )}, (1.4) A1 ∓ iA2 = −i(∂1 ± i∂2 ) ln φ 2 is a smooth, finite energy solution of (1.1)–(1.2) the on R , satisfying φ(pj ) = 0(with s corresponding multiplicity mj ∈ N) and E = R2 Eδ (A, φ) = ± R2 F12 = 2π j=1 mj = 2πN , where F12 = ∂1 A2 − ∂2 A1 is the magnetic field (the curvature of A). In connection with Abrikosov’s mixed states in superconductivity [1], it is also of physical interest to analyze (1.1)–(1.2) on the flat torus T2 ≡ R2 /(aZ × bZ), where a, b > 0. Such a case has been considered by Wang and Yang in [20]. It is shown in [20] that solutions of (1.1)–(1.2) with δ = 1 exist for any given set of vortex points pj ∈ T2 , s j = 1, 2, . . . , s, with multiplicity mj ∈ N, if and only if N = j=1 mj < |T2 |/(4π). In particular, on T2 the total number of vortices N cannot be arbitrarily large. Similarly as on R2 , denoting Ω = (0, a) × (0, b), solutions of (1.1)–(1.2) on T2 correspond to solutions for the singular elliptic problem s −Δu = δ −2 (1 − eu ) − 4π j=1 mj δpj in Ω, (1.5) u doubly periodic on ∂Ω.
The periodic boundary conditions are justified by certain more general gauge invariant conditions on the configuration (A, φ) introduced by ’t Hooft [11]. Such conditions force the magnetic flux through a lattice cell to be a “quantized” value proportional to the number of vortices confined. Namely, s the ’t Hooft boundary conditions imply the topological constraint ± Ω F12 = 2π j=1 mj = 2πN on the solutions of (1.5), exactly as for finite energy solutions on R2 . Integrating (1.5) on the periodic cell Ω, we obtain that a necessary condition to the solvability of (1.5) is δ 2 < |Ω|/(4πN ). This is obviously satisfied for any finite vortex number N provided δ > 0 is sufficiently small. Our aim in this note is to show that a “shadowing-type lemma” as introduced in the context of elliptic PDEs by Angenent [5] (see also Nolasco [15]) may be adapted to
ASYMPTOTICS FOR SELFDUAL VORTICES
3
elliptic equations with singular sources in order to construct solutions for the following more general equation: (1.6) mj δpj in R2 , −Δu = δ −2 (1 − eu ) − 4π j∈P
where the set of indices P may be either finite or countable, and the vortex points pj , j ∈ N, are arbitrarily distributed in the plane with the only constraint that (1.7)
d := inf |pj − pk | > 0 k=j
and
m := sup mj < +∞. j∈P
The solution we obtain for (1.6) coincides with the solution obtained by Taubes for problem (1.3) when P is finite and with the solution obtained by Wang and Yang for problem (1.5) when P is infinite and the vortex points are periodically arranged in R2 . In fact, unlike the previous approaches, our method provides a unified analysis of (1.3) and (1.5). It should be mentioned that suitable modifications to the method described in [5] are necessary due to the singular sources appearing in (1.6). The case where P is countable and the vortex points are arbitrarily arranged in R2 does not seem to have been considered before. Of course, if P is countable, the energy of such a solution is infinite and only locally bounded. On the other hand, our “gluing” technique is, particularly, convenient and natural to analyze the asymptotics as δ → 0+ . In particular, as a by-product of our construction, we derive a rigorous proof of the following approximate product formula:
x − pj + ηδ , Φmj φ(x) = (1.8) δ j∈P
where ηδ L∞ (R2 ) ≤ Ce−c/δ with C, c > 0 independent of δ. Here (Amj , Φmj ) is the unique, up to gauge transformation, single vortex (or antivortex) solution with multiplicity mj to (1.1)–(1.2) with δ = 1 on R2 . We note that in the small δ > 0 regime, a product formula of the form (1.8) is a widely used ansatz in the physics literature, in particular in the study of the dynamics of vortices in the Ginzburg– Landau model; see, e.g., E [7], Neu [14], and Weinstein and Xin [21]. However, we have found a rigorous proof of (1.8) only for the case N = 2 on R2 in Stuart [17]. The asymptotic behavior of solutions of (1.1)–(1.2) as δ → 0+ is readily derived from formula (1.8) as well as the convergence rates. In fact, in the case of T2 , our asymptotic description improves the previous result obtained (for general compact Riemann surfaces) by Hong, Jost, and Struwe [10] (see Corollary 2.1). Although we have chosen to consider the Abelian Higgs model for the sake of simplicity, we will show in a forthcoming note that our method may be adapted to other selfdual gauge theories as considered, e.g., in the monographs [18, 22]. 2. Main results and outline of the proof. In order to state precisely our results, we denote by UN the unique radial solution for the problem (see [12]) −ΔUN = 1 − eUN − 4πN δ0 in R2 , (2.1) as |x| → +∞. UN (x) → 0 Our main result is the following theorem. Theorem 2.1. Let pj ∈ R2 , mj ∈ N, j ∈ P ⊆ N, and assume that conditions (1.7) hold. Then there exists a constant δ1 > 0 (depending on d and m only) such
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M. MACR`I, M. NOLASCO, AND T. RICCIARDI
that for every δ ∈ (0, δ1 ) there exists a solution uδ for (1.6). Furthermore, uδ satisfies the approximate superposition rule
|x − pj | + ωδ , (2.2) Umj uδ (x) = δ j∈P
where the error term ωδ satisfies ωδ ∞ ≤ Ce−c/δ for some C, c > 0 independent of δ. In particular, uδ satisfies the following properties: (i) 0 ≤ euδ < 1, euδ vanishes exactly at pj , j ∈ P; (ii) for every compact subset K of R2 \ ∪j∈P {pj }, there exist C, c > 0 such that + supK (1 − euδ ) ≤ Ce−c/δ as δ → 0 ; 1 uδ (iii) ±F12 = 2δ2 (1 − e ) → 2π j∈P mj δpj in the sense of distributions as δ → 0+ . In the case that P is countable, we say that the vortex points pj , j ∈ P, are doubly periodically arranged in R2 if there exists s ∈ N such that (2.3)
{pk }k∈P = {pj + me1 + ne2 : j = 1, . . . , s; m, n ∈ Z},
where e1 and e2 are the unit vectors in R2 defining the periodic cell domain Ω (for simplicity, we assume a = b = 1). Under this condition, solving (1.6) is equivalent to solving (1.5). Namely, we deal with the physically relevant case of a finite number of vortex points s p1 , . . . , ps ∈ Ω, with the corresponding multiplicity mj , j = 1, . . . , s, such that j=1 mj = N , where N is the vortex number and Ω is the periodic cell domain. As a consequence of Theorem 2.1, and proving in addition that if (2.3) is satisfied, then the solution uδ for (1.6) is in fact doubly periodic with periodic cell domain Ω, we derive the following result. Corollary 2.1. If the pj ’s are doubly periodically arranged in R2 , there exists a constant δ1 > 0 (depending on N only) such that for every δ ∈ (0, δ1 ) the solution uδ , given in Theorem 2.1, is a solution for (1.5). Furthermore, the corresponding vortex configurations (Aδ , φδ ) satisfy the approximate factorization rule
s x − pj + ηδ , Φmj x ∈ Ω, φδ (x) = δ j=1 where the error term ηδ satisfies ηδ ∞ ≤ Ce−c/δ for some C, c > 0 independent of δ, and (Amj , Φmj ) is the unique, up to gauge transformation, single vortex (or antivortex) solution with multiplicity mj , to (1.1)–(1.2) with δ = 1 on R2 . In particular, we have the following: (i) 0 ≤ |φδ |2 < 1, φδ vanishes exactly at pj , j = 1, . . . , s; (ii) for every compact subset K of Ω \ {p1 , . . . , ps }, there exist C, c > 0 such that + 0 ≤ supK (1 − |φδ |2 ) ≤ Ce−c/δ as δ → 0s ; (iii) ±F12 (Aδ , φδ ) = 2δ12 (1 − |φδ |2 ) → 2π j=1 mj δpj in the sense of distributions + (on Ω) as δ → 0 ; (iv) Ω Eδ (Aδ , φδ ) = ± Ω F12 (Aδ , φδ ) = 2πN . An outline of the proof is as follows. Our starting point in proving Theorem 2.1 is to consider δ > 0 as a scaling parameter. Setting u ˆ(x) = u(δx), we have that u ˆ satisfies −Δˆ u = 1 − euˆ − 4π (2.4) mj δpˆj in R2 , j∈P
ASYMPTOTICS FOR SELFDUAL VORTICES
5
where pˆj = pj /δ. Note that the vortex points pˆj “separate” as δ → 0+ . Section 3 contains some properties of the radial solutions UN to (2.1). We rely on the results of Taubes [19] for the existence and uniqueness of UN as well as for the exponential decay properties at infinity. We also prove a nondegeneracy property of UN . The exponential decay of solutions justifies the following approximate superposition picture for small values of δ, i.e., for vortex points pˆj which are “far apart”: u ˆ(x) ≈
Umj (|x − pˆj |) .
j∈P
In fact, we take the following preliminary form of the superposition rule: u ˆ= (2.5) ϕˆj Umj (x − pˆj ) + z j∈P
as an ansatz for u ˆδ . Here, radial solutions centered at pˆj are “glued” together by the functions ϕˆj , which belong to a suitable locally finite partition of unity. Section 4 contains the definition and the main properties of the partition as well as of ˆ δ , Yˆδ , which are also obtained by “gluing” H 2 (R2 ) the appropriate functional spaces X 2 2 and L (R ), respectively. Hence, we are reduced to show that for small values of δ > 0, there exists an exponentially small “error” z such that u ˆ defined by (2.5) ˆ δ is the aim of section 5 (see is a solution of (2.4). The existence of such a z ∈ X Proposition 5.1). To this end, we use the shadowing lemma. We characterize z ˆ δ → Yˆδ is suitably defined. The nonby the property Fδ (z) = 0, where Fδ : X degeneracy property of UN is essential in order to prove that the operator DFδ (0) is invertible, and that its inverse is bounded independently of δ > 0 (Lemma 5.3). At this point, the Banach fixed point argument applied to I − (DFδ (0))−1 Fδ yields the existence of the desired error term z. In section 6 we show that (2.5) implies (2.2) and we derive the asymptotic behavior of solutions, thus concluding the proof of Theorem 2.1. Finally, we derive Corollary 2.1 by showing that periodically arranged vortex points lead to periodic solutions. Henceforth, unless otherwise stated, we denote by C, c > 0 general constants independent of δ > 0 and of j ∈ P. 3. Single vortex point solutions. In this section, we consider the solution UN to the radially symmetric equation (2.1). For every r > 0, we denote Br = {x ∈ R2 : |x| < r}. The following lemma contains some properties of UN that will be needed in the following. The proof is a consequence of the results of Taubes [12, 19] on the existence, uniqueness, and the exponential decay of UN together with standard elliptic theory as in, e.g., [9]. Therefore, it is omitted. Lemma 3.1. The following properties hold: (i) eUN (x) < 1 for any x ∈ R2 ; (ii) for every r > 0 there exist constants CN > 0 and αN > 0, depending on r and N only, such that |1 − eUN (x) | + |∇UN (x)| + |UN (x)| ≤ CN e−αN |x| for all x ∈ R2 \ Br . We consider the bounded linear operator LN = −Δ + eUN : H 2 (R2 ) → L2 (R2 ).
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M. MACR`I, M. NOLASCO, AND T. RICCIARDI
In order to apply the shadowing lemma, we also need the following nondegeneracy property of UN . Lemma 3.2. The operator LN is invertible and for every N > 0 there exists CN > 0 such that L−1 N ≤ CN . Proof. It is readily seen that LN is injective. Indeed, suppose LN u = 0 for some u ∈ H 2 (R2 ). Multiplying by u and integrating on R2 , we have |∇u|2 + eUN u2 = 0. Therefore, u = 0. Now, we claim that LN is a Fredholm operator. Indeed, we write LN = (−Δ + 1)(I − T ) with T = (−Δ + 1)−1 (1 − eUN ) : H 2 (R2 ) → H 2 (R2 ). Clearly, T is continuous. Let us check that T is compact. To this end, let un ∈ H 2 (R2 ), un H 2 = 1. We have to show that T un has a convergent subsequence. Note that by the Sobolev embedding (3.1)
uL∞ (R2 ) ≤ CS uH 2 (R2 ) ,
for all u ∈ H 2 (R2 ), we have un ∞ ≤ C , for some C > 0 independent of n, and there exists u∞ , u∞ H 2 ≤ 1, such that unk → u∞ strongly in L2loc for a subsequence unk . Now, by Lemma 3.1, for any fixed ε > 0, there exists R > 0 such that 1 − eUN L2 (R2 \BR ) ≤ ε. Consequently, (1 − eUN )(unk − u∞ )L2 (R2 \BR ) ≤ 2C ε. On the other hand, (1 − eUN )(unk − u∞ )L2 (BR ) → 0. We conclude that (1 − eUN )(unk − u∞ ) → 0 in L2 . In turn, we have T (unk − u∞ ) = (−Δ + 1)−1 (1 − eUN )(unk − u∞ ) → 0 in H 2 , which implies that T is compact. It follows that LN is a Fredholm operator. Consequently, LN is also surjective. At this point, the open mapping theorem concludes the proof. 4. A partition of unity. In this section, we introduce a partition of unity and we prove some technical results which will be needed in the following. Let pj ∈ R2 (j ∈ P ⊆ N) be the vortex points. By assumption (1.7), r0 = d/8 = inf j=k |pj − pk |/8 > 0. We consider the set K = (− 34 r0 , 34 r0 ) × (− 34 r0 , 34 r0 ). Then, for any n ∈ Z2 , we introduce Kn = K + nr0 . The collection of sets {Kn }n∈Z2 is a locally finite an associated partition of unity defined as follows: let covering of R2 . We consider 0 ≤ ζ ∈ Cc∞ (K) be such that n∈Z2 ζn = 1 pointwise on R2 , where ζn (x) = ζ(x−nr0 ) for all x ∈ R2 . Then, for any j ∈ P, we introduce the set
r0 2 . Pj = n ∈ Z : d(pj , Kn ) < 4 Note that the cardinality of Pj is uniformly bounded, namely, |Pj | ≤ 4 for any j ∈ P. We set Pj = Kn , ϕj = ζn n∈Pj
n∈Pj
for any j ∈ P. Since, by choice of r0 , Z2 \ j∈P Pj is a countable set (even in the case that P is countable), there exists a bijection I : N → Z2 \ j∈P Pj . For convenience of notation, we denote by Q the countable set of indices defined by
−1 2 Z \ Q=I Pj . j∈P
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ASYMPTOTICS FOR SELFDUAL VORTICES
We set Qk = KI(k) ,
∀k ∈ Q.
ψk = ζI(k)
Then, {Pj , Qk }(j,k)∈P×Q is a locally finite open covering of R2 with the property that Pj ∩ Pj = ∅ for every j = j. Moreover, {ϕj , ψk }(j,k)∈P×Q is a partition of unity associated with {Pj , Qk }(j,k)∈P×Q such that supp ϕj ⊂ Pj ,
supp ψk ⊂ Qk
and such that sup{∇ϕj ∞ , D2 ϕj ∞ } < +∞,
sup {∇ψk ∞ , D2 ψk ∞ } < +∞.
j∈P
In particular, 0 ≤ ϕj , ψk ≤ 1
and
k∈Q
ϕj +
j∈P
ψk =
ζn = 1.
n∈Z2
k∈Q
We define a rescaled covering Pˆj = Pj /δ,
ˆ k = Qk /δ. Q
Then, {ϕˆj , ψˆk }(j,k)∈P×Q defined by ψˆk (x) = ψk (δx)
ϕˆj (x) = ϕj (δx),
ˆ k }(j,k)∈P×Q . It will also be convenient is a partition of unity associated with {Pˆj , Q to define the sets Cˆj = {x ∈ Pˆj : ϕˆj (x) = 1},
j ∈ P.
Note that supp{∇ϕˆj , D2 ϕˆj } ⊂ Pˆj \ Cˆj and (4.1) sup
{∇ϕˆj ∞ + ∇ψˆk ∞ } ≤ Cδ,
sup
(j,k)∈P×Q
{D2 ϕˆj ∞ + D2 ψˆk ∞ } ≤ Cδ 2 .
(j,k)∈P×Q
For every fixed x ∈ R , we define the following subsets of indices: 2
K(x) = {k ∈ Q : ψˆk (x) = 0}.
J(x) = {j ∈ P : ϕˆj (x) = 0}, Note that, for every x ∈ R2 , (4.2)
|J(x)| ≤ 1,
|K(x)| ≤ 4,
where |J(x)| and |K(x)| denote the cardinality of J(x) and K(x), respectively. We shall use the following Banach spaces:
ˆk uH 2 (R2 ) < +∞ , ˆ δ = u ∈ H 2 (R2 ) : 2 (R2 ) , ψ X ϕ ˆ sup u j H loc ˆ Yδ = f ∈ L2loc (R2 ) :
(j,k)∈P×Q
sup
ϕˆj f
L2 (R2 )
ˆ 2 2 , ψk f L (R ) < +∞ .
(j,k)∈P×Q
We collect in the following lemma some estimates that will be used in what follows.
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
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ˆ δ and j ∈ P, Lemma 4.1. There exists a constant C > 0 such that for any u ∈ X we have (i) uH 2 (Pˆj ) ≤ CuXˆ δ , (ii) uL∞ (R2 ) ≤ CuXˆ δ . Proof. (i) For every fixed j ∈ P, let J (j) = {k ∈ Q : supp ϕˆj ∩ supp ψˆk = ∅}. Then, supj∈P |J (j)| < +∞, and we estimate uH 2 (Pˆj )
=ϕˆj u + ψˆk u k∈J (j)
≤ ϕˆj uH 2 (Pˆj ) +
H 2 (Pˆj )
ψˆk uH 2 (Pˆj )
k∈J (j)
≤ (1 + |J (j)|) uXˆ δ ≤ CuXˆ δ . (ii) For any fixed x ∈ R2 , we have in view of (3.1) and (4.2) |u(x)| =
ϕˆj (x)|u(x)| +
j∈P
=
≤
ψˆk (x)|u(x)|
k∈Q
ϕˆj (x)|u(x)| +
j∈J(x)
ψˆk (x)|u(x)|
k∈K(x)
CS ϕˆj uH 2 (R2 ) +
j∈J(x)
CS ψˆk uH 2 (R2 )
k∈K(x)
≤ sup (|J(x)| + |K(x)|)CS uXˆ δ = CuXˆ δ . x∈R2
Hence, (ii) is established. We shall also need the following family of functions: ϕˆj (x) 1/2 , 2 + ˆ2 ϕ ˆ ψ k∈P k k∈Q k
gˆj (x) =
ψˆk (x) 1/2 . 2+ ˆ2 ϕ ˆ ψ j∈P j j∈Q j
ˆ k (x) = h
In view of (4.1), it is readily checked that the following lemma follows. ˆ k }(j,k)∈P×Q satisfies supp gˆj ⊂ Pˆj , supp h ˆk ⊂ Q ˆk Lemma 4.2. The family {ˆ gj , h and furthermore, ˆ 2 (x) = 1 ∀x ∈ R2 ; (i) ˆj2 (x) + k∈Q h k j∈P g ˆ k (x) ≤ C ψˆk (x) ∀x ∈ R2 ; (ii) C −1 ϕˆj (x) ≤ gˆj (x) ≤ C ϕˆj (x) and C −1 ψˆk (x) ≤ h ˆ (iii) sup(j,k)∈P×Q {∇ˆ gj ∞ + ∇hk ∞ } ≤ Cδ and sup(j,k)∈P×Q {D2 gˆj ∞ + ˆ k ∞ } ≤ Cδ 2 . D2 h 5. The shadowing lemma. Recall from the introduction that pˆj = pj /δ, j ∈ P. For every j ∈ P we define ˆj (x) = Um (x − pˆj ). U j We make the following ansatz for solutions u ˆ to (2.4): (5.1)
u ˆ=
ˆj + z. ϕˆj U
j∈P
Our aim in this section is to prove the following.
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ASYMPTOTICS FOR SELFDUAL VORTICES
Proposition 5.1. There exists δ1 > 0 such that for all δ ∈ (0, δ1 ) there exists ˆ δ such that u ˆj + zδ is a solution of (2.4). Moreover, zδ ∈ X ˆδ defined by u ˆδ = j∈P ϕˆj U zδ Xˆ δ ≤ Ce−c/δ . ˆ δ → Yˆδ given by We note that the functional Fδ : X ˆ ˆ ˆj Fδ (z) = −Δz + ϕˆj (1 − eUj ) − (1 − e j∈P ϕˆj Uj +z ) − [ϕˆj , Δ]U j∈P
j∈P
ˆδ is well defined as well as C 1 . Here [Δ, ϕˆj ] = Δϕˆj + 2∇ϕˆj ∇. Moreover, if z ∈ X satisfies Fδ (z) = 0, then u ˆ defined by (5.1) is a solution of (2.4). Lemma 5.1. For δ > 0 sufficiently small, we have Fδ (0)Yˆδ ≤ Ce−c/δ
(5.2)
as δ → 0+
for some constants C, c > 0 independent of δ. Proof. Let ˆ ˆ R= ϕˆj (1 − eUj ) − (1 − e j∈P ϕˆj Uj ), j∈P
C=
ˆj . [ϕˆj , Δ]U
j∈P
Note that {supp R, supp C} ⊂ ∪j∈P Pˆj \ Cˆj . We fix x ∈ ∪j∈P Pˆj . We estimate ˆ
ˆ
|R(x)| ≤ sup ϕˆj (1 − eUj )L∞ (Pˆj \Cˆj ) + sup 1 − eϕˆj Uj L∞ (Pˆj \Cˆj ) j∈P
j∈P
ˆj ∞ ˆ ˆ ≤ C1 e−c1 /δ . ≤C sup U L ( Pj \C j ) j∈P
On the other hand, in view of (4.1) and Lemma 3.1, for x ∈ ∪j∈P Pˆj , we have ˆj ∞ ˆ ˆ |C(x)| ≤ sup [Δ, ϕˆj ]U L ( Pj \C j ) j∈P
ˆj Δϕˆj ∞ ˆ ˆ + sup |∇U ˆj | |∇ϕˆj | ∞ ˆ ˆ ≤ C sup U L ( Pj \C j ) L ( Pj \C j ) j∈P
≤ C2 e−c2 /δ .
j∈P
Here and above, c1 , C1 , c2 , C2 > 0 are positive constants independent of δ > 0. Hence, we conclude that, as δ → 0+ , Fδ (0)Yˆδ ≤ C sup RL2 (Pˆj ) + CL2 (Pˆj ) ≤ Ce−c/δ j∈P
for some constants C, c > 0 independent of δ > 0. ˆ δ → Yˆδ given by Now, we consider the operator Lδ ≡ DFδ (0) : X Lδ = −Δ + e
j∈P
ˆj ϕ ˆj U
For every j ∈ P, we define the operators ˆ j = −Δ + eUˆj . L
.
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
10
It will also be convenient to define ˆ 0 = −Δ + 1. L The following lemma holds. ˆ δ , we have Lemma 5.2. There exist C, c > 0 such that for any u ∈ X ˆ j )ϕˆj uL2 ≤ Ce−c/δ ϕˆj uL2 , (Lδ − L ˆ 0 )ψˆk uL2 ≤ Ce−c/δ ψˆk uL2 , (Lδ − L
j ∈ P, k ∈ Q.
Proof. For any j ∈ P, by Lemma 3.1, we have, as δ → 0+ , ˆ j )ϕˆj uL2 ≤ 1 − eUˆj ∞ ˆ ˆ + 1 − eϕˆj Uˆj ∞ ˆ ˆ ϕˆj uL2 (Lδ − L L ( Pj \C j ) L ( Pj \C j ) ≤ C1 − eUj L∞ (Pˆj \Cˆj ) ϕˆj uL2 ≤ Ce−c/δ ϕˆj uL2 . ˆ
Similarly, as δ → 0+ , ˆ 0 )ψˆk uL2 ≤(1 − e (Lδ − L
j∈P
ˆj ϕ ˆj U
)ψˆk uL2
ˆ ≤ sup 1 − eϕˆj Uj L∞ (Pˆj \Cˆj ) ψˆk uL2 ≤ Ce−c/δ ψˆk uL2 . j∈P
Now, we prove an essential nondegeneracy property of Lδ . Lemma 5.3. There exists δ0 > 0 such that for any δ ∈ (0, δ0 ), the operator Lδ is ˆ ˆ invertible. Moreover, L−1 δ : Yδ → Xδ is uniformly bounded with respect to δ ∈ (0, δ0 ). Proof. Following a gluing technique introduced in [5], we construct an “approxiˆ δ for L−1 as follows: mate inverse” Sδ : Yˆδ → X δ ˆkL ˆk, ˆ −1 gˆj + ˆ −1 h Sδ = gˆj L h 0 j j∈P
k∈Q
ˆ k are the functions introduced in section 4. We claim that the operator where gˆj and h Sδ is well defined and uniformly bounded with respect to δ. That is, we claim that Sδ f Xˆ δ ≤ Cf Yˆδ
(5.3)
ˆ δ and of δ > 0. for some C > 0 independent of f ∈ X Indeed, for any f ∈ Yˆδ , we have Sδ f Xˆ δ = and ϕˆj Sδ f
H2
sup
{ ϕˆj Sδ f H 2 , ψˆk Sδ f H 2 }
(j,k)∈P×Q
−1 −1 ˆ ˆ ˆ ˆ 2 ≤ ϕˆj gˆj Lj gˆj f H + ϕˆj hk L0 hk f
−1 ˆ ˆ ˆ 2 ψk Sδ f H ≤ ψk gˆj Lj gˆj f j∈P
H2
,
H2
k∈Q
−1 ˆ ˆ ˆ ˆ + ψk hj L0 hj f j∈Q
H2
.
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ASYMPTOTICS FOR SELFDUAL VORTICES
We estimate, recalling the properties of ϕˆj and gˆj as in Lemma 4.2, and in view of Lemma 3.2 ˆ −1 gˆj f H 2 ≤ Cˆ ˆ −1 gˆj f H 2 ≤ CL gj f L2 ≤ Cϕˆj f L2 ≤ Cf Yˆδ . ϕˆj gˆj L j j We have −1 ˆ ˆ ˆ ϕˆj hk L0 hk f
H2
k∈Q
−1 ˆ ˆ ˆ ≤ ϕˆj hk L0 hk f
H2
k∈J (j)
≤
ˆkL ˆ k f H 2 , ˆ −1 h ϕˆj h 0
k∈J (j)
where J (j) = {k ∈ Q : supp ψˆk ∩ supp ϕˆj = ∅} satisfies supj∈P |J (j)| < +∞. In view of Lemmas 4.2 and 3.2, we estimate ˆ k f L2 ˆkL ˆ k f H 2 ≤ C ˆ k f H 2 ≤ C ˆ −1 h ˆ −1 h ϕˆj h L h 0 0 k∈J (j)
≤C
k∈J (j)
k∈Q
k∈J (j)
Therefore,
−1 ˆ ˆ ˆ sup ϕ ˆ f h h L k 0 k j
j∈P
Similarly, we obtain that −1 ˆ ˆ sup ψk gˆj Lj gˆj f k∈Q
k∈J (j)
ψˆk f L2 ≤ C|J (j)| sup ψˆk f L2 ≤ Cf Yˆδ .
H2
j∈P
H2
k∈Q
≤ Cf Yˆδ ,
≤ Cf Yˆδ .
−1 ˆ ˆ ˆ ˆ sup ψk hj L0 hj f
k∈Q
j∈Q
H2
≤ Cf Yˆδ ,
and (5.3) follows. Now, we claim that there exists δ0 such that for any δ ∈ (0, δ0 ), the operator Sδ Lδ : ˆδ → X ˆ δ is invertible, and furthermore, Sδ Lδ ≤ C for some C > 0 independent X ˆk : X ˆ j )ˆ ˆ δ → Yˆδ and (Lδ − L ˆ 0 )h ˆ δ → Yˆδ are of δ > 0. We note that (Lδ − L gj : X well-defined bounded linear operators. Thus, we decompose ˆ k Lδ − L ˆkL ˆk) ˆ j gˆj ) + ˆ0h ˆ −1 (ˆ ˆ −1 (h Sδ Lδ = IXˆ δ + gˆj L gj Lδ − L h 0 j j∈P
(5.4)
= IXˆ δ +
k∈Q
ˆ −1 (Lδ gˆj L j
ˆ j )ˆ −L gj +
j∈P
+
ˆ −1 [Δ, gˆj ] gˆj L j
+
j∈P
ˆk ˆkL ˆ 0 )h ˆ −1 (Lδ − L h 0
k∈Q
ˆ k ]. ˆkL ˆ −1 [Δ, h h 0
k∈Q
Hence, it suffices to prove that the last four terms in (5.4) are sufficiently small, in the operator norm, provided δ > 0 is sufficiently small. By Lemmas 5.2 and 4.2, we ˆδ , have, for any u ∈ X −1 ˆ ˆ −1 (Lδ − L ˆ j )ˆ ˆ ≤ C sup L g ˆ (L − L )ˆ g u gj uH 2 L j j δ j j j j∈P
ˆδ X
j∈P
ˆ j )ˆ ≤ C sup (Lδ − L gj uL2 j∈P
≤ Ce−c/δ sup ϕˆj uL2 ≤ Ce−c/δ uXˆ δ . j∈P
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
12
ˆ δ , we have Similarly, for u ∈ X −1 ˆ −1 [Δ, gˆj ]uH 2 ≤ C sup [Δ, gˆj ]uL2 . ˆ gˆj Lj [Δ, gˆj ]u sup L j ˆ ≤ C j∈P j∈P Xδ
j∈P
Recalling that [Δ, gˆj ]u = 2∇u∇ˆ gj + uΔˆ gj , by Lemmas 4.2 and 4.1(i) we derive that [Δ, gˆj ]uL2 ≤ CδuH 1 (Pˆj ) ≤ CδuXˆ δ . The remaining terms are estimated similarly. Hence, Sδ Lδ − IXˆ δ → 0 as δ → 0+ . −1 Now, we observe that L−1 Sδ . It follows that for any f ∈ Yˆδ , we have δ = (Sδ Lδ ) −1 L−1 Sδ f Xˆ δ ≤ CSδ f Yˆδ ≤ Cf Yˆδ ˆ δ = (Sδ Lδ ) δ f X
with C > 0 independent of δ. Hence, Lδ is invertible and its inverse is bounded independently of δ, as asserted. Now we can provide the following proof. Proof of Proposition 5.1. We use the Banach fixed point argument. For any δ ∈ (0, δ0 ), with δ0 > 0 given by Lemma 5.3, we introduce the nonlinear map Gδ ∈ ˆδ , X ˆ δ ) defined by C 1 (X Gδ (z) = z − L−1 δ Fδ (z). Then, fixed points of Gδ correspond to solutions of the functional equation Fδ (z) = 0. First, note that DGδ (0) = 0 and that DF (z) = −Δ + e
j∈P
ˆj +z ϕ ˆj U
.
ˆ δ and u ∈ X ˆ δ , we have By Lemma 5.3, for any z ∈ X DGδ (z)uXˆ δ = (DGδ (z) − DGδ (0))uXˆ δ = L−1 ˆδ δ (DFδ (z) − Lδ )uX ≤ C(DFδ (z) − Lδ )uYˆδ = Ce
j∈P
ˆj ϕ ˆj U
(ez − 1)uYˆδ ≤ C(ez − 1)uYˆδ .
By the elementary inequality et − 1 ≤ Ctet , for all t > 0, where C > 0 does not depend on t, and in view of Lemma 4.1, we have ez − 1∞ ≤ ez∞ − 1 ≤ Cz∞ ez∞ ≤ CzXˆ δ e
zX ˆ
δ
.
Hence, DGδ (z)uXˆ δ ≤ C(ez − 1)uYˆδ ≤ CzXˆ δ e
zX ˆ
δ
uYˆδ ≤ CzXˆ δ e
zX ˆ
δ
Consequently, there exists R0 > 0 such that for every R ∈ (0, R0 ), we have DGδ (z)
0, where ˆ δ : u ˆ < R}. BR = {u ∈ X Xδ
uXˆ δ .
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ASYMPTOTICS FOR SELFDUAL VORTICES
Now, for every R ∈ (0, R0 ), Gδ (z)Xˆ δ ≤ Gδ (z) − Gδ (0)Xˆ δ + Gδ (0)Xˆ δ 1 ≤ zXˆ δ + L−1 ˆδ . δ Fδ (0)X 2 By Lemmas 5.3 and 5.1, there exist C0 , c0 > 0 independent of δ > 0 such that −c0 /δ . L−1 ˆ δ ≤ CFδ (0)Yˆδ ≤ C0 e δ Fδ (0)X
Choosing R = Rδ = 2C0 e−c0 /δ , we obtain that Gδ (BRδ ) ⊂ BRδ . Hence, Gδ is a strict contraction in BRδ , for any δ ∈ (0, δ1 ), with δ1 = c0 /(ln(2C0 /R0 )). By the Banach fixed-point theorem, for any δ ∈ (0, δ1 ), there exists a unique zδ ∈ BRδ such that Fδ (zδ ) = 0. 6. Proof of the main results. In this section, we finally provide the proof of Theorem 2.1 and derive Corollary 2.1. In view of Proposition 5.1, the function u ˆδ defined by ˆ j + zδ u ˆδ = ϕˆj U j∈P
is a solution of (2.4). Consequently, uδ defined by
x x x − pj (6.1) = + zδ ˆδ ϕj (x)Umj uδ (x) = u δ δ δ j∈P
is a solution of (1.6). Lemma 6.1. The solution uδ defined in (6.1) satisfies the approximate superposition rule
x − pj + ωδ (x) Umj uδ (x) = δ j∈P
with ωδ ∞ ≤ Ce−c/δ . Proof. In view of (6.1) and of the definition of J(x) in section 4, we have
x − pj +ω ˜ δ (x), uδ (x) = Umj δ j∈J(x)
where ω ˜ δ (x) = −
(1 − ϕj (x))Umj
j∈J(x)
In view of Lemma 3.1, we estimate
x − p j (1 − ϕj (x))Umj δ j∈J(x)
∞
≤
x − pj δ
j∈J(x)
+ zδ
x δ
.
x − pj −c/δ sup Umj . ≤ Ce δ 2
R \Cj
14
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
On the other hand, by Proposition 5.1 and Lemma 4.1(ii), we have zδ · = zδ ∞ ≤ Ce−c/δ . δ ∞ Therefore, ˜ ωδ ∞ ≤ Ce−c/δ . We have to show that
x − pj ≤ Ce−c/δ . Umj δ j∈P\J(x) ∞
To this end, we fix x ∈ R2 and for every M ∈ N we define BM = {y ∈ R2 : |y − x| < dM }. Then,
x − pj x − pj = . Umj Umj δ δ j∈P\J(x)
M ∈N pj ∈BM +1 \BM
Since inf j=k |pj −pk | = d > 0, there exists C > 0 independent of M ∈ N and of x ∈ R2 such that |{pj ∈ BM +1 \ BM }| ≤ CM. Hence, we estimate
x − p j ≤C U M e−cM/δ ≤ Ce−c/δ . m j δ j∈P\J(x) M ∈N This implies the statement of the lemma. We are left to analyze the asymptotic behavior of uδ as δ → 0+ . Such behavior is a direct consequence of (6.1). Lemma 6.2. Let uδ be given by (6.1). The following properties hold: (i) euδ < 1 on R2 and vanishes exactly at pj with multiplicity 2mj , j ∈ P; (ii) for every compact subset K of R2 \ ∪j∈P {pj }, there exist C, c > 0 such that + 1 − euδ ≤ Ce−c/δ as δ → 0 ; −2 uδ (iii) δ (1 − e ) → 4π j∈P mj δpj in the sense of distributions as δ → 0+ . Proof. (i) Since uδ is a solution of (1.6), euδ < 1 follows by the maximum principle. Moreover, since
x − pj = ln |x − pj |2mj + vj Umj (6.2) δ with vj a continuous function (see [12]), we have near pj that euδ = |x − pj |2mj fj,δ (x) with fj,δ (x) a continuous strictly positive function. Hence, (i) is established. (ii) Let K be a compact subset of R2 \ ∪j∈P {pj }. In view of Lemma 3.1 and Proposition 5.1, we have as δ → 0+ · sup 1 − eϕj (x)Umj ((x−pj )/δ) ≤ Ce−c/δ zδ δ ∞ x∈K∩Pj ≤ Czδ Xˆ δ ≤ CRδ ≤ Ce−c/δ .
ASYMPTOTICS FOR SELFDUAL VORTICES
15
Therefore, we have that for any compact set K ⊂ R2 \ ∪j∈P {pj }, 0 ≤ sup (1 − euδ ) ≤ C sup sup (1 − euδ ) ≤ Ce−c/δ . x∈K
j∈P x∈K∩Pj
(iii) Let ϕ ∈ Cc∞ (R2 ). Then, − uδ Δϕ = δ −2 (1 − eu )ϕ − 4π mj ϕ(pj ). R2
We claim that
R2
j∈P
(6.3) R2
uδ Δϕ → 0
as δ → 0.
Indeed, let jk ∈ P, k = 1, . . . , n, be such that supp ϕ ⊂ ∪nk=1 Pjk ∪ K with K a compact subset of R2 \ ∪j∈P {pj }. Since supK |uδ | ≤ Ce−c/δ , we have uδ Δϕ ≤ CΔϕ∞ e−c/δ → 0. K
On the other hand, in view of Lemma 6.1, in Pjk we have uδ (x) = Umjk (|x − pjk |/δ) + O(e−c/δ ). Note that Umjk ∈ L1 (R2 ) in view of (6.2) and Lemma 3.1. Therefore,
x − pjk sup Δϕ + O(e−c/δ ) uδ Δϕ ≤ sup Umjk δ 1≤k≤n 1≤k≤n Pj Pj k
k
≤ δ 2 sup Δϕ∞ Umjk L1 + O(e−c/δ ) ≤ Cδ 2 → 0. 1≤k≤n
Hence, (6.3) follows, and (iii) is established. Proof of Theorem 2.1. For every δ ∈ (0, δ1 ), where δ1 is given in Proposition 5.1, we obtain a solution uδ of (1.6). Furthermore, uδ satisfies (2.2) in view of Lemma 6.1. Finally, uδ satisfies the asymptotic behavior as in (i)–(iii) in view of Lemma 6.2. Hence, Theorem 2.1 is completely established. Proof of Corollary 2.1. To begin, we want to prove that if pj ’s are doubly periodically arranged in R2 , then uδ is in fact a doubly periodic solution of (1.5). Recall that the pj ’s are doubly periodically arranged in R2 if (2.3) holds. We define ˆek = ek /δ, k = 1, 2. Equivalently, we show u ˆδ (x + ˆek ) = u ˆδ (x) for any x ∈ R2 and for k = 1, 2. Indeed, we may assume that ϕˆj (x + ˆek ) = ϕˆj (x), ψˆj (x + ˆek ) = ψˆj (x) for any j ∈ N, x ∈ R2 , k = 1, 2. Then, ˆj (x) + zδ (x + ˆe ). ϕˆj (x)U u ˆδ (x + ˆek ) = k j∈N
Hence, it is sufficient to prove that zδ (x + ˆek ) = zδ (x) for every x ∈ R2 and for k = 1, 2. First, we claim that zδ ( · + ˆek ) ∈ BRδ . Indeed, for every j ∈ N there exists exactly one j ∈ N such that (6.4)
ϕˆj zδ ( · + ˆek )H 2 = ϕˆj zδ H 2 .
Hence, we obtain (6.5)
zδ ( · + ˆek )Xˆ δ = zδ Xˆ δ ≤ Rδ .
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M. MACR`I, M. NOLASCO, AND T. RICCIARDI
Moreover, if Fδ (zδ ) = 0 we also have Fδ (zδ ( · + ˆek )) = 0. Therefore, zδ ( · + ˆek ) is a fixed point of Gδ in BRδ . By uniqueness, we conclude that zδ ( · +ˆek ) = zδ , k = 1, 2, as asserted. At this point, the remaining statements follow recalling that in the periodic cell domain Ω, (Aδ , φδ ) is given, up to gauge transformations, by (1.4). REFERENCES [1] A.A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP, 5 (1957), pp. 1174–1182. [2] A. Aftalion, E. Sandier, and S. Serfaty, Pinning phenomena in the Ginzburg-Landau model of superconductivity, J. Math. Pures Appl., 80 (2001), pp. 339–372. [3] S. Alama and L. Bronsard, Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains, Comm. Pure Appl. Math., to appear. ´, P. Bauman, and D. Phillips, Vortex pinning with bounded fields for the Ginzburg[4] N. Andre Landau equation, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 20 (2003), pp. 705–729. [5] S. Angenent, The shadowing lemma for elliptic PDE, in Dynamics of Infinite-Dimensional Systems, NATO Adv. Sci. Inst. Ser. F Comput. Syst. Sci. 37, Springer, Berlin, 1987, pp. 7–22. ´lein, Ginzburg-Landau Vortices, Birkh¨ [6] F. Bethuel, H. Brezis, and F. He auser, Boston, MA, 1994. [7] W. E, Dynamics of vortices in the Ginzburg-Landau theories with applications to superconductivity, Phys. D, 77 (1994), pp. 383–404. [8] O. Garc´ia-Prada, A direct existence proof for the vortex equations over a compact Riemann surface, Bull. London Math. Soc., 26 (1994), pp. 88–96. [9] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, in Classics in Mathematics, Springer, Berlin, 2001. [10] M.Y. Hong, J. Jost, and M. Struwe, Asymptotic limits of a Ginzburg Landau type functional, in Geometric Analysis and Calculus of Variations, International Press, Cambridge, MA, 1996, pp. 99–123. [11] G. ’t Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nuclear Phys. B, 153 (1979), pp. 141–160. [12] A. Jaffe and C. Taubes, Vortices and Monopoles, Birkh¨ auser, Boston, MA, 1980. [13] F.H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), pp. 323–442. [14] J.C. Neu, Vortices in the complex scalar field, Phys. D, 43 (1990), pp. 385–406. [15] M. Nolasco, Non topological n-vortex condensates for the selfdual Chern-Simons theory, Comm. Pure Appl. Math., 56 (2003), pp. 1752–1780. [16] J. Rubinstein and P. Sternberg, On the slow motion of vortices in the Ginzburg-Landau heat flow, SIAM J. Math. Anal., 26 (1995), pp. 1452–1466. [17] D. Stuart, Dynamics of Abelian Higgs vortices in the near Bogomolny regime, Comm. Math. Phys., 159 (1994), pp. 51–91. [18] G. Tarantello, The Analysis of Selfdual Gauge Field Vortices, Progr. Nonlinear Differential Equations Appl., Birkh¨ auser, Boston, MA, in preparation. [19] C. Taubes, Arbitrary n-vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys., 72 (1980), pp. 277–292. [20] S. Wang and Y. Yang, Abrikosov’s vortices in the critical coupling, SIAM J. Math. Anal., 23 (1992), pp. 1125–1140. [21] M.I. Weinstein and J. Xin, Dynamic stability of vortex solutions of Ginzburg-Landau and nonlinear Schr¨ odinger equations, Comm. Math. Phys., 180 (1996), pp. 389–428. [22] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monogr. Math., Springer, New York, 2001.
SIAM J. MATH. ANAL. Vol. 37, No. 1, pp. 1–16
ASYMPTOTICS FOR SELFDUAL VORTICES ON THE TORUS AND ON THE PLANE: A GLUING TECHNIQUE∗ MARTA MACR`I† , MARGHERITA NOLASCO‡ ,
AND TONIA RICCIARDI†
Abstract. We consider multivortex solutions for the selfdual Abelian Higgs model, as the ratio of the vortex core size to the separation distance between vortex points (the scaling parameter) tends to zero. To this end, we use a gluing technique (a shadowing lemma) for solutions to the corresponding semilinear elliptic equation on the plane, allowing any number (finite or countable) of arbitrarily prescribed singular sources. Our approach is particularly convenient and natural for the study of the asymptotics. In particular, in the physically relevant cases where the vortex points are either finite or periodically arranged in the plane, we prove that a frequently used factorization ansatz for multivortex solutions is rigorously satisfied, up to an error which is exponentially small. Key words. selfdual Abelian Higgs model, elliptic equation, shadowing lemma AMS subject classifications. 35J60, 58E15, 81T13 DOI. 10.1137/040619843
1. Introduction. We consider the energy density for the static, two-dimensional selfdual Abelian Higgs model in the following form: Eδ (A, φ) = δ 2 |dA|2 + |Dφ|2 +
2 1 2 |φ| − 1 , 2 4δ
where A = A1 dx1 + A2 dx2 , A1 (x), A2 (x) ∈ R is a gauge potential (a connection over a principal U (1) bundle), φ, φ(x) ∈ C is a Higgs matter field (a section over an associated complex line bundle), D = d − iA is the covariant derivative, and δ > 0 is the scaling parameter. Eδ is a rescaling of E1 = Eδ |δ=1 , which coincides with the two-dimensional Ginzburg–Landau energy density in the so-called Bogomol’nyi limit. Such a limit describes the borderline between type I and type II superconductors; see, e.g., Jaffe and Taubes [12]. By the selfdual structure, solutions to the Euler–Lagrange equations of Eδ may be obtained from solutions to the first order system: (1.1)
(D1 ± iD2 )φ = 0,
(1.2)
F12 = ∂1 A2 − ∂2 A1 = ±
1 (1 − |φ|2 ). 2δ 2
The vortex-type critical points for the energy associated with Eδ , namely, the solutions of (1.1)–(1.2), have received considerable attention in recent years, in view of both their physical and geometrical interest; see, e.g., Garc´ia-Prada [8], Hong, Jost, and Struwe [10], Stuart [17], Taubes [19], Wang and Yang [20], and the references therein. In particular, Hong, Jost, and Struwe [10] consider (1.1)–(1.2) on a compact ∗ Received by the editors November 30, 2004; accepted for publication (in revised form) April 13, 2005; published electronically August 17, 2005. This research was supported in part by the M.I.U.R. Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari.” http://www.siam.org/journals/sima/37-1/61984.html † Dipartimento di Matematica e Applicazioni, Universit` a di Napoli Federico II, Via Cintia, 80126 Napoli, Italy ([email protected], [email protected]). The last author’s research was supported in part by Regione Campania L.R. 5/02. ‡ Dipartimento di Matematica Pura ed Applicata, Universit` a di L’Aquila, Via Vetoio, Coppito, 67010 L’Aquila, Italy ([email protected]).
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2
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
Riemannian surface and perform a detailed analysis of the asymptotics as δ → 0+ . Indeed, the small δ > 0 regime, corresponding to the limit of small vortex core size with respect to the separation distance between vortices, is an appropriate approximation for the analysis near the vortex points of solutions of (1.1)–(1.2) with δ = 1. This type of asymptotics is also relevant in the context of Ginzburg–Landau vortices; see, e.g., Aftalion, Sandier, and Serfaty [2], Alama and Bronsard [3], Andr´e, Bauman, and Phillips [4], Bethuel, Brezis, and H´elein [6], Lin [13], Rubinstein and Sternberg [16], just to mention a few. It has also been widely investigated in the context of other selfdual gauge theories; see the monographs of Tarantello [18] and Yang [22]. The fundamental results concerning finite energy solutions of (1.1)–(1.2) on R2 were obtained by Taubes [12, 19]. In particular, Taubes showed that such solutions are completely determined by solutions to the singular elliptic problem s −Δu = δ −2 (1 − eu ) − 4π j=1 mj δpj on R2 , (1.3) u(x) → 0 as |x| → +∞. Here s ∈ N, and for j = 1, 2, . . . , s, pj ∈ R2 are the vortex points, mj ∈ N is the multiplicity of pj , and δpj is the Dirac measure at pj . By variational methods, Taubes proved that, for any δ > 0, there exists a unique solution of (1.3) such that the configuration (A, φ) defined in complex notation by s φ(z) = exp{ 12 u(z) ± i j=1 mj arg(z − pj )}, (1.4) A1 ∓ iA2 = −i(∂1 ± i∂2 ) ln φ 2 is a smooth, finite energy solution of (1.1)–(1.2) the on R , satisfying φ(pj ) = 0(with s corresponding multiplicity mj ∈ N) and E = R2 Eδ (A, φ) = ± R2 F12 = 2π j=1 mj = 2πN , where F12 = ∂1 A2 − ∂2 A1 is the magnetic field (the curvature of A). In connection with Abrikosov’s mixed states in superconductivity [1], it is also of physical interest to analyze (1.1)–(1.2) on the flat torus T2 ≡ R2 /(aZ × bZ), where a, b > 0. Such a case has been considered by Wang and Yang in [20]. It is shown in [20] that solutions of (1.1)–(1.2) with δ = 1 exist for any given set of vortex points pj ∈ T2 , s j = 1, 2, . . . , s, with multiplicity mj ∈ N, if and only if N = j=1 mj < |T2 |/(4π). In particular, on T2 the total number of vortices N cannot be arbitrarily large. Similarly as on R2 , denoting Ω = (0, a) × (0, b), solutions of (1.1)–(1.2) on T2 correspond to solutions for the singular elliptic problem s −Δu = δ −2 (1 − eu ) − 4π j=1 mj δpj in Ω, (1.5) u doubly periodic on ∂Ω.
The periodic boundary conditions are justified by certain more general gauge invariant conditions on the configuration (A, φ) introduced by ’t Hooft [11]. Such conditions force the magnetic flux through a lattice cell to be a “quantized” value proportional to the number of vortices confined. Namely, s the ’t Hooft boundary conditions imply the topological constraint ± Ω F12 = 2π j=1 mj = 2πN on the solutions of (1.5), exactly as for finite energy solutions on R2 . Integrating (1.5) on the periodic cell Ω, we obtain that a necessary condition to the solvability of (1.5) is δ 2 < |Ω|/(4πN ). This is obviously satisfied for any finite vortex number N provided δ > 0 is sufficiently small. Our aim in this note is to show that a “shadowing-type lemma” as introduced in the context of elliptic PDEs by Angenent [5] (see also Nolasco [15]) may be adapted to
ASYMPTOTICS FOR SELFDUAL VORTICES
3
elliptic equations with singular sources in order to construct solutions for the following more general equation: (1.6) mj δpj in R2 , −Δu = δ −2 (1 − eu ) − 4π j∈P
where the set of indices P may be either finite or countable, and the vortex points pj , j ∈ N, are arbitrarily distributed in the plane with the only constraint that (1.7)
d := inf |pj − pk | > 0 k=j
and
m := sup mj < +∞. j∈P
The solution we obtain for (1.6) coincides with the solution obtained by Taubes for problem (1.3) when P is finite and with the solution obtained by Wang and Yang for problem (1.5) when P is infinite and the vortex points are periodically arranged in R2 . In fact, unlike the previous approaches, our method provides a unified analysis of (1.3) and (1.5). It should be mentioned that suitable modifications to the method described in [5] are necessary due to the singular sources appearing in (1.6). The case where P is countable and the vortex points are arbitrarily arranged in R2 does not seem to have been considered before. Of course, if P is countable, the energy of such a solution is infinite and only locally bounded. On the other hand, our “gluing” technique is, particularly, convenient and natural to analyze the asymptotics as δ → 0+ . In particular, as a by-product of our construction, we derive a rigorous proof of the following approximate product formula:
x − pj + ηδ , Φmj φ(x) = (1.8) δ j∈P
where ηδ L∞ (R2 ) ≤ Ce−c/δ with C, c > 0 independent of δ. Here (Amj , Φmj ) is the unique, up to gauge transformation, single vortex (or antivortex) solution with multiplicity mj to (1.1)–(1.2) with δ = 1 on R2 . We note that in the small δ > 0 regime, a product formula of the form (1.8) is a widely used ansatz in the physics literature, in particular in the study of the dynamics of vortices in the Ginzburg– Landau model; see, e.g., E [7], Neu [14], and Weinstein and Xin [21]. However, we have found a rigorous proof of (1.8) only for the case N = 2 on R2 in Stuart [17]. The asymptotic behavior of solutions of (1.1)–(1.2) as δ → 0+ is readily derived from formula (1.8) as well as the convergence rates. In fact, in the case of T2 , our asymptotic description improves the previous result obtained (for general compact Riemann surfaces) by Hong, Jost, and Struwe [10] (see Corollary 2.1). Although we have chosen to consider the Abelian Higgs model for the sake of simplicity, we will show in a forthcoming note that our method may be adapted to other selfdual gauge theories as considered, e.g., in the monographs [18, 22]. 2. Main results and outline of the proof. In order to state precisely our results, we denote by UN the unique radial solution for the problem (see [12]) −ΔUN = 1 − eUN − 4πN δ0 in R2 , (2.1) as |x| → +∞. UN (x) → 0 Our main result is the following theorem. Theorem 2.1. Let pj ∈ R2 , mj ∈ N, j ∈ P ⊆ N, and assume that conditions (1.7) hold. Then there exists a constant δ1 > 0 (depending on d and m only) such
4
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
that for every δ ∈ (0, δ1 ) there exists a solution uδ for (1.6). Furthermore, uδ satisfies the approximate superposition rule
|x − pj | + ωδ , (2.2) Umj uδ (x) = δ j∈P
where the error term ωδ satisfies ωδ ∞ ≤ Ce−c/δ for some C, c > 0 independent of δ. In particular, uδ satisfies the following properties: (i) 0 ≤ euδ < 1, euδ vanishes exactly at pj , j ∈ P; (ii) for every compact subset K of R2 \ ∪j∈P {pj }, there exist C, c > 0 such that + supK (1 − euδ ) ≤ Ce−c/δ as δ → 0 ; 1 uδ (iii) ±F12 = 2δ2 (1 − e ) → 2π j∈P mj δpj in the sense of distributions as δ → 0+ . In the case that P is countable, we say that the vortex points pj , j ∈ P, are doubly periodically arranged in R2 if there exists s ∈ N such that (2.3)
{pk }k∈P = {pj + me1 + ne2 : j = 1, . . . , s; m, n ∈ Z},
where e1 and e2 are the unit vectors in R2 defining the periodic cell domain Ω (for simplicity, we assume a = b = 1). Under this condition, solving (1.6) is equivalent to solving (1.5). Namely, we deal with the physically relevant case of a finite number of vortex points s p1 , . . . , ps ∈ Ω, with the corresponding multiplicity mj , j = 1, . . . , s, such that j=1 mj = N , where N is the vortex number and Ω is the periodic cell domain. As a consequence of Theorem 2.1, and proving in addition that if (2.3) is satisfied, then the solution uδ for (1.6) is in fact doubly periodic with periodic cell domain Ω, we derive the following result. Corollary 2.1. If the pj ’s are doubly periodically arranged in R2 , there exists a constant δ1 > 0 (depending on N only) such that for every δ ∈ (0, δ1 ) the solution uδ , given in Theorem 2.1, is a solution for (1.5). Furthermore, the corresponding vortex configurations (Aδ , φδ ) satisfy the approximate factorization rule
s x − pj + ηδ , Φmj x ∈ Ω, φδ (x) = δ j=1 where the error term ηδ satisfies ηδ ∞ ≤ Ce−c/δ for some C, c > 0 independent of δ, and (Amj , Φmj ) is the unique, up to gauge transformation, single vortex (or antivortex) solution with multiplicity mj , to (1.1)–(1.2) with δ = 1 on R2 . In particular, we have the following: (i) 0 ≤ |φδ |2 < 1, φδ vanishes exactly at pj , j = 1, . . . , s; (ii) for every compact subset K of Ω \ {p1 , . . . , ps }, there exist C, c > 0 such that + 0 ≤ supK (1 − |φδ |2 ) ≤ Ce−c/δ as δ → 0s ; (iii) ±F12 (Aδ , φδ ) = 2δ12 (1 − |φδ |2 ) → 2π j=1 mj δpj in the sense of distributions + (on Ω) as δ → 0 ; (iv) Ω Eδ (Aδ , φδ ) = ± Ω F12 (Aδ , φδ ) = 2πN . An outline of the proof is as follows. Our starting point in proving Theorem 2.1 is to consider δ > 0 as a scaling parameter. Setting u ˆ(x) = u(δx), we have that u ˆ satisfies −Δˆ u = 1 − euˆ − 4π (2.4) mj δpˆj in R2 , j∈P
ASYMPTOTICS FOR SELFDUAL VORTICES
5
where pˆj = pj /δ. Note that the vortex points pˆj “separate” as δ → 0+ . Section 3 contains some properties of the radial solutions UN to (2.1). We rely on the results of Taubes [19] for the existence and uniqueness of UN as well as for the exponential decay properties at infinity. We also prove a nondegeneracy property of UN . The exponential decay of solutions justifies the following approximate superposition picture for small values of δ, i.e., for vortex points pˆj which are “far apart”: u ˆ(x) ≈
Umj (|x − pˆj |) .
j∈P
In fact, we take the following preliminary form of the superposition rule: u ˆ= (2.5) ϕˆj Umj (x − pˆj ) + z j∈P
as an ansatz for u ˆδ . Here, radial solutions centered at pˆj are “glued” together by the functions ϕˆj , which belong to a suitable locally finite partition of unity. Section 4 contains the definition and the main properties of the partition as well as of ˆ δ , Yˆδ , which are also obtained by “gluing” H 2 (R2 ) the appropriate functional spaces X 2 2 and L (R ), respectively. Hence, we are reduced to show that for small values of δ > 0, there exists an exponentially small “error” z such that u ˆ defined by (2.5) ˆ δ is the aim of section 5 (see is a solution of (2.4). The existence of such a z ∈ X Proposition 5.1). To this end, we use the shadowing lemma. We characterize z ˆ δ → Yˆδ is suitably defined. The nonby the property Fδ (z) = 0, where Fδ : X degeneracy property of UN is essential in order to prove that the operator DFδ (0) is invertible, and that its inverse is bounded independently of δ > 0 (Lemma 5.3). At this point, the Banach fixed point argument applied to I − (DFδ (0))−1 Fδ yields the existence of the desired error term z. In section 6 we show that (2.5) implies (2.2) and we derive the asymptotic behavior of solutions, thus concluding the proof of Theorem 2.1. Finally, we derive Corollary 2.1 by showing that periodically arranged vortex points lead to periodic solutions. Henceforth, unless otherwise stated, we denote by C, c > 0 general constants independent of δ > 0 and of j ∈ P. 3. Single vortex point solutions. In this section, we consider the solution UN to the radially symmetric equation (2.1). For every r > 0, we denote Br = {x ∈ R2 : |x| < r}. The following lemma contains some properties of UN that will be needed in the following. The proof is a consequence of the results of Taubes [12, 19] on the existence, uniqueness, and the exponential decay of UN together with standard elliptic theory as in, e.g., [9]. Therefore, it is omitted. Lemma 3.1. The following properties hold: (i) eUN (x) < 1 for any x ∈ R2 ; (ii) for every r > 0 there exist constants CN > 0 and αN > 0, depending on r and N only, such that |1 − eUN (x) | + |∇UN (x)| + |UN (x)| ≤ CN e−αN |x| for all x ∈ R2 \ Br . We consider the bounded linear operator LN = −Δ + eUN : H 2 (R2 ) → L2 (R2 ).
6
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
In order to apply the shadowing lemma, we also need the following nondegeneracy property of UN . Lemma 3.2. The operator LN is invertible and for every N > 0 there exists CN > 0 such that L−1 N ≤ CN . Proof. It is readily seen that LN is injective. Indeed, suppose LN u = 0 for some u ∈ H 2 (R2 ). Multiplying by u and integrating on R2 , we have |∇u|2 + eUN u2 = 0. Therefore, u = 0. Now, we claim that LN is a Fredholm operator. Indeed, we write LN = (−Δ + 1)(I − T ) with T = (−Δ + 1)−1 (1 − eUN ) : H 2 (R2 ) → H 2 (R2 ). Clearly, T is continuous. Let us check that T is compact. To this end, let un ∈ H 2 (R2 ), un H 2 = 1. We have to show that T un has a convergent subsequence. Note that by the Sobolev embedding (3.1)
uL∞ (R2 ) ≤ CS uH 2 (R2 ) ,
for all u ∈ H 2 (R2 ), we have un ∞ ≤ C , for some C > 0 independent of n, and there exists u∞ , u∞ H 2 ≤ 1, such that unk → u∞ strongly in L2loc for a subsequence unk . Now, by Lemma 3.1, for any fixed ε > 0, there exists R > 0 such that 1 − eUN L2 (R2 \BR ) ≤ ε. Consequently, (1 − eUN )(unk − u∞ )L2 (R2 \BR ) ≤ 2C ε. On the other hand, (1 − eUN )(unk − u∞ )L2 (BR ) → 0. We conclude that (1 − eUN )(unk − u∞ ) → 0 in L2 . In turn, we have T (unk − u∞ ) = (−Δ + 1)−1 (1 − eUN )(unk − u∞ ) → 0 in H 2 , which implies that T is compact. It follows that LN is a Fredholm operator. Consequently, LN is also surjective. At this point, the open mapping theorem concludes the proof. 4. A partition of unity. In this section, we introduce a partition of unity and we prove some technical results which will be needed in the following. Let pj ∈ R2 (j ∈ P ⊆ N) be the vortex points. By assumption (1.7), r0 = d/8 = inf j=k |pj − pk |/8 > 0. We consider the set K = (− 34 r0 , 34 r0 ) × (− 34 r0 , 34 r0 ). Then, for any n ∈ Z2 , we introduce Kn = K + nr0 . The collection of sets {Kn }n∈Z2 is a locally finite an associated partition of unity defined as follows: let covering of R2 . We consider 0 ≤ ζ ∈ Cc∞ (K) be such that n∈Z2 ζn = 1 pointwise on R2 , where ζn (x) = ζ(x−nr0 ) for all x ∈ R2 . Then, for any j ∈ P, we introduce the set
r0 2 . Pj = n ∈ Z : d(pj , Kn ) < 4 Note that the cardinality of Pj is uniformly bounded, namely, |Pj | ≤ 4 for any j ∈ P. We set Pj = Kn , ϕj = ζn n∈Pj
n∈Pj
for any j ∈ P. Since, by choice of r0 , Z2 \ j∈P Pj is a countable set (even in the case that P is countable), there exists a bijection I : N → Z2 \ j∈P Pj . For convenience of notation, we denote by Q the countable set of indices defined by
−1 2 Z \ Q=I Pj . j∈P
7
ASYMPTOTICS FOR SELFDUAL VORTICES
We set Qk = KI(k) ,
∀k ∈ Q.
ψk = ζI(k)
Then, {Pj , Qk }(j,k)∈P×Q is a locally finite open covering of R2 with the property that Pj ∩ Pj = ∅ for every j = j. Moreover, {ϕj , ψk }(j,k)∈P×Q is a partition of unity associated with {Pj , Qk }(j,k)∈P×Q such that supp ϕj ⊂ Pj ,
supp ψk ⊂ Qk
and such that sup{∇ϕj ∞ , D2 ϕj ∞ } < +∞,
sup {∇ψk ∞ , D2 ψk ∞ } < +∞.
j∈P
In particular, 0 ≤ ϕj , ψk ≤ 1
and
k∈Q
ϕj +
j∈P
ψk =
ζn = 1.
n∈Z2
k∈Q
We define a rescaled covering Pˆj = Pj /δ,
ˆ k = Qk /δ. Q
Then, {ϕˆj , ψˆk }(j,k)∈P×Q defined by ψˆk (x) = ψk (δx)
ϕˆj (x) = ϕj (δx),
ˆ k }(j,k)∈P×Q . It will also be convenient is a partition of unity associated with {Pˆj , Q to define the sets Cˆj = {x ∈ Pˆj : ϕˆj (x) = 1},
j ∈ P.
Note that supp{∇ϕˆj , D2 ϕˆj } ⊂ Pˆj \ Cˆj and (4.1) sup
{∇ϕˆj ∞ + ∇ψˆk ∞ } ≤ Cδ,
sup
(j,k)∈P×Q
{D2 ϕˆj ∞ + D2 ψˆk ∞ } ≤ Cδ 2 .
(j,k)∈P×Q
For every fixed x ∈ R , we define the following subsets of indices: 2
K(x) = {k ∈ Q : ψˆk (x) = 0}.
J(x) = {j ∈ P : ϕˆj (x) = 0}, Note that, for every x ∈ R2 , (4.2)
|J(x)| ≤ 1,
|K(x)| ≤ 4,
where |J(x)| and |K(x)| denote the cardinality of J(x) and K(x), respectively. We shall use the following Banach spaces:
ˆk uH 2 (R2 ) < +∞ , ˆ δ = u ∈ H 2 (R2 ) : 2 (R2 ) , ψ X ϕ ˆ sup u j H loc ˆ Yδ = f ∈ L2loc (R2 ) :
(j,k)∈P×Q
sup
ϕˆj f
L2 (R2 )
ˆ 2 2 , ψk f L (R ) < +∞ .
(j,k)∈P×Q
We collect in the following lemma some estimates that will be used in what follows.
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
8
ˆ δ and j ∈ P, Lemma 4.1. There exists a constant C > 0 such that for any u ∈ X we have (i) uH 2 (Pˆj ) ≤ CuXˆ δ , (ii) uL∞ (R2 ) ≤ CuXˆ δ . Proof. (i) For every fixed j ∈ P, let J (j) = {k ∈ Q : supp ϕˆj ∩ supp ψˆk = ∅}. Then, supj∈P |J (j)| < +∞, and we estimate uH 2 (Pˆj )
=ϕˆj u + ψˆk u k∈J (j)
≤ ϕˆj uH 2 (Pˆj ) +
H 2 (Pˆj )
ψˆk uH 2 (Pˆj )
k∈J (j)
≤ (1 + |J (j)|) uXˆ δ ≤ CuXˆ δ . (ii) For any fixed x ∈ R2 , we have in view of (3.1) and (4.2) |u(x)| =
ϕˆj (x)|u(x)| +
j∈P
=
≤
ψˆk (x)|u(x)|
k∈Q
ϕˆj (x)|u(x)| +
j∈J(x)
ψˆk (x)|u(x)|
k∈K(x)
CS ϕˆj uH 2 (R2 ) +
j∈J(x)
CS ψˆk uH 2 (R2 )
k∈K(x)
≤ sup (|J(x)| + |K(x)|)CS uXˆ δ = CuXˆ δ . x∈R2
Hence, (ii) is established. We shall also need the following family of functions: ϕˆj (x) 1/2 , 2 + ˆ2 ϕ ˆ ψ k∈P k k∈Q k
gˆj (x) =
ψˆk (x) 1/2 . 2+ ˆ2 ϕ ˆ ψ j∈P j j∈Q j
ˆ k (x) = h
In view of (4.1), it is readily checked that the following lemma follows. ˆ k }(j,k)∈P×Q satisfies supp gˆj ⊂ Pˆj , supp h ˆk ⊂ Q ˆk Lemma 4.2. The family {ˆ gj , h and furthermore, ˆ 2 (x) = 1 ∀x ∈ R2 ; (i) ˆj2 (x) + k∈Q h k j∈P g ˆ k (x) ≤ C ψˆk (x) ∀x ∈ R2 ; (ii) C −1 ϕˆj (x) ≤ gˆj (x) ≤ C ϕˆj (x) and C −1 ψˆk (x) ≤ h ˆ (iii) sup(j,k)∈P×Q {∇ˆ gj ∞ + ∇hk ∞ } ≤ Cδ and sup(j,k)∈P×Q {D2 gˆj ∞ + ˆ k ∞ } ≤ Cδ 2 . D2 h 5. The shadowing lemma. Recall from the introduction that pˆj = pj /δ, j ∈ P. For every j ∈ P we define ˆj (x) = Um (x − pˆj ). U j We make the following ansatz for solutions u ˆ to (2.4): (5.1)
u ˆ=
ˆj + z. ϕˆj U
j∈P
Our aim in this section is to prove the following.
9
ASYMPTOTICS FOR SELFDUAL VORTICES
Proposition 5.1. There exists δ1 > 0 such that for all δ ∈ (0, δ1 ) there exists ˆ δ such that u ˆj + zδ is a solution of (2.4). Moreover, zδ ∈ X ˆδ defined by u ˆδ = j∈P ϕˆj U zδ Xˆ δ ≤ Ce−c/δ . ˆ δ → Yˆδ given by We note that the functional Fδ : X ˆ ˆ ˆj Fδ (z) = −Δz + ϕˆj (1 − eUj ) − (1 − e j∈P ϕˆj Uj +z ) − [ϕˆj , Δ]U j∈P
j∈P
ˆδ is well defined as well as C 1 . Here [Δ, ϕˆj ] = Δϕˆj + 2∇ϕˆj ∇. Moreover, if z ∈ X satisfies Fδ (z) = 0, then u ˆ defined by (5.1) is a solution of (2.4). Lemma 5.1. For δ > 0 sufficiently small, we have Fδ (0)Yˆδ ≤ Ce−c/δ
(5.2)
as δ → 0+
for some constants C, c > 0 independent of δ. Proof. Let ˆ ˆ R= ϕˆj (1 − eUj ) − (1 − e j∈P ϕˆj Uj ), j∈P
C=
ˆj . [ϕˆj , Δ]U
j∈P
Note that {supp R, supp C} ⊂ ∪j∈P Pˆj \ Cˆj . We fix x ∈ ∪j∈P Pˆj . We estimate ˆ
ˆ
|R(x)| ≤ sup ϕˆj (1 − eUj )L∞ (Pˆj \Cˆj ) + sup 1 − eϕˆj Uj L∞ (Pˆj \Cˆj ) j∈P
j∈P
ˆj ∞ ˆ ˆ ≤ C1 e−c1 /δ . ≤C sup U L ( Pj \C j ) j∈P
On the other hand, in view of (4.1) and Lemma 3.1, for x ∈ ∪j∈P Pˆj , we have ˆj ∞ ˆ ˆ |C(x)| ≤ sup [Δ, ϕˆj ]U L ( Pj \C j ) j∈P
ˆj Δϕˆj ∞ ˆ ˆ + sup |∇U ˆj | |∇ϕˆj | ∞ ˆ ˆ ≤ C sup U L ( Pj \C j ) L ( Pj \C j ) j∈P
≤ C2 e−c2 /δ .
j∈P
Here and above, c1 , C1 , c2 , C2 > 0 are positive constants independent of δ > 0. Hence, we conclude that, as δ → 0+ , Fδ (0)Yˆδ ≤ C sup RL2 (Pˆj ) + CL2 (Pˆj ) ≤ Ce−c/δ j∈P
for some constants C, c > 0 independent of δ > 0. ˆ δ → Yˆδ given by Now, we consider the operator Lδ ≡ DFδ (0) : X Lδ = −Δ + e
j∈P
ˆj ϕ ˆj U
For every j ∈ P, we define the operators ˆ j = −Δ + eUˆj . L
.
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
10
It will also be convenient to define ˆ 0 = −Δ + 1. L The following lemma holds. ˆ δ , we have Lemma 5.2. There exist C, c > 0 such that for any u ∈ X ˆ j )ϕˆj uL2 ≤ Ce−c/δ ϕˆj uL2 , (Lδ − L ˆ 0 )ψˆk uL2 ≤ Ce−c/δ ψˆk uL2 , (Lδ − L
j ∈ P, k ∈ Q.
Proof. For any j ∈ P, by Lemma 3.1, we have, as δ → 0+ , ˆ j )ϕˆj uL2 ≤ 1 − eUˆj ∞ ˆ ˆ + 1 − eϕˆj Uˆj ∞ ˆ ˆ ϕˆj uL2 (Lδ − L L ( Pj \C j ) L ( Pj \C j ) ≤ C1 − eUj L∞ (Pˆj \Cˆj ) ϕˆj uL2 ≤ Ce−c/δ ϕˆj uL2 . ˆ
Similarly, as δ → 0+ , ˆ 0 )ψˆk uL2 ≤(1 − e (Lδ − L
j∈P
ˆj ϕ ˆj U
)ψˆk uL2
ˆ ≤ sup 1 − eϕˆj Uj L∞ (Pˆj \Cˆj ) ψˆk uL2 ≤ Ce−c/δ ψˆk uL2 . j∈P
Now, we prove an essential nondegeneracy property of Lδ . Lemma 5.3. There exists δ0 > 0 such that for any δ ∈ (0, δ0 ), the operator Lδ is ˆ ˆ invertible. Moreover, L−1 δ : Yδ → Xδ is uniformly bounded with respect to δ ∈ (0, δ0 ). Proof. Following a gluing technique introduced in [5], we construct an “approxiˆ δ for L−1 as follows: mate inverse” Sδ : Yˆδ → X δ ˆkL ˆk, ˆ −1 gˆj + ˆ −1 h Sδ = gˆj L h 0 j j∈P
k∈Q
ˆ k are the functions introduced in section 4. We claim that the operator where gˆj and h Sδ is well defined and uniformly bounded with respect to δ. That is, we claim that Sδ f Xˆ δ ≤ Cf Yˆδ
(5.3)
ˆ δ and of δ > 0. for some C > 0 independent of f ∈ X Indeed, for any f ∈ Yˆδ , we have Sδ f Xˆ δ = and ϕˆj Sδ f
H2
sup
{ ϕˆj Sδ f H 2 , ψˆk Sδ f H 2 }
(j,k)∈P×Q
−1 −1 ˆ ˆ ˆ ˆ 2 ≤ ϕˆj gˆj Lj gˆj f H + ϕˆj hk L0 hk f
−1 ˆ ˆ ˆ 2 ψk Sδ f H ≤ ψk gˆj Lj gˆj f j∈P
H2
,
H2
k∈Q
−1 ˆ ˆ ˆ ˆ + ψk hj L0 hj f j∈Q
H2
.
11
ASYMPTOTICS FOR SELFDUAL VORTICES
We estimate, recalling the properties of ϕˆj and gˆj as in Lemma 4.2, and in view of Lemma 3.2 ˆ −1 gˆj f H 2 ≤ Cˆ ˆ −1 gˆj f H 2 ≤ CL gj f L2 ≤ Cϕˆj f L2 ≤ Cf Yˆδ . ϕˆj gˆj L j j We have −1 ˆ ˆ ˆ ϕˆj hk L0 hk f
H2
k∈Q
−1 ˆ ˆ ˆ ≤ ϕˆj hk L0 hk f
H2
k∈J (j)
≤
ˆkL ˆ k f H 2 , ˆ −1 h ϕˆj h 0
k∈J (j)
where J (j) = {k ∈ Q : supp ψˆk ∩ supp ϕˆj = ∅} satisfies supj∈P |J (j)| < +∞. In view of Lemmas 4.2 and 3.2, we estimate ˆ k f L2 ˆkL ˆ k f H 2 ≤ C ˆ k f H 2 ≤ C ˆ −1 h ˆ −1 h ϕˆj h L h 0 0 k∈J (j)
≤C
k∈J (j)
k∈Q
k∈J (j)
Therefore,
−1 ˆ ˆ ˆ sup ϕ ˆ f h h L k 0 k j
j∈P
Similarly, we obtain that −1 ˆ ˆ sup ψk gˆj Lj gˆj f k∈Q
k∈J (j)
ψˆk f L2 ≤ C|J (j)| sup ψˆk f L2 ≤ Cf Yˆδ .
H2
j∈P
H2
k∈Q
≤ Cf Yˆδ ,
≤ Cf Yˆδ .
−1 ˆ ˆ ˆ ˆ sup ψk hj L0 hj f
k∈Q
j∈Q
H2
≤ Cf Yˆδ ,
and (5.3) follows. Now, we claim that there exists δ0 such that for any δ ∈ (0, δ0 ), the operator Sδ Lδ : ˆδ → X ˆ δ is invertible, and furthermore, Sδ Lδ ≤ C for some C > 0 independent X ˆk : X ˆ j )ˆ ˆ δ → Yˆδ and (Lδ − L ˆ 0 )h ˆ δ → Yˆδ are of δ > 0. We note that (Lδ − L gj : X well-defined bounded linear operators. Thus, we decompose ˆ k Lδ − L ˆkL ˆk) ˆ j gˆj ) + ˆ0h ˆ −1 (ˆ ˆ −1 (h Sδ Lδ = IXˆ δ + gˆj L gj Lδ − L h 0 j j∈P
(5.4)
= IXˆ δ +
k∈Q
ˆ −1 (Lδ gˆj L j
ˆ j )ˆ −L gj +
j∈P
+
ˆ −1 [Δ, gˆj ] gˆj L j
+
j∈P
ˆk ˆkL ˆ 0 )h ˆ −1 (Lδ − L h 0
k∈Q
ˆ k ]. ˆkL ˆ −1 [Δ, h h 0
k∈Q
Hence, it suffices to prove that the last four terms in (5.4) are sufficiently small, in the operator norm, provided δ > 0 is sufficiently small. By Lemmas 5.2 and 4.2, we ˆδ , have, for any u ∈ X −1 ˆ ˆ −1 (Lδ − L ˆ j )ˆ ˆ ≤ C sup L g ˆ (L − L )ˆ g u gj uH 2 L j j δ j j j j∈P
ˆδ X
j∈P
ˆ j )ˆ ≤ C sup (Lδ − L gj uL2 j∈P
≤ Ce−c/δ sup ϕˆj uL2 ≤ Ce−c/δ uXˆ δ . j∈P
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
12
ˆ δ , we have Similarly, for u ∈ X −1 ˆ −1 [Δ, gˆj ]uH 2 ≤ C sup [Δ, gˆj ]uL2 . ˆ gˆj Lj [Δ, gˆj ]u sup L j ˆ ≤ C j∈P j∈P Xδ
j∈P
Recalling that [Δ, gˆj ]u = 2∇u∇ˆ gj + uΔˆ gj , by Lemmas 4.2 and 4.1(i) we derive that [Δ, gˆj ]uL2 ≤ CδuH 1 (Pˆj ) ≤ CδuXˆ δ . The remaining terms are estimated similarly. Hence, Sδ Lδ − IXˆ δ → 0 as δ → 0+ . −1 Now, we observe that L−1 Sδ . It follows that for any f ∈ Yˆδ , we have δ = (Sδ Lδ ) −1 L−1 Sδ f Xˆ δ ≤ CSδ f Yˆδ ≤ Cf Yˆδ ˆ δ = (Sδ Lδ ) δ f X
with C > 0 independent of δ. Hence, Lδ is invertible and its inverse is bounded independently of δ, as asserted. Now we can provide the following proof. Proof of Proposition 5.1. We use the Banach fixed point argument. For any δ ∈ (0, δ0 ), with δ0 > 0 given by Lemma 5.3, we introduce the nonlinear map Gδ ∈ ˆδ , X ˆ δ ) defined by C 1 (X Gδ (z) = z − L−1 δ Fδ (z). Then, fixed points of Gδ correspond to solutions of the functional equation Fδ (z) = 0. First, note that DGδ (0) = 0 and that DF (z) = −Δ + e
j∈P
ˆj +z ϕ ˆj U
.
ˆ δ and u ∈ X ˆ δ , we have By Lemma 5.3, for any z ∈ X DGδ (z)uXˆ δ = (DGδ (z) − DGδ (0))uXˆ δ = L−1 ˆδ δ (DFδ (z) − Lδ )uX ≤ C(DFδ (z) − Lδ )uYˆδ = Ce
j∈P
ˆj ϕ ˆj U
(ez − 1)uYˆδ ≤ C(ez − 1)uYˆδ .
By the elementary inequality et − 1 ≤ Ctet , for all t > 0, where C > 0 does not depend on t, and in view of Lemma 4.1, we have ez − 1∞ ≤ ez∞ − 1 ≤ Cz∞ ez∞ ≤ CzXˆ δ e
zX ˆ
δ
.
Hence, DGδ (z)uXˆ δ ≤ C(ez − 1)uYˆδ ≤ CzXˆ δ e
zX ˆ
δ
uYˆδ ≤ CzXˆ δ e
zX ˆ
δ
Consequently, there exists R0 > 0 such that for every R ∈ (0, R0 ), we have DGδ (z)
0, where ˆ δ : u ˆ < R}. BR = {u ∈ X Xδ
uXˆ δ .
13
ASYMPTOTICS FOR SELFDUAL VORTICES
Now, for every R ∈ (0, R0 ), Gδ (z)Xˆ δ ≤ Gδ (z) − Gδ (0)Xˆ δ + Gδ (0)Xˆ δ 1 ≤ zXˆ δ + L−1 ˆδ . δ Fδ (0)X 2 By Lemmas 5.3 and 5.1, there exist C0 , c0 > 0 independent of δ > 0 such that −c0 /δ . L−1 ˆ δ ≤ CFδ (0)Yˆδ ≤ C0 e δ Fδ (0)X
Choosing R = Rδ = 2C0 e−c0 /δ , we obtain that Gδ (BRδ ) ⊂ BRδ . Hence, Gδ is a strict contraction in BRδ , for any δ ∈ (0, δ1 ), with δ1 = c0 /(ln(2C0 /R0 )). By the Banach fixed-point theorem, for any δ ∈ (0, δ1 ), there exists a unique zδ ∈ BRδ such that Fδ (zδ ) = 0. 6. Proof of the main results. In this section, we finally provide the proof of Theorem 2.1 and derive Corollary 2.1. In view of Proposition 5.1, the function u ˆδ defined by ˆ j + zδ u ˆδ = ϕˆj U j∈P
is a solution of (2.4). Consequently, uδ defined by
x x x − pj (6.1) = + zδ ˆδ ϕj (x)Umj uδ (x) = u δ δ δ j∈P
is a solution of (1.6). Lemma 6.1. The solution uδ defined in (6.1) satisfies the approximate superposition rule
x − pj + ωδ (x) Umj uδ (x) = δ j∈P
with ωδ ∞ ≤ Ce−c/δ . Proof. In view of (6.1) and of the definition of J(x) in section 4, we have
x − pj +ω ˜ δ (x), uδ (x) = Umj δ j∈J(x)
where ω ˜ δ (x) = −
(1 − ϕj (x))Umj
j∈J(x)
In view of Lemma 3.1, we estimate
x − p j (1 − ϕj (x))Umj δ j∈J(x)
∞
≤
x − pj δ
j∈J(x)
+ zδ
x δ
.
x − pj −c/δ sup Umj . ≤ Ce δ 2
R \Cj
14
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
On the other hand, by Proposition 5.1 and Lemma 4.1(ii), we have zδ · = zδ ∞ ≤ Ce−c/δ . δ ∞ Therefore, ˜ ωδ ∞ ≤ Ce−c/δ . We have to show that
x − pj ≤ Ce−c/δ . Umj δ j∈P\J(x) ∞
To this end, we fix x ∈ R2 and for every M ∈ N we define BM = {y ∈ R2 : |y − x| < dM }. Then,
x − pj x − pj = . Umj Umj δ δ j∈P\J(x)
M ∈N pj ∈BM +1 \BM
Since inf j=k |pj −pk | = d > 0, there exists C > 0 independent of M ∈ N and of x ∈ R2 such that |{pj ∈ BM +1 \ BM }| ≤ CM. Hence, we estimate
x − p j ≤C U M e−cM/δ ≤ Ce−c/δ . m j δ j∈P\J(x) M ∈N This implies the statement of the lemma. We are left to analyze the asymptotic behavior of uδ as δ → 0+ . Such behavior is a direct consequence of (6.1). Lemma 6.2. Let uδ be given by (6.1). The following properties hold: (i) euδ < 1 on R2 and vanishes exactly at pj with multiplicity 2mj , j ∈ P; (ii) for every compact subset K of R2 \ ∪j∈P {pj }, there exist C, c > 0 such that + 1 − euδ ≤ Ce−c/δ as δ → 0 ; −2 uδ (iii) δ (1 − e ) → 4π j∈P mj δpj in the sense of distributions as δ → 0+ . Proof. (i) Since uδ is a solution of (1.6), euδ < 1 follows by the maximum principle. Moreover, since
x − pj = ln |x − pj |2mj + vj Umj (6.2) δ with vj a continuous function (see [12]), we have near pj that euδ = |x − pj |2mj fj,δ (x) with fj,δ (x) a continuous strictly positive function. Hence, (i) is established. (ii) Let K be a compact subset of R2 \ ∪j∈P {pj }. In view of Lemma 3.1 and Proposition 5.1, we have as δ → 0+ · sup 1 − eϕj (x)Umj ((x−pj )/δ) ≤ Ce−c/δ zδ δ ∞ x∈K∩Pj ≤ Czδ Xˆ δ ≤ CRδ ≤ Ce−c/δ .
ASYMPTOTICS FOR SELFDUAL VORTICES
15
Therefore, we have that for any compact set K ⊂ R2 \ ∪j∈P {pj }, 0 ≤ sup (1 − euδ ) ≤ C sup sup (1 − euδ ) ≤ Ce−c/δ . x∈K
j∈P x∈K∩Pj
(iii) Let ϕ ∈ Cc∞ (R2 ). Then, − uδ Δϕ = δ −2 (1 − eu )ϕ − 4π mj ϕ(pj ). R2
We claim that
R2
j∈P
(6.3) R2
uδ Δϕ → 0
as δ → 0.
Indeed, let jk ∈ P, k = 1, . . . , n, be such that supp ϕ ⊂ ∪nk=1 Pjk ∪ K with K a compact subset of R2 \ ∪j∈P {pj }. Since supK |uδ | ≤ Ce−c/δ , we have uδ Δϕ ≤ CΔϕ∞ e−c/δ → 0. K
On the other hand, in view of Lemma 6.1, in Pjk we have uδ (x) = Umjk (|x − pjk |/δ) + O(e−c/δ ). Note that Umjk ∈ L1 (R2 ) in view of (6.2) and Lemma 3.1. Therefore,
x − pjk sup Δϕ + O(e−c/δ ) uδ Δϕ ≤ sup Umjk δ 1≤k≤n 1≤k≤n Pj Pj k
k
≤ δ 2 sup Δϕ∞ Umjk L1 + O(e−c/δ ) ≤ Cδ 2 → 0. 1≤k≤n
Hence, (6.3) follows, and (iii) is established. Proof of Theorem 2.1. For every δ ∈ (0, δ1 ), where δ1 is given in Proposition 5.1, we obtain a solution uδ of (1.6). Furthermore, uδ satisfies (2.2) in view of Lemma 6.1. Finally, uδ satisfies the asymptotic behavior as in (i)–(iii) in view of Lemma 6.2. Hence, Theorem 2.1 is completely established. Proof of Corollary 2.1. To begin, we want to prove that if pj ’s are doubly periodically arranged in R2 , then uδ is in fact a doubly periodic solution of (1.5). Recall that the pj ’s are doubly periodically arranged in R2 if (2.3) holds. We define ˆek = ek /δ, k = 1, 2. Equivalently, we show u ˆδ (x + ˆek ) = u ˆδ (x) for any x ∈ R2 and for k = 1, 2. Indeed, we may assume that ϕˆj (x + ˆek ) = ϕˆj (x), ψˆj (x + ˆek ) = ψˆj (x) for any j ∈ N, x ∈ R2 , k = 1, 2. Then, ˆj (x) + zδ (x + ˆe ). ϕˆj (x)U u ˆδ (x + ˆek ) = k j∈N
Hence, it is sufficient to prove that zδ (x + ˆek ) = zδ (x) for every x ∈ R2 and for k = 1, 2. First, we claim that zδ ( · + ˆek ) ∈ BRδ . Indeed, for every j ∈ N there exists exactly one j ∈ N such that (6.4)
ϕˆj zδ ( · + ˆek )H 2 = ϕˆj zδ H 2 .
Hence, we obtain (6.5)
zδ ( · + ˆek )Xˆ δ = zδ Xˆ δ ≤ Rδ .
16
M. MACR`I, M. NOLASCO, AND T. RICCIARDI
Moreover, if Fδ (zδ ) = 0 we also have Fδ (zδ ( · + ˆek )) = 0. Therefore, zδ ( · + ˆek ) is a fixed point of Gδ in BRδ . By uniqueness, we conclude that zδ ( · +ˆek ) = zδ , k = 1, 2, as asserted. At this point, the remaining statements follow recalling that in the periodic cell domain Ω, (Aδ , φδ ) is given, up to gauge transformations, by (1.4). REFERENCES [1] A.A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP, 5 (1957), pp. 1174–1182. [2] A. Aftalion, E. Sandier, and S. Serfaty, Pinning phenomena in the Ginzburg-Landau model of superconductivity, J. Math. Pures Appl., 80 (2001), pp. 339–372. [3] S. Alama and L. Bronsard, Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains, Comm. Pure Appl. Math., to appear. ´, P. Bauman, and D. Phillips, Vortex pinning with bounded fields for the Ginzburg[4] N. Andre Landau equation, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 20 (2003), pp. 705–729. [5] S. Angenent, The shadowing lemma for elliptic PDE, in Dynamics of Infinite-Dimensional Systems, NATO Adv. Sci. Inst. Ser. F Comput. Syst. Sci. 37, Springer, Berlin, 1987, pp. 7–22. ´lein, Ginzburg-Landau Vortices, Birkh¨ [6] F. Bethuel, H. Brezis, and F. He auser, Boston, MA, 1994. [7] W. E, Dynamics of vortices in the Ginzburg-Landau theories with applications to superconductivity, Phys. D, 77 (1994), pp. 383–404. [8] O. Garc´ia-Prada, A direct existence proof for the vortex equations over a compact Riemann surface, Bull. London Math. Soc., 26 (1994), pp. 88–96. [9] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, in Classics in Mathematics, Springer, Berlin, 2001. [10] M.Y. Hong, J. Jost, and M. Struwe, Asymptotic limits of a Ginzburg Landau type functional, in Geometric Analysis and Calculus of Variations, International Press, Cambridge, MA, 1996, pp. 99–123. [11] G. ’t Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nuclear Phys. B, 153 (1979), pp. 141–160. [12] A. Jaffe and C. Taubes, Vortices and Monopoles, Birkh¨ auser, Boston, MA, 1980. [13] F.H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), pp. 323–442. [14] J.C. Neu, Vortices in the complex scalar field, Phys. D, 43 (1990), pp. 385–406. [15] M. Nolasco, Non topological n-vortex condensates for the selfdual Chern-Simons theory, Comm. Pure Appl. Math., 56 (2003), pp. 1752–1780. [16] J. Rubinstein and P. Sternberg, On the slow motion of vortices in the Ginzburg-Landau heat flow, SIAM J. Math. Anal., 26 (1995), pp. 1452–1466. [17] D. Stuart, Dynamics of Abelian Higgs vortices in the near Bogomolny regime, Comm. Math. Phys., 159 (1994), pp. 51–91. [18] G. Tarantello, The Analysis of Selfdual Gauge Field Vortices, Progr. Nonlinear Differential Equations Appl., Birkh¨ auser, Boston, MA, in preparation. [19] C. Taubes, Arbitrary n-vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys., 72 (1980), pp. 277–292. [20] S. Wang and Y. Yang, Abrikosov’s vortices in the critical coupling, SIAM J. Math. Anal., 23 (1992), pp. 1125–1140. [21] M.I. Weinstein and J. Xin, Dynamic stability of vortex solutions of Ginzburg-Landau and nonlinear Schr¨ odinger equations, Comm. Math. Phys., 180 (1996), pp. 389–428. [22] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monogr. Math., Springer, New York, 2001.
