Doubly connected V-states for the planar Euler equations

Doubly connected V-states for the planar Euler equations

Taoufik Hmidi, Francisco de la Hoz, Joan Mateu and Joan Verdera

Abstract

We prove existence of doubly connected V-states for the planar Euler equations which are not annuli. The proof proceeds by bifurcation from annuli at simple

“eigenvalues”. The bifurcated V -states we obtain enjoy a m-fold symmetry for some m ≥ 3. The existence of doubly connected V -states of strict 2-fold symmetry remains open.

1 Introduction

The Euler system in the plane, which governs the motion of a two dimensional inviscid incompressible fluid, is equivalent, under mild smoothness assumptions on the velocity field, to the vorticity equation







∂_tω(z, t) + v(z, t) · ∇ ω(z, t) = 0, z ∈ C, t > 0, v(z, t) = ∇^⊥4⁻¹ω(z, t),

ω(z, 0) = ω₀(z).

(1)

Here v(z, t) = v₁(z, t) + iv₂(z, t) is the velocity field at the point z ∈ C and time t and the vorticity is given by the scalar

ω(z, t) = ∂₁v₂(z, t) − ∂₂v₁(z, t), z ∈ C, t ≥ 0.

The known function ω₀(z) is the initial condition. The Biot-Savart law tells us how to recover velocity from vorticity. For a fixed time t one has

v(z, t) = ∇^⊥4⁻¹ω(z, t), z ∈ C, or, in complex notation,

v(z, t) = i 2π

Z

C

ω(ζ, t)

z − ζ dA(ζ), z ∈ C, (2)

with dA being two dimensional Lebesgue measure. The first equation in (1) simply means that the vorticity is constant along particle trajectories. A convenient reference for these matters is [BM, Chapter 2].

Yudovich Theorem asserts that the vorticity equation has a unique global solution in the weak sense provided the initial vorticity ω0 lies in L¹∩ L^∞. See, for instance, [BM,

Chapter 8]. A vortex patch is the solution of (1) with initial condition the characteristic function of a bounded domain D₀. Since the vorticity is transported along trajectories, we conclude that ω(z, t) is the characteristic function of a domain D_t. In fact, D_t = X(D₀, t) is the image of D₀ under the flow. Recall that the flow X is the solution of the ordinary differential equation

d

dtX(z, t) = v(X(z, t), t), X(z, 0) = z. (3) If D₀ is the unit disc, then the particle trajectories are circles centered at the origin and thus D_t = D₀, t ≥ 0. A remarkable fact discovered by Kirchhoff is that when the initial condition is the characteristic function of an ellipse centered at the origin, then the domain D_t is a rotation of D₀. Indeed, D_t = e^itΩD₀, where the angular velocity Ω is determined by the semi-axis a and b of the initial ellipse through the formula Ω = ab/(a + b)². See, for instance, [BM, p.304] or [L, p.232]. Kirchhoff’s result can also be checked readily using (8) below.

A rotating vortex patch or V-state is a domain D₀ such that if χ_D0 is the initial condition of the vorticity equation, then the region of vorticity 1 rotates with constant angular velocity around its center of mass, which we assume to be the origin. In other words, D_t= e^itΩD₀ or, equivalently, the vorticity at time t is given by

ω(z, t) = χ_D0(e^−itΩz), z ∈ C, t > 0.

Here the angular velocity Ω is a real number associated with D₀.

Deem and Zabusky [DZ] discovered numerically that there exist simply connected V- states with m−fold symmetry for any integer m ≥ 2. A domain D₀ is m-fold symmetric if e^2πi/mD₀ = D₀. In other words, if it is invariant by the m−th dihedral group, that is, the set of planar isometries leaving invariant a regular polygon of m sides. A few years later Burbea [B] gave an analytic proof by bifurcation at simple “eigenvalues”. See also [HMV], where the C^∞ boundary regularity of bifurcated V −states close to the disc of bifurcation was proven. Incidentally, we mention that whether the boundary of bifurcated V −states is real analytic is an open question.

In this paper we study doubly connected V -states. Recall that a planar domain is doubly connected if its complement in the Riemann sphere has two connected compo- nents. For example, an annulus is doubly connected. Because of rotation invariance, it is easy to ascertain that an annulus is a V -state. Indeed, if the annulus is

A = {z ∈ C : b < |z| < 1},

for some inner radius b, 0 < b < 1, then the vector field with vorticity χ_A is v(z) = i

2(z −b²

z )χ_A(z) + i 2

1 − b²

z χ_C\A(z).

The trajectories satisfying (3) are clearly circles centered at the origin. Hence vorticity is conserved along trajectories and ω(z, t) = χA(z) is a steady solution to equation (1).

Therefore A is a V -state rotating with any angular velocity.

No other explicit doubly connected V -state is known. In [HMV2] one proved that there do not exist doubly connected Kirchhoff like examples. In other words, the domain

between two ellipses is a V -state only if it is an annulus.

Our main result reads as follows (a more detailed statement will be given later in this section).

Theorem A. There exist doubly connected V -states which are not annuli.

The proof shows that there exist, like in the simply connected case, doubly connected V -states with m-fold symmetry for any integer m ≥ 3. See figure 1, obtained from a numerical simulation. It is remarkable that our proof breaks down for m = 2. In fact, we do not know if there are V -states with strict 2-fold symmetry, in the sense that they are 2-fold symmetric but do not have a m-fold symmetry for an even m larger than 2. This is very likely connected to the non existence of doubly connected Kirchhoff like examples.

The difficulty for m = 2 is that either the space of “eigenfunctions” is two dimensional or it is one dimensional but the transversality condition in Crandall-Rabinowitz’s theorem fails [CR] (see the statement of this basic result in section 4 below; the transversality condition is (d)).

The proof follows the general scheme of [B] and [HMV]. We first find a system of two equations, each corresponding to a boundary component of the patch, which describes doubly connected V -states. Each equation is a differentiated form of Burbea’s equation (3.1) in [B] (see also (13) in [HMV]). This differentiated form was already found useful in [HMV, (53)] in proving boundary regularity of V -states. Next step is to use conformal mapping to transport the system into the unit circle T. We then consider the Banach space of bounded holomorphic functions on {z ∈ C : |z| > 1} with derivative satisfying a H¨older condition of order α up to the boundary, and whose extension to the unit circle has real Fourier coefficients. Here α is any number satisfying 0 < α < 1. We check the hypothesis of Crandall-Rabinowitz’s Theorem for this Banach space and the transported system. This requires a lengthy but nice technical work. In particular, we find all possible

“eigenvalues” of the system, namely, those values of the bifurcation parameter (which is the angular velocity of rotation) for which the differential of the mapping giving the system has a non-zero kernel.

This paper is simpler that [HMV] from the technical point of view. The reason is that the use of the differentiated form of Burbea’s equation for V -states smoothes away technical issues. We also found a much more direct way to deal with complex functions having real Fourier coefficients, which was unnecessarily involved in [HMV]. Although throughout the present paper we work in the doubly connected context, all our proofs apply to the simply connected case, as the reader will easily realize.

We close this introductory section by stating a more precise form of Theorem A.

Theorem B. Given 0 < b < 1, let m be a positive integer such that 1 + b^m−1 − b²

2 m < 0.

Then there exists a curve of non-annular m-fold symmetric doubly connected V -states bifurcating from the annulus {z : b < |z| < 1} at each of the angular velocities

Ω_m = 1 − b²

4 ± 1

2m r

m(1 − b²) 2 − 12

− b^2m.

A remark on the meaning of Ω_m is in order. As we showed before an annulus is a V -state rotating with any angular velocity. The angular velocity plays the role of a bifurcation parameter and Ω_m is the “eigenvalue” at which bifurcation takes place.

Remark that for each frequency there are two eigenvalues Ω_m associated with the ± signs in the previous formula. The reader will find a discussion on the different behavior of the V-states bifurcating at each of the two values of Ω_min Subsection 9.3. Another way of understanding Ω_m is the following. If the curve of V -states is given by a (continuous) mapping

ξ ∈ (−, ) 7→ D(ξ),

where  is a positive number and D(ξ) is a V -state rotating with angular velocity Ω(ξ), then D(0) is the annulus {z : b < |z| < 1} and Ω(0) = Ω_m.

Dritschel found in [DR, (4.1), p. 162] a similar expression for eigenvalues in studying stability of vortices which are a perturbation of an annulus by an eigenfunction associated with a specific mode. His σ(m, a) is exactly mΩ_m−1 with b replaced by a.

2 The equation of a doubly connected V -state

Let D be a bounded doubly connected domain with boundary of class C¹. Equivalently, the boundary of D has two connected components which are Jordan curves of class C¹. Call Γ₁ the exterior curve and Γ₂ the interior one. The goal of this section is to deduce an equation which is equivalent to D being a V -state. Indeed, the equation can be thought of as a system of two equations, one for each Γ_j, j = 1, 2.

Consider the vortex patch with initial condition the characteristic function of the domain D. At time t the region of vorticity 1 is a domain D_t, which we also assume to be of class C¹. The two closed boundary curves of D_t are denoted by Γ_{1, t} and Γ_{2, t}. We know that the boundary of D_t is advected by the flow (3). It is folklore (see,for instance, [B], [HMV] and [HMV2]) that this condition can be expressed by the equation

∂z

∂t(α, t) · ~n = v(z(α, t), t) · ~n, (4) where the dot stands for the scalar product of vectors in the plane and z(α, t) is a proper parametrization of any of the curves Γ_{j, t}, j = 1, 2. By this we mean that z(α, t) is continuously differentiable in α and t and, for fix t, is a homeomorphism of the interval of parameters α with the extremes identified onto the closed curve Γj, t. The interpretation of the left hand side of (4) is the normal component of the motion of the boundary curve and the right hand side is the normal component of the motion of a particle on the curve.

Tangential components do not contribute to the motion of the boundary and are ignored.

The simplest minded argument for (4) is as follows. Let z(α, t) and w(β, t) two proper parametrizations of one of the boundary components of D_t. Then there exists a change of parameters α(β, t) such that w(β, t) = z(α(β, t), t) for all β and t. Thus

∂w

∂t(β, t) = ∂z

∂α(α, t)∂α(β, t)

∂t +∂z

∂t(α, t).

Since _∂α^∂z(α, t) is a tangent vector at the boundary at the point z(α, t) and ^∂α_∂t(β, t) is a scalar we conclude that

∂w

∂t(β, t) · ~n = ∂z

∂t(α, t) · ~n (5)

where ~n is the exterior unit normal vector at the point z(α, t) = w(β, t). Now apply (5) with w(β, t) the lagrangian parametrization, that is,

w(β, t) = X(w₀(β), t),

where X(z, t) is the flow (3) and w₀(β) is any proper parametrization of one of the boundary components of D.

Let us add to the vortex patch condition (4) the V -state requirement that Dt is a rotation of D around its center of mass, which we assume to be the origin. This amounts to say that if z₀(α) is a proper parametrization of one of the boundary components of D, then z(α, t) = e^iΩtz0(α) is a proper parametrization of the corresponding boundary component of D_t. Since the scalar product of the vectors z and w in the plane is just the real part of zw, (4) yields

Re(−i Ω z(α, t) ~n) = Re(v(z(α, t), t) ~n), (6) which can be rewritten without resorting to parametrizations as

Re(−i Ω z ~n) = Re(v(z, t) ~n), z ∈ ∂Dt,

where ~n is the exterior unit normal vector to the boundary of D_t at the point z.

By the Biot-Savart law (2)

v(z, t) = − i 2π

Z

Dt

dA(ζ)

z − ζ, z ∈ ∂D_t and by Green-Stokes

−1 π

Z

Dt

dA(ζ) z − ζ = 1

2πi Z

∂Dt

ζ − z

ζ − z dζ, z ∈ D_t.

The last identity remains true also for z ∈ ∂D_t, because both sides are continuous func- tions of z ∈ C. Therefore

Re



2Ω z + 1 2πi

Z

∂Dt

ζ − z ζ − z dζ



~τ



= 0, z ∈ ∂D_t, (7)

~

τ being the unit tangent vector to ∂D_t, positively oriented.

Notice that the left hand side of the above identity is invariant by rotations. Hence (7) holds if and only if it holds for t = 0. We conclude that the domain D is a V -state if and only if

Re(2Ω z + I(z)) ~τ = 0, z ∈ ∂D, (8)

where

I(z) = 1 2πi

Z

∂D

ζ − z

ζ − z dζ, z ∈ C.

A final remark is that the argument we have discussed gives that equation (8) charac- terizes V -states among domains with C¹ boundary, regardless of the number of boundary components. If the domain is simply connected there is only one boundary component and so only one equation. If the domain is doubly connected (8) gives actually two equa- tions, one per each boundary component. Of course, in each equation the other boundary component is present through the operator I(z).

3 Conformal mapping

In this section we transform (8) in a system of two equations on the unit circle T. Living on T has the advantage that the system can be posed in a Banach space, so that functional analysis tools become available.

Recall that our doubly connected bounded domain D has two boundary components Γj, j = 1, 2, which are Jordan curves of class C¹. Let Dj be the domain enclosed by the Jordan curve Γ_j. Let ∆ denote the open unit disc {z ∈ C : |z| < 1}. The domains C \ Dj

are simply connected and thus there are conformal mappings Φ_j : C \ ∆ → C \ Dj fixing the point at ∞. We can normalize Φ1 so that its expansion at ∞ has coefficient 1 in z, namely,

Φ₁(z) = z + a₀ +a₁

z + · · · ≡ z + f (z), (9)

valid for z outside a large disc. Here f plays the role of an analytic perturbation of the identity. The expansion of Φ₂ at ∞ is

Φ₂(z) = bz + b₀+ b₁

z + · · · ≡ bz + g(z), (10)

where 0 < b < 1. We can assume the coefficient b to be positive by making a rotation in

∆. The inequality b < 1 follows from Schwarz Lemma applied to the mapping 1/(Φ⁻¹₂ ◦ Φ₁)(1/z), |z| < 1. As before, g should be viewed as an analytic perturbation of bz.

The domain D can be written as D = D₁\ D₂ =

C \ Φ1(C \ ∆)

∩ Φ₂(C \ ∆)
. (11)

Notice that if f = g = 0, then D is the annulus {z ∈ C : b < |z| < 1}.

Set

I_j(z) = 1 2πi

Z

Γj

ζ − z

ζ − zdζ, z ∈ C, j = 1, 2, (12)

where Γ_j is oriented in the counterclockwise direction for j = 1, 2. Clearly Γ_j can be parametrized by Φ_j on T. Here appears a subtle issue related to the smoothness of Φj

and we pause momentarily to discuss it.

Assume that A is a Jordan domain with C¹ boundary, that is, a bounded simply connected domain whose boundary is a C¹ Jordan curve Γ = ∂A. It is well known that the conformal mapping Φ of C \ 4 into C \ A extends to a homeomorphism of T onto Γ and that this homeomorphism is not necessarily continuously differentiable. This is related to the mapping properties of the conjugation operator on T, concretely to the fact that it does not preserve C¹(T). The Kellogg-Warschawski theorem [P, Theorem 3.6, p.49] asserts that if Γ is of class C^1+α then Φ is of class C^1+α(T) (see next section for a precise definition of this space).

Thus we assume throughout the paper that D is a doubly connected domain with boundary of class C^1+α, for some α satisfying 0 < α < 1. The two boundary components Γ_j, j = 1, 2 are then Jordan curves of class C^1+α.

Coming back to our previous discussion we conclude that Γ_j can be parametrized by Φ_j on T and that ~τ(Φj(w)) = iwΦ⁰_j(w), |w| = 1. Notice that the preceding equation makes sense at all points w ∈ T because Φj is of class C^1+α(T) ⊂ C¹(T). Thus, taking into

account that Re(iz) = − Im(z), the single equation (8) is transformed into the system of two equations on T

Imh

2Ω Φ₁(w) + I₁(Φ₁(w)) − I₂(Φ₁(w))

w Φ⁰₁(w)i

= 0, |w| = 1 Imh

2Ω Φ₂(w) + I₁(Φ₂(w)) − I₂(Φ₂(w))

w Φ⁰₂(w)i

= 0, |w| = 1.

(13)

The functions (Ij)²_j=1 introduced in (12) take the form

I_j(z) = 1 2iπ

Z

T

φ_j(w) − z

φ_j(w) − zφ⁰_j(w)dw, z ∈ C.

It will be useful in later calculations to replace in the preceding system the angular velocity Ω by the parameter λ = 1 − 2Ω. The left hand sides of the two equations in (13) can be thought of as functions of f and g, as defined in (9) and (10). Define functions F₁(λ, f, g) and F₂(λ, f, g) on T by

F_j(λ, f, g)(w) = Im h

(1 − λ) Φ_j(w) + I(Φ_j(w))



w Φ⁰_j(w) i

, |w| = 1, (14) and a function F (λ, f, g) by

F (λ, f, g)(w) = (F₁(λ, f, g)(w), F₂(λ, f, g)(w)) , |w| = 1. (15) Hence the system (13) is equivalent to the single equation

F (λ, f, g) = 0. (16)

Therefore we have shown that if D is a bounded doubly connected V -state of class C^1+α, then equation (16) is satisfied. Conversely, if f and g are appropriate functions in C^1+α(T), then Φ1(z) = z + f (z) and Φ₂(z) = bz + g(z) can be extended to conformal mappings of C\∆ and the domain D defined by (11) is a V -state provided (16) is satisfied.

For example, if f is the boundary values of a function analytic in C \ ∆ with Lipschitz norm

sup{|f (z) − f (w)|

|z − w| : |z| > 1|w| > 1} ≡ δ < 1 (17) then Φ₁(z) = z + f (z) is conformal on C \ ∆, because

|Φ₁(z) − Φ₁(w)| ≥ |z − w| − |f (z) − f (w)| ≥ (1 − δ)|z − w|, |z| > 1|w| > 1.

Condition (17) is satisfied provided f belongs to the open unit ball of C^1+α(T). Thus if f and g are boundary values of analytic functions on C \ ∆, f belongs to the open unit ball of C^1+α(T) and g belongs to the open ball with center 0 and radius b in C^1+α(T), then Φ₁(z) = z + f (z) and Φ₂(z) = bz + g(z) are conformal on C \ ∆ and the domain D defined by (11) is a V -state provided (16) is satisfied.

In the next section we establish the precise conditions one needs to require to f and g so that V -states are produced via (16).

4 The Banach spaces for Crandall-Rabinowitz’s The- orem

In this section we discuss the Banach spaces involved in our application of Crandall- Rabinowitz’s Theorem. Its original statement in [CR, p.325] is included below for the reader’s convenience. For a linear mapping L we let N (L) and R(L) stand for the kernel and the range of L respectively. If Y is a vector space and R is a subspace, then Y /R denotes the quotient space.

Crandall-Rabinowitz’s Theorem. Let X, Y be two Banach spaces, V be a neighbor- hood of 0 in X and

F : (−1, 1) × V → Y have the properties

(a) F (λ, 0) = 0 for any |λ| < 1.

(b) The partial derivatives F_λ, F_x and F_λx exist and are continuous.

(c) N (F_x(0, 0)) and Y /R(F_x(0, 0)) are one-dimensional.

(d) F_λx(0, 0)x₀ ∈ R(F/ _x(0, 0)), where

N (F_x(0, 0)) = span{x₀}.

If Z is any complement of N (F_x(0, 0)) in X, then there is a neighborhood U of (0, 0) in R × X, an interval (−a, a), and continuous functions ϕ : (−a, a) → R, ψ : (−a, a) → Z such that ϕ(0) = 0, ψ(0) = 0 and

F⁻¹(0) ∩ U =n

(ϕ(ξ), ξx₀+ ξψ(ξ)) : |ξ| < a ∪ /big{(λ, 0) : (λ, 0) ∈ Uo .

We proceed now to define the spaces X and Y to which the above theorem will be applied. Let E be a subset of C and 0 < α < 1. We denote by C^α(E) the space of continuous functions f such that

kf k_C^α_(E):= kf k_L^∞+ kf k_α where kf k_L^∞ stands for the supremum norm of f on E and

kf k_α = sup

x6=y∈E

|f (x) − f (y)|

|x − y|^α ·

The space C^1+α(T) is the set of continuously differentiable functions f on the unit circle T whose derivatives satisfy a H¨older condition of order α, endowed with the norm

kf k_C^1+α_(T) = kf k_L^∞ + kdf

dwk_L^∞+

df dw

α

.

A word on the operator d/dw is in order. For a smooth function f we set df

dw = −ie^−iθdf (e^iθ) dθ .

It will be more convenient in the sequel, in estimating norms in C^1+α(T), to work with d/dw instead of d/dθ. This is legitimate because they differ only by a smooth factor.

Notice that we have the identity

d{f }

dw = − 1 w²

df dw.

Let C^∞ = C ∪ {∞} stand for the Riemann sphere (the one point compactification of C). Let Ca^1+α(∆^c) be the space of analytic functions on C^∞\ 4 whose derivatives satisfy a H¨older condition of order α up to T. This is also the space Ca^1+α(T) of functions in C^1+α(T) whose Fourier coefficients of positive frequency vanish. In other words,

C_a^1+α(∆^c) = C_a^1+α(T) = {f ∈ C^1+α(T) : f (w) =X

n≥0

a_nwⁿ, |w| = 1}.

Let C_ar^1+α(T) be the subspace of Ca^1+α(T) consisting of those functions in Ca^1+α(T) with real Fourier coefficients. This requirement is due to the fact that the “simple eigenvalues”

assumption in condition (c) of Crandall-Rabinobitz’s Theorem could not be proved in our context if we had worked with the full complex Banach space C_a^1+α(T). At the geometric level this assumption implies that the V -states we will find have the real line as axis of symmetry.

Define the Banach space X as

X = C_ar^1+α(T) × Car^1+α(T). (18) Given 0 < b < 1, let V stand for B(0, r₀) × B(0, r₀), where B(0, r₀) is the open ball of center 0 and radius r₀ = ¹₂min(b, 1 − b) in C_ar^1+α(T). From the above discussion is clear that if (f, g) ∈ V, then then Φ₁(z) = z + f (z) and Φ₂(z) = bz + g(z) are conformal on C \ ∆, the Jordan curves Γj = Φ_j(T) are of class C^1+α and Γ₂ is in the domain enclosed by Γ₁.

Set

H = {h ∈ C^α(T) : h(e^iθ) =X

n≥1

β_nsin(nθ), β_n ∈ R, n ≥ 1}, and define Y as

Y = H × H.

We now have the basic elements in Crandall-Rabinowitz’s Theorem : the Banach spaces X and Y, the function F (defined in (15)) and its domain R × V. We have already mentioned that F is well defined on R × V, because for (f, g) ∈ V Φ1(z) = z + f (z) and Φ₂(z) = bz + g(z) are conformal mappings on C \ ∆. It is rather easy to show that F maps R × V into Y. Discussing the details is the goal of the next section.

5 F maps into Y

Recall that F was defined in (15) as F = (F₁, F₂), where F_j(λ, f, g)(w) = Imh

(1 − λ) Φ_j(w) + I(Φ_j(w))

w Φ⁰_j(w)i

, |w| = 1. (19)

To show that F_j ∈ C^α(T) we observe that there are three relevant terms in the right hand side of the above identity : Φ⁰_j(w), which is in C^α(T), Φj(w), which is in C^1+α(T), and I(Φ_j(w)), which is in C^1+β(T), 0 < β < α, as was shown in [HMV, equation(61)].

Indeed, the fact that I(Φ_j(w)) is in C^α(T), 0 < α < 1, follows from the following simple lemma ([HMV, Lemma 4, p.191]), which we state now for future reference.

Lemma 1. Let K(w, τ ) be a measurable function on T × T \ {(w, τ ) ∈ T × T : w 6= τ } satisfying, for some positive constant C₀,

|K(w, τ )| ≤ C₀, w 6= τ,

and that for each τ ∈ T the function w → K(w, τ ) is differentiable for w 6= τ and 

∂

∂wK(w, τ )    

≤ C0

1

|w − τ |. Then the integral operator

T ϕ(w) = Z

|τ |=1

K(w, τ ) ϕ(τ ) dτ, (20)

satisfies

kT ϕkα ≤ CαC0kϕk∞, ϕ ∈ L^∞(T), 0 < α < 1, where C_α depends only on α.

The proof of the lemma follows from standard arguments (see, for example, [MOV, p.419]).

Proving that the image of F lies in Y is now reduced to ascertaining that the Fourier series expansion of Fj(λ, f, g) is of the form P

n≥1βnsin(nθ), βn ∈ R, n ≥ 1. A function h on T has a Fourier expansion of that form if and only if

h(w) = Im(X

n∈Z

β_nwⁿ), w ∈ T,

with real coefficients β_n, n ∈ Z. Therefore we have to prove that G_j(λ, f, g)(w) :=

(1 − λ) Φ_j(w) + I(Φ_j(w))

w Φ⁰_j(w), |w| = 1, (21) has real Fourier coefficients for j = 1, 2. Notice that a continuous function H defined on the circle T has real Fourier coefficients if and only if

H(w) = H(w), |w| = 1.

Owing to the definition of the space X (18) the mappings Φ_j, j = 1, 2, have real Fourier coefficients. Hence all terms appearing in the right-hand side of (21) have clearly real Fourier coefficients, except, perhaps, I ◦ Φ_j = I₁◦ Φ_j − I₂ ◦ Φ_j, j = 1, 2. Let us deal, for

example, with I_j ◦ Φ_j. One simply has to write (Ij ◦ Φj)(w) = − 1

2πi Z

|τ |=1

Φ_j(τ ) − Φ_j(w)

Φ_j(τ ) − Φ_j(w)Φ⁰_j(τ ) dτ

= 1 2πi

Z

|ζ|=1

Φ_j(ζ) − Φ_j(w)

Φj(ζ) − Φj(w)Φ⁰_j(ζ) dζ

= 1 2πi

Z

|ζ|=1

Φ_j(ζ) − Φ_j(w)

Φ_j(ζ) − Φ_j(w)Φ⁰_j(ζ) dζ

= (I_j◦ Φ_j)(w).

The other terms are treated similarly.

6 Differentiability properties of F (λ, f, g)

Recall that F (λ, f, g) is the function that gives the equation of doubly connected V - states (16). The goal of this section is to check the differentiability properties of the function F (λ, f, g) required by Crandall-Rabinowitz’s Theorem. Notice that F (λ, f, g) depends linearly on λ, so that we only have to care about the (joint) differentiability in (f, g) keeping λ fixed. Differentiability is understood in the Fr´echet sense. By definition, F = (F₁, F₂) (see (15)). Hence we will work with F_j(λ, f, g), j = 1, 2, as defined in (14).

Examining the definition of Fj one realizes that the only difficult terms are I(Φ_j(w)) = I₁(Φ_j(w)) − I₂(Φ_j(w)), |w| = 1, j = 1, 2,

and thus we have to show that the four functions I_k(Φ_j(w)), j, k = 1, 2, are continuously differentiable with respect to the variable (f, g) in the domain V . Recall that

V = B(0, r₀) × B(0, r₀), r₀ = 1

2min(b, 1 − b), (22)

where B(0, r₀) is the open ball of center 0 and radius r₀ in the Banach space C_ar^1+α(T).

Let G(f, g) be a function of (f, g) defined on V and taking values in Y. We now describe a convenient way to prove that G is differentiable on V, that later on will be applied to I_j ◦ Φ_k, j, k = 1, 2. One first shows the existence of Gˆateaux derivatives in certain particular directions. The Gˆateaux derivative of G in the direction (h, 0), h ∈ C_ar^1+α(T) at (f, g) is

d_fG(f, g)(h) := lim

t→0

G(f + th, g) − G(f, g)

t ,

where the limit is required to exist in Y (that is, in C^α(T)). We will eventually show that d_fG(f, g) is the standard partial derivative D_fG(f, g), but for now we use the notation involving the lower case d. Then one checks that d_fG(f, g)(h) is linear and bounded as a function of h, that is, that d_fG(f, g) ∈ L(C_ar^1+α(T), Y ). The next step is to prove that d_fG(f, g) is continuous as a mapping of (f, g) ∈ V into the Banach space L(C_ar^1+α(T), Y ).

In particular, this shows that, for a fixed g, the mapping f → d_fG(f, g) ∈ L(C_ar^1+α(T), Y )

is continuous of f. It is a well-known elementary fact that then the partial derivative D_fG(f, g) exists for (f, g) ∈ V and D_fG(f, g) = d_fG(f, g).

One argues similarly for the second variable g and shows that the limit d_gG(f, g)(k) := lim

t→0

G(f, g + tk) − G(f, g)

t ,

exists in Y for each k ∈ C_ar^1+α(T), that d^gG(f, g) ∈ L(C_ar^1+α(T), Y ) and that d^gG(f, g) is continuous as a function of (f, g) ∈ V into L(C_ar^1+α(T), Y ). The conclusion is that the partial derivative D_g(f, g) exists for (f, g) ∈ V and D_g(f, g) = d_gG(f, g).

Therefore the partial derivatives DfG(f, g) and DgG(f, g) exist for (f, g) ∈ V and they are continuous functions on V. Thus G(f, g) is continuously differentiable on V ([D, Chapter VIII, section 9]).

6.1 Existence of the Gˆ ateaux derivatives of F (λ, f, g)

We first compute the Gˆateaux derivative d_f(I₁ ◦ Φ₁)(f, g)(h) of I₁ ◦ Φ₁ at (f, g) in the direction (h, 0), h ∈ C_ar^1+α(T). To simplify the writing we introduce the following notation :

∆Φ₁ = Φ₁(τ ) − Φ₁(w), ∆h = h(τ ) − h(w), and

Q(t, τ, w) = ∆Φ₁+ t∆h

∆Φ₁+ t∆h(Φ⁰₁(τ ) + th⁰(τ )),

where t is a real number that is close enough to 0 to ensure that the denominator does not vanish. We claim that

d_f(I₁◦ φ₁)(f, g)(h)(w) = 1 2πi

Z

|τ |=1

∂

∂tQ(0, τ, w) dτ, (23) or, equivalently, that

Z

|τ |=1

 Q(t, τ, w) − Q(0, τ, w)

t − ∂

∂tQ(0, τ, w)

 dτ

tends to 0 in C^α(T) as t tends to 0. A straightforward computation gives

∂

∂tQ(0, τ, w) = −(∆h)(∆Φ₁) (∆Φ₁)² Φ⁰₁(τ ) + ∆h

∆Φ₁ Φ⁰₁(τ ) + ∆Φ₁

∆Φ1

h⁰(τ ),

(24)

which shows that the right hand-side of (23) is linear as a function of h. Appealing to Lemma 1 we see that this linear mapping is bounded from C_ar^1+α(T) into C^α(T). But this

fact is a consequence of the proof of (23) we are going to present. Indeed, (23) follows from Lemma 1 applied to the kernel

Kt(τ, w) := Q(t, τ, w) − Q(0, τ, w)

t − ∂

∂tQ(0, τ, w), after checking that the constant of K_t(τ, w), namely,

C₀(t) := sup

τ 6=w

|K_t(τ, w)| + sup

τ 6=w

|τ − w| | ∂

∂wK_t(τ, w)|

tends to 0 with t.

If τ 6= w, then

K_t(τ, w) = 1 t

Z t 0

 ∂

∂uQ(u, τ, w) − ∂

∂uQ(0, τ, w)

 du and thus

|K_t(τ, w)| ≤ sup

|u|<|t|

| ∂²

∂u²Q(u, τ, w)| |t|. (25)

The derivative of Q(t, τ, w) with respect to t is given by the sum

∂

∂tQ(t, τ, w) = −∆h ∆Φ₁+ t∆h

(∆Φ₁+ t∆h)² (Φ⁰₁(τ ) + th⁰(τ ))

+ ∆h

∆Φ₁ + t∆h(Φ⁰₁(τ ) + th⁰(τ )) + ∆Φ₁+ t∆h

∆Φ₁+ t∆hh⁰(τ ) and the second derivative is described by the sum

∂²

∂t²Q(t, τ, w) = 2(∆h)² ∆Φ₁+ t∆h

(∆Φ1+ t∆h)³ (Φ⁰₁(τ ) + th⁰(τ ))

− ∆h ∆h

(∆Φ₁+ t∆h)² (Φ⁰₁(τ ) + th⁰(τ )) − ∆h ∆Φ1+ t∆h

(∆Φ₁+ t∆h)² h⁰(τ )

− ∆h∆h

(∆Φ₁+ t∆h)² (Φ⁰₁(τ ) + th⁰(τ )) + ∆h

∆Φ₁+ t∆hh⁰(τ )

− ∆h ∆Φ₁+ t∆h

(∆Φ₁+ t∆h)² h⁰(τ ) + ∆h

∆Φ₁+ t∆hh⁰(τ ).

(26)

Each of the seven terms in (26) can be easily estimated by a constant C(f, h) depending only on f and h. Here we are taking t so small that

|∆Φ₁+ t∆h| ≥ |τ − w| − r₀|τ − w| − tkhk_C^1+α_(T)|τ − w| ≥ 1

3|τ − w|.

Therefore, by (25),

|K_t(τ, w)| ≤ C(f, h) |t|,

which means that the first constant of the kernel tends to 0 with t.

We now argue similarly to get an estimate for the derivative of K_t(τ, w) with respect to w. We have

| ∂

∂wK_t(τ, w)| ≤ sup

0<u<t

| ∂²

∂u²

∂

∂wQ(u, τ, w)| |t| (27)

and

| ∂²

∂u²

∂

∂wQ(u, τ, w)| ≤ C(f, h) 1

|τ − w|, (28)

for sufficiently small t. For (28) just differentiate with respect to w in (26) and notice that that the absolute value of each term one obtains can be estimated by C/|τ − w|.

The proof of (23) is now complete.

Since I₁◦ Φ₁ does not depend on g, one easily sees that d_g(I₁◦ Φ₁)(f, g) = 0.

The Gˆateaux derivatives of the remaining functions I1◦ Φ2, I2◦ Φ1 and I2 ◦ Φ2 are shown to exist as bounded linear operators from C_ar^1+α(T) into C^α(T) in the same way.

We omit the details.

6.2 Continuity of D

F (λ, f, g) and D

F (λ, f, g)

We first discuss the continuity of d_fF (λ, f, g) with respect to (f, g). Similar arguments apply for the continuity of d_gF (λ, f, g). As in the previous subsection we present the complete details of just one case. The other cases are dealt with via straightforward variations of the case considered.

Take d_f(I₁◦ φ₁)(f, g)(h)(w), which is the integral in τ, divided by 2πi, of the three terms in (24). Consider, for example, the integral of the third one

T (f, g)(h)(w) := 1 2πi

Z

|τ |=1

∆Φ₁

∆Φ₁ h⁰(τ ) dτ,

where Φ₁(w) = w + f (w) and ∆Φ₁ = Φ₁(τ ) − Φ₁(w). One has to show continuity of T (f, g) at the point (f₀, g₀) ∈ V as a mapping from V into L(C_ar^1+α(T), Y ). This case is particularly simple because T (f, g) does not depend on g. Set Φ_1,0(w) = w + f₀(w). To estimate T (f, g)(h)(w) − T (f₀, g₀)(h)(w) we just add and subtract inside the integral the term

∆Φ_1,0

∆Φ₁ h⁰(τ ) to obtain

T (f, g)(h)(w) − T (f₀, g₀)(h)(w) = 1 2πi

Z

|τ |=1

∆Φ1− ∆Φ1,0

∆Φ₁ h⁰(τ ) dτ + 1

2πi Z

|τ |=1

∆Φ_1,0 ∆Φ1,0− ∆Φ1

∆Φ₁∆Φ_1,0 h⁰(τ ) dτ

= A(w) + B(w),

where the last identity is a definition of the terms A(w) and B(w). We estimate A and B in C^α(T) by Lemma 1. Think of the integrands of A(w) and B(w) as kernels KA(τ, w) and K_B(τ, w), so that A(w) and B(w) are the integrals of the respective kernels in τ against the bounded function 1. The straightforward estimate of the absolute value of K_A is

|K_A(τ, w)| ≤ k 1

Φ⁰₁k∞kf − f₀k_C^1+α_(T)kh⁰k∞. For the kernel of B(w) we have

|K_B(τ, w)| ≤ k 1

Φ⁰₁k∞k 1

Φ⁰_1,0k∞kf − f₀k_C^1+α_(T)kf₀k_C^1+α_(T)kh⁰k∞. Since kf₀k_C^1+α_(T) ≤ 1 and k1/Φ⁰₁k∞≤ 2, because of the definition of V, we get

|K_A(τ, w)| + |K_B(τ, w)| ≤ 6 kf − f₀k_C^1+α_(T)khk_C^1+α_(T). Similar estimates yield

|∂_wK_A(τ, w)| + |∂_wK_B(τ, w)| ≤ Ckf − f₀k_C^1+α_(T)khk_C^1+α_(T)

|τ − w| ,

where C is a n absolute constant.

Thus, by Lemma 1,

kT (f, g) − T (f₀, g₀)k_L(C^1+α

ar (T),Y )≤ C kf − f₀k_C^1+α_(T).

6.3 Second order derivatives

In this subsection we remark that

∂

∂λDF (λ, f, g) (29)

exists and is a continuous function of its variables. This is straightforward because F (λ, f, g) depends linearly on λ. We easily get

∂

∂λDF₁(λ, f, g)(h, k)(w) = − Imw Φ⁰₁(w) h(w) + w Φ₁(w) h⁰(w)

(30)

and ∂

∂λDF₂(λ, f, g)(h, k)(w) = − Imw Φ⁰₂(w) k(w) + w Φ₂(w) k⁰(w). (31) It is then clear that (29) is a continuous function of (λ, f, g) ∈ R × X into the space of bounded linear mappings L(X × X, Y ).

7 Spectral study

By an eigenvalue we understand a real number λ such that the kernel of DF (λ, 0, 0) is non-trivial. Our plan is to apply Crandall-Rabinowitz’s Theorem to the equation of V -states F (λ, f, g) = 0. Hence we need to perform a spectral study of the linearized operator at the annular solution (λ, 0, 0). In particular we shall identify the ”eigenvalues”

corresponding to one-dimensional kernels and determine when the linearized operator is a Fredholm operator of zero index. Since F = (F₁, F₂), given (h, k) ∈ X, we have

DF (λ, 0, 0)(h, k) = DfF1(λ, 0, 0)h + DgF1(λ, 0, 0)k D_fF₂(λ, 0, 0)h + D_gF₂(λ, 0, 0)k

!

:= L_λ(h, k).

Before stating the main result of this section we shall introduce the following set describing the dispersion relation.

S =n

λ ∈ R : ∆n(λ, b) = 0 for some non-negative integer no

, (32)

with

∆_n(λ, b) :=

1 − λ
+ b² + n(b²− λ)

n(1 − λ) − λ

+ b²ⁿ⁺².

The meaning of ∆_n(λ, b) will become clear in (45). The implementation of Crandall- Rabinowitz theorem is connected to the following theorem which is the cornerstone of the proof of Theorem B.

Theorem 1. The following assertions hold true.

1. The kernel of L_λ : X → Y is non trivial if and only if λ ∈ S. If in addition λ 6= ^1+b₂² then the kernel is the one-dimensional vector space generated by

w ∈ T 7→ (m(1 − λ) − λ) w^m, −b^mw^m
, where m the unique integer such that ∆_m(λ, b) = 0.

2. If λ = ^1+b₂², then dim Ker L_λ ∈ {1, 2}. The kernel has dimension 2 if and only if there exists n ≥ 2 such that ∆n(^1+b₂², b) = 0.

3. For λ ∈ S\1, b²,^1+b₂² the range of L_λ is closed and is of codimension one.

4. For λ ∈1, b² , the codimension of the range is infinite.

5. For λ = ^1+b₂², the codimension of the range is 1 or 2. It is 2 if and only if there exists n ≥ 2 such that ∆_n(^1+b₂², b) = 0.

6. The transversality assumption is satisfied if and only if λ ∈ S\_1+b²

2 .

Remark 1. The transversality assumption is automatically satisfied when λ ∈ S and the associated wave number m is zero. However for m ≥ 1, since the function λ 7→ ∆_m(λ, b) is polynomial of degree 2, the transversality condition holds if and only if the discriminant is strictly positive, that is,

1 + b^m+1−1 − b²

2 (1 + m) < 0.

The proof of this theorem will be presented in several steps spread out in several sub- sections. The first step is to have at our disposal an explicit expression for the functions F₁ and F₂ which is suitable for the computations one needs to perform to describe the linearized operator.

7.1 More explicit expressions for F

and F

The non-explicit terms in the definition of F_j in (14) are I_j(Φ_k(w)), k = 1, 2. For I1(Φ1(w)), set Φ1(τ ) = τ + f (τ ) and

J₁(Φ₁(w)) = 1 2πi

Z

|τ |=1

f (τ ) − f (w)

Φ₁(τ ) − Φ₁(w)Φ⁰₁(τ ) dτ, |w| = 1.

We get, using τ − w = −w(τ − w)/τ for |τ | = |w| = 1, I₁(φ₁(w)) = 1

2πi Z

|τ |=1

τ − w

Φ₁(τ ) − Φ₁(w)Φ⁰₁(τ )dτ + J₁(Φ₁(w))

= −w 1 2πi

Z

|τ |=1

τ − w

Φ₁(τ ) − Φ₁(w)Φ⁰₁(τ )dτ

τ + J₁(Φ₁(w))

= −w + J₁(Φ₁(w)).

To check that the integral in the second line above is 1 one should realize that the expansion at ∞ of the integrand is 1/τ + a₂/τ²+ ... Similarly

I₂(φ₂(w)) = −b w + J₂(Φ₂(w)) where

J₂(Φ₂(w)) = 1 2πi

Z

|τ |=1

g(τ ) − g(w)

Φ2(τ ) − Φ2(w)Φ⁰₂(τ ) dτ, |w| = 1.

For I₁(Φ₂(w)) one sets

Ie₁(Φ₂(w)) = 1 2πi

Z

|τ |=1

f (τ )

Φ₁(τ ) − Φ₁(w)Φ⁰₁(τ )dτ.

We get

I₁(φ₂(w)) = 1 2πi

Z

|τ |=1

Φ1(τ ) − Φ2(w)

Φ₁(τ ) − Φ₂(w)Φ⁰₁(τ )dτ

= −Φ₂(w) + 1 2πi

Z

|τ |=1

Φ⁰₁(τ ) Φ₁(τ ) − Φ₂(w)

dτ

τ + eI₁(Φ₂(w))

= −Φ₂(w) + eI₁(Φ₂(w)),

where in the last identity we used that the integral over the unit circle vanishes because the integrand has a double zero at ∞.

For I₂(Φ₁(w)), one sets Φ₂(w) = bw + g(w) and

Ie₂(Φ₁(w)) = 1 2πi

Z

|τ |=1

g(τ )

Φ₂(τ ) − Φ₁(w)Φ⁰₂(τ )dτ.

We get

I₂(φ₁(w)) = 1 2πi

Z

|τ |=1

Φ₂(τ ) − Φ₁(w)

Φ₂(τ ) − Φ₁(w)Φ⁰₂(τ )dτ

= b 2πi

Z

|τ |=1

Φ⁰₂(τ ) Φ₂(τ ) − Φ₁(w)

dτ

τ + eI₂(Φ₁(w))

− φ₁(w) 1 2πi

Z

|τ |=1

Φ⁰₂(τ )

Φ2(τ ) − Φ1(w)dτ,

= b 2πi

Z

|τ |=1

Φ⁰₂(τ ) Φ₂(τ ) − Φ₁(w)

dτ

τ + eI₂(Φ₁(w))

because the winding number of Γ2 = Φ2(T) with respect to Φ¹(w) is 0. Take p with |p| > 1 such that Φ₁(w) = Φ₂(p). Then, by the residue theorem, the factor of b in the first term above is

1 2πi

Z

|τ |=1

Φ⁰₂(τ ) Φ₂(τ ) − Φ₂(p)

dτ

τ = −1 p

= − 1

Φ⁻¹₂ (Φ₁(w)). By (14) we have

2iF_j(λ, f, g) = G_j(λ, f, g) − G_j(λ, f, g), j = 1, 2, where

Gj(λ, f, g)(w) =



(1 − λ) Φj(w) + I(Φj(w))



w Φ⁰_j(w), j = 1, 2.

Therefore

G₁(λ, f, g)(w) =



−λw + (1 − λ)f (w) + J₁(Φ₁(w))

+ b

Φ⁻¹₂ (Φ₁(w))− eI₂(Φ₁(w))



w 1 + f⁰(w)

(33)

and

G₂(λ, f, g)(w) =



(1 − λ)bw − λg(w) − J₂(Φ₂(w)) + eI₁(Φ₂(w))



w b + g⁰(w)
.

7.2 Computation of DF (λ, 0, 0)

Since F_j is the imaginary part of G_j and we have the explicit expressions (33) and (7.1) for G_j, our plan is to compute the derivatives with respect to f and g at the point (λ, 0, 0) of all terms appearing in (33) and (7.1). We first show that

D_fJ₁(Φ₁(·))(λ, 0, 0) = 0.

If h ∈ C_ar^1+α(T), then

D_fJ₁(Φ₁(·))(λ, 0, 0)(h)(w) = d dt

  t=0

1 2πi

Z

|τ |=1

t h(τ ) − h(w)

τ − w + t(h(τ ) − h(w)) 1 + t h⁰(τ )
dτ

= 1 2πi

Z

|τ |=1

h(τ ) − h(w)

τ − w dτ = 0,

where the last identity is due to the fact that the integrand is a bounded analytic function in the unit disc {τ ∈ C : |τ | < 1}.

Since J1(Φ1(·)) does not depend on g,

D_gJ₁(Φ₁(·))(λ, 0, 0) = 0.

By similar arguments

D_fJ₂(Φ₂(·))(λ, 0, 0) = D_gJ₂(Φ₂(·))(λ, 0, 0) = 0. (34) Next we show that

D_f

 b

Φ⁻¹₂ ◦ Φ1



(λ, 0, 0)(h)(w) = −b²w²h(w), h ∈ C_ar^1+α(T), |w| = 1 (35) and

D_g

 b

Φ⁻¹₂ ◦ Φ₁



(λ, 0, 0)(k)(w) = b²w²k(w

b), k ∈ C_ar^1+α(T), |w| = 1. (36) For (35), take Φ1(w) = w + th(w) and Φ2(w) = bw, so that

Φ⁻¹₂ (Φ₁(w)) = 1

b(w + th(w)) and thus

d dt

   t=0

b

w + th(w) = −bw²h(w).

For (36), take Φ₁(w) = w and Φ₂(w) = bw + tk(w). Set ψ(w, t) = Φ⁻¹₂ (w). Then w = Φ2(ψ(w, t)) = bψ(w, t) + tk(ψ(w, t)).

Taking derivative with respect to t and evaluating at 0 yields 0 = b∂ψ

∂t(w, 0) + k(ψ(w, 0))

or ∂ψ

∂t(w, 0) = −1 bk(w

b).

Hence

d dt

   t=0

1

(bw + tk(w))⁻¹ = −

∂ψ

∂t(w, 0)

ψ(w, 0)² = bw²k(w b).

Clearly

D_fIe₂(Φ₁(·))(λ, 0, 0) = 0 (37) because eI₂(Φ₁(·)) vanishes if g = 0. We also have

DgIe2(Φ1(·))(λ, 0, 0) = 0.

To see that, let k ∈ C_ar^1+α(T). Then d

dt   t=0

1 2πi

Z

|τ |=1

t k(τ )

bτ + tk(τ ) − w b + tk⁰(τ )
dτ = b 2πi

Z

|τ |=1

k(τ ) bτ − wdτ

= 0.

The last identity is due to the fact that the integrand is analytic in the open unit disc and continuous up to the closed unit disc.

On the one hand,

D_gIe₁(Φ₂(·))(λ, 0, 0) = 0 (38) because eI₂(Φ₁(·)) vanishes for f = 0. On the other hand, setting

h(w) = X

n≥0

α_n 1

wⁿ, |w| ≥ 1, we get

D_fIe₁(Φ₂(·))(λ, 0, 0)(h)(w) = d dt

  _t=0

1 2πi

Z

|τ |=1

t h(τ )

τ + t h(τ ) − bw 1 + t h⁰(τ )
dτ

= 1 2πi

Z

|τ |=1

h(τ ) τ − b wdτ

=X

n≥0

α_nbⁿwⁿ

= h(w/b).

(39)

We are now ready to gather all previous calculations to compute DF (λ, 0, 0). The expression (33) of G₁, (35), (37) and the product rule for differentiation yield

DfG1(λ, 0, 0)(h)(w) =



(1 − λ)h(w) − b²w²h(w)



w + −λw + b²

w
wh⁰(w)

= (1 − λ)w h(w) − b²w h(w) + (b²− λ) h⁰(w)

and

D_fF₁(λ, 0, 0)(h)(w) = Im−((1 − λ) + b²)w h(w) + (b²− λ)h⁰(w) . Similarly

D_gG₁(λ, 0, 0)(h)(w) = b²w k(w b) and

D_gF₁(λ, 0, 0)(h)(w) = Imh

b²w k(w b)i

. By (34) and (39) we get

D_fG₂(λ, 0, 0)(h)(w) = b w h(w b ) and

D_fF₂(λ, 0, 0)(h)(w) = − Imh

b w h(w b)i

. Finally, by (34) and (38)

D_gG₂(λ, 0, 0)(h)(w) = b

−λ w k(w) + (1 − λ)k⁰(w) and

D_gF₂(λ, 0, 0)(h)(w) = b Im [λ w k(w) + (1 − λ)k⁰(w)] . Therefore

DF (λ, 0, 0)(h, k)(w) =





 Imh

− 1 − λ+b²
w h(w)+(b²− λ) h⁰(w) + b²w k(^w_b)i Imh

−b w h(^w_b) + b λw k(w) + (1 − λ) k⁰(w)
i







,

L¹_λ(h, k)(w) L²_λ(h, k)(w)

!

, (40)

which gives a convenient expression for the linearized operator DF (λ, 0, 0). To understand its kernel and range it is useful to expand the components of (40) in Fourier series. Set

h(w) =X

n≥0

α_nwⁿ and k(w) =X

n≥0

β_nwⁿ. Then by straightforward computations we obtain

L¹_λ(h, k)(e^iθ) =X

n≥0



(1 − λ) + b²+ n(b² − λ)
α_n− bⁿ⁺²β_n

sin((n + 1)θ) (41) and

L²_λ(h, k)(e^iθ) =X

n≥0



bⁿ⁺¹α_n+ b n(1 − λ) − λ
β_n

sin((n + 1)θ). (42) Therefore

DF (λ, 0, 0)(h, k)(e^iθ) =X

n≥0

M_nα_n β_n



sin (n + 1)θ

(43) with

M_n :=





(1 − λ) + b² + n(b²− λ) −bⁿ⁺² bⁿ⁺¹ b n(1 − λ) − λ



. This completes the computation of DF (λ, 0, 0).

7.3 The kernel of DF (λ, 0, 0)

Our next goal is to derive the dispersion relation which gives the relationship between the wave number n and the angular velocity Ω = ^1−λ₂ in order to get a non trivial kernel.

This will be easily follow from (41) and (42). Indeed, the couple of functions (h, k) is in the kernel of DF (λ, 0, 0) if and only if all Fourier coefficients in (41) and (42) vanish, namely,

(1 − λ) + b²+ n(b²− λ)
α_n− bⁿ⁺²β_n= 0 bⁿα_n+ n(1 − λ) − λ
β_n= 0

(44)

for n = 0, 1, 2, . . . Thus, for each non-negative frequency n, we have a linear homogeneous system of two equations in the unknowns αnand βn. The determinant of the system (44) is

∆_n = ∆_n(λ, b) =

1 − λ
+ b²+ n(b²− λ)

n(1 − λ) − λ

+ b²ⁿ⁺². (45) Thus the only way the kernel of DF (λ, 0, 0) can be non-trivial is that for some fre- quency m ≥ 0 one has ∆_m(λ, b) = 0. This non-trivial kernel is one dimensional if and only if

∆m(λ, b) = 0 and ∆n(λ, b) 6= 0, 0 ≤ n 6= m. (46) In this case a generator of Ker DF (λ, 0, 0) is the pair of functions

(m(1 − λ) − λ) w^m, −b^mw^m
, w ∈ T. (47) We pause to discuss the frequencies m = 0 and m = 1, which turn out to be specially challenging.

7.4 Eigenvalues associated with the frequencies m = 0, 1

For m = 0 the determinant of the system (44) is

∆₀ = λ²− (1 + b²)λ + b², which vanishes for λ = 1 and λ = b².

For λ = 1 the determinant ∆n is (1 − b²)



n − b²(1 + b²+ · · · + b²⁽ⁿ⁻¹⁾)



≥ n(1 − b²)² (48) and thus ∆n does not vanish for n ≥ 1. Hence the kernel of DF (λ, 0, 0) for λ = 1 is one dimensional and is generated by (h, k) = (1, 1). Therefore λ = 1 is a simple eigenvalue.

We will show in subsection 8.1 below that the codimension of the range of DF (1, 0, 0) is infinite, so that Crandall-Rabinowitz’s theorem cannot be applied. It is easily seen that for λ = 1 or, which is the same, for Ω = 0, equation (8) is translation invariant. Thus the translations Φ_ξ1(z) = z + ξ and Φ_ξ2(z) = bz + ξ give obvious solutions to (8) : translated annuli.

For λ = b² the determinant ∆_n is again the left hand side of (48) and so it does not vanish for n ≥ 1. The kernel of DF (b², 0, 0) is one dimensional and is generated by (b², 1).

Thus λ = b² is a simple eigenvalue. As in the previous case, the codimension of the range