IOP PUBLISHING
NONLINEARITY
Nonlinearity 20 (2007) 1193–1213
doi:10.1088/0951-7715/20/5/008
Generalized Bernstein property and gravitational strings in Born–Infeld theory Lesley Sibner1 , Robert Sibner2 and Yisong Yang1 1
Department of Mathematics, Polytechnic University, Brooklyn, New York 11201, USA Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, New York 11210, USA
2
Received 22 February 2006, in final form 8 March 2007 Published 17 April 2007 Online at stacks.iop.org/Non/20/1193 Recommended by A Chenciner Abstract Motivated by Born–Infeld geometric electromagnetic theory, we consider a series of nonlinear equations which extend the minimal surface equations and the related, generalized, Bernstein problems. We study the relation of these equations and the conditions which lead to the triviality of the solutions. We also study a non-Abelian extension of these equations and establish a gap theorem for the Yang–Mills–Born–Infeld fields. We then couple the Born–Infeld electromagnetism with a Higgs scalar field and obtain an existence theorem for the self-dual multiple cosmic string solutions on a closed surface characterized jointly by the first Chern class and the Thom class formulated over the hosting complex line bundle. Mathematics Subject Classification: 58, 35, 81E
1. Introduction A minimal hypersurface in R n+1 , given as the graph of the function x n+1 = u(x 1 , . . . , x n ), satisfies the differential equation ∇u = 0. ∇· 1 + |∇u|2
(1.1)
It was Bernstein [2] who showed that the solutions of (1.1) are all affine linear functions when n = 2. In other words, nonparametric minimal surfaces in R3 are simply planes. In [10], Calabi considered the maximal space-like hypersurfaces in the Minkowskispace Rn,1 with the coordinates (x 1 , . . . , x n , t) and the fundamental form ds 2 = dt 2 − nj=1 (dx j )2 . If a 0951-7715/07/051193+21$30.00 © 2007 IOP Publishing Ltd and London Mathematical Society Printed in the UK
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maximal space-like hypersurface is given as the graph of the function t = v(x 1 , · · · , x n ), then the function v satisfies the equation ∇v ∇· = 0. (1.2) 1 − |∇v|2 Calabi showed that equations (1.1) and (1.2) are equivalent over any simply connected domain in R2 . As a consequence, the classical Bernstein theorem for (1.1) implies that the solutions of (1.2) in R2 are all affine linear as well. Calabi further remarked that, for n 3, there exists no known relation between the solutions of (1.1) and (1.2). In fact, regarding whether the Bernstein property holds, i.e. whether any solution of the equation over the full Rn is affine linear, (1.1) and (1.2) give different answers in general. In the case of (1.1), the dimensionality n plays an important role. It is known through the work of Federer, Fleming, de Giorgi, Almgren and Simons and de Giorgi, Giusti and Bombieri that the Bernstein property only holds in lower dimensions, n 7, and fails in all higher dimensions, n 8. In the case of (1.2), inspired by the work of Calabi in two dimensions, Cheng and Yau [12] proved that the Bernstein property holds in all dimensions. Hence, it is seen from the Bernstein property that (1.1) and (1.2) indeed behave rather differently. In fact, it is not hard to show [46, 47] along the lines of Calabi [10] that, for n = 3, equations (1.1) and (1.2) over a simply connected domain are, respectively, equivalent instead to the vector equations ∇ ×A ∇× = 0, (1.3) 1 ∓ |∇ × A|2 which arise in the nonlinear electromagnetic theory of Born and Infeld [4–7, 25]. This observation leads to a generalized treatment [46] of the equations of the types (1.1) and (1.2) expressed in terms of differential forms and a reformulation of Calabi’s equivalence theorem in arbitrary n dimensions. Note that Born–Infeld theory is of contemporary interest due to its relevance in string theory [1, 8, 11, 15, 17, 20, 21, 24, 29, 33, 36, 37]. Our study here however is mainly inspired by its rich geometric structure [18]. In section 2, we write down the extended Born–Infeld equations in terms of differential forms on an n-dimensional manifold M and establish a general equivalence theorem which says that the obstruction to equivalence lies in the de Rham cohomology. In section 3, we prove a Bernstein type theorem by showing the triviality of a solution over Rn under some integrability conditions motivated by a finite-energy requirement. The classical Born–Infeld action is for the gradient of a real-valued function ϕ or, equivalently, as the action on a scalar valued 1-form ω = dϕ. Formally replacing ω by the field strength F (the electromagnetic field) of a gauge potential 1-form A (the electromagnetic potential), one obtains (for the Minkowski spacetime) the Born–Infeld theory of electromagnetism (in vacua) for the electric and magnetic fields. See [46] for a further discussion and references. In this setup, the gauge group is U (1), which is Abelian, but as suggested by Otway [34], one can study other gauge groups, for example, the group SU (n) of Yang–Mills theory (see also [35] for an earlier work of Otway along these lines). In section 4, we consider these non-Abelian gauge fields over R4 governed by a Born– Infeld action functional and show that there do not exist nontrivial self-dual solutions. We also establish a gap theorem for the Yang–Mills–Born–Infeld fields over S 4 which states that there is an L2 neighbourhood of the origin in which there do not exist nontrivial solutions. In section 5, we consider coupled systems of Born–Infeld type equations suggested by the mixed electrostatic and magnetostatic interactions and we establish an equivalence theorem for these coupled systems. In particular, we extend Calabi’s equivalence theorem for scalar equations over R2 to a coupled system situation. We also prove a triviality result over R2 under some
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sufficiency conditions. In section 6, we consider the Born–Infeld magnetism generated from a Higgs field over a compact surface whose metric is to be determined by equalizing the Gauss curvature with the energy density function. Such an equation arises from the existence problem of self-dual cosmic strings [13, 26–28, 38, 45–47] as a consequence of a suitable symmetry reduction from the full Einstein gravitational equations. In order to recognize the associated topological invariants, we study the problem in the framework of a complex line bundle for which the magnetic field is the curvature induced from a connection. We will obtain an existence theorem for a prescribed distribution of cosmic strings and antistrings topologically characterized by the first Chern class and Thom class [43]. As a direct implication, we show that the solutions obtained may be used to generate solutions over R2 . In section 7, we prove this existence theorem. 2. Born–Infeld equations in differential forms Let (M, g) be a Riemannian manifold and use p (M) to denote the vector bundle of p-forms on M. We use ∗ to denote the Hodge dual induced from the metric g. Then ∗ : p (M) → n−p (M)
(2.1)
satisfies ∗ ∗ = (−1)p(n−p) ,
∗1 = dVg ≡
and the natural inner product, ·, ·, on each fibre is related to ∗ by the well-known equation σ ∧ (∗τ ) = σ, τ ∗ 1,
|g|dx,
p x (M)
(2.2)
(x ∈ M) induced from the metric g
σ, τ ∈ px (M).
(2.3)
In (2.2), dVg is the canonical volume element of the Riemannian manifold (M, g). From (2.3) p n−p it is easily seen that ∗ : x (M) → x (M) is an isometry: σ, τ = ∗σ, ∗τ ,
σ, τ ∈ px (M).
(2.4)
Thus, using |σ |2 = σ, σ to denote the induced pointwise squared norm of a differential form σ , we can rewrite equation (1.1) in the form dω d∗ = 0, ω ∈ p (M). (2.5) 2 1 + |dω| Of course, the minimal hypersurface equation (1.1) or the Born–Infeld equation (1.3) (with the plus sign) is recovered from (2.5) when p = 0 and M = Rn or p = 1 and M = R3 . Likewise, the space-like maximal hypersurface equation (1.2) or the Born–Infeld equation (1.3) (with the minus sign) is a special case of the equation dσ d∗ = 0, σ ∈ q (M), (2.6) 1 − |dσ |2 for q = 0 and M = Rn or q = 1 and M = R3 . We now establish the following generalized equivalence theorem. Theorem 2.1.. Equations (2.5) with 0 p n − 2 and (2.6) with q = n − p − 2 are equivalent provided that both the (n − p − 1)st and the (p + 1)st de Rham cohomologies of the manifold M vanish. In this case, the following relations hold: dω = ±
∗dσ 1−
|dσ |2
,
dσ = ±
∗dω 1 + |dω|2
.
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Proof. Assume that the stated condition on the values of p and q holds. Let ω be a solution of (2.5). Then dω (2.7) τ =±∗ 1 + |dω|2 is a closed (n − p − 1)-form. Since the (n − p − 1)st de Rham cohomology of M vanishes, τ must be exact. That is, there is an (n − p − 2)-form σ such that τ = dσ or dω dσ = ± ∗ . (2.8) 1 + |dω|2 Since ∗ is an isometry, we see from (2.8) that dσ is space-like, |dσ |2 < 1 and |dσ |2 . 1 − |dσ |2 Using (2.2), (2.8) and (2.9), we have dσ . dω = ±(−1)p(n−p)+n−1 ∗ dσ 1 + |dω|2 = ±(−1)p(n−p)+n−1 ∗ 1 − |dσ |2 |dω|2 =
(2.9)
(2.10)
Applying the Bianchi identity d2 ω = 0 in (2.10), we arrive at equation (2.6) with q = n−p−2. Conversely, we can follow the same path to show that (2.6) implies (2.5) with p = n−q −2 by using the condition that the (n−q −1)st or the (p+1)st de Rham cohomology of M vanishes. Hence the theorem follows. Remarks. (a) If M is compact, the (n − p − 1)st and the (p + 1)st de Rham cohomology groups are in fact identical as a consequence of the Poincar´e duality and the two topological conditions stated in theorem 2.1 become a single condition. (b) We would like to explain, in more detail, how the conditions of theorem 2.1 arise. First of all, equations (2.5) and (2.6) appear in the work of the first two authors [41, 42] where the ‘density functions’ (1+s 2 )−1/2 and (1−s 2 )−1/2 are special cases of a more general density function ρ(s) which is required to satisfy conditions ensuring positivity and ellipticity of the equations (the density functions given above satisfy these conditions) for |s| < ∞ and |s| < 1, respectively. More precisely, if the equation considered is dν = 0,
d ∗ ρ(ν)ν = 0,
where ρ(ν) = (1 + |ν|2 )−1/2 ,
(2.11)
d ∗ µ(τ )τ = 0,
where µ(τ ) = (1 − |τ |2 )−1/2 ,
(2.12)
then the equation dτ = 0,
is a ‘dual’ (conjugate) problem for a form τ whose degree is dim(M) − deg(ν). More precisely, τ = ∗ρ(ν)ν and the two problems are seen to be equivalent. Now, writing ωp for a p-form solution of (2.5) and setting ν p+1 = dωp , one sees that ν p+1 is a solution of (2.11). But, as we have just observed, (2.11) is equivalent to the dual problem (2.12) for a closed (n − (p + 1))-form τ n−p−1 . If the (n − p − 1) cohomology vanishes, then τ n−p−1 is in fact exact and we can write τ n−p−1 = dσ n−p−2 so that σ n−p−2 is a solution of (2.6). By the same reasoning, a solution of (2.6) gives a solution of (2.5) if the (p + 1)st cohomology vanishes. From this point of view, one sees that Calabi’s particular equivalence result [10] for 0forms φ and ψ satisfying d ∗ ρdφ = 0 and d ∗ µdψ = 0, respectively, is a consequence of the star (∗) of a 1-form in dimension n = 2 being again a 1-form and the cohomology of the simply connected domain R2 vanishing.
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3. A Bernstein property It will be useful to identify and establish various Bernstein type properties for the generalized equations (2.5) and (2.6). Naturally, the triviality property dω = 0 or dσ = 0 could be one of the simplest possible such properties. In this section, we show that certain conditions are sufficient to conclude that the p-form solutions of (2.5) and (2.6) on Rn are trivial. Note that in the original Bernstein problem in which the minimal surface is given by the graph of a scalar function u(x 1 , . . . , x n ) over Rn , the property that the surface is a plane [2] is equivalent to the property that ∇u is constant (and vanishes if it is required to be in Lp ). Although the classical Bernstein property is only valid for n 7, a result of Moser [31] states that the property remains valid for all n under the additional assumption that ∇u be bounded. Recall that we can define a global inner product, (·, ·), on the vector space p (M) by (σ, τ ) = σ ∧ (∗τ ) = σ, τ ∗ 1, σ, τ ∈ p (M). M
M
∗
As usual, let d be the co-differential operator, d∗ : p (M) → p−1 (M),
d∗ = (−1)n(p+1)+1 ∗ d ∗ .
Then the following Green’s formula holds: (dσ, τ ) = (σ, d∗ τ ) + boundary terms for σ ∈ p−1 (M), τ ∈ p (M). √ Setting ρ(t) = 1/ 1 + t and γ = dω, we rewrite (2.5) as d∗ (ρ(|γ |2 )γ ) = 0.
(3.1)
(3.2)
Having done this, it is now clear that the natural problem is to look for a closed (not necessarily exact) form γ satisfying (3.2). Note that (3.2) is the Euler–Lagrange equation of the action functional |γ |2 1 E(γ ) = ( 1 + |γ |2 − 1) dVg = ρ(t) dt dVg . (3.3) 0 M M 2 Setting ϕ(ε) = E(γ + ε dτ ), we find that ϕ (0) = (ρ(|γ |2 )γ , dτ ) = (d∗ ρ(|γ |2 )γ , τ ), where τ is a test (p − 1)-form with compact support. Various existence and regularity questions for the equations of the type (3.2), on compact manifolds and manifolds with boundary, have been studied by the first two authors [39,41,42]. The particular ‘density function’ ρ taken above is the minimal surface density. The results in this and the next few sections also hold for these more general density functions. Remark. On a compact Riemannian manifold, it is an easy exercise, using Green’s theorem, to show that if a p-form γ is a solution of (3.2), then (i) ργ is orthogonal to the space of exact p-forms and (ii) if γ is exact then it vanishes identically. To see (ii), one should observe that ρ must be positive almost everywhere. Then, writing γ = dν, one has 0 = (d∗ ργ , ν) = (ργ , dν) = (ργ , γ ) = ρ|γ |2 dVg , which gives γ ≡ 0. Note also that (ii) implies that a solution ω of (2.5) (d∗ ρ dω = 0) is automatically closed. We now consider solutions on the simplest noncompact manifolds, Rn , n 3. To that end, choose a sequence of cutoff functions {fk } such that fk → 1
and ∇fk n → 0
as k → ∞.
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(Such a sequence is constructed in, for example, [19, 40].) If ω is a p-form with ∇ω ∈ L2 (Rn ), then ωk = fk ω has compact support and ∇ωk → ∇ω in L2 (Rn ) as k → ∞. This follows from the inequality
∇ωk − ∇ω 2 (fk − 1)∇ω 2 + (∇fk )ω 2 (fk − 1)∇ω 2 + ∇fk n ω 2n/(n−2) (fk − 1)∇ω 2 + C(n) ∇fk n ∇ω 2 , (3.4) where the right-hand side tends to zero as k → ∞. We will also use the fact that if γ is a closed form, γ ∈ L2 (Rn ), we can find ω such that γ = dω with d∗ ω = 0. This follows from the (orthogonal) Hodge decomposition γ = dd∗ + d∗ d where the potential ∈ W 1,2 (Rn ). Since γ is closed, ω = d∗ has the right behaviour (see Morrey [30] for more details). Theorem 3.1.. If γ ∈ L2 (Rn ) (n 3) is closed (and hence exact) and satisfies (3.2), then γ ≡ 0. Proof. Let γ = dω with d∗ ω = 0. Then ∇ω 22 ≡ dω 22 + d∗ ω 22 = γ 22 < ∞. As shown above, ωk = fk ω has compact support and ∇ωk → ∇ω in L2 (Rn ). Therefore, 0 = (d∗ ργ , ωk ) = (ργ , dωk ) → (ργ , dω) = (ργ , γ ), which implies γ ≡ 0. When n = 2, the above argument fails and a stronger hypothesis is needed. We assume now that, in addition, γ is in the Hardy space H1 . To define the norm on this space, we take a peaked kernel with compact support such that Rn = 1. For t > 0, set t = t −n (x/t) and for γ ∈ L1 (Rn ), let γˆ (x) = sup |( t ∗ γ )(x)|, t>0
where ∗ denotes the convolution product. Then, by definition,
γ H1 = γ 1 + γˆ 1 . The classical theory of Calderon–Zygmund [22] implies that if − = γ , then
W 1,2 C γ H1 . We have, using this result, the following theorem. Theorem 3.2.. If γ ∈ L2 ∩ H1 on R2 , γ is closed, and satisfies equation (3.2), then γ ≡ 0. Proof. As in the previous theorem, γ = dd∗ , where, by the previous remark, W 2,1 C γ H1 . Taking ω = d∗ , one has d∗ ω = 0, dω = γ . By Sobolev’s inequality,
ω 2 = ω n/(n−1) K ∇ω 1 K ∇d∗ 1 K1 γ H1 < ∞. As before, ∇ω 2 = γ 2 < ∞. With the same sequence of cutoff functions {fk }, let ωk = fk ω. Then 0 = (d∗ ργ , ωk ) = (ργ , dωk ) = (ργ , fk dω) + (ργ , ω dfk ). Since |ργ | < 1, we have |(ργ , ω dfk )|
|∇fk | |ω| ∇fk 2 ω 2 ∇fk 2 (K1 γ H1 ).
Thus, we find in the limit, as k → ∞, 0 = (ργ , dω) = (ργ , γ ) and hence γ ≡ 0.
To conclude this section, we remark √ on the condition, in theorems 3.1 and 3.2, that γ ∈ L2 (Rn ). In view of the inequality 1 + s 2 − 1 s 2 /2, it is clear that being in L2 ensures
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finite-energy E(γ ). On the other hand, only with the additional assumption that γ ∈ L∞ does the reverse inequality hold (with a constant depending on the bound). It is an open question whether the conclusion that γ vanishes, for finite-energy solutions, can be obtained without this additional assumption. 4. Yang–Mills–Born–Infeld fields We now consider a variational problem which is a generalization of the (scalar valued) Born– Infeld model and at the same time a quasilinear generalization of the Yang–Mills equations. The geometric framework is familiar from Yang–Mills theory. Let G be a general compact matrix Lie group such as SU (n). The Lie algebra of G is denoted by G . If A, B are G -valued p-forms, the pointwise inner product ·, · for the pair A, B may be expressed as A, B = − ∗ Tr(A ∧ (∗B)).
(4.1)
The pointwise norm of A is then the induced one, |A| = A, A. Let A be a G -valued connection 1-form and FA its induced curvature 2-form: 2
FA = dA + A ∧ A.
(4.2)
The covariant derivative on a G -valued p-form α is defined by Dα = dα + A ∧ α − (−1)p α ∧ A ≡ dα + [A, α]. Then its adjoint is D ∗ β = d∗ β + ∗[A, ∗β] and there holds the useful Bianchi identity: DFA = dFA + A ∧ FA − FA ∧ A = 0.
(4.3)
Given a covariant derivative D, we first consider the (generalized) Born–Infeld type energy for the curvature F = FA = dA + 21 [A, A] over R4 : |F |2 1 2 E(A) = ( 1 + |F | − 1) dx = ρ(s) ds dx, (4.4) 2 R4 R4 0 where we have written ρ(s) = (1 + s)−1/2 as in the scalar case studied in section 3. If A is a critical point of the energy functional E(·), then it also satisfies the Euler–Lagrange equation of (4.4), D ∗ (ρ(|F |2 )F ) = D ∗ ((1 + |F |2 )−1/2 F ) = 0,
(4.5)
which may be derived in a standard way as follows. Consider a one-parameter deformation of A: A → At = A + tB, FA → FAt = d(A + tB) + (A + tB) ∧ (A + tB), where B is a compactly supported G -valued 1-form. Set K = dFAt /dt t=0 = dB +B ∧A+A∧B. Therefore, assuming that A is a critical point of (4.4) and skipping the domain of the integral R4 and the volume element dx, then in view of (4.1), d ∗Tr(K ∧ (∗FA ) + FA ∧ (∗K)) Tr(K ∧ (∗FA ) + FA ∧ (∗K)) E(At ) =− =− 2 dt 1 + |FA | 1 + |FA |2 t=0 Tr((dB + B ∧ A + A ∧ B) ∧ (∗FA )) = −2 1 + |FA |2 ∗FA A ∧ (∗FA ) − (∗FA ) ∧ A = −2 Tr B ∧ d + 1 + |FA |2 1 + |FA |2 FA = 0. (4.6) = −2 Tr B ∧ D ∗ 1 + |FA |2
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Since the trial 1-form B is arbitrary, we find from (4.6) the Euler–Lagrange equation of the action principle (4.4) as follows: FA D ∗ = 0, (4.7) 1 + |FA |2 which coincides with (4.5) and is a Born–Infeld type generalization of the classical Yang–Mills equation, D(∗FA ) = 0. This equation may be hard to solve. It is tempting to try to reduce (4.5) or (4.7) to a Bogomol’nyi equation by requiring a ‘self-dual’ condition and using the Bianchi identity as one does for the Yang–Mills equations. The space of the Yang–Mills fields (the solutions of second-order equations) is invariant under the star operation; if F is a solution of the second-order Yang–Mills equations, then so is ∗F . The fixed point set of the operator ∗ consists of the (self-dual) instantons. In view of this observation and the Bianchi identity, we see that we can achieve a significant reduction of (4.5) or (4.7) as in the classical Yang–Mills theory by imposing the following ‘generalized self-dual’ condition: FA ∗ = kFA , (4.8) 1 + |FA |2 where k is a constant. It can be immediately checked that any solution of (4.8) automatically satisfies the original equation, (4.5) or (4.7), because of the Bianchi identity (4.3). However, unlike Yang–Mills theory, we can show that (4.8) gives no information about nontrivial solutions of (4.7). Theorem 4.1.. For any constant k, the self-dual equation (4.8) over R4 has no finite-energy solution except the trivial solution, FA = 0. Proof. Since ∗ is an isometry, we find from (4.8) that |FA |2 = k 2 |FA |2 . 1 + |FA |2 We can solve the above equation to see that |FA |2 must be a constant. Using this in the energy (4.4), we see that FA = 0 as claimed. We now turn to the problem over S 4 . With the pointwise inner product defined for G valued forms, we define the induced global inner product by (α, β) = S 4 α, β ∗ 1. We have Green’s formula: (Dα, β) = (α, D ∗ β).
(4.9)
The following gap theorem for the Yang–Mills curvature form F ∈ L2 (S 4 ) has been obtained by Nakajima [32] under the assumption that F is smooth and by Donaldson and Kronheimer [16] under the assumption that F ∈ W 1,2 (S 4 ). We give here a new proof without these additional assumptions, assuming only that F ∈ L2 (S 4 ). Theorem 4.2.. There is a constant ε > 0, such that if F ∈ L2 (S 4 ) is a curvature form satisfying D ∗ F = 0 with F 2 < ε, then F ≡ 0. For the Born–Infeld equation (4.5) or (4.7), we have the gap theorem. Theorem 4.3.. There is a constant ε > 0, such that if the curvature form F ∈ L∞ (S 4 ) (and hence also in L2 (S 4 )) satisfies D ∗ ρF = 0 with F 2 < ε, then F ≡ 0. To prove the theorems, we need the following modification of Uhlenbeck’s theorem on ‘good gauges’ (cf proposition 2.3.13 in [16]).
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Proposition 4.4.. Let D be a connection on a trivial bundle over S 4 . Then there is a constant κ > 0 such that if FA 2 < κ, there is a gauge in which FA = dA + A ∧ A with A satisfying (i) d∗ A = 0, (ii) A W 2,1 C FA 2 . To prove theorem 4.2, we assume that the Coulomb gauge has been chosen and write F = FA with d∗ A = 0 and D ∗ FA = DFA = 0. Then A satisfies the nonlinear elliptic equation: T A ≡ (dd∗ + d∗ d)A + 21 d∗ [A, A] + ∗[A, ∗FA ] = 0. We have seen that if we take a one-parameter deformation (variation) At = A + tB and write Ft = FA+tB , then (d/dt)Ft |t=0 = DB. The linearization LB of T A is then given by d LB ≡ T At = (dd∗ + d∗ d)B + d∗ [A, B] + ∗[B, ∗FA ] + ∗[A, DB]. (4.10) dt t=0
4
On S , the Weitzenbach formula (cf [16]) then takes the form (dd∗ + d∗ d)B = ∇ ∗ ∇B + 3B.
(4.11)
Taking the inner product with B and integrating by parts, we have (B, (dd∗ + d∗ d)B) = (∇B, ∇B) + 3(B, B) B 2W 1,2 . Returning to (4.10), taking an inner product with B, and then using (4.11), H¨older’s inequality and the observation that (B, ∗[A, DB]) = (B, ∗[A, dB]) + (B, ∗[A, [A, B]]) 2 AB 2 ∇B 2 + 4 A2 2 B 2 2 , (4.12) we have, with C denoting a generic positive constant, the estimate
B 2W 1,2 C( B 2 LB 2 + AB 2 ∇B 2 + B 2 2 ∇A 2 + B 2 2 F 2 + AB 2 ∇B 2 + A2 2 B 2 2 ) C( B W 1,2 LB 2 + A 4 B 4 B W 1,2 + B 24 A W 1,2 + B 24 F 2 + A 4 B 4 B W 1,2 + A 24 B 24 ). Consequently, the use of the Sobolev inequality u 4 C u W 1,2 shows
B 2W 1,2 C( LB 2 B W 1,2 + A W 1,2 B 2W 1,2 + B 2W 1,2 A W 1,2 + B 2W 1,2 F 2 + A W 1,2 B 2W 1,2 + A 2W 1,2 B 2W 1,2 ) C( LB 2 B W 1,2 + A W 1,2 B 2W 1,2 + A 2W 1,2 B 2W 1,2 + F 2 B 2W 1,2 ). Hence, the inequality stated in proposition 4.4 and the assumption on the size of F 2 result in
B W 1,2 C LB 2 + ε B W 1,2 , with C a positive constant. This shows that the linear operator L is injective and, consequently, that the only solution of the original (nonlinear) equation T A = 0 (with A in the Coulomb gauge d∗ A = 0) is A ≡ 0. Then F ≡ 0 as claimed. Next, we prove theorem 4.3. Let A be the connection 1-form in the Coulomb gauge of proposition 4.4 and FA = dA + A ∧ A. Again, we take ρ = ρ(|FA |2 ) = (1 + |FA |2 )−1/2 and suppose that D ∗ (ρFA ) = 0 so that FA satisfies the (nonlinear) equation T A ≡ dd∗ A + D ∗ (ρFA ) = 0.
(4.13)
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As in the previous proof, we take a one-parameter variation At = A + tB. Then, with ρt = ρ(|Ft |2 ), T At = dd∗ At + d∗ (ρt FAt ) + ∗[At , ρt Ft ] and
d d d . (T At ) = dd∗ B + d∗ (ρt Ft ) + ∗[B, ρFA ] + ∗ A, ∗ (ρt Ft ) dt dt dt t=0 t=0 t=0 To compute (d/dt)(ρt Ft ) , we use Morrey’s mean value formula (lemma 1.1 in [39]): t=0 1 Ft − FA 1 2 2 IJ M (µ) ds , (ρ(|Ft | )Ft − ρ(|FA | )FA )I = t t 0 J where I, J are the multiple indices for the scalar components of the forms, µ = FA + s(Ft − FA ), and LB ≡
ˆ
M I J (µ) = ρ(|µ|2 )δIJ + 2ρ g J J (µ)I (µ)Jˆ . Taking the limit as t → 0 and using the fact that µ → F , we obtain dFt d IJ (ρ(|Ft |2 )Ft )I = M I J (FA ) = M (FA )(DB)J . dt dt t=0 J t=0 The linear operator M on 2-forms defined by (Mω)I = J M I J ωJ is bounded since FA is bounded. A computation (cf [39]) shows that M is elliptic and hence c|ω|2 Mω, ω C|ω|2 .
(4.14)
Finally, we obtain the linearization of the operator T : LB = dd∗ B + d∗ MdB + d∗ [A, MB] + ∗[B, ∗ρFA ] + ∗[A, MDB]. Taking an inner product with B and integrating by parts, we have (LB, B) = (d∗ B, d∗ B) + (MdB, dB) + , where consists of the lower order terms. Using (4.14) and integrating by parts, we have (LB, B) K((dd∗ + d∗ d)B, B) + = K{(∇B, ∇B) + 3(B, B)} + K B 2W 1,2 + . The last step is to estimate the terms in so that they can be absorbed. Since = ([A, MB], dB) + (∗[B, ∗ρFA ], B) + (∗[A, MdB], B) + (∗[A, M[A, B]], B), hence || C( AB 2 ∇B 2 + FA 2 B 24 + |A|2 |B|2 22 ). Using the Sobolev inequality and the smallness of FA 2 , we have || C( A 4 B 4 ∇B 2 + FA 2 ∇B 22 + A 24 B 24 ) C FA 2 B 2W 1,2 εC B 2W 1,2 . From this we obtain C B W 1,2 LB 2 which proves the injectivity of L and the conclusion of the theorem that FA ≡ 0. Note. The Born–Infeld type Yang–Mills model studied in this section is the simplest one. More complicated Yang–Mills models based on an action density of the form det(gµν + κFµν ), (4.15) where gµν is the metric form of spacetime, Fµν is a non-Abelian gauge field strength tensor and κ > 0 is a coupling parameter, have been the subject of many recent papers in the physics literature. See [44] for one of the earlier papers.
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5. Mixed interaction and coupled systems In this section, we study the static solutions of the Born–Infeld equations without assuming that either the electric or the magnetic field is absent. The electric field E and magnetic field B are now represented by a scalar field φ and a vector field A, respectively, such that E = ∇φ and B = ∇ × A. Hence, we are led to the following (static) Born–Infeld equations governing the mixed interaction of the electric and magnetic fields: ∇φ ∇· = 0, (5.1) 1 + |∇ × A|2 − |∇φ|2 ∇ ×A ∇× = 0. (5.2) 1 + |∇ × A|2 − |∇φ|2 As in section 2, these equations suggest the following coupled system for differential forms over an n-dimensional Riemannian manifold (M, g): dω1 d∗ = 0, (5.3) 1 + |dω2 |2 − |dω1 |2 dω2 d∗ = 0, (5.4) 1 + |dω2 |2 − |dω1 |2 where ω1 ∈ p (M)ω2 ∈ q (M) and p, q are arbitrary nonnegative integers. We can show as before that, if (ω1 , ω2 ) is a solution of (5.3) and (5.4) and M is compact, then ω1 and ω2 are both closed, dω1 = 0, dω2 = 0. In (5.3) and (5.4), set dω1 dω2 ∗ = τ1 , = τ2 . ∗ (5.5) 1 + |dω2 |2 − |dω1 |2 1 + |dω2 |2 − |dω1 |2 Then dτ1 = 0, dτ2 = 0. If the (n − p − 1)st and (n − q − 1)st de Rham cohomologies of M are trivial, there are an (n − p − 2)-form σ1 and an (n − q − 2)-form σ2 such that τ1 = dσ1 and τ2 = dσ2 . Hence, inserting (5.5) into (5.3) and (5.4), we obtain ∗dω1 = 1 + |dω2 |2 − |dω1 |2 dσ1 , (5.6) ∗dω2 = 1 + |dω2 |2 − |dω1 |2 dσ2 . (5.7) From these equations, we can derive the simple relation: 1 + |dω2 |2 − |dω1 |2 =
1 |2
1 + |dσ1 − |dσ2 |2
.
Substituting (5.8) into (5.6) and (5.7), we arrive at the following coupled system: dσ1 d∗ = 0, 1 + |dσ1 |2 − |dσ2 |2 dσ2 d∗ = 0, 1 + |dσ1 |2 − |dσ2 |2
(5.8)
(5.9)
(5.10)
σ1 ∈ n−p−2 (M), σ2 ∈ n−q−2 (M). Conversely, if the (p + 1)st and (q + 1)st de Rham cohomologies of M are trivial, we can arrive at (5.3) and (5.4) from (5.9) and (5.10).
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Finally, we remark that the generalization of the static Born–Infeld electromagnetic equations (5.1) and (5.2) in R3 into (5.3) and (5.4) or (5.9) and (5.10) is not unique. In fact, (5.2) says that there is a real scalar field ψ such that
∇ ×A 1 + |∇ × A|2 − |∇φ|2
= ∇ψ,
(5.11)
|∇ψ|2 . (1 − |∇ψ|2 )
(5.12)
which gives us the relation |∇ × A|2 = (1 − |∇φ|2 )
Substituting (5.12) into (5.11), we have 1 − |∇φ|2 . ∇ × A = ∇ψ 1 − |∇ψ|2
(5.13)
Using the Bianchi identity ∇ · (∇ × A) = 0 in (5.13) and (5.12) in (5.1), we see that the system of equations (5.1) and (5.2) are equivalent to the following system governing two coupled scalar fields: 1 − |∇ψ|2 ∇ · ∇φ = 0, (5.14) 1 − |∇φ|2 1 − |∇φ|2 ∇ · ∇ψ = 0. 1 − |∇ψ|2 Of course, these equations suggest the generalization: 1 − |dω2 |2 d ∗ dω1 = 0, 1 − |dω1 |2
d ∗ dω2
1 − |dω1 |2
1 − |dω2 |2
(5.15)
(5.16)
= 0,
(5.17)
for which the obvious significance is that ω1 and ω2 are two differential forms of the same order, say p, on the n-dimensional Riemannian manifold M. Sometimes we can transform the negative sign under the radical roots in (5.16) and (5.17) into a positive sign. To see this, we again assume that the (n − p − 1)st de Rham cohomology of M is trivial. Then, there are two (n − p − 2)-forms, σ1 and σ2 , such that 1 − |dω2 |2 1 − |dω1 |2 ∗ dω1 = dσ1 , = dσ2 , ∗ dω2 1 − |dω1 |2 1 − |dω2 |2 which leads to
dσ1 1 + |dσ2 |2 d∗ = 0, 1 + |dσ1 |2 dσ2 1 + |dσ1 |2 = 0. d∗ 1 + |dσ2 |2
(5.18)
(5.19)
Conversely, if the (p + 1)st de Rham cohomology of M is trivial, we can arrive at (5.16) and (5.17) from (5.18) and (5.19).
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In the special case when M is a simply connected subdomain of R2 and p = 0, (5.16) and (5.17) become (5.14) and (5.15). Hence, the two scalar equations (5.14) and (5.15) are equivalent to the following two scalar equations: 1 + |∇v|2 = 0, (5.20) ∇ · ∇u 1 + |∇u|2 1 + |∇u|2 ∇ · ∇v = 0, (5.21) 1 + |∇v|2 which is another generalization of the equivalence theorem of Calabi [10] for the minimal and maximal surface equations stated in section 1. Of course, this system possesses the obvious class of trivial affine linear solutions as before. However, there is an additional class of new trivial (nonlinear) solutions of the form u = v = any harmonic function. It will be interesting to know whether there exist other types of trivial solutions. It will be interesting to derive a certain condition under which the solutions of (5.20) and (5.21) over the full space R2 become trivial, ∇u = ∇v = 0. For this purpose, we observe that (5.20) and (5.21) are the Euler–Lagrange equations of the action functional E(u, v) = ( 1 + |∇u|2 1 + |∇v|2 − 1) dx, (5.22) R2
and it will be natural to impose the finite-action condition E(u, v) < ∞.
(5.23)
Suppose that (u, v) is a finite-action solution of (5.20) and (5.21). Since (u, v) is a critical point of (5.22), we see by rescaling (u, v) and the fact that (uλ (x), vλ (x)) = (u(λx), v(λx)) is a critical point of (5.22) at λ = 1 that d = 0. (5.24) E(uλ , vλ ) dλ λ=1
However, in (5.24), we may absorb the parameter λ in the integral by x → xλ , dx → λ−2 dxλ , and return to the original notation for the independent variable, xλ → x. Thus, we derive from (5.24) Derrick’s identity [14]: 2 2 2 1 + |∇v| 2 1 + |∇u| |∇u| dx = 2E(u, v). (5.25) + |∇v| 1 + |∇u|2 1 + |∇v|2 R2 This relation suggests that the finiteness of the energy (5.22) is ensured by the assumption |∇u|, |∇v| ∈ L2 (R2 ), 1 + |∇v|2 1 + |∇u|2 , ∈ L∞ (R2 ). 1 + |∇u|2 1 + |∇v|2
(5.26)
In the rest of this section, we prove that (5.26) implies the triviality of a solution pair of (5.20) and (5.21). Theorem 5.1.. Suppose that u and v solve (5.20) and (5.21) and satisfy the (stronger) finiteenergy condition (5.26). Then u and v are constant over R2 . Proof. Let R > 0 be a given number and uR satisfy uR = u,
|u| < R;
uR = R,
u R;
Then |∇uR | is measurable and |∇uR | = 0 for |u| R.
uR = −R,
u −R.
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Introduce a smooth function ξ over R2 satisfying ξ(x) = 1,
|x| 1;
ξ(x) = 0,
|x| 2;
0 ξ(x) 1,
x ∈ R2 ,
and put ξr (x) = ξ(x/r) for r > 0. It is clear that ∇ξr is supported within {x | r |x| 2r} and |∇ξr | C/r where C > 0 is an absolute constant. Then multiplying (5.20) by ξρ uR and integrating, we have uR (∇ξr · ∇u)ρ dx + ξr (∇uR · ∇u)ρ dx = 0, R2
R2
where now ρ = 1 + |∇v|2 / 1 + |∇u|2 . Hence, we obtain from the above relation the following estimate: 1/2 1/2 2 2 2 ξr |∇u| ρ dx R ρ ∞ |∇ξr | dx |∇u| dx |u|
r|x|2r
√
r|x|2r
CR ρ ∞ 3π
1/2 |∇u|2 dx
r|x|2r
(since the area of the annulus {x | r |x| 2r} is 3π r 2). Letting r → ∞ in the above and using the assumption that |∇u| ∈ L2 (R2 ), we see that |u| 0 is the Born parameter [4–7, 25], the second term governs the dynamics of a (bosonic) spin-zero particle and V is a potential term giving rise to the Higgs mechanism and is also of Born–Infeld type:
1 |u|2 − 1 2 2 2 V (|u| ) = b 1 − 1 − 2 . (6.2) b |u|2 + 1 The Euler–Lagrange equations of (6.1) are FA uDA u − uDA u d∗ = jA ≡ 2i , (6.3) F (1 + |u|2 )2 DA u ∗(DA u ∧ ∗DA u) 1 ∂V DA∗ = , (6.4) + 2 2 (1 + |u| ) (1 + |u|2 )3 2 ∂u where F = 1 + b−2 ∗ (FA ∧ ∗FA ) is a scalar. Of course, equation (6.3) defines the Born– Infeld electromagnetism induced from a matter source current, jA .
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We first observe that (6.3) and (6.4) have a self-dual reduction as in Abelian Higgs theory [3, 23, 43, 46]. To see this, we use the notation
1 |u|2 − 1 2 2 U = 1− 2 . V = b (1 − U ), b |u|2 + 1 After a lengthy calculation, we can extend the argument of Shiraishi and Hirenzaki [38] to obtain the following interesting decomposition for the Hamiltonian (6.1): 2 1 |u| − 1 2 −1 1 H = FA ± ∗F F + |DA u ± i ∗ DA u|2 2 |u|2 + 1 (1 + |u|2 )2 b2 + (F U − 1)2 F −1 ± ∗FA ± ∗JA , (6.5) 2 where JA is defined by 2|u|2 i FA + (DA u ∧ DA u − ∗DA u ∧ ∗DA u), (6.6) |u|2 + 1 (1 + |u|2 )2 so that the topological invariant 1 1 ∗JA dVg = JA = P (6.7) τ (L∗ ) = 4π S 4π S is the Thom class of the dual bundle L∗ → S [43] for which the integer P is the algebraic number of poles of the section u. Recall that 1 1 ∗FA dVg = FA = N − P (6.8) c1 (L) = 2π S 2π S is the classical first Chern class for which the integer N is the algebraic number of zeros of u. With (6.7) and (6.8), we see that we are led from (6.5) to the topological lower bound for the energy (6.9) E = H dVg 2π|c1 (L) + τ (L∗ )| = 2π(N + P ). JA = −
S
The equality is attained if (u, A) satisfies the equations 2 |u| − 1 = 0, FA ± ∗F |u|2 + 1
(6.10)
DA u ± i ∗ DA u = 0,
(6.11)
F U − 1 = 0.
(6.12)
We may identify the scalar BA = ∗FA with the magnetic field. Then we see that (6.10)–(6.12) can be compressed into
2 |u| − 1 1 |u|2 − 1 2 1− 2 , (6.13) BA = ∓ |u|2 + 1 b |u|2 + 1 DA u = ∓i ∗ DA u. (6.14) Note that the underlying (Riemannian) metric g over the surface S defines the Hodge dual ∗. Motivated by theoretic cosmology (cosmic strings, see [45]), we are interested in the solutions of (6.13) and (6.14) over an unknown surface (S, g) coupled with the Gauss curvature equation: Kg = 8πG H,
(6.15)
where G is Newton’s gravitational constant and the Hamiltonan density H is given as in (6.1).
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Equation (6.13) says that at the zeros and poles of u, respectively, the magnetic field BA attains its absolute maximum and minimum values over S: 1 BA = ± √ . (6.16) 1 − b−2 Thus, these zeros and poles give rise to magnetic strings and antistrings and the Born parameter b may be adjusted so that these strings at the zeros and poles may achieve arbitrarily large or small magnetic penetrations (depending on whether b is close to 1 or infinity). However, the global geometry (i.e. the total curvature) and hence the topology (i.e. the Euler characteristic) of (S, g) in view of (6.9) and (6.13)–(6.15) only depend on the sum of the number of strings (zeros) N and the number of antistrings (poles) P : 2πχ (S) = Kg dVg = 8π G H dVg = 16π 2 G(N + P ). (6.17) S
S
Hence we have χ (S) = 2 (thus topologically S must be a 2-sphere which will be an observed assumption throughout the rest of the paper) and this fact and (6.17) imply the necessary condition 4π G(N + P ) = 1.
(6.18)
We are interested in constructing a solution (u, A, g) of (6.13)–(6.15) so that u has N zeros and P poles over S which represent a distribution of N cosmic strings and P antistrings on S under condition (6.18). Here is our main existence theorem concerning such types of solutions. Theorem 6.1.. Let Q = {q} and P = {p} be two finite sets of points in S and {mq }q∈Q and {np }p∈P be two sets of positive integers with N = q∈Q nq and P = p∈P np . (i) Necessary condition: a necessary condition for the existence of a multistring solution (u, A, g) to the system of equations (6.13)–(6.15) so that Q and P are the sets of zeros and poles of the section u with the respective corresponding sets of algebraic multiplicities {mq }q∈Q and {np }p∈P is that (6.18) holds. (ii) Sufficient condition: a sufficient condition for the multistring solution with the properties stated in (i) to exist under the condition (6.18) is that 1 1 mq < (N + P ), q ∈ Q and np < (N + P ), p ∈ P . (6.19) 2 2 In particular, if N + P 3, there always exists a solution (u, A, g) to (6.13)–(6.15) with arbitrarily prescribed arrays of N single zeros and P single poles for the section u. Note. In the solution triplet (u, A, g), the metric g may always be chosen to be globally conformal to a standard metric. In [26], a similar existence problem is studied in an R2 -setting. The main existence theorem there states that a solution with a complete metric and representing N zeros and P poles of arbitrary multiplicities exists if and only if 1 N +P . (6.20) 8πG It will be interesting to know whether there exist solutions with noncomplete metrics when (6.19) is violated. As an application of theorem 6.1, we can show that this is indeed the case. Theorem 6.2.. In the range 1 1 max 2,
(6.21)
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the system of equations (6.13)–(6.15) over R2 has a solution (u, A, g) so that u has N prescribed single zeros and P prescribed single poles but the metric g on R2 is noncomplete for which (R2 , g) compactifies smoothly into (S 2 , g) where the metric g on the unit 2-sphere S 2 is globally conformal to the standard metric. Proof. Consider a solution (u, A, g) produced from part (ii) in theorem 6.1 over S = S 2 . Subtracting a point from S 2 which may or may not be a zero or a pole of u on S 2 , corresponding to either N + P < 1/4πG or N + P = 1/4π G, respectively, we obtain a desired solution in the stated R2 -setting. Note. Since the gravitational constant G is a tiny quantity so that 1/8π G 2 is trivially satisfied, condition (6.21) may be compressed in practice into 1 1
(6.22)
In the next section, we sketch a proof of theorem 6.1. 7. Sketch of proof of theorem 6.1 We will construct a solution triplet (u, A, g) under condition (6.18) so that Q = {q} and P = {p} are the sets of zeros and poles of u with the respective corresponding sets of multiplicities {mq } and {np }. For this purpose, we set v = ln |u|2 and reduce (6.13) and (6.14) into 2 f (v) − 4π np δp + 4π m q δq on S, (7.1) g v = −2 2 1 − b f (v) p∈P q∈Q where g is the Laplace–Beltrami operator on the 2-surface (S, g), δp is the Dirac distribution concentrated at p ∈ S with respect to the measure dVg and ev − 1 . ev + 1 On the other hand, in terms of v, we can rewrite the Hamiltonian density H as [26] 1 v H = g ln(1 + e ) − v + 2π np δp + m q δq . 2 p∈P q∈Q f (v) =
(7.2)
(7.3)
We assume that the unknown metric g is globally conformal to a standard metric g0 (say), then g = eη g0 , g = e−η g0 and (7.3) becomes 1 η v 0 0 (7.4) np δ p + mq δ q , e H = g0 ln(1 + e ) − v + 2π 2 p∈P q∈Q where δp0 is the Dirac distribution concentrated at p ∈ S with respect to the measure dVg0 . Let u0 be a solution of the equation 2π(N + P ) 0 0 (7.5) + 2π np δ p + mq δ q . g u0 = − |S|g0 p∈P q∈Q Inserting (7.5) into (7.4), we get 2π(N + P ) 1 eη H = g0 ln(1 + ev ) − v + u0 + . 2 |S|g0
(7.6)
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Recall that the Gaussian curvatures Kg on (S, g) and Kg0 on (S, g0 ) are related through the conformal exponent η by the equation − g0 η + 2Kg0 = 2Kg eη .
(7.7)
Without loss of generality, we may assume that Kg0 = constant = 1 (standard unit sphere). Hence |S|g0 = |S 2 | = 4π and (7.7) becomes −g0 η + 2 = 2Kg eη . Using this result and (7.6) in (6.15), we have 1 η 1 1 = − (N + P ) = 0, (7.8) g0 ln(1 + ev ) − v + u0 + 2 16π G 8π G 2 in view of condition (6.18). Hence 1 η = an arbitrary constant. (7.9) ln(1 + ev ) − v + u0 + 2 16π G In other words, we have seen that the Gaussian curvature equation (6.15) can be resolved by (7.9) and the conformal exponent η is given by v 1 (7.10) η = 16πG − u0 + 2 v − ln(1 + e ) + c. Therefore, equation (7.1) takes the form a f (v) ev −av0 0 n δ + 4π mq δq0 , − 4π g0 v = λe p p (1 + ev )2 1 − b−2 f 2 (v) p∈P q∈Q
(7.11)
where λ > 0 is a free parameter, v0 = 2u0 and a= 8π G. Note that v0 satisfies the (normalized) equation g0 v0 = −(N + P ) + 4π( np δp0 + mq δq0 ). We can prove the existence of a solution of the nonlinear equation (7.11) following the method of [48] for a slightly simpler equation (without the square root term arising from the Born–Infeld model). Since the proof here is analogous, we will only point out the key steps. First we consider a version of (7.11) so that the poles are switched into zeros: a f (v) ev 0 n δ + 4π mq δq0 . (7.12) g0 v = λe−av0 + 4π p p −2 2 (1 + ev )2 1 − b f (v) p∈P q∈Q With v = v0 + w, (7.12) becomes a f (v0 + w) ew g 0 w = λ + (N + P ). (1 + ev0 +w )2 1 − b−2 f 2 (v0 + w)
(7.13)
In order to solve (7.13), we consider the Chern–Simons equation g0 w = λev0 +w (ev0 +w − 1) + (N + P ).
(7.14)
This equation is well studied in [9] where it is shown that, when λ is sufficiently large, there exists a solution satisfying v0 + w < 0 on S. We now consider a perturbed version of (7.13) first. For each point p ∈ P (or q ∈ Q), let (Up , (x j )) (or (Uq , (x j ))) be a small isothermal coordinate chart around p (or q). Then v0 has the representation v0 (x) = np ln |x|2 + wp0 (x) in (Up , (x j )), where wp0 is smooth on Up . For any σ > 0 so that z ∈ Up whenever |x(z)| < 3σ , choose a function ρ ∈ C ∞ (S) satisfying 0 ρ 1, ρ(z) = 1 for |x(z)| < σ , ρ(z) = 0 for |x(z)| > 2σ . Choose σ small so that v0δ (x) = np ln(|x|2 + δρ(x)) + wp0 (x) in (Up , (x j ))
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for Up around p (δ > 0) extends to a smooth function on the full surface S and v0δ = v0 in S \ [(∪p∈P Up ) ∪ (∪q∈Q Uq )]; v0 v0δ in S. Now we write down our perturbed equation as a f (v0 + w) ev0 +w −av0δ g0 w = λe + (N + P ). (7.15) (1 + ev0 +w )2 1 − b−2 f 2 (v0 + w) We see that (7.15) reduces to (7.13) when δ = 0. We can rewrite the necessary condition (6.18) as a = 2/(N + P ). Then the assumption N + P 2 implies a 1. Let w1 be a solution of (7.14) with λ = λ1 and v0 + w1 < 0. Then (ev0 +w1 )a ev0 +w1 . Hence w1 satisfies g0 w1 λ1 (ev0 +w1 )a (ev0 +w − 1) + (N + P ). Of course, we may assume that e
−av01
e
−av0δ
v01
v0δ
(7.16)
(0 < δ 1). Thus
0 < δ 1.
,
(7.17)
Choose λ large so that λe−av0 · 1
1 (1 + ev0 +w1 )2a
·
(1 + ev0 +w1 )−1 1 − b−2 f 2 (v0 + w1 )
λ1
on S.
Combining (7.16) and (7.18), we get a f (v0 + w1 ) ev0 +w1 −av01 g0 w1 λe + (N + P ). +w v 2 0 1 (1 + e ) 1 − b−2 f 2 (v0 + w1 )
(7.18)
(7.19)
Using (7.17) in (7.19), we see that w1 is a subsolution of the perturbed equation (7.15) for all 0 < δ 1. Using the subsolution w1 and the supersolution v1 = −v0 > w1 for (7.15), we may obtain a solution of (7.15), say wδ , satisfying v1 wδ w1 for 0 < δ 1. Define a f (t) et g(t) = (7.20) (1 + et )2 1 − b−2 f 2 (t) and rewrite (7.15) as g0 w δ = λe−av0 g(v0 + w δ ) + (N + P ). δ
(7.21)
In order to take the δ → 0 limit in (7.21), we will use the L -theory of elliptic equations to control the sequence {w δ }. For this purpose, we first note in view of v1 wδ w1 that the sequence {wδ } has a uniform Lγ bound over S. Besides, definition (7.20) implies that g(v0 + w δ )(·) has a uniform Lγ -bound as well. Hence it now becomes crucial to bound the δ coefficient e−av0 in (7.21). Around each p ∈ P with multiplicity np , we have γ
e−av0 (x) e−av0 (x) = e−awp (x) |x|−2anp . 0
δ
(7.22) γ
It is seen that the right-hand side of (7.22) belongs to L (S) if anp γ < 1. Therefore, if anp < 1
or
np <
1 (N + P ), 2
(7.23)
then there is γ > 1 satisfying anp γ < 1. For such γ > 1, we conclude from (7.22) that e−av0 has a uniform Lγ -bound, which gives us a uniform Lγ -bound for the right-hand side of (7.21). Using Lγ -estimates, we see that {w δ } is uniformly bounded in W 2,γ (S). From the embedding W k,γ (S) → C α (S) for 0 α < k − 2/γ with k = 2 and γ > 1, we obtain the bound δ
sup |w δ |C α (S) C 0<δ<1
(7.24)
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for some α > 0. Now rewrite (7.21) as a δ f (v0 + w δ ) ew δ g0 w δ = λea(v0 −v0 ) + (N + P ). δ (1 + ev0 +w )2 1 − b−2 f 2 (v0 + w δ )
(7.25)
Recall the definition of v0 and v0δ . We see that ea(v0 −v0 ) is pointwise uniformly bounded with respect to δ. Hence, the right-hand side of (7.25) has uniform Lγ -bound for any γ > 1, which implies that {wδ } has uniform W 2,γ (S)-bound for any γ > 1. Using the compact embedding W 2,γ (S) → W 1,γ (S), we obtain a subsequence {w δn } (δn → 0 as n → ∞) so that δ
w δn w
in W 2,γ (S),
w δn → w
in
W 1,γ (S).
(7.26)
Inserting (7.26) into (7.25) and using the well-known Trudinger–Moser inequality K2 ∇w 2 2 w L (S) e dVg0 K1 e , w dVg0 = 0, K1 , K2 > 0, S
S
we see that we can take the δ = δn → 0 limit in (7.25) to show that w defined in (7.26) is a solution to (7.13). Returning to the function v = v0 + w, we get a solution of equation (7.12). Recall that w satisfies v1 > w w1 . Hence, v = v0 + w = −v1 + w < 0 which is an important property to be utilized below. Indeed, we see from (7.12) that the positive function −v satisfies the inequality a e−v f (−v) np δp0 − 4π mq δq0 g0 (−v) = λe−av0 − 4π −v 2 −2 2 (1 + e ) 1 − b f (−v) p∈P q∈Q a −v e f (−v) λe−av0 np δp0 + 4π mq δq0 − 4π (1 + e−v )2 1 − b−2 f 2 (−v) p∈P q∈Q in the sense of distribution, which says that v+ ≡ −v is a (positive) supersolution of (7.11). Similarly, v− ≡ v is a (negative) subsolution of (7.11). Of course, v− < v+ . Consequently, (7.11) has a solution v satisfying v− < v < v+ and the proof of theorem 6.1 is complete. Acknowledgment The first and second authors (LS and RS) acknowledge the support from the de Giorgi Institute, Pisa, Italy, the second author (RS) also acknowledges the support by grants PSC-CUNY36 and PSC-CUNY37 and the third author (YY) acknowledges partial support by the NSF under grant DMS-0406446. References [1] Barbashov B M and Nesterenko V V 1990 Introduction to the Relativistic String Theory (Singapore: World Scientific) [2] Bernstein S 1915–1917 Sur un theoreme de geometrie et ses applications aux equations aux derivees partielles du type elliptique Commun. Soc. Math. Khorkov 15 38–45 [3] Bogomol’nyi E B 1976 The stability of classical solutions Sov. J. Nucl. Phys. 24 449–54 [4] Born M 1933 Modified field equation with a finite radius of the electron Nature 132 282 [5] Born M 1934 On the quantum theory of the electromagnetic field Proc. R. Soc. Lond. A 143 410–37 [6] Born M and Infeld L 1933 Foundation of the new field theory Nature 132 1004 [7] Born M and Infeld L 1934 Foundation of the new field theory Proc. R. Soc. Lond. A 144 425–51 [8] Brecher D 1998 BPS states of the non-Abelian Born–Infeld action Phys. Lett. B 442 117–24 [9] Caffarelli L and Yang Y 1995 Vortex condensation in the Chern–Simons Higgs model: an existence theorem Commun. Math. Phys. 168 321–36
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