H-bubbles in a perturbative setting: the finite-dimensional reduction’s method∗ Paolo Caldiroli1 and Roberta Musina2 1
Dipartimento di Matematica Universit`a di Torino
via Carlo Alberto, 10 – 10123 Torino, Italy e-mail:
[email protected] 2
Dipartimento di Matematica ed Informatica Universit`a di Udine via delle Scienze, 206 – 33100 Udine, Italy e-mail:
[email protected] Abstract
Given a regular function H: R3 → R, we look for H-bubbles, i.e, regular surfaces in R3 parametrized by the sphere S2 with mean curvature H at every point. Here we study the case H(u) = H0 + εH1 (u) =: Hε (u), where H0 is a nonzero constant, ε is the smallness parameter, and H1 is any C 2 function. We prove that if p¯ ∈ R3 is a “good” extremal point for the Melnikov-type function Γ(p) = R − |q−p|<|H0 |−1 H1 (q) dq then for |ε| small there exists an Hε -bubble ω ε that converges to a sphere of radius |H0 |−1 centered at p¯, as ε → 0.
Introduction In recent years the study of so-called H-surfaces has seen a renewed and growing interest from the viewpoint of the Calculus of Variations. Among the most recent contributions on this topic, let us quote, for instance, [7], [8], [9], [10], [11], [14], [15]. ∗
Work supported by M.U.R.S.T. progetto di ricerca “Metodi Variazionali ed Equazioni Differenziali Nonlineari” (cofin. 2001/2002)
1
In this paper we are interested in the existence of H-bubbles, namely compact surfaces, without boundary, of S2 -type, whose mean curvature at every point equals a prescribed smooth function H: R3 → R. The study of these objects is motivated by some models for capillarity phenomena (see [13]). The geometrical problem we are considering admits an analytical formulation as follows: given H ∈ C 1 (R3 ) find a nonconstant function ω: R2 → R3 , smooth and conformal as a map on S2 , solving ( ∆ω = 2H(ω)ωx ∧ ωy in R2 R (0.1) 2 R2 |∇ω| < +∞ . Notice that problem (0.1) is invariant under the action of the conformal group of S2 ≈ R2 ∪ {∞}. This invariance means that in fact we deal with a problem on the image of the unknown ω, rather than on the mapping ω itself. When the prescribed mean curvature is a nonzero constant H(u) ≡ H0 , Brezis and Coron [6] proved that the only nonconstant solutions to (0.1) are spheres of radius |H0 |−1 anywhere placed in R3 . Only recently, the problem of H-bubbles has been investigated for H variable, using tools from Calculus of Variations. In particular, in [7] we studied the case of a bounded function H ∈ C 1 (R3 ) asymptotic to a constant at infinity and, under some global assumptions, we prove the existence of an H-bubble having “minimal energy”. In addition, in [8] we showed that this existence result persists under small L∞ perturbations of H. But in [8], the case of perturbations of a nonzero constant is excluded, since the unperturbed problem is asked to satisfy a variationaltype condition that forces the minimal H-bubbles to be uniformly bounded in L∞ . Clearly this fact fails when H is constant because in this case problem (0.1) is invariant also with respect to translations on the image. Here we study the perturbative case H(u) ≡ Hε (u) := H0 + εH1 (u)
(0.2)
where H0 is a nonzero constant, ε ∈ R is the smallness parameter, and H1 : R3 → R is any smooth function. In the article [4], Bethuel and Rey studied the Plateau’s problem for disc type surfaces with mean curvature Hε as in (0.2). In this paper they also made some 2
considerations about S2 -type parametric surfaces in the perturbative case (0.2) and, more precisely, they state that for every point p ∈ R3 and for |ε| small, there exists an Hε -bubble, which is an embedded sphere passing through p. Actually, this result is incorrect, since we can exhibit an example of perturbation H1 such that for every ε 6= 0 there is no regular Hε -bubble (see Proposition 4.1). In our nonexistence example, the perturbation H1 is a smooth function without stationary points. This suggests us that the existence of H-bubbles, even in the perturbative case (0.2), is strongly affected by the behaviour of the perturbation H1 which essentially has to possess stationary points (compare with Proposition 4.3). In this work we find some conditions on H1 ensuring the existence of regular Hε -bubbles, for |ε| small. Furthermore, we give the asymptotic behaviour of these Hε -bubbles when ε → 0, obtaining some information on the location of the solutions. Before stating our results, let us introduce the following notation: let ω be a (1-degree) H0 -bubble parametrizing a sphere centered at 0, with radius |H0 |−1 . Let Γ: R3 → R be the Melnikov-type function defined by Z H1 (q) dq . Γ(p) := − |q−p|< |H1
0|
Clearly Γ(p) measures the H1 weighted volume of a ball centered at p with radius |H0 |−1 . In our first result we consider the case in which Γ admits nondegenerate stationary points. Theorem 0.1 Let H0 ∈ R \ {0}, H1 ∈ C 2 (R3 ), and Hε as in (0.2). Assume: (Γ∗ ) there exists a nondegenerate stationary point p¯ ∈ R3 for Γ. Then there exists a curve ε 7→ ω ε of class C 1 from a neighborhood I ⊂ R of 0 into C 1,α (S2 , R3 ) such that ω 0 = ω + p¯ and ω ε is a smooth Hε -bubble without branch points for every ε ∈ I. In the next result we consider just the case of extremal points for Γ. In particular we assume no nondegeneracy but we ask for a weaker, topological stability condition. More precisely, we have the following result. Theorem 0.2 Let H0 ∈ R \ {0}, H1 ∈ C 2 (R3 ) and Hε be as in (0.2). Assume: 3
(Γ∗∗ ) there exists a nonempty compact set K ⊂ R3 such that max Γ(p) < max Γ(p) or min Γ(p) > min Γ(p) . p∈K
p∈∂K
p∈∂K
p∈K
Then for |ε| small enough there exists a smooth Hε -bubble ω ε , without branch points, and such that kω ε − (ω + pε )kC 1,α (S2 ,R3 ) → 0 as ε → 0, where pε ∈ K is such that Γ(pε ) → maxK Γ (or Γ(pε ) → minK Γ, respectively). Our assumptions on Γ in Theorems 0.1 and 0.2 can be made explicit in terms of H1 when |H0 | is large. In particular, as a first consequence of the above existence theorems we obtain the following result, which says that nondegenerate critical points as well as topologically stable extremal points of the perturbation term H1 are concentration points of Hε -bubbles, in the double limit ε → 0 and |H0 | → ∞. Theorem 0.3 Let H1 ∈ C 2 (R3 ) and assume that one of the following conditions is satisfied: (H1∗ ) there exists a nondegenerate stationary point p¯ ∈ R3 for H1 ; (H1∗∗ ) there exists a nonempty compact set K ⊂ R3 such that max H1 (p) < max H1 (p) or min H1 (p) > min H1 (p) . p∈K
p∈∂K
p∈∂K
p∈K
Then, for every H0 ∈ R with |H0 | large, there exists εH0 > 0 such that for every ε ∈ [−εH0 , εH0 ] there is a smooth Hε -bubble ω H0 ,ε without branch points. Moreover lim
lim kω H0 ,ε − pε kC 1,α (S2 ,R3 ) = 0
|H0 |→∞ ε→0
where pε ≡ p¯ if (H1∗ ) holds, or pε ∈ R3 is such that pε ∈ K and H1 (pε ) → maxK H1 , or H1 (pε ) → minK H1 if (H1∗∗ ) holds. In addition, under the condition (H1∗ ), the map ε 7→ ω H0 ,ε defines a C 1 curve in C 1,α (S2 , R3 ). As a further application of Theorem 0.2, we consider a perturbation H1 having some decay at infinity.
4
Theorem 0.4 Let H0 ∈ R \ {0} and H1 ∈ C 2 (R3 ). If H1 ∈ Lr (R3 ) for some r ∈ [1, 2], then for |ε| small enough there exist pε ∈ R3 and a smooth Hε -bubble ω ε , without branch points, such that kω ε − (ω + pε )kC 1,α (S2 ,R3 ) → 0 as ε → 0, and (pε ) is uniformly bounded with respect to ε. To prove Theorems 0.1 and 0.2 we adopt a variational-perturbative method introduced by Ambrosetti and Badiale in [1] and subsequently used with success to get existence and multiplicity results for a wide class of variational problems in some perturbative setting (see, e.g., [2] and [3]). Roughly speaking, this method can be applied to study a class of variational problems of the following type: one looks for critical points of some functional of the form f (u, ε) = f0 (u) + g(u, ε), where u lies in some Banach space, ε is the perturbation parameter, and f0 is the functional associated to an unperturbed basic problem which exhibits invariance with respect to some (possibly) noncompact group and which satisfies some nondegeneracy condition. The perturbation g(u, ε), in general, breaks all the invariances of the unperturbed problem. But the problem considered here exhibits a substantial difference in comparison with the general situation described above. Indeed, in our case, also the perturbed problem is invariant with respect to the action of a noncompact group, the group of conformal diffeomorphisms of R2 ∪ {∞}, and this invariance cannot be removed in any way. In order to make clearer the problem, let us describe the strategy followed to prove Theorems 0.1 and 0.2. First, one observes that, taking H = Hε as in (0.2), solutions to problem (0.1) can be obtained as critical points of the energy functional Eε (u) = E0 (u) + 2εVH1 (u) where 1 E0 (u) = 2
Z
2H0 |∇u| + 3 2 R 2
Z u · ux ∧ uy , R2
Z VH1 (u) =
Q1 (u) · ux ∧ uy , R2
being Q1 : R3 → R3 any vectorfield such that div Q1 = H1 . The first difficulty concerns the functional space on which Eε can be defined. Since, in general, no 5
boundedness or decay conditions on H1 are asked, we cannot define the functional Eε on H 1 (S2 , R3 ) that, according to the shape of the equation, could be considered as the natural space to set up the variational problem. Hence we are forced to take a smaller Sobolev space and we can work on W 1,s (S2 , R3 ) with s > 2 fixed (actually we take s = 3). With this choice, since for s > 2 the space W 1,s (S2 , R3 ) is embedded into L∞ , the energy functional Eε turns out to be of class C 2 . Notice that the regularity of the energy functional is an essential condition for our argument. Now, one also notices that the unperturbed problem ( ∆ω = 2H0 ωx ∧ ωy in R2 R 2 R2 |∇ω| < +∞ admits a fundamental solution ω (parametrizing the sphere centered at 0 with radius |H0 |−1 ) and a corresponding family of solutions of the form ω ◦ φ + p where φ is any conformal diffeomorphism of R2 ∪ {∞} and p runs in R3 . Hence, this set of solutions defines a manifold Z of critical points for the unperturbed functional E0 that can be parametrized by G × R3 , where G is the conformal group, having dimension 6. Thanks to some key results already known in the literature, in particular, due to Isobe [14], Z is a nondegenerate manifold, that is the tangent space of Z at any u ∈ Z equals the kernel of E000 (u). This allows us to apply the implicit function theorem to get, taking account also of the G-invariance of Eε , for |ε| small, a 3-dimensional manifold Zε close to Z, constituting a natural constraint for the perturbed functional Eε . This means that we can look for free critical points for Eε in correspondence to the critical points for Eε constrained to Zε . This step gives the finite dimensional reduction of the problem. We emphasize the fact that while the invariance with respect to translations on the image is broken in case of nonconstant H, the invariance with respect to the conformal group G cannot be eliminated, that is, any H-system is conformally invariant. As a consequence, in performing the finite dimensional reduction, the dependence on the 6-dimensional conformal group can be essentially neglected, and we are allowed to look for critical points for Eε constrained to a 3-dimensional manifold Zε which just depend on the variable p ∈ R3 . Indeed, we point out that Zε is close to the submanifold of Z defined by {ω + p | p ∈ R3 }. 6
The paper is structured as follows: in Sec. 1 we fix some notation and we set up the variational problem. In particular we introduce the energy functional, we discuss its regularity properties and we study the nondegeneracy condition for the unperturbed problem. Then, in Sec. 2 we apply the finite dimensional reduction’s method. In particular, we construct a 3-dimensional manifold Zε which defines a natural constraint for the perturbed energy functional Eε . In Sec. 3 we give the proofs of Theorems 0.1–0.4. Finally in Sec. 4 we present a nonexistence example and we show that essentially if there exists a sequence of Hε -bubbles approaching a sphere centered at some p¯ as ε → 0, then p¯ must be a stationary point for the Melnikov function Γ. In the appendix we collect some technical results. Acknowledgments. We wish to thank the Referee for his meaningful and helpful remarks.
1
Notation and preliminaries
From now on, we will take H0 = 1. This is not restrictive, by changing H1 (u) into ˜ 1 (u) = H0 H1 (H0 u). Moreover, we will always write H Hε (u) = 1 + εH(u) where H: R3 → R is a given function of class C 2 . Let ω: R2 → S2 be defined by µx 2 , ω(z) = µy , µ = µ(z) = 1 + |z|2 1−µ
(1.1)
being z = (x, y) and |z|2 = x2 + y 2 . Observe that ω is the inverse of the standard stereographic projection, it is a conformal parametrization of the unit sphere (centered at 0), and it solves problem ( ∆ω = 2ωx ∧ ωy on R2 R (1.2) 2 R2 |∇ω| < +∞ .
7
1.1
The variational space
For s ∈ (1, +∞) we set Ls := Ls (S2 , R3 ). Notice that to every map v ∈ Ls one can associate the map v¯ = v ◦ ω: R2 → R3 , that satisfies Z Z s 2 |¯ v| µ = |v|s R2
S2
In the following, we shall always use the same notation for a map v on S2 and for its composition with the stereographic projection ω, which is a map on R2 . Thus, for example, the norm in Ls is given by Z 1/s s 2 kvks = |v| µ . R2
Next, we simply write W 1,s instead of W 1,s (S2 , R3 ) and H 1 instead of W 1,2 (S2 , R3 ). Using the above identification, the norm of a map v ∈ W 1,s is given by kvkW 1,s
= kdvks + kvks Z 1/s Z s 2−s = |∇v| µ +
s 2
1/s
|v| µ
R2
R2
where “d” denotes differentiation on the sphere, while “∇” is the gradient for func0 tions defined on R2 . In the same spirit, the duality product between W 1,s and W 1,s can be written as Z Z (v|ϕ) = ∇v · ∇ϕ + v · ϕµ2 . R2
for every v ∈ W 1,s , ϕ ∈ W
1.2
1,s0
R2
.
The energy functional
Let H ∈ C 0 (R3 ) be a given curvature, and let Q: R3 → R3 be any smooth vectorfield such that div Q = H. For every u ∈ H 1 ∩ L∞ let us set Z VH (u) = Q(u) · ux ∧ uy . R2
If u is smooth enough, the integral defined by VH (u) has the meaning of an algebraic volume enclosed by the surface parametrized by u, with weight H, and it is independent of the choice of the vectorfield Q. For example, one can choose Z u 1 Z u2 Z u1 1 Q(u) = H(s, u2 , u3 ) ds, H(u1 , s, u3 ) ds, H(u1 , u2 , s) ds . 3 0 0 0 8
Notice that in particular, for H = 1, 1 V1 (u) = 3
Z u · ux ∧ uy . R2
For several results and remarks on the volume functional VH we refer to [5] for the case H ≡ const., and [16] for variable H. We start to point out some regularity properties of the H-volume functional VH on the space W 1,3 . Lemma 1.1 Assume that H ∈ C 1 (R3 ). Then the functional VH is of class C 1 on W 1,3 . In particular, the Fr´echet differential of VH at u ∈ W 1,3 is the functional Z dVH (u)ϕ = H(u)ϕ · ux ∧ uy (1.3) R2
for every ϕ ∈ W 1,3 . Proof. The directional derivative of VH at u ∈ W 1,3 along a smooth direction ϕ is given by Z ∂ϕ VH (u) = H(u)ϕ · ux ∧ uy , R2
(see Lemma A.7 in the Appendix). Now, the linear functional ϕ 7→ ∂ϕ VH (u) is continuous from W 1,3 into W 1,3/2 since Z 1 2 2 H(u)ϕ · ux ∧ uy ≤ 2 kHk∞ kϕk∞ kduk2 R ≤ CkϕkW 1,3 kuk2W 1,3 thanks to the embeddings W 1,3 ,→ H 1 and W 1,3 ,→ L∞ . Hence, the functional VH turns out to be Gˆateaux differentiable on W 1,3 and the Gˆateaux differential of VH at any u ∈ W 1,3 is given by (1.3). Now, let us prove that the mapping u 7→ dVH (u) is continuous on W 1,3 . Let (uk ) be a sequence in W 1,3 strongly converging in W 1,3 to some map u. One has Z Z dVH (uk )ϕ − dVH (u)ϕ = H(uk )ϕ · θxk ∧ θyk + H(uk )ϕ · θxk ∧ uy 2 2 R R Z Z k k + H(u )ϕ · ux ∧ θy + (H(uk ) − H(u))ϕ · ux ∧ uy R2
R2
9
where θk = uk − u. One can estimate each term in the above decomposition, according to the following inequality: Z H(u)ϕ · vx ∧ wy ≤ sup |H(ξ)| kϕk∞ kdvk2 kdwk2 R2
|ξ|≤kuk∞
≤ C
sup
|H(ξ)| kϕkW 1,3 kvkW 1,3 kwkW 1,3
|ξ|≤kuk∞
for arbitrary u, v, w and ϕ in W 1,3 . Since θk → 0 strongly in W 1,3 and uniformly on S2 ≈ R2 ∪ {∞}, and (uk ) is uniformly bounded, one easily obtains that dVH (uk )ϕ − dVH (u)ϕ ≤ o(1)kϕkW 1,3 for every ϕ ∈ W 1,3 , with o(1) → 0 as k → +∞. This concludes the proof. Lemma 1.2 Assume that H ∈ C 1 (R3 ). Then the Fr´echet differential of VH at every point u ∈ W 1,3 admits a (unique) continuous and linear extension on W 1,3/2 , 0 (u) ∈ W 1,3 such that defined by (1.3). Moreover, for every u ∈ W 1,3 there exists VH Z 0 (VH (u)|ϕ) = H(u)ϕ · ux ∧ uy for every ϕ ∈ W 1,3/2 . R2
Proof. For every u ∈ W 1,3 and ϕ ∈ L3 we have that Z 1 |H(u)ϕ · ux ∧ uy | ≤ sup |H(ξ)| kϕk3 kduk23 . 2 |ξ|≤kuk∞ R2 Hence the mapping Z ϕ 7→
H(u)ϕ · ux ∧ uy R2
is a continuous, linear functional on L3 . Since W 1,3/2 is (compactly) embedded into L3 , one immediately gets the first statement. Since W 1,3/2 and W 1,3 are dual spaces, also the second statement follows. Finally, we are going to compute the second derivative of the H-volume functional. The proof of the next Lemma is similar to that of Lemma 1.1. 0 : W 1,3 → W 1,3 is of class Lemma 1.3 Assume that H ∈ C 2 (R2 ). Then the map VH C 1 , and for every u, η ∈ W 1,3 and ϕ ∈ W 1,3/2 one has Z Z 00 (VH (u) · η|ϕ) = H(u)ϕ · (ηx ∧ uy + ux ∧ ηy ) + (∇H(u) · η)ϕ · (ux ∧ uy ) . R2
R2
10
Now we are in position to define the energy functional. First, it is convenient to introduce the Dirichlet’s integral Z Z 1 1 2 D(u) = |du| = |∇u|2 . 2 S2 2 R2 Then, for ε ∈ R, and for H ∈ C 2 (R3 ) we set Eε (u) = D(u) + 2V1 (u) + 2εVH (u) = E0 (u) + 2εVH (u) . By the results above, we have that Eε is well defined on W 1,3 , and it is of class C 2 . In particular, for all u ∈ W 1,3 one has Eε0 (u) ∈ W 1,3 and Z Z Z 0 ϕ · ux ∧ uy + 2ε H(u)ϕ · ux ∧ uy , (Eε (u)|ϕ) = ∇u · ∇ϕ + 2 R2
R2
R2
for every ϕ ∈ W 1,3/2 . Remark 1.4 Note that critical points of the energy functional Eε on W 1,3 , give rise to bounded, weak solutions to problem ( ∆ω = 2(1 + εH(ω))ωx ∧ ωy on R2 R (1.4) 2 R2 |∇ω| < +∞ . In fact it is known by the regularity theory for H-systems (see [12]) that any bounded weak solution ω to (1.4) is a classical, conformal solution. More precisely ω is of class C 3,α as a map on S2 .
1.3
Nondegeneracy condition for the unperturbed problem
Here we make some considerations on the unperturbed functional Z Z 1 2 2 E0 (u) = |∇u| + u · ux ∧ uy . 2 R2 3 R2 Notice that the functional E0 inherits all the invariances of unperturbed problem (1.2). More precisely, one can see that problem (1.2) is invariant with respect to conformal diffeomorphisms of the domain R2 ∪{∞}, and with respect to translations of the images (in R3 ).
11
According to a result by Brezis and Coron [6], the mapping ω defined by (1.1) generates all the solutions u of (1.2), which can be written as P (z) u(z) = ω + p =: ωP,Q (z) + p Q(z) where p ∈ R3 is a constant vector, and P, Q are irreducible polynomials in the (complex) variable z. Moreover one has D (ωP,Q + p) = 4kπ , E0 (ωP,Q + p) = where k = max{deg P, deg Q}. Hence the values levels of E0 and the set
4kπ 3
4kπ 3
(k ∈ N∪{0}) are all the critical
Z = {u ∈ W 1,3 : E00 (u) = 0, E0 (u) =
4π } 3
defines a manifold that, following Isobe [14], can be characterized more explicitely as follows: Z = Rω ◦ Lλ,ζ + p : R ∈ SO(3), λ > 0, ζ ∈ R2 , p ∈ R3 where for λ > 0 and ζ ∈ R2 we set Lλ,ζ z = λ(z − ζ). Since SO(3) ≈ S3 , the above expression shows that Z is a manifold of dimension 9. Let us recall that the manifold Z is said to be nondegenerate if Tu Z = ker E000 (u) for all u ∈ Z, where Tu Z denotes the tangent space of Z at u. The nondegeneracy of the unperturbed critical manifold Z constitutes a key step in performing the Ambrosetti-Badiale finite-dimensional reduction’s method (see [1]), based on the implicit function theorem. In the rest of this section we will see that Z is nondegenerate. The tangent space to Z at ω can be written in an explicit way as stated by the following result. Lemma 1.5 Tω Z = {(a · ω)ω + b ∧ ω + c : a, b, c ∈ R3 }. Proof. Let e1 , e2 , and e3 be the constant mappings corresponding to the canonical basis in R3 . We have to show that Tω Z is spanned by the maps (ei · ω)ω, ei ∧ ω and
12
ei , (i = 1, 2, 3). To this aim, we compute ∂ (ω + p) = ei (i = 1, 2, 3) ∂pi p=0 ∂ ω ◦ Lλ,0 = xωx + yωy = −(e3 · ω)ω + e3 ∂λ λ=1 ∂ ω ◦ L1,ζ = ωx = −(e1 · ω)ω − e2 ∧ ω + e1 ∂x ζ=0 ∂ ω ◦ L1,ζ = ωy = −(e2 · ω)ω + e1 ∧ ω + e2 . ∂y ζ=0 Next, we show that the maps ei ∧ ω (for i = 1, 2, 3) are tangent to Z at ω, by differentiating with respect to the variable R ∈ SO(3). To make this computation it is convenient to notice that the Lie algebra of SO(3) admits as a basis the set of matrices {ξ1 , ξ2 , ξ3 } defined by: 0 0 0 0 0 1 0 −1 0 ξ1 = 0 0 −1 , ξ2 = 0 0 0 , ξ3 = 1 0 0 , 0 1 0 −1 0 0 0 0 0 and one has ξ1 ω = e1 ∧ ω ,
ξ2 ω = e2 ∧ ω ,
ξ3 ω = e3 ∧ ω .
Remark 1.6 For g = (R, λ, ζ) ∈ SO(3) × R+ × R2 and for p ∈ R3 we introduce the following notation: ω g = Rω ◦ Lλ,ζ , ω g,p = ω g + p. Then the tangent space to Z at ω g,p is given by Tωg,p Z = {(a · ω g )ω g + b ∧ ω g + c : a, b, c ∈ R3 } and in particular it does not depend on p, i.e., Tωg,p Z = Tωg Z for all p ∈ R3 . Notice that, by Lemma 1.3, for any η ∈ W 1,3 and ϕ ∈ W 1,3/2 one has Z Z 00 (E0 (ω) · η|ϕ) = ∇η · ∇ϕ + 2 ϕ · (ωx ∧ ηy + ηx ∧ ωy ). R2
R2
13
(1.5)
We remark that in fact E0 ∈ C 2 (H 1 , R) and (1.5) holds also for every η, ϕ ∈ H 1 . Moreover (E000 (ω) · η|ϕ) = (E000 (ω) · ϕ|η), because of Lemma A.3. In particular, if ϕ is constant, one has that (E000 (ω) · η|ϕ) = 0. Now, define Z (Tω Z)⊥ := {η ∈ H 1 | ∇η · ∇u = 0 ∀u ∈ Tω Z} . R2
Notice that, by direct computation, ω ∈ (Tω Z)⊥ . The following result, due to Isobe [14], plays a key role in order to show the nondegeneracy of the manifold Z. Lemma 1.7 There exists a constant C > 0 such that Z Z (E000 (ω) · η|η) = |∇η|2 + 4 ω · ηx ∧ ηy ≥ Ckdηk22 R2
for all η ∈ (Tω Z)⊥ with
R
R2
R2
∇η · ∇ω = 0.
Let us introduce the Hilbert space H0 = {u ∈ H 1 |
Z u=0} S2
endowed with the inner product Z hu1 , u2 i =
∇u1 · ∇u2 .
(1.6)
R2
As an important consequence of Lemma 1.7, we finally obtain that Z is nondegenerate. Corollary 1.8 Tω Z = ker E000 (ω). Proof. The inclusion Tω Z ⊆ ker E000 (ω) always holds true. Now, let η ∈ H 1 be such that (E000 (ω)·η|ϕ) = 0 for all ϕ ∈ H 1 . Since ker E000 (ω) contains constant maps, we can assume η ∈ H0 . Now we use Lemma A.6 and E00 (ω) = 0 to get 0 = (E000 (ω) · η|ω) = R − R2 ∇η · ∇ω, that is, η is orthogonal to ω. Therefore, we can decompose η = v + τ with v ∈ H0 ∩ (Tω Z)⊥ orthogonal to ω, and τ ∈ Tω Z ⊆ ker E000 (ω). Then 0 = (E000 (ω) · η|v) = (E000 (ω) · v|v) ≥ Ckdvk22 thanks to Lemma 1.7. Hence v = 0 and, consequently, η = τ belongs to Tω Z. Thus, ker E000 (ω) ⊆ Tω Z, and this concludes the proof. 14
As a final consequence, we have that: Corollary 1.9 A map η ∈ H 1 is a weak solution to the equation ∆η = 2(ωx ∧ ηy + ηx ∧ ωy ) on R2 if and only if η = (a · ω)ω + b ∧ ω + c for some a, b, c ∈ R3 .
2
Construction of a natural constraint
In this section we perform a (local) construction, for |ε| small, of a smooth 3dimensional manifold Zε with the following properties: Zε is close to Z and actually locally diffeomorphic to a submanifold of Z, and Zε is a natural constraint for the perturbed energy functional Eε , that is, if u ∈ Zε is such that dEε |Zε (u) = 0, then Eε0 (u) = 0. Lemma 2.1 Let R > 0 be fixed. Then there exist ε¯ > 0, and a (unique) C 1 -map η = η ε (p) ∈ W 1,3 defined on a neighbourhood of [−¯ ε, ε¯] × BR ⊂ R × R3 , such that η 0 (p) = 0 and Eε0 (ω + p + η ε (p)) ∈ Tω Z Z
(2.1)
η ε (p) ∈ (Tω Z)⊥
(2.2)
η ε (p) = 0.
(2.3)
S2
Moreover, for every fixed ε ∈ [−¯ ε, ε¯] the set ZεR := {ω + p + η ε (p) | |p| ≤ R} is a natural constraint for Eε . Proof. First, we choose τ1 , . . . , τ6 in Tω Z such that Z Z ∇τi · ∇τj = δij , τi = 0 R2
S2
where i, j = 1, . . . , 6 and δij is the Kronecker symbol. Thus, Tω Z is spanned by τ1 , . . . , τ6 , e1 , e2 , e3 (remind that e1 , e2 , and e3 are constant maps that correspond to the canonical basis in R3 ). Our goal is to construct η ε (p) by applying the implicit function theorem to the map F = (F1 , F2 ): R × R3 × W 1,3 × (R6 × R3 ) → W 1,3 × (R6 × R3 ) 15
defined by (F1 (ε, p, η; λ, α)|ϕ) = (Eε0 (ω + p + η)|ϕ) Z Z 6 X − λi ∇ϕ · ∇τi + α · ϕ R2
i=1
Z F2 (ε, p, η; λ, α) =
for every ϕ ∈ W 1,3/2
S2
∇η · ∇τ6 ;
∇η · ∇τ1 , . . . , R2
Z
Z
η
.
S2
R2
Notice that F(0, p, 0; 0, 0) = 0 since E00 (ω) = 0. Moreover, by Lemma 1.3, F is of class C 1 and for every v ∈ W 1,3 , ϕ ∈ W 1,3/2 , µ ∈ R6 and β ∈ R3 we have that (∂η F1 (0, p, 0; 0, 0) · v|ϕ) = (E000 (ω) · v|ϕ) Z 6 X (∂λ F1 (0, p, 0; 0, 0) · µ|ϕ) = − µi ∇ϕ · ∇τi R2
i=1
Z (∂α F1 (0, p, 0; 0, 0) · β|ϕ) = −β · ϕ S2 Z Z ∂η F2 (0, p, 0; 0, 0) · v = ∇v · ∇τ1 , . . . , R2
Z ∇v · ∇τ6 ;
R2
v S2
∂λ F2 (0, p, 0; 0, 0) = ∂α F2 (0, p, 0; 0, 0) = 0. Let us consider the linear, continuous operator L: W 1,3 ×(R6 ×R3 ) → W 1,3 ×(R6 ×R3 ) defined as L(v, µ, β) = ∂(η,λ,α) F(0, p, 0; 0, 0) · (v, µ, β) . More explicitely, one has L = (L1 , L2 ), with Z 6 X (L1 (v, µ, β)|ϕ) = (E000 (ω) · v|ϕ) − µi
R2
i=1
Z L2 (v, µ, β) =
Z ∇ϕ · ∇τi − β ·
Z ∇v · ∇τ6 ; R2
(ϕ ∈ W 1,3/2 )
Z
∇v · ∇τ1 , . . . , R2
ϕ S2
v
.
S2
We need to prove that L is an isomorphism. First, we show that L is injective. In fact, if L(v, µ, β) = 0 then (v, µ, β) ∈ W 1,3 × R6 × R3 satisfies Z Z 6 X (E000 (ω) · v|ϕ) = µi ∇ϕ · ∇τi + β · ϕ (ϕ ∈ W 1,3/2 ) (2.4) i=1
R2
S2
Z ∇v · ∇τi = 0 ZR
(i = 1, . . . , 6)
(2.5)
2
v = 0.
(2.6)
S2
16
Taking ϕ ≡ const. in (2.4) we get β = 0 by Lemma A.3. Now we take ϕ = τj in (2.4), we use (E000 (ω) · v|τj ) = (E000 (ω) · τj |v) = 0 since τj ∈ Tω Z = ker E000 (ω) (see Corollary 1.8), and we get µj = 0 for each j = 1, . . . , 6. Hence, (2.4) becomes (E000 (ω) · v|ϕ) = 0 for all ϕ ∈ W 1,3/2 , namely v ∈ ker E000 (ω). Then, by Corollary 1.8, v ∈ Tω Z which, together with (2.5) and (2.6), implies that also v = 0 that is, L is injective. Secondly, we show that L is a surjective. Fix (u, ν, γ) ∈ W 1,3 × R6 × R3 . We have to find (v, µ, β) ∈ W 1,3 × R6 × R3 such that: Z Z 6 X 00 (E0 (ω) · v|ϕ) = µi ∇ϕ · ∇τi + β · ϕ + (u|ϕ) (ϕ ∈ W 1,3/2 ) (2.7) i=1
R2
S2
Z ∇v · ∇τi = νi
(i = 1, . . . , 6)
(2.8)
2 ZR
v = γ.
(2.9)
S2
First, taking ϕ = const. in (2.7), we see that β has to be choosen in the following way: Z 1 β=− u. 4π S2 Next, we choose ϕ = τj in (2.7) to see that µ has to satisfy µj = −(u|τj )
(j = 1, . . . , 6).
In order to determine v, let us consider the following decomposition: v = tω + τ + ψ¯ with t ∈ R, τ ∈ Tω Z and ψ¯ ∈ Xω ∩ W 1,3 , being Z Xω := {ψ ∈ H0 | ∇ψ · ∇ω = 0} ∩ (Tω Z)⊥ . R2
Using (2.8) and (2.9), we infer that τ=
6 X
νi τi +
i=1
1 γ. 4π
In this way, (2.8) and (2.9) are fulfilled. Taking ϕ = ω in (2.7) and using the previous decomposition of v, we deduce that t=−
1 (u|ω) . 8π 17
¯ Notice that taking ϕ ∈ Xω in (2.7), we obtain the following condition for ψ: ¯ (E000 (ω) · ψ|ϕ) = (u|ϕ)
for every ϕ ∈ Xω .
(2.10)
This is equivalent to say that ψ¯ is a critical point of the functional J: Xω → R defined by Z Z 1 2 |∇ψ| + 2 ω · ψx ∧ ψy − (u|ϕ) (ψ ∈ Xω ) . J(ψ) = 2 R2 R2 Here we consider Xω as a Hilbert space endowed with the scalar product h·, ·i defined in (1.6). Indeed, one has h∇J(ψ), ϕi = (E000 (ω) · ψ|ϕ) − (u|ϕ) for every ϕ ∈ Xω . In addition, J is weakly lower semicontinuous on Xω thanks to the compactness Lemma A.4. Moreover J is bounded from below and coercive by the Isobe’s Lemma 1.7. Therefore, its minimum is attained, and hence there exists ψ¯ ∈ Xω solving (2.10). In order to show that ψ¯ ∈ W 1,3 , since ψ¯ ∈ Xω , it is enough to check that Z ¯ ∇ψ · ∇ϕ ≤ CkϕkW 1,3/2 for every ϕ ∈ Xω R2
where C is a positive constant independent of ϕ. Using (2.10) and the fact that |∇ω(z)| = µ(z), we have that Z Z ¯ ¯ ¯ ϕ · (ωx ∧ ψy + ψx ∧ ωy ) ∇ψ · ∇ϕ = (u|ϕ) − 2 R2
R2
Z ≤ kukW 1,3 kϕkW 1,3/2 + 2
¯2 |∇ψ|
1/2 Z
R2
|ϕ|2
1/2
S2
¯ 2 kϕk 1,3/2 . ≤ kukW 1,3 kϕkW 1,3/2 + Ck∇ψk W Hence ψ¯ ∈ W 1,3 . Finally, it remains to check that the function v = tω + τ + ψ¯ (with t, τ and ψ¯ defined as above) satisfies (2.7). This can be done taking into account R that τ ∈ ker E000 (ω) and that (E000 (ω) · ω|ϕ) = (E000 (ω) · ϕ|ω) = − R2 ∇ω · ∇ϕ for all ϕ ∈ W 1,3/2 , since E00 (ω) = 0. Therefore L(v, µ, β) = (u, ν, γ) and the surjectivity is proved. Now we are in position to apply Dini’s theorem. More precisely, for every R > 0 there exists ε¯ > 0 and, for |ε| ≤ ε¯ unique maps η ε (p) ∈ W 1,3 , λε (p) ∈ R6 and αε (p) ∈ R3 defined in a neighborhood of BR , and of class C 1 with respect to the
18
variables (ε, p), such that for every ϕ ∈ W 1,3/2 and for every p ∈ BR (Eε0 (ω
ε
+ p + η (p))|ϕ) =
6 X
λεi (p)
i=1
Z
∇η ε (p) · ∇τi = 0
Z
ε
Z
∇ϕ · ∇τi + α (p) · R2
ϕ
(2.11)
S2
(i = 1, . . . , 6)
(2.12)
R2
Z
η ε (p) = 0 .
(2.13)
S2
Notice that (2.11)–(2.13) are equivalent to (2.1)–(2.3) and thus the first part of the Lemma’s statement is proved. For future convenience, we notice that taking ϕ ≡ const. in (2.11) we get Z αε (p) = ε(2π)−1 H(ω + p + η ε (p)) (ω + η ε (p))x ∧ (ω + η ε (p))y . (2.14) R2
Moreover, we remark that the function fε : BR → R, fε (p) := Eε (ω + p + η ε (p))
(2.15)
is of class C 1 and satisfies ∇fε (p) = 4παε (p)
for every p ∈ BR .
(2.16)
Indeed, using (2.11), for k = 1, 2, 3 one has ∂k fε (p) = (Eε0 (ω + p + η ε (p))|ek + ∂k η ε (p)) Z Z 6 X ∂k η ε (p) . = λεi (p) ∇(∂k η ε (p)) · ∇τi + 4παkε (p) + αε (p) · R2
i=1
S2
But (2.12) and (2.13) yield, respectively, Z Z ε ∇(∂k η (p)) · ∇τi = ∂k ∇(η ε (p)) · ∇τi = 0 2 2 R ZR Z ε ε ∂k η (p) = ∂k η (p) = 0 S2
S2
and then (2.16) follows. Now, let us show that ZεR := {ω + p + η ε (p) : |p| ≤ R} is a natural constraint for Eε . Now, let ω + p¯ + η ε (¯ p) ∈ ZεR be a constrained critical point for Eε on ZεR , with 19
|¯ p| < R. We have that ∇fε (¯ p) = 0 and then (2.16) implies αε (¯ p) = 0. Let us show ε that for |ε| small, λ (¯ p) = 0. Indeed, let g1 (t), . . . , g6 (t) be smooth paths of conformal transforms such that gi (0) = Id and (ω ◦ gi )0 (0) = τi (i = 1, . . . , 6). Since for every conformal transform g, and for every u, v ∈ H 1 there results hu ◦ g, v ◦ gi = hu, vi, in particular one has Z d ∇(τj ◦ gi ) · ∇(η ε ◦ gi ) = 0 dt R2 where gi = gi (t) and η ε = η ε (¯ p). Therefore, after computations, we obtain Z Z ε ε 0 Ai,j := − ∇τj · ∇((η ◦ gi ) (0)) = ∇((τj ◦ gi )0 (0)) · ∇η ε R2
R2
for every i, j = 1, . . . , 6. Consequently |Aεi,j | ≤ Ck∇η ε k2 = o(1) .
(2.17)
Now we use the invariance of Eε with respect to the conformal group, that is, Eε (u) = Eε (u ◦ g) for every u ∈ W 1,3 and for every conformal mapping g. Writing ω ε = ω + p¯ + η ε (¯ p) we have that (ω ε ◦ gi )0 (0) = τi + (η ε ◦ gi )0 (0) and, by (2.11), d 0 = Eε ((ω ε ◦ gi )(t)) dt t=0 = (Eε0 (ω ε )|τi ) + (Eε0 (ω ε )|(η ε ◦ gi )0 (0)) Z 6 X ε ε = λi (¯ p) + λj (¯ p) ∇τj · ∇((η ε ◦ gi )0 (0)) , j=1
R2
because αε (¯ p) = 0. Then, by definition of Aεi,j , λε (¯ p) solves the system Aε λ = λ where Aε := (Aεi,j )i,j=1,...,6 is a 6×6 matrix. If for a sequence ε → 0 it were λε (¯ p) 6= 0, ε then the value 1 would be an eigenvalue of A , in contradiction with (2.17). Hence, taking a smaller ε¯ > 0, if necessary, one has λε (¯ p) = 0 for all |ε| ≤ ε¯. Therefore, by 0 ε (2.11), Eε (ω ) = 0. This completes the proof.
3 3.1
Proofs Proof of Theorem 0.1
Let Γ: R3 → R be defined by Z Γ(p) = VH (ω + p) = −
H(ξ + p) dξ. B1
20
(3.1)
The last equality in (3.1) can be easily obtained by applying the Gauss-Green theorem. Let p¯ ∈ R3 be a nondegenerate critical point of Γ and let R > |¯ p|. Let fε = Eε |ZεR as in (2.15). We look for a smooth path ε 7→ pε of critical points of fε , with p0 = p¯. Using (2.16) and (2.14), we can write ∇fε (p) = 2εG(ε, p) where
Z
(3.2)
H(ω + p + η ε (p))(ω + η ε (p))x ∧ (ω + η ε (p))y .
G(ε, p) = R2
Notice that the function G is of class C 1 , and that, by Gauss-Green Theorem, Z G(0, p¯) = H(ω + p¯) ωx ∧ ωy = ∇Γ(¯ p) = 0. R2
Denoting by ∂k the k th -partial derivative in R3 and using the Gauss-Green theorem, since div (H(· + p)ek ) = ∂k H(· + p), we have that Z Gk (0, p) = ek · H(ω + p) ωx ∧ ωy = ∂k VH(·+p) (ω) = ∂k Γ(p) . R2
Hence G(0, p¯) = 0, because p¯ is a stationary point of Γ. Moreover, we also obtain 2 ∂i Gk (0, p) = ∂ik Γ(p) .
Since p¯ is nondegenerate, ∇p G(0, p¯) is invertible. Therefore by the implicit function theorem, there exists a neighborhood I of 0 (in R) and a C 1 mapping ε 7→ pε ∈ R3 defined on I, such that p0 = p¯ and G(ε, pε ) = 0 for all ε ∈ I. Hence, by (3.2) and by the definition (2.15) of fε and by Lemma 2.1, we obtain that the function ε 7→ ω ε := ω + pε + η ε (pε )
(ε ∈ I)
defines a C 1 curve from I into W 1,3 of Hε -bubbles, passing through ω+ p¯ when ε = 0. It remains to prove that the curve ε 7→ ω ε is of class C 1 from I into C 1,α (S2 , R3 ). This can be obtained by a boot-strap argument. Indeed ω ε solves ∆ω ε = F ε on R2 , where F ε = 2(1 + εH(ω ε ))ωxε ∧ ωyε . Since ε 7→ ω ε is of class C 1 from I into W 1,3 we have that ε 7→ F ε is of class C 1 from I into L3/2 . Now, regularity theory yields that the mapping ε 7→ ω ε turns out of class C 1 from I into W 2,3/2 . This implies that ε 7→ dω ε is C 1 from I into L6 , by Sobolev embedding. Hence ε 7→ F ε 21
belongs to C 1 (I, L3 ). Consequently, again by regularity theory, ε 7→ ω ε is of class C 1 from I into W 2,3 . By the embedding of W 2,3 into C 1,α (S2 , R3 ), the conclusion follows. Lastly, we point out that ω ε has no branch points because ω ε → ω + p¯ in C 1,α (S2 , R3 ) as ε → 0, and ω is conformal on R2 . R Remark 3.1 By Lemma 1.7, E000 (ω) is positive definite on {η ∈ (Tω Z)⊥ : R2 ∇η · ∇ω = 0}, whereas (E000 (ω) · ω|ω) = −8π. In particular, we deduce that for every p ∈ R3 , ω + p is a nondegenerate critical point for E0 restricted to (Tω Z)⊥ , with Morse index 1. Now, following the arguments used in Theorem 3.2, [1], under the assumptions of Theorem 0.1, one can prove that ω ε is a nondegenerate critical point for Eε with Morse index 1 + d, where d equals the Morse index of Γ at p.
3.2
Proof of Theorem 0.2
First we point out that, since η ε (p) is of class C 1 with respect to the pair (ε, p), and η 0 (p) = 0, we have that kη ε (p)kW 1,3 = O(ε) uniformly for p ∈ BR , as ε → 0 .
(3.3)
Now we show that Eε (ω + p + η ε (p)) = E0 (ω) + 2εΓ(p) + O(ε2 ) as ε → 0, uniformly for p ∈ BR . (3.4) Indeed, set Rε (p) := Eε (ω + p + η) − E0 (ω) − 2εΓ(p) =
E0 (ω + η) − E0 (ω) + 2ε (VH (ω + p + η) − VH (ω + p)) ,
where Γ is as in (3.1). Using E00 (ω) = 0 and the decomposition Z Z V1 (ω + η) = V1 (ω) + V1 (η) + ω · ηx ∧ ηy + η · ωx ∧ ωy R2
R2
we compute Z E0 (ω + η) − E0 (ω) = D(η) + 2V1 (η) + 2 R2
22
ω · ηx ∧ ηy = O(kdηk23 )
Therefore, using also (3.3), we infer that Rε (p)ε−2 = O(kdηk23 )ε−2 + 2 (VH (ω + p + η) − VH (ω + p)) ε−1 = O(1) + 2(dVH (ω + p)η + kηkW 1,3 o(1))ε−1 = O(1), and (3.4) follows. Now, let K be given according to the hypothesis (Γ∗∗ ) and take R > 0 so large that K ⊂ BR . The assumption (Γ∗∗ ) and (3.4) imply that for |ε| small, there exists pε ∈ K such that ω ε := ω + pε + η ε (pε ) is a stationary point for Eε constrained to ZεR . According to Lemma 2.1, Eε0 (ω ε ) = 0 ε
(3.5)
ε
kη (p )kW 1,3 → 0 as ε → 0
(3.6)
Γ(pε ) → max Γ (or Γ(pε ) → min Γ) as ε → 0. K
K
By Remark 1.4 and by (3.5), ω ε turns out to be a smooth Hε -bubble. To prove that kω ε − (pε + ω)kC 1,α (S2 ,R3 ) → 0 as ε → 0 one can follow a boot-strap argument, as in the last part of the proof of Theorem 0.1.
3.3
Proof of Theorem 0.3
First, define Hε (u) = H0 + εH1 (u), as in (0.2). Let ω H0 = H10 ω a (1-degree) H0 bubble parametrizing a sphere centered at 0, with radius |H0 |−1 , and consider the Melnikov function Z ΓH0 (p) = − H1 (q) dq . B|H
−1 (p) 0|
Assume that (H1∗ ) holds, and set Z ˜ Γ(δ, p) = H1 (p + δq) dq
for every (δ, p) ∈ R × R3 .
B1
By an application of the implicit function theorem, using (H1∗ ), one can prove the existence of a C 1 path δ 7→ pδ , defined in a neighborhood J ⊂ R of 0 such that p0 = p¯ ˜ ·). Then the and, for every δ ∈ J, pδ is a nondegenerate stationary point for Γ(δ, −1 ˜ conclusion follows by Theorem 0.1, observing that ΓH0 (p) = −|H0 |−3 Γ(|H 0 | , p). Now, assume that (H1∗∗ ) holds, and notice that ΓH0 (p) ∼ −|H0 |3 H1 (p) as |H0 | → ∞. Then, by (H1∗∗ ), for |H0 | large, the condition (Γ∗∗ ) is fulfilled and Theorem 0.2 23
implies that there exists εH0 > 0 such that for every ε ∈ [−εH0 , εH0 ] there is a smooth Hε -bubble ω H0 ,ε without branch points and kω H0 ,ε − (ω H0 + pε )kC 1,α (S2 ,R3 ) → 0 as ε → 0 ,
(3.7)
where pε ∈ K satisfies ΓH0 (pε ) → maxK ΓH0 , or ΓH0 (pε ) → minK ΓH0 , as ε → 0. Now we pass to the limit as |H0 | → ∞. Thus we obtain that H1 (pε ) → maxK H1 , or H1 (pε ) → minK H1 , respectively, as ε → 0. Moreover, by (3.7) and since ω H0 = H10 ω, we have that lim|H0 |→∞ limε→0 kω H0 ,ε − pε kC 1,α (S2 ,R3 ) = 0.
3.4
Proof of Theorem 0.4
Since H1 ∈ Lr (R3 ), one has that Γ(p) → 0 as |p| → ∞. If kΓk∞ > 0 then the thesis immediately follows by Theorem 0.2. Now we prove that if Γ ≡ 0 then H1 ≡ 0 and consequently also in this case the thesis trivially follows. If Γ ≡ 0 then H1 ∗χ ≡ 0, where χ is the characteristic function of the ball B1/|H0 | , and “∗” denotes the convolution operator. Since H1 ∈ Lr (R3 ) with 1 ≤ r ≤ 2, applying the Fourier ˆ 1χ transform, one obtains that H ˆ ≡ 0. Noting that χ ˆ is an entire nonzero function, 3 ˆ 1 = 0 almost everywhere. and then χ ˆ 6= 0 almost everywhere on R , one infers that H Consequently H1 ≡ 0. This concludes the proof.
4
Nonexistence results
In this section we exhibit an example of H-system which admits no H-bubble diffeomorphic to the sphere S2 . This example is nonperturbative, but it also shows that, in the perturbative case, the fact that the Melnikov function Γ defined by (3.1) admits critical points constitutes, in general, a necessary condition in order that (1.4) admits S2 -type solutions without branch points. Proposition 4.1 Let H ∈ C 1 (R3 ) be such that H(u) = f (ν · u) as |u| < R, where R > 0, ν is a unit versor in R3 and f : (−R, R) → R is a strictly monotone function of class C 1 . Then there is no H-bubble ω diffeomorphic to the sphere S2 , with kωk∞ < R.
24
Proof. Let ω: R2 → R3 be a solution to (0.1) such that kωk∞ < R, and let EH = D + 2VH be the energy functional associated to problem (0.1). Since ω solves (0.1), dEH (ω)ν = 0. On the other hand, by the results of subsection 1.2, Z d EH (ω + tν) =2 Qν (ω) · ωx ∧ ωy dEH (ω)ν = dt 2 R t=0 where Qν (u) = (∇Q1 (u) · ν, ∇Q2 (u) · ν, ∇Q3 (u) · ν) . 2 Since div Qν = ∂H ∂ν , and since ω is assumed to be diffeomorphic to the sphere S , by Gauss-Green theorem, one has Z Z Z ∂H ν Q (ω) · ωx ∧ ωy = − (ξ) dξ = − f 0 (ξ · ν) dξ ∂ν 2 R Ω Ω
where Ω is the region in R3 enclosed by the surface parametrized by ω. Hence, Z 0= f 0 (ξ · ν) dξ Ω
in contrast with the fact that f is strictly monotone. Remark 4.2 It is known (see [16]) that for every u ∈ H 1 ∩L∞ there exists an integer R valued BV function iu : R3 → R such that VK (u) = R3 iu K for every K ∈ L∞ (R3 ). Thus, a mapping u can be interpretated as an integer valued current. Hence, the conclusion of Proposition 4.1 holds true more generally if ω is a negative valued (or positive valued) current, that is iω ≤ 0 (or iω ≥ 0) almost everywhere on R3 . In the next result we show that the existence of a stationary point for the Melnikov function Γ is in essence a necessary condition in order to have the existence of ω ε bubbles approaching a sphere, as in the conclusions of Theorems 0.1 and 0.2. Proposition 4.3 Let H1 ∈ C 1 (R3 ), Hε = 1 + εH1 , and assume that there exists a sequence ω εk of Hεk -bubbles, with εk → 0, and a point p ∈ R3 such that kω εk − (ω + p)kC 1 (S2 ,R3 ) → 0 as k → ∞. Then p is a stationary point for Γ. 25
Proof. The maps ω εk solve ∆ω εk = 2ωxεk ∧ ωyεk + 2εk H(ω εk )ωxεk ∧ ωyεk . Testing with the constant functions ei (i = 1, 2, 3) and using Lemma A.2 we get Z Z εk εk εk 0= H(ω )ei · ωx ∧ ωy = o(1) + H(ω + p)ei · ωx ∧ ωy = o(1) + ∂i Γ(p), R2
R2
and the Proposition is readily proved.
A
Auxiliary results on the volume functional
We start with a density Lemma (for scalar-valued functions) which has already been used by Brezis-Coron in [5]. Lemma A.1 Let u ∈ H 1 (S2 ). Then there exists a sequence (uk ) ⊂ C ∞ (S2 ) such that uk ◦ω ≡ uk : R2 → R is constant far out for every k ∈ N , and uk → u in H 1 (S2 ). From now on we shall deal with vector-valued functions. Recall that W 1,s = W 1,s (S2 , R3 ) and H 1 = H 1 (S2 , R3 ). Lemma A.2 There exists C > 0 such that for every v ∈ H 1 ∩ L∞ and θ ∈ H 1 one has Z 1/2 (A.1) 2 v · θx ∧ θy ≤ CD(v) D(θ). R
Proof. The inequality (A.1) is almost standard. For completeness, we skech here the proof. By Lemma A.1, we can select a sequence θk of smooth functions that are constant in a neighbourhood of the noth pole, and such that θk → θ in H 1 . Notice R R that in particular ∇θk → ∇θ in L2 (R2 , R6 ). Therefore, R2 v ·θxk ∧θyk → R2 v ·θx ∧θy . On the other hand, the standard Bononcini-Wente isoperimetric inequality applies R (see for example [5], Lemma A.8) to control R2 v · θxk ∧ θyk , and then the conclusion is achieved by a limit procedure.
26
0
Lemma A.3 Assume θ, v ∈ H 1 ∩ L∞ , or θ ∈ W 1,s for some s > 2 and v ∈ W 1,s . Then Z Z 2 v · θx ∧ θy = θ · (vx ∧ θy + θx ∧ vy ) . (A.2) R2
R2
Proof. In the first case, we first use Lemma A.1. Hence we can assume that v has a compact support as a function on R2 . Then one notice that the integrals in (A.2) are invariant with respect to dilations in R2 . Therefore, we just have to consider the case: v ∈ L∞ ∩ H01 (D, R3 ). But then Lemma A.5 in [5] applies, and leads to 0 the conclusion. In the second case, namely, θ ∈ W 1,s for some s > 2 and v ∈ W 1,s , 0 we first approximate v in W 1,s with a sequence v k of smooth maps on the sphere. Then equality (A.2) applies to the pair v k , θ, and gives Z Z k 2 v · θx ∧ θy = θ · (vxk ∧ θy + θx ∧ vyk ) . (A.3) R2
R2
Now, since v k → v in L3 by Sobolev embedding theorem, then Z 2 k k s (v − v) · θ ∧ θ = o(1) . x y ≤ kdθks kv − vk s−2 2 R
On the other hand, using the embedding of W 1,s into L∞ and H¨older inequality, we R get R2 θ · (v k − v)x ∧ θy ≤ kθk∞ kdθks kd(v k − v)ks0 = o(1) and, in a similar way, R k R2 θ · θx ∧ (v − v)y = o(1). Then, we can pass to the limit in (A.3) to conclude the proof of the Lemma. Lemma A.4 Let v ∈ H 1 ∩ L∞ , and let (θk ) ⊂ H 1 be such that θk → 0 weakly in H 1 . Then Z v · θxk ∧ θyk → 0. R2
Proof. By Lemma A.1 we can assume that v is constant far out as a map on R2 , hence the support of ∇v is contained in some bounded set Ω ⊂ R2 . Since by Rellich embedding theorem we have that θk → 0 strongly in L2 (Ω), while vx ∧θy , θx ∧vy → 0 weakly in L2 (Ω), from (A.2) we get Z Z k k 2 v · θx ∧ θy = θk · (vx ∧ θyk + θxk ∧ vy ) = o(1), R2
R2
that concludes the proof. 27
Corollary A.5 Let (v k ) ⊂ H 1 ∩ L∞ be a sequence such that v k → v uniformly on S2 , for some v ∈ H 1 ∩ L∞ , and let (uk ) ⊂ H 1 be such that uk → u weakly in H 1 . Then Z Z k k k v · ux ∧ uy → v · ux ∧ uy . R2
R2
Finally, arguing as in Lemma A.3 one can prove the following result, which is useful in computing the derivative of the volume functional VH . 0
Lemma A.6 Assume that u, v ∈ W 1,s and ϕ ∈ W 1,s for some s > 2. Then Z Z ϕ · (ux ∧ vy + vx ∧ uy ) = u · (vx ∧ ϕy + ϕx ∧ vy ) . (A.4) R2
R2
If M is any square matrix of order 3, and a, b and c are vectors in R3 , then the following identity holds: a · b ∧ (M c) + a · (M b) ∧ c + (M a) · b ∧ c = (trM )a · b ∧ c .
(A.5)
(One can check (A.5) taking (a, b, c) = (ei , ej , ek ) for any triple (i, j, k)). The above results allow us to compute the directional derivative of VH at u ∈ 1,3 W along a smooth direction ϕ: Lemma A.7 Let u, ϕ ∈ W 1,3 . If H ∈ C 1 , then Z ∂ϕ VH (u) = H(u)ϕ · ux ∧ uy . R2
Proof. Direct computation leads to Z Z 0 ∂ϕ VH (u) = (Q (u) · ϕ) · ux ∧ uy + R2
Q(u) · (ux ∧ ϕy + ϕx ∧ uy ).
R2
Then one applies Lemma A.4 with v = Q(u) and the identity (A.5) to conclude.
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