Advances in Differential Equations

Volume 12, Number 6 (2007), 601–622

HELICOIDAL TRAJECTORIES OF A CHARGE IN A NONCONSTANT MAGNETIC FIELD Paolo Caldiroli and Michela Guida Dipartimento di Matematica, Universit` a di Torino via Carlo Alberto, 10 – 10123 Torino, Italy (Submitted by: Jean Mawhin)

1. Introduction In this note we investigate the existence of helicoidal trajectories for a charged particle in a magnetic field. More precisely, denoting by p = p(t) the position in R3 at the time t of the particle, we say that it moves along a helicoidal trajectory if there exists a versor n in R3 such that the component of p(t) in the direction of n describes a uniform right motion, whereas the projection p⊥ (t) of p(t) on a plane orthogonal to n is periodic. In particular, if the closed curve supported by p⊥ is simple; i.e., it has no self-intersections, the helicoidal trajectory p(t) will be called simple. From classical physics, in the presence of an external magnetic field B, the motion of a particle of mass m and charge e is driven by the Lorentz force, namely p(t) is a solution of m¨ p = ep˙ ∧ B.

(1.1)

When B is a uniform, constant field, namely B = b0 n for some versor n and nonzero constant b0 , one can explicitly solve (1.1) and deduce that the particle admits helicoidal trajectories p(t) which are coaxial with the magnetic field B. In particular the projection p⊥ (t) of p(t) on a plane orthogonal to B moves on a circle of radius r with constant angular speed ν. The values of ν and r are given respectively by ν=

|eb0 | , m

r=

|p˙⊥ (0)| . ν

(1.2)

Accepted for publication: November 2006. AMS Subject Classifications: 53A04, 34B15. The first author is supported by MIUR-PRIN Project “Metodi Variazionali ed Equazioni Differenziali Nonlineari”. 601

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The situation can drastically change if one switches a perturbation on in the magnetic field, even considering the simplest situation B(p) = Bε (p) = (b0 + εb(p))n

(1.3)

R3

where b is some scalar function defined on and ε is a smallness parameter. Notice that the perturbation does not affect the direction of the magnetic field but only its modulus. In this case again the component of p(t) in the direction of n follows a uniform motion, but the projection p⊥ (t) does not necessarily describe a closed (or also bounded) orbit, even for small nonzero |ε|. For instance the following nonexistence result holds: Proposition 1.1. Assume B(p1 , p2 , p3 ) = (b0 +εβ(p1 ))e3 where e3 = (0, 0, 1) and β : R → R is of class C 1 and strictly monotone. Then for every ε = 0 there is no simple helicoidal trajectory. Our goal is to provide conditions on the perturbative term b ensuring the existence of helicoidal trajectories, at least for small |ε|. We will assume that b is constant in the direction of n, namely ∂b (1.4) (p) = 0 for all p ∈ R3 . ∂n Moreover, changing sign in n, b0 and b in (1.3) if necessary, we can assume that eb0 < 0. Hereinafter we will denote by p⊥ the projection of a vector p ∈ R3 on the plane p · n = 0. Hence p⊥ identifies with a vector in R2 ≈ C and, according to (1.4), b depends just on the two components of p⊥ , so that we can write b(p) = b(p⊥ ), considering b as a mapping on R2 . We will always assume b is at least of class C 1 . Noting that, by (1.3), |p˙⊥ (t)| = |p˙⊥ (0)| for all t ∈ R, the problem of helicoidal trajectories consists in studying the existence of solutions of  m¨ p = e(b0 + εb(p⊥ ))p˙ ∧ n (P )ε |p˙⊥ | = v, p⊥ periodic where v > 0 is given. In order to state our main results let us introduce the function M : R2 → R defined by  b(ζ) dζ for every z ∈ R2 . (1.5) M (z) = Dr (z)

Here Dr (z) denotes the two-dimensional disc centered at z and with radius r given by (1.2). The mapping M can be interpreted as the Poincar´e-Melnikov

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function associated to the problem. In fact the existence of (branches of) helicoidal trajectories for small |ε| turns out to be strongly related to the existence of critical points for M and to their stability properties. Firstly let us state a necessary condition for the existence of a sequence of helicoidal trajectories converging in a suitable sense for ε → 0. Theorem 1.2. Let (εn ) ⊂ R \ {0} with εn → 0, and for every n ∈ N let p(t, εn ) be a solution of problem (P )εn for some fixed v > 0. Let μn be the minimal period of p⊥ (t, εn ) and set  μn 1 p⊥ (t, εn ) dt. zn = μn 0 If the sequence (μn ) is bounded in R and far from 0, and zn → z¯ then z¯ is a critical point of M . On the contrary, the presence of “stable” critical points of M constitutes a sufficient condition for the existence of helicoids. More precisely, we have the following: Theorem 1.3. If z¯ ∈ R2 is a nondegenerate critical point of M , then for every v > 0 and for small |ε| (depending on v), problem (P )ε admits a solution p(t, ε) drawing a simple helicoidal trajectory. Moreover, the mapping ε → p⊥ (·, ε) is of class C 1 (in the space of C 2 periodic functions) and, for ε = 0, p⊥ (t, 0) = z¯ + reiνt , where ν and r are given by (1.2). In the presence of extremal points for M we can consider a weaker stability condition, as follows. Theorem 1.4. If there exists a nonempty, open, bounded set A ⊂ R2 such that max∂A M < supA M (or min∂A M > inf A M ), then for every v > 0 and for small |ε| (depending on v), problem (P )ε admits a solution p(t, ε) drawing a simple helicoid. Moreover, denoting by zε ∈ R2 the average of p⊥ (t, ε) over its minimal period, one has that zε ∈ A, M (zε ) → supA M (or M (zε ) → inf A M , respectively) and p⊥ (t, ε) − zε → reiνt in the C 2 topology, as ε → 0, where ν and r are given by (1.2). We can state further existence results by making explicit assumptions on b (rather than on M ) when the ratio |b0 |/v is sufficiently large. In particular we have that: Theorem 1.5. Assume that one of the following conditions is satisfied: (i) b is of class C 2 and there exists a nondegenerate critical point z¯ ∈ R2 of b;

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(ii) there exists a nonempty, open, bounded set A ⊂ R2 such that max b < sup b ∂A

A

(or min b > inf b). ∂A

A

Then for ρ := |b0 |/v large enough and for small |ε| (depending on ρ), problem (P )ε admits a solution p(t, ε, ρ) drawing a simple helicoidal trajectory. Moreover, lim lim p⊥ (·, ε, ρ) − zε,ρ C 2 → 0, ρ→+∞ ε→0

where zε,ρ ≡ z¯ if (i) holds, or, in case (ii), zε,ρ is the average of p⊥ (·, ε, ρ) and satisfies: zε,ρ ∈ A and lim lim b(zε,ρ ) = sup b (or

ρ→+∞ ε→0

A

lim lim b(zε,ρ ) = inf b, respectively).

ρ→+∞ ε→0

A

In addition, in case (i), for every large ρ, the map ε → p⊥ (·, ε, ρ) is of class C 1 in the space of C 2 periodic functions. As a consequence of Theorem 1.4 we can also prove the existence of helicoidal trajectories under some decay assumption on b, as follows. Theorem 1.6. If b ∈ L1 (R2 ) + L2 (R2 ) then for every v > 0 and for small |ε| (depending on v), problem (P )ε admits a solution p(t, ε) corresponding to a simple helicoidal trajectory. Moreover, denoting by zε ∈ R2 the average of p⊥ (t, ε) over its minimal period, one has that (zε ) is bounded with respect to ε, and p⊥ (t, ε) − zε → reiνt in the C 2 topology, as ε → 0, where ν and r are given by (1.2). The main tool in the proof of Theorems 1.3 and 1.4 is the LyapunovSchmidt reduction method. In fact one can take advantage of the variational character of the problem (see Section 2) and follow a procedure introduced by Ambrosetti and Badiale [1]. This is developed in Section 3, devoted to the study of the “unperturbed” problem (P )0 , and in Section 4 where we make the finite-dimensional reduction of the “perturbed” problem (P )ε . In Section 5 we give the proofs of the above results. Finally in Section 6 we point out some geometrical problems, concerning closed curves in the plane with prescribed curvature, and (right) cylinders in R3 with prescribed mean curvature. These problems, in some cases, share the same analytical formulation of the problem of helicoids; hence in this geometrical frame we can state analogous versions of the previously stated theorems. We conclude by observing that the method of the finite-dimensional reduction we follow has been widely and successfully used for a large class

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of variational-perturbative problems. We quote especially the recent monograph [2] and the references therein. We also mention the paper [3] which exhibits significant similarities with the problem discussed here. 2. Preliminaries Let B : R3 → R3 be as in (1.3), with eb0 < 0, n a versor in R3 and b of class C 1 satisfying (1.4). Let us introduce a coordinate system such that n = (0, 0, 1), let us denote by pj the j-th component of p (j = 1, 2, 3) and let us set p⊥ := (p1 , p2 ). According to (1.4), the function b depends just on p⊥ and the problem (P )ε can be equivalently written as follows:  m¨ p⊥ = −ie(b0 + εb(p⊥ ))p˙⊥ (2.1) |p˙⊥ | = v, p⊥ periodic, where v > 0 is given and, making the usual identification between R2 and C, the product by the imaginary unit i acts as a counterclockwise rotation of π/2. Now set r := −

mv , eb0

1 b(rz) b0

κ(z) :=

for z ∈ R2

(2.2)

and observe that a function p⊥ (t) solves (2.1) if and only if ζ(t) = r−1 p⊥ ( rt v) solves  ζ¨ = i(1 + εκ(ζ))ζ˙ (2.3) ˙ = 1, ζ periodic. |ζ| Since the period of ζ(t) (or of p⊥ (t)) is a priori unknown, one makes a rescaling in order to include the unknown period in the equation. In particular one can consider the following problem: ⎧ ⎪ ¨ = i u ˙ 2 (1 + εκ(u))u˙ in [0, 1] ⎨u (2.4) u(0) − u(1) = 0 = u(0) ˙ − u(1) ˙ ⎪ ⎩ u nonconstant, where u ˙ 2 :=



1

|u| ˙2

1/2

.

0

The relationship between problems (P )ε , (2.3) and (2.4) is expressed by the next lemma.

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Lemma 2.1. If u ∈ C 2 ([0, 1], R2 ) solves (2.4), then setting ζ(t) = u(t/ u ˙ 2) for t ∈ [0, u ˙ 2 ] and extending ζ by periodicity on R, the mapping ζ belongs to C 2 (R, R2 ) and solves (2.3). Hence problem (P )ε admits a helicoidal solution p(t) with p⊥ (t) = rζ(vt/r). Conversely, if problem (P )ε admits a helicoidal solution p(t) and μ > 0 is the (minimal) period of p⊥ (t), then the mapping u(t) = r−1 p⊥ (μt) for t ∈ [0, 1] solves (2.4). The proof of Lemma 2.1 is trivial. According to Lemma 2.1 we are led to search for solutions of problem (2.4). This problem is variational in nature and its solutions can be found as critical points of a suitable energy functional associated to (2.4). More precisely, let us introduce the following functional setting: let H := {u ∈ H 1 ([0, 1], R2 ) : u(0) = u(1)} be the standard Sobolev space of 1-periodic mappings, endowed with the inner product u, v := [u] · [v] + (u, ˙ v) ˙ 2,

1 where [u] := 0 u is the average of u, and (·, ·)2 is the standard inner product in L2 ([0, 1], R2 ). It is well known that H is a Hilbert space with the above inner product and H is compactly embedded into the space of continuous, 1-periodic functions taking values in R2 . We set  u := u, u for every u ∈ H and we point out that · is a norm in H equivalent to the standard norm of H 1 ([0, 1], R2 ), because of the Poincar´e-Wirtinger inequality. Given κ ∈ C 0 (R2 ), let Qκ : R2 → R2 be a differentiable vector field such that div Qκ (z) = κ(z) for all z ∈ R2 . (2.5) A possible choice is  y  1 x κ(s, y) ds, κ(x, s) ds for z = (x, y) ∈ R2 . Qκ (x, y) = 2 0 0 Then, for every u ∈ H set  1 iQκ (u) · u. ˙ Sκ (u) := 0

We point out that Sκ is well defined in H because if u ∈ H then Qκ ◦ u ∈ L∞ and u˙ ∈ L2 .

Helicoidal trajectories of a charge

Notice that in the case κ(z) ≡ 1 one can choose Q1 (z) = obtains  1 1 S1 (u) = iu · u. ˙ 2 0

607 1 2z

and one

When u ∈ H is regular and one-to-one, S1 (u) measures (up to a sign) the area enclosed by the bounded component of R2 \ u([0, 1]). More generally, the functional Sκ can be interpreted as the κ-weighted algebraic area of the inner domain bounded by u([0, 1]). In particular, if ω(t) = e2πit ,

(2.6)

using (2.5) and the divergence theorem, for every z ∈ R2 and ρ > 0 one has that  Sκ (ρω + z) = κ(q) dq (2.7) Dρ (z)

where Dρ (z) denotes the two-dimensional disc centered at z and with radius ρ. Finally, let Eκ : H → R be the functional defined by ˙ 2 − Sκ (u) Eκ (u) := u

(2.8)

and let Ω = {u ∈ H : u is nonconstant}. Lemma 2.2. If κ ∈ C 1 (R2 ), then Eκ ∈ C 2 (Ω, R) and for every u ∈ Ω the first and the second derivative of Eκ at u are given respectively by:  1  1 1  ˙ Eκ (u)h = u˙ · h + h · iκ(u)u˙ for all h ∈ H (2.9) u ˙ 2 0 0  1    1 1  1 1  ˙ ˙ ˙ ˙ Eκ (u)[h, k] = h·k− u ˙ · h u ˙ · k (2.10) u ˙ 2 0 u ˙ 32 0 0  1 k · (iκ(u)h˙ + (∇κ(u) · h)iu) ˙ for all h, k ∈ H. + 0

Moreover, a mapping u : [0, 1] → R2 is a classical solution of (2.4) if and only if u is a critical point for E1+κ in Ω. Remark 2.3. In case κ ≡ 1, after integration by parts one finds  1  1 1 h˙ · iu for all u ∈ Ω and h ∈ H. E1 (u)h = u˙ · h˙ − u ˙ 2 0 0

(2.11)

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Proof. It is well known that the functional u → u ˙ 2 is of class C ∞ in Ω. Let us study the regularity of the functional Sκ . Fixing u, h ∈ H and taking ε = 0 one has that  1  1 Qκ (u + εh) − Qκ (u) Sκ (u + εh) − Sκ (u) ˙ Qκ (u + εh) · ih. =− · iu˙ − ε ε 0 0 Since Qκ ∈ C 1 (R2 , R2 ) and, in particular, Qκ and ∇Qκ are locally uniformly continuous, by standard arguments, using also the embedding of H into C 0 ([0, 1], R2 ), one can prove that Qκ (u + εh) − Qκ (u) = ∇Qκ (u)h and ε lim Qκ (u + εh) = Qκ (u) uniformly in [0, 1]. lim

ε→0

ε→0

Therefore, by the Lebesgue dominated convergence theorem, we infer that  1  1 Sκ (u + εh) − Sκ (u) lim ∇Qκ (u)[h, iu] ˙ − Qκ (u) · ih˙ =− ε→0 ε 0 0  1  1 =− ∇Qκ (u)[h, iu] ˙ + ∇Qκ (u)[u, ˙ ih] (2.12) 0 0  1 =− κ(u)h · iu, ˙ (2.13) 0

where (2.12) is obtained by integration by parts and (2.13) follows from (2.5) and from the algebraic formula Av · iw − Aw · iv = (tr A)v · iw

for every v, w ∈ R2 ,

where A is any 2 × 2 matrix and tr A denotes its trace. Fixing u ∈ H, the mapping  1 h → − κ(u)h · iu˙ (2.14) 0

is linear and continuous from H into R. Hence Sκ is Gateaux-differentiable in H and for every u ∈ H the Gateaux derivative of Sκ at u is the functional defined by (2.14). If (un ) is a sequence in H converging to u, then (un ) is bounded both in H and in L∞ and, since κ is locally uniformly continuous, κ ◦ un → κ ◦ u uniformly in [0, 1]. Then  1 (κ(u )h · i u ˙ − κ(u)h · i u) ˙ n n 0  1  1 ≤ |κ(un ) − κ(u)| |h · iu˙ n | + |κ(u)| |h · i(u˙ n − u)| ˙ 0

0

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≤ C κ ◦ un − κ ◦ u ∞ h + C u˙ n − u ˙ 2 h . Therefore, the Gateaux derivative is a continuous operator from H into H  and consequently Sκ is of class C 1 in H and  1  Sκ (u)h = − κ(u)h · iu˙ for every u, h ∈ H. (2.15) 0

Now let us prove that the mapping Sκ : H → H  defined by (2.15) is of class C 1 . Fixing u, h ∈ H and taking ε ∈ R, ε = 0, we have that

 Sκ (u + εh) − Sκ (u) κ(u + εh) − κ(u) ˙ ·)2 . =− iu, ˙ · − (iκ(u + εh)h, ε ε 2 Since κ ∈ C 1 (R2 ) and, in particular, ∇κ and κ are locally uniformly continuous, one obtains that κ(u + εh) − κ(u) = ∇κ(u) · h and lim κ(u + εh) = κ(u) ε→0 ε→0 ε uniformly in [0, 1]. As before, one infers that  S  (u + εh) − Sκ (u) ˙ · . lim κ = − (∇κ(u) · h)iu˙ + κ(u)ih, ε→0 ε 2 The operator  ˙ · =: Lu h h → − (∇κ(u) · h)iu˙ + κ(u)ih, (2.16) lim

2

H

from H into is linear and continuous, because the mappings ∇κ ◦ u, κ ◦ u are bounded and because of the embedding of H into L∞ . Thus Sκ is Gateaux-differentiable at any u ∈ H and its Gateaux derivative at u is the operator Lu : H → H  defined in (2.16). Now let (un ) ⊂ H be such that un → u in H. We have to prove that Lun → Lu in the uniform topology of the space of linear, continuous operators from H into H  . Writing Lun h − Lu h = ((∇κ(u) − ∇κ(un )) · h)iu˙ n , ·)2  ˙ · , + ((∇κ(u) · h)i(u˙ − u˙ n ), ·)2 + (κ(u) − κ(un ))ih, 2

using again the embedding of H into L∞ , one infers that Lun − Lu ≤ C ∇κ ◦ u − ∇κ ◦ un ∞ u˙ n 2 + C u˙ − u˙ n 2 + κ ◦ u − κ ◦ un ∞ . Since (un ) is bounded in H, un → u uniformly, and κ and ∇κ are locally uniformly continuous mappings, we derive that Lun − Lu → 0. Finally let us prove the last part of the lemma. By (2.9) one can plainly check that any classical solution of (2.4) is a critical point of E1+κ . On the contrary, let

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 u ∈ Ω be such that E1+κ (u) = 0. Setting v = i(1 + εκ(u)) u ˙ 2 u, ˙ one has 2 that v ∈ L and  1  1 u˙ · h˙ = − v · h for every h ∈ H; 0

0

namely, u˙ ∈ H and its weak derivative equals v. Hence u˙ is an absolutely continuous function, with u(0) ˙ = u(1) ˙ and  t u(t) ˙ = u(0) ˙ + v for every t ∈ [0, 1]. 0

¨ = v; namely u is a classical Since v is continuous, u˙ is of class C 1 and u 1-periodic solution of u ¨ = i(1 + εκ(u)) u ˙ 2 u. ˙ Furthermore u is nonconstant since u ∈ Ω. Hence u solves (2.4) and the proof is complete.  Throughout the rest of this paper, with a slight abuse of notation, for every functional F ∈ C 2 (Ω, R) and for every u ∈ Ω we will identify the differential F  (u) ∈ H  with its unique representative in H and similarly we will consider F  (u) as a continuous linear operator in H, writing F  (u)h = F  (u), h and F  (u)[h, k] = F  (u)h, k for every h, k ∈ H. 3. The unperturbed problem In this section we study the problem ⎧ ⎪ in [0, 1] ¨ = i u ˙ 2 u˙ ⎨u u(0) − u(1) = 0 = u(0) ˙ − u(1) ˙ ⎪ ⎩ u nonconstant

(3.1)

and the properties of the linearized problem at any solution of (3.1). ˙ 2 −S1 (u). One has that F0 = E1 (compare For every u ∈ Ω let F0 (u) := u with (2.8)) and thus, according to Lemma 2.2, F0 ∈ C 2 (Ω, R) and solutions to (3.1) correspond to critical points of F0 in Ω. In fact solutions of (3.1) parametrize unit circles anywhere placed in the plane. Indeed the following result holds true. Lemma 3.1. The solutions of (3.1) are given by u(t) = z + e2πin(t+s) where z ∈ R2 , s ∈ R and n ∈ N. Proof. A solution u of (3.1) satisfies the linear equation u ¨ = iu˙ with  = u ˙ 2 > 0. Hence u(t) = z + qei t with z, q ∈ C. Imposing the periodicity condition u(0) = u(1), since u is nonconstant, we get ei = 1, which yields  = 2πn with n ∈ N. Finally,  = u ˙ 2 implies |q| = 1, so that we can write q = e2πins with s ∈ R. 

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Let us introduce some notation. First of all, for every s ∈ R, set τs ω := ω(· + s), where ω is defined in (2.6). Now, set Z := {z + τs ω : z ∈ R2 , s ∈ R}.

(3.2)

Z is a manifold in H diffeomorphically parametrized by R2 × S1 (indeed τs+n ω = τs ω if n ∈ Z). Moreover, using Lemma 3.1, one has Z = {u ∈ Ω : F0 (u) = 0, F0 (u) = π}.

(3.3)

F0 (ω)

is a Fredholm operator of index zero. Notice Now we show that that, by Lemma 2.2 and by definition of ω one has ˙ 2 ˙ k) ˙ 2 (ω, ˙ 2 (ω, ˙ k) (h, ˙ h) ˙ ik)2 for every h, k ∈ H. (3.4) F0 (ω)h, k = − (h, − 2π (2π)3 In particular F0 (ω)ω = −

1 ω. 2π

(3.5)

Lemma 3.2. One has that ker F0 (ω) = Tω Z = R2 ⊕ Riω, where Tω Z denotes the tangent space of Z at ω. Proof. Considering the definition of Z, one has that ej = ∂zj (z + τs ω) (z,s)=(0,0) (j = 1, 2) and iω = ∂s (z + τs ω) (z,s)=(0,0) namely, Tω Z = R2 ⊕ Riω. By explicit computation one can check that R2 ⊕ Riω ⊆ ker F0 (ω). It remains to prove the opposite inequality. Let h ∈ ker F0 (ω). By (3.5) and since [ω] = 0 and F0 (ω) is symmetric, one has ˙ 2 = 0. Then, substituting into (3.4), one finds that that (ω, ˙ h) ˙ k) ˙ 2 (h, ˙ k)2 = 0 for every k ∈ H, + (ih, 2π ¨ = 2πih, ˙ and hence h(t) = z + qe2πit namely h is a weak solution in H of h ˙ with z, q ∈ C. Since (ω, ˙ h)2 = 0, it must be that q ∈ iR. In conclusion h ∈ R2 ⊕ Riω.  ⊥ Let (Tω Z) = {u ∈ H : u, h = 0 for every h ∈ Tω Z}. Note that ω ∈ (Tω Z)⊥ (use the characterization of Tω Z given by Lemma 3.2). Setting Hω := {u ∈ (Tω Z)⊥ : u, ω = 0} one has that (Tω Z)⊥ = Hω ⊕ Rω. Lemma 3.3. One has that im F0 (ω) = (Tω Z)⊥ bijection of (Tω Z)⊥ onto itself.

(3.6) and F0 (ω) (Tω Z)⊥ is a

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Proof. By Lemma 3.2 F0 (ω) is one-to-one in (Tω Z)⊥ . F0 (ω) being a selfadjoint operator, for every u ∈ H we have F0 (ω)u, h = h, F0 (ω)u = 0 for all h ∈ ker F0 (ω) and then im F0 (ω) ⊆ (Tω Z)⊥ . Now let us prove the opposite inclusion. Fix v ∈ (Tω Z)⊥ . According to (3.6) we can write v = w + sω for some w ∈ Hω and s ∈ R and we have to find u ∈ Hω and σ ∈ R such that F0 (ω)(u + σω) = w + sω. (3.7) Multiplying (3.7) by ω and using (3.5), the fact that ω ⊥ Hω and u, w ∈ Hω , we infer that σ = −2πs. Hence, by (3.5), (3.7) reduces to solving F0 (ω)u = w,

u ∈ Hω .

(3.8)

More explicitly, by (3.4) and since (u, ˙ ω) ˙ 2 = 0 and [w] = 0, (3.8) becomes ⎧ ⎨ 1 ˙ 2 = (w, ˙ 2 for every h ∈ H ˙ 2 − (iu, h) ˙ h) (u, ˙ h) (3.9) 2π ⎩u ∈ H . ω Observe that, by Lemma 3.2 and by (3.6), one has Hω = {u ∈ H : [u] = 0, (u, ˙ ω) ˙ 2 = (u, ˙ iω) ˙ 2 = 0}. We can write any w ∈ Hω according to its Fourier expansion in Hω with respect to the basis {ωn }n∈Z\{0,1} , where ωn (t) := e2πint /(2πn), namely:  w= cn ωn . n∈Z\{0,1}

Notice that the coefficients cn can be complex numbers and the product cn ωn is meant as a product in C. Setting  2πn u := cn ωn , n−1 n∈Z\{0,1}

we can easily recognize that u satisfies (3.9). This concludes the proof.



4. The finite-dimensional reduction Fixing a mapping κ ∈ C 1 (R2 ) and a value ε ∈ R, we are now interested in finding solutions to the problem (2.4) when |ε| is small. According to Lemma 2.2, solutions to (2.4) can be sought as critical points of the functional Fε (u) := E1+εκ (u) which is defined and of class C 2 on Ω. Let us observe that Fε (u) = F0 (u) − εSκ (u) for u ∈ Ω. Here F0 plays the role of an unperturbed functional and Sκ can be viewed as a perturbation.

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As proved in Section 3, the functional F0 admits a three-dimensional, nondegenerate, critical manifold Z. Our goal is to construct, for every R > 0 and for |ε| small, a “perturbed” three-dimensional manifold ZεR which is close and diffeomorphic to Z0R := {ω + z : z ∈ R2 , |z| < R} and which constitutes a so-called natural constraint for the perturbed functional Fε , namely every stationary point for Fε restricted to ZεR is in fact a free critical point for Fε . This yields the finite-dimensional reduction of the problem and it can be accomplished by means of a suitable adaptation, due to Ambrosetti and Badiale [1], of the Lyapunov-Schmidt reduction method. As a first result, we state the existence of the perturbed manifold ZεR . Lemma 4.1. For every R > 0 there exist εR > 0 and a unique C 1 mapping (ε, z) → ηε (z) ∈ H, defined in (−εR , εR ) × DR (here DR = {z ∈ R2 : |z| < R}), such that η0 (z) = 0

(4.1) ⊥

ηε (z) ∈ (Tω Z) and ηε (z) < ω ω + z + ηε (z) ∈ Ω with derivative in H Fε (ω + z + ηε (z)) ∈ Tω Z

(4.2) (4.3) (4.4)

for every ε ∈ (−εR , εR ) and z ∈ DR . For every R > 0 and for ε ∈ (−εR , εR ) the perturbed manifold is defined using the previous lemma, as ZεR := {ω + z + ηε (z) : |z| < R}. Proof. Let us consider (Tω Z)⊥ as a Hilbert space endowed with the inner product induced by H. Set O := {η ∈ (Tω Z)⊥ : η < ω } and let F : R × R2 × O × R × R2 → H be defined by F(ε, z, η, λ, α) := Fε (ω + z + η) − λiω − α. We want to apply the implicit function theorem to the equation F(ε, z, η, λ, α) = 0 in order to obtain (η, λ, α) as a function of (ε, z). Note that F(0, z, 0, 0, 0) = F0 (ω + z) = 0 for every z ∈ R2 , thanks to (3.2) and (3.3). Moreover, by Lemma 2.2, F is of class C 1 in its domain and ∂F (ε, z, η, λ, α)[v, μ, β] = Fε (ω + z + η)v − μiω − β ∂(η, λ, α) for every (v, μ, β) ∈ (Tω Z)⊥ × R × R2 . In particular, ∂F (0, z, 0, 0, 0)[v, μ, β] = F0 (ω)v − μiω − β, ∂(η, λ, α)

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since, by direct computation, F0 (ω + z) = F0 (ω). We have to show that the continuous, linear operator ∂F T := (0, z, 0, 0, 0) : (Tω Z)⊥ × R × R2 → H ∂(η, λ, α) is bijective. Injectivity of T : Let (v, μ, β) ∈ (Tω Z)⊥ ×R×R2 be such that T (v, μ, β) = 0, namely F0 (ω)v = μiω + β. By Lemma 3.3, F0 (ω)v ∈ (Tω Z)⊥ , whereas, by Lemma 3.2, μiω + β ∈ Tω Z. Hence μ = 0, β = 0, and, again by Lemma 3.3, v = 0. Surjectivity of T : Fixing u ∈ H we have to find (v, μ, β) ∈ (Tω Z)⊥ ×R×R2 such that T (v, μ, β) = u. By Lemma 3.2 we have that H = R2 ⊕ Riω ⊕ (Tω Z)⊥ and thus we can write u = [u] + isω + h with s ∈ R and h ∈ (Tω Z)⊥ . By Lemma 3.3, there exists v ∈ (Tω Z)⊥ such that F0 (ω)v = h. Hence, T (v, −s, −[u]) = F0 (ω)v + isω + [u] = u. Now we are in position to apply the implicit function theorem which ensures that for every R > 0, since DR is compact, there exist εR > 0, a neighbourhood UR of DR and a C 1 map φ : (−εR , εR ) × UR → O × R × R2 such that φ(0, z) = (0, 0, 0) and F(ε, z, φ(ε, z)) = 0

(4.5)

for every z ∈ UR and ε ∈ (−εR , εR ). Set φ(ε, z) =: (ηε (z), λε (z), αε (z)) ∈ O × R × R2

for (ε, z) ∈ (−εR , εR ) × UR . (4.6) Hence (4.1) and (4.2) are fulfilled. Moreover the fact that ηε (z) < ω implies that ω + z + ηε (z) ∈ Ω. In addition, writing explicitly (4.5) by means of (4.6), one obtains Fε (ω + z + ηε (z)) = αε (z) + λε (z)iω.

(4.7)

Notice that (4.7) is equivalent to (4.4), by Lemma 3.2. Fixing (ε, z) ∈ (−εR , εR ) × DR and setting u = ω + z + ηε (z), it remains to prove that u˙ ∈ H. Since ˙ 2 (u, ˙ h) Fε (u)h = + ((1 + εκ(u))iu, ˙ h)2 for every h ∈ H, u ˙ 2 from (4.7) it follows that ˙ 2 (u, ˙ h) ˙ 2 = −((1 + εκ(u))iu, ˙ h)2 + [αε (z)] · [h] + λε (z)(iω, ˙ h) u ˙ 2 = −((1 + εκ(u))iu, ˙ h)2 + (αε (z), h)2 + 2πλε (z)(ω, ˙ h)2 ,

(4.8)

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where (4.8) is obtained by an integration by parts and because iω˙ = −2πω. Hence we get ˙ 2 = −(v, h)2 for every h ∈ H, (u, ˙ h) ˙ . Since v ∈ L2 , we infer where v := u ˙ 2 ((1 + εκ(u))iu˙ − αε (z) − 2πλε (z)ω) that u˙ ∈ H. This completes the proof.  R As a next step, we show that Zε is a natural constraint for Fε . More precisely, define fε : DR → R by setting fε (z) := Fε (ω + z + ηε (z)) for z ∈ DR .

(4.9)

The regularity of Fε and ηε (z) ensures that fε ∈ C 1 (DR ). Lemma 4.2. If z¯ ∈ DR is a critical point for fε , for some ε ∈ (−εR , εR ), then Fε (ω + z¯ + ηε (¯ z )) = 0. Proof. By the definition (4.9) one has ∂zj fε (z) = Fε (ω+z+ηε (z)), ej +∂zj ηε (z) = Fε (ω+z+ηε (z)), ej (j = 1, 2) (4.10) ⊥  because, by (4.2), also ∂zj ηε (z) ∈ (Tω Z) whereas Fε (ω + z + ηε (z)) ∈ Tω Z (see (4.4)). Let z¯ ∈ DR be a critical point for fε , for some ε ∈ (−εR , εR ). Setting u = ω + z¯ + ηε (¯ z ), by (4.10) one has Fε (u), ej = 0 for j = 1, 2. Therefore, by (4.4) and by Lemma 3.2, one has Fε (u) = λiω for some λ ∈ R. By (4.3) u˙ ∈ H and we can compute ˙ = λiω, u ˙ = (λiω, ˙ u ¨)2 = −(λi¨ ω , u) ˙ 2 = 2πλ(ω, ˙ u) ˙ 2 = 2πλω, u . Fε (u), u Using the explicit expression of Fε (u) we also have Fε (u)u˙ =

(u, ˙ u ¨)2 + ((1 + εκ(u))iu, ˙ u) ˙ 2=0 u ˙ 2

by easy computations. Hence λω, u = 0. In fact ω, u = ω 2 + ω, ηε (¯ z ) ≥ ω ( ω − ηε (¯ z ) ) > 0 by (4.2). In conclusion λ = 0 and Fε (u) = 0.  As a last result of this section we provide the expansion of fε (z) with respect to ε in a neighbourhood of ε = 0. Lemma 4.3. For every R > 0 one has that fε (z) = F0 (ω) − εM0 (z) + o(ε) as ε → 0, uniformly on compact sets of DR , where for every z ∈ R2  M0 (z) = κ(q) dq. (4.11) D1 (z)

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1 Proof. Since the mapping (ε, z) →  fε (z) belongs to C ((−εR , εR ) × DR ) one has that fε (z) = f0 (z)+ε ∂ε fε (z) ε=0 +o(ε) as ε → 0, uniformly on compact sets of DR . From the definition of fε it follows that f0 (z) = F0 (ω+z) = F0 (ω) and

∂ε fε (z) = F0 (ω+z+ηε (z))∂ε ηε (z)−Sκ (ω+z+ηε (z))−εSκ (ω+z+ηε (z))∂ε ηε (z). Noting that, by (4.2), ∂ε ηε (z) ∈ (Tω Z)⊥ whereas F0 (ω + z + ηε (z)) ∈ Tω Z, and, fixing ε = 0, one infers that   ∂ε fε (z) ε=0 = −Sκ (ω + z) 

and the conclusion follows using (2.7). 5. Proof of the main results

Proof of Proposition 1.1. By contradiction, assume that there exists a simple helicoidal trajectory for some ε = 0. Hence, there is a C 2 mapping p⊥ : R → R2 periodic with some positive period μ, injective on [0, μ) and solving m¨ p⊥ = −ie(b0 + εβ(p1 ))p˙⊥ (here p1 is the first component of p⊥ ). Testing the equation with e1 and integrating over [0, μ] one gets  μ β(p1 )e1 · ip˙⊥ . (5.1) 0=ε 0

Using the Gauss-Green theorem, since divp (β(p1 )e1 ) = β  (p1 ), one can write   μ β(p1 )e1 · ip˙⊥ = β  (x) dx dy , (5.2) 0

D

R2

where D is the bounded domain in enclosed by range p⊥ . Since ε =  0 and β is strictly monotone, (5.1) and (5.2) yield a contradiction.  Remark 5.1. By the definitions (1.5) and (4.11) of M and M0 respectively, and by (2.2), one has 1 M0 (z) = M (rz). (5.3) b0 r 2 Moreover, by (2.7), it turns out that M0 (z) = Sκ (ω + z). Hence M0 and consequently also M are of class C 2 (see Lemma 2.2). In particular, ∂zj M0 (z) = Sκ (ω + z)ej

for all z ∈ R2 and j = 1, 2.

(5.4)

Proof of Theorem 1.2. Let εn , p(t, εn ), μn and zn be given as in the statement of the theorem. Set 1 un (t) = p⊥ (μn t, εn ). r

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One has that

zn vμn , |u˙ n | = . (5.5) r r According to Lemma 2.1, un solves problem (2.4) with ε = εn . Hence 0 = Fεn (un ) = F0 (un ) − εn Sκ (un ). By (2.11) one has that F0 (un )ej = 0 and then, since εn = 0, un ∈ Ω,

Sκ (un )ej = 0

[un ] =

for all n ∈ N and j = 1, 2.

(5.6)

By (5.5) and by the assumptions on zn and μn , the sequence (un ) is bounded in H. After extracting subsequences, we may assume μn → μ,

un → u weakly in H,

(5.7)

for some μ ≥ 0 and u ∈ H. In particular, since the sequence (μn ) is assumed to be bounded away from 0, one has μ > 0. Set μ ¯n = vμn /r and μ ¯ = vμ/r. Since ˙ 2+μ (u˙ n , h) ¯n ((1 + εn κ(un ))h, iu˙ n )2 = 0 for all h ∈ H, (5.8) passing to the limit, one obtains that u solves u ¨ = μ ¯iu. ˙ Moreover, using Rellich’s theorem, one finds that (un , iu˙ n )2 → (u, iu) ˙ 2 , and then, taking h = un in (5.8), one infers that u˙ n 22 → u ˙ 22 . Hence un → u strongly in H. As a consequence, u is a nonconstant 1-periodic solution of u ¨ = i u ˙ 2 u. ˙ Therefore, using Lemma 3.1, u(t) = [u]+e2πi¯n(t+s) for some n ¯ ∈ N and s ∈ R. In addition, one has that [u] = z¯/r. Finally, passing to the limit in (5.6) and making easy computations, one obtains 0 = Sκ (u)ej = Sκ (ω + z¯/r)ej for j = 1, 2, namely ∇M0 (¯ z /r) = 0, because of (5.4). Hence, by (5.3), ∇M (¯ z ) = 0.  2 Proof of Theorem 1.3. Let z¯ ∈ R be a nondegenerate critical point of M . Then z0 := z¯/r is a nondegenerate critical point of M0 . Fix R > |z0 | and set uε,z := ω + z + ηε (z) for any ε ∈ (−εR , εR ) and z ∈ DR , where εR and ηε (z) are given by Lemma 4.1. Note that by (4.1) u0,z = ω + z. Define G : (−εR , εR ) × DR → R2 by setting: Gj (ε, z) := Sκ (uε,z )ej

(j = 1, 2).

In particular, by (5.4) one has G(0, z) = ∇M0 (z)

for all z ∈ DR .

(5.9)

Moreover, by (2.11) and (4.10) one has that −εGj (ε, z) = Fε (uε,z )ej = ∂zj fε (z)

for all (ε, z) ∈ (−εR , εR ) × DR . (5.10)

In addition, observe that G is of class C 1 on its domain, because of the C 1 regularity of Sκ (see Lemma 2.2), and of the mapping (ε, z) → uε,z (see

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Lemma 4.1). Since z0 is a nondegenerate critical point of M0 , by (5.9), G(0, z0 ) = 0 and ∂z G(0, z0 ) = ∇2 M0 (z0 ) is invertible. Therefore, by the implicit function theorem, there exist an open neighbourhood I ⊂ (−εR , εR ) of 0 and a C 1 mapping ε ∈ I → zε ∈ DR such that G(ε, zε ) = 0 for every ε ∈ I. Hence, by (5.10) we get ∇fε (zε ) = 0 for every ε ∈ I. Then Lemmata 2.2 and 4.2 imply that uε := uε,zε is a solution of (2.4) for every ε ∈ I. Since the mapping ε → zε is of class C 1 , also ε → uε belongs to C 1 (I, H). Then a standard boot-strap argument provides the C 1 regularity from I into C 2 ([0, 1], R2 ). Finally, using Lemma 2.1, for ε ∈ I problem (P )ε admits a helicoidal solution p(t, ε) with p⊥ (t, ε) = ruε (t/με ), being με = r u˙ ε 2 /v. In particular one can check that p⊥ (t, 0) = z¯ + reiνt where ν = |eb0 |/m and r = mv/|eb0 |. Notice that the convergence in the C 2 topology (but C 1 is enough) also ensures that the helicoidal trajectory corresponding to p⊥ (t, ε) is simple, for small |ε|.  Proof of Theorem 1.4. Let A be a nonempty open, bounded subset of R2 such that supz∈A M (z) > maxz∈∂A M (z). By (5.3) one has that supz∈A0 M0 (z) > maxz∈∂A0 M0 (z), where A0 = 1r A. Fix R > 0 such that A0 ⊂ DR and take εR and ηε (z) according to Lemma 4.1. Using Lemma 4.3, for |ε| small, ε = 0, one obtains that supz∈A0 fε (z) > maxz∈∂A0 fε (z) or inf z∈A0 fε (z) < minz∈∂A0 fε (z) according to the sign of ε. In both cases there exists z¯ε ∈ A0 which is an extremal point of fε in A0 . Hence by Lemmata 4.2 and 2.2 the mapping uε = ω + z¯ε + ηε (¯ zε ) is a solution of (2.4) for every ε in a neigbourhood of 0. As a consequence, by Lemma 2.1, for small |ε| problem (P )ε admits a helicoidal solution p(t, ε) with p⊥ (t, ε) = ruε (t/με ), and με = r u˙ ε 2 /v. Now, let zε be the average of p⊥ (t, ε) over [0, με ]. One has that zε = r[uε ] = r ([ω] + z¯ε + [ηε (¯ zε )]) = r¯ zε

(5.11)

(see (4.2)). In particular zε ∈ A. Moreover, by Lemma 4.3, one has M0 (¯ zε ) → supA0 M0 and then M (zε ) → supA M as ε → 0. Now let us prove that uε − z¯ε − ω C 2 → 0

as ε → 0.

(5.12)

Firstly, since the mapping (ε, z) → ηε (z) ∈ H is of class C 1 and z¯ε runs through a bounded set, by (4.1) one has that uε − z¯ε − ω = ηε (¯ zε ) → 0 as ε → 0. Then u ¨ε = i u˙ ε 2 (1 + εκ(uε ))u˙ ε → i ω ˙ 2 ω˙ = 2πiω˙ = ω ¨ strongly in L2 as ε → 0. Hence uε − z¯ε → ω strongly in H 2 as ε → 0. By standard arguments, we have convergence in C 2 and (5.12) is proved. Finally, recalling

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(5.11), observe that ρε (t) := p⊥ (t, ε) − zε − reiνt = ruε (t/με ) − r¯ zε − reiu˙ ε 2 t/με and then, by (5.12), ρε → 0 in the C 2 topology, as ε → 0. This convergence also implies that the helicoidal trajectory corresponding to p⊥ (t, ε) is simple, for small |ε|.  Proof of Theorem 1.5. Let ρ := |b0 |/v and r := m/(|e|ρ). Moreover, set  1 M (z, r) = b(z + rq) dq for z ∈ R2 and r > 0. π D1 Observe that M (z, r) = M (z)/(πr2 ), where M (z), defined in (1.5), actually depends also on r. Notice that in the above definition of M (z, r) in fact we can take any r ∈ R and b ∈ C 2 (R2 ) implies M ∈ C 2 (R2 × R). Moreover 2 M (z, 0) = ∇2 b(z) for all z ∈ R2 . M (z, 0) = b(z), ∂z M (z, 0) = ∇b(z) and ∂zz First, assume we are in the case (i), namely b admits a nondegenerate critical 2 M (¯ ¯ (¯ z , 0) is invertible. By the z , 0) = 0 and ∂zz point z¯ ∈ R2 . Then ∂z M implicit function theorem there exists a mapping r → zr ∈ R2 of class C 1 on [0, r¯) such that z0 = z¯ and zr is a nondegenerate critical point of M (·, r), and consequently of M , for all r ∈ (0, r¯). Hence we can apply Theorem 1.3, according to which, for every r ∈ (0, r¯) there exist εr > 0 and, for all ε ∈ (−εr , εr ) a helicoidal trajectory p(t, ε, r) such that the mapping ε → p⊥ (·, ε, r) is of class C 1 from (−εr , εr ) into the space of C 2 periodic functions and moreover p⊥ (t, 0, r) = zr + reiνt . Hence limr→0 limε→0 p⊥ (·, ε, r) − z¯ C 2 = limr→0 zr − z¯ +reiνt C 2 = 0 which is the desired conclusion. Clearly one can pass from the parameter r into ρ according to the definition given at the beginning of the proof. Now let us discuss the case (ii), assuming that there exists a nonempty open bounded set A ⊂ R2 such that maxz∈∂A b(z) < supz∈A b(z). Since M (·, r) → b as r → 0, uniformly on compact sets, for r > 0 sufficiently small one has that maxz∈∂A M (z, r) < supz∈A M (z, r). Hence we can apply Theorem 1.4: for every r > 0 small enough there exist εr > 0 and, for all ε ∈ (−εr , εr ), a helicoidal trajectory p(t, ε, r). Moreover, denoting by zε,r the average of p⊥ (·, ε, r) over its minimal period, we know that zε,r ∈ A, limε→0 (p⊥ (t, ε, r) − zε,r ) = reiνt in the C 2 topology, and limε→0 M (zε,r , r) = supz∈A M (z, r). From the estimate  1 |b(zε,r ) − M (zε,r , r)| ≤ 2 |b(z) − b(zε,r )| dz ≤ r sup |∇b(z)| πr Dr (zε,r ) z∈Dr (zε,r )

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lim lim (b(zε,r ) − M (zε,r , r)) = 0.

r→0 ε→0

Moreover, since M (·, r) → b as r → 0, uniformly on compact sets, we have lim sup M (z, r) = sup b(z).

r→0 z∈A

z∈A

In conclusion limr→0 limε→0 b(zε,r ) = supz∈A b(z).  Proof of Theorem 1.6. Since b ∈ L1 (R2 ) + L2 (R2 ), also the function κ defined by (2.2) belongs to the same space. This easily implies that the mapping M0 defined by (4.11) is such that M0 (z) → 0 as |z| → +∞. If M0 is not identically null, then the assumptions of Theorem 1.4 are satisfied with A = DR for some R > 0. Hence the thesis follows. Now suppose that M0 ≡ 0. We shall show that κ ≡ 0 and then also in this case the conclusion trivially follows. Denoting by χ the characteristic function of the unit disc, for all z ∈ R2 we have that κ ∗ χ = M0 , where ∗ is the standard convolution operator. Hence κ ∗ χ ≡ 0. Since κ ∈ L1 (R2 ) + L2 (R2 ) its Fourier transform κ ˆ belongs to L2loc (R2 ) and κ ∗χ = κ ˆ χ. ˆ One knows that χ(q) ˆ = J1 (2π|q|)/|q| for q ∈ R2 \ {0}, where J1 is the Bessel function of the first kind. In particular J1 and consequently also χ ˆ admit only a countable set of zeroes. Since κ ˆχ ˆ = 0, we infer that κ ˆ = 0 almost everywhere on R2 and then κ ≡ 0. This concludes the proof.  6. A related geometric problem Let us consider a cylinder C in R3 with infinite length and circular section of radius 1. The mean curvature of C at any point p ∈ C is 1/2. After introducing a Cartesian coordinate system in R3 such that the third axis coincides with the rotational axis of the cylinder C, the mapping U : R2 \ {0} → R3 defined, in polar coordinates, by U (r, θ) = (cos θ, sin θ, log r) provides a conformal parametrization of C. Given a regular mapping H : R3 → R, we call H-cylinder a surface M in R3 having mean curvature H(p) at every point p ∈ M and admitting a conformal parametrization U : R2 \ {0} → R3 of class C 2 , diffeomorphic to U . Such a mapping U has to satisfy the equation of the prescribed mean curvature together with the conformality conditions. In polar coordinates they are respectively: 1 ΔU = 2H(U ) Ur ∧ Uθ (6.1) r

Helicoidal trajectories of a charge

Ur · Uθ = 0 = r2 |Ur |2 − |Uθ |2 .

621

(6.2)

Now assume H of class C 1 and depending just on two variables; namely there exists a unit vector n ∈ R3 such that ∂H (p) = 0 for every p ∈ R3 . ∂n Without loss of generality, up to a rotation of the coordinate system, we can take n = e3 , so that H depends only on the two first Cartesian coordinates p1 = p·e1 and p2 = p·e2 (here (e1 , e2 , e3 ) is the canonical basis in R3 ). In this situation we can look for H-cylinders admitting conformal parametrizations of the form U (r, θ) = (v1 (θ), v2 (θ), log r) (6.3) with v1 , v2 : R → R periodic. Observe that H-cylinders of this kind are invariant under translation with respect to the third axis and we will call them p3 -invariant H-cylinders. Clearly there could exist also H-cylinders which are not p3 -invariant; let us think, e.g., of the Delaunay surfaces (see [4] and also [5]). We can see that for prescribed mean curvatures H depending only on the first two components p1 and p2 , the problem of p3 -invariant H-cylinders is equivalent to the problem of closed curves in the plane with prescribed curvature K(p1 , p2 ) = 2H(p1 , p2 ). Indeed, using complex notation and setting v = (v1 , v2 ) = v1 + iv2 , the equations (6.1) and (6.2) for U of the form (6.3) are equivalent, respectively, to v¨ = 2iH(v)v˙ |v| ˙ = 1.

(6.4) (6.5)

Notice that in the argument of H we can omit the entry corresponding to the third component, since we are assuming that ∂p3 H ≡ 0. In fact, the system (6.4)–(6.5) describes analytically the problem of curves in the plane which are parametrized by arclength and with prescribed curvature K = 2H. Now we introduce the following definition. Given a regular mapping K : R2 → R let us call K-loop a closed curve Γ in 2 R such that for every z ∈ Γ the curvature of Γ at z equals K(z). Hence, if C is a p3 -invariant H-cylinder, then the intersection of C with the plane p3 = 0 is a 2H-loop. Conversely, if Γ is a K-loop and v is a parametrization of Γ by arclength, then the mapping U defined by (6.3) parametrizes a p3 -invariant (K/2)-cylinder.

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Let us focus on the perturbative case Hε (p1 , p2 , p3 ) =

1 2

+ εκ(p1 , p2 )

for (p1 , p2 , p3 ) ∈ R3 .

(6.6)

Under this assumption on H, according to the previous discussion, the problem of p3 -invariant Hε -cylinders reduces to problem (2.3) and any result obtained in the previous sections and stated in terms of helicoids can be equivalently phrased in the geometrical frame considered here. For instance, the analogue of Theorem 1.3 is the following: Theorem 6.1. Let Hε be as in (6.6) with κ ∈ C 1 (R2 ) and let M0 be as in (4.11). If z ∈ R2 is a nondegenerate critical point of M0 , then, for small |ε|, there exists a p3 -invariant Hε -cylinder Cε . In particular, Γ0 := C0 ∩ {p3 = 0} is a unit circle centered at z. In addition, for small |ε|, Γε := Cε ∩ {p3 = 0} is a 2Hε -loop admitting a uniform, 1-periodic parametrization uε such that the mapping ε → uε is of class C 1 in the space of C 2 functions. Similarly, in correspondence of Theorem 1.4 we have the following: Theorem 6.2. Let Hε and M0 be as in Theorem 6.1. If there exists a nonempty, open, bounded set A ⊂ R2 such that max∂A M0 < supA M0 (or min∂A M0 > inf A M0 ), then, for small |ε|, there exists a p3 -invariant Hε cylinder Cε . Moreover, for every small |ε| there exist a point zε ∈ A and a uniform, 1-periodic parametrization uε of the 2Hε -loop Γε = Cε ∩ {p3 = 0} satisfying: M0 (zε ) → supA M0 (or M0 (zε ) → inf A M0 , respectively) and uε (t) − (zε + e2πit ) → 0 in C 2 . Clearly, analogous results to Theorems 1.2, 1.5, 1.6 and Proposition 1.1 can be stated. References [1] A. Ambrosetti and M. Badiale, Variational perturbative methods and bifurcation of bound states from the essential spectrum, Proc. Roy. Soc. Edinburgh, A 128 (1998), 1131–1161. [2] A. Ambrosetti and A. Malchiodi, “Perturbation Methods and Semilinear Elliptic Problems on Rn ,” Progress in Mathematics, vol. 240, Birkh¨ auser (2005) [3] P. Caldiroli, R. Musina, H-bubbles in a perturbative setting: the finite-dimensional reduction method, Duke Math. J., 122 (2004), 457–484. [4] Ch. Delaunay, Sur la surface de r´evolution dont la courbure moyenne est constante, J. Math. Pures Appl., S´er. I, 6 (1841), 309–320. [5] R. Mazzeo, F. Pacard, D. Pollack, The conformal theory of constant mean curvature surfaces in R3 , in: Global Theory of Minimal Surfaces, Proceedings of the Clay Summer Institute on the Global Theory of Minimal Surfaces (David Hoffman, ed.)