Stationary motion of a self gravitating toroidal incompressible liquid layer

0.24 MB·English

by  Giorgio Fusco

Stationary motion of a self-gravitating toroidal
incompressible liquid layer
Giorgio Fusco, Piero Negrini and Waldyr M. Oliva
∗ † ‡
Abstract
2
1 We consider an incompressible fluid contained in a toroidal stratum which is only
0
subjectedtoNewtonianself-attraction. Undertheassumptionofinfinitesimaltickness
2
of the stratum we show the existence ofstationary motions during which the stratum

n
is approximatly a round torus (with radii r, R and R >> r) that rotates around
a
its axis and at the same time rolls on itself. Therefore each particle of the stratum
J
describes an helix-like trajectory around the circumference of radius R that connects
5
1 the centers of the cross sections of the torus.

]
h
p
1 Introduction
-
h
t
a Theproblemofthedeterminationofthefiguresofequilibriumofaself-gravitating rotating
m
mass has received a lot of attention in the classical literature beginning with the work of
[
Newton on the oblateness of the earth [2],[1] and [3]. The method of canals introduced by

1 NewtonwasexploitedbyMaclaurinandJacobiwhodiscoveredseveralfamiliesofellipsoids
v
ofequilibrium. Thesearchforstationarymotionsinellipsoidalregions,includingellipsoids
6
0 of equilibrium, of self-gravitating incompressible fluids was completed by Riemann [4]
1 who developed ideas of Dirichlet [5]. Poincar´e [6] adopted a global point of view and
3
introducedtheconceptofbifurcationintheattemptofdescribingthewholesetoffiguresof
.
1
equilibriumofrotatingself-gravitatingmasses. Morerecentlyseveralpapershaveappeared
0
2 approaching this classical problem with variational techniques [8] [9] [10]. The case of a
1 rotating solid torus was first considered by Poincar´e [6], see also [7], and revisited in
: T
v [8]. It is natural to expect that the motion studied in these papers belongs to a family
i of stationary motions where, beside rotating around its axis, the torus rolls on itself so
X
that the fluid particles describe a helix-like path around the circumference of radius
r
a R that connects the centers of the cross sections of the torus. We can alsoCconjecture
that beside this class of stationary motions of solid self-gravitating torii, there is also the
possibility of similar stationary motions of self-gravitating toroidal strata. This last class
of motions however will not contain as a special case the case of relative equilibrium since
for a toroidal stratum the pressure cannot compensate the self attraction that tries to
collapse the torus to the circumference . Assuming that this set of motions does exist,
C
then rigid helicoidal motions of cylindrical strata should be in the closure of the set in
∗DipartimentodiMatematica PuraedApplicata, Universit`adegli Studidell’Aquila, ViaVetoio,67010
Coppito, L’Aquila, Italy; e-mail:[email protected]
†DipartimentodiMatematica,Sapienza,Universit`adiRoma,PiazzaleAldoMoro5,00185Rome,Italy;
e-mail: [email protected]
‡CAMGSD and ISR,InstitutoSuperior Tcnico, UTL - Lisboa;e-mail:[email protected]
1
e ✻
3
✄✗ǫ2 ✡✣e
✄ ✡ 2
✄ ✡
✄ ✡
✄ ✡
✄ ✡ ǫθ
✑✸
✄ ✡ ✑
✄φ✡ ✑
✑
✄✡ r ✑
O✡✄PPPφR ✬✑ ✩θ ǫ✲1
P
t PP t
P
P
P
P
✫✪P PPqe1
Figure 1: The geometry of .
R
T
the sense that, in the singular limit R + , the motion of the toroidal stratum should
→ ∞
converge to the motion of a cylindrical stratum. In this note we adopt this point of view
and focus on the case of an incompressible self-gravitating toroidal stratum of very
R
T
small (infinitesimal) thickness and prove the existence of stationary motions of of the
R
T
type alluded to above for R >> 1 (cfr. Theorem 1.1).
We represent in the form, see Figure1
R
T
= x 3 : x = Rǫ (φ)+(r(θ)+λ)ǫ (θ,φ),
R 1 r
T { ∈R
(θ,φ) S1 S1,λ [ s(θ), s(θ)]
 ∈ × ∈ − 2 2 }
 ǫ1(φ) = cosφe1+sinφe2,

ǫ (θ,φ) = cosθǫ (φ)+sinθe ,
r 1 3


where e ,j =1,2,3 are the vector of the standard basis of 3 , θ r(θ) r is the polar
j 0
R → ≃
representation of the cross section of the middle fiber 0 = x : λ = 0 of and
TR { ∈ TR } TR
s(θ) s is the thickness of the stratum. We let r the average of r(θ) and consider the
0 0
≃
case where ε := r0 << 1. Under this assumption we regard the stationary motion of the
R
toroidal stratum as a perturbation of the limit helicoidal motion of a cylindrical stratum
that we can ideally associate to ε = 0. Since cylinder and torus are different topological
objects, the problem of continuing the motion of the fluid in the cylindrical stratum into
a stationary motion in the toroidal stratum is a singular perturbation problem. From
R
T
a mathematical point of view the singularity cylinder-torus manifest itself in the fact that
r(θ)ands(θ)andtheotherfunctionsω(θ),Ω2(θ)thatweintroducetodescribethevelocity
field on can not be expanded in powers of ε but terms of the form εklog 1, k N,
TR ε ∈
must be included.
Set
(1.1) X(θ,φ) = Rǫ (φ)+r(θ)ǫ (θ,φ).
1 r
Under the standing assumption of infinitesimal thickness the velocity field v = v(θ,φ,λ)
on does not depend on λ and can be computed on the middle fiber 0 = x =
X(θT,Rφ),(θ,φ) S1 S1 . Therefore we have v = X θ˙ +X φ˙ where subscTrRipts d{enote
θ φ
∈ × }
partialdifferentiation and˙timedifferentiation. Sincewelookforstationarymotionswhich
2
are invariant under rotations around the symmetry axis of we have
R
T
θ˙ = ω(θ),
(1.2)
φ˙ = Ω(θ),
(cid:26)
for some 2π periodic functions Ω,ω and therefore the velocity and acceleration vector
−
fields on 0 are
TR
v = X ω+X Ω,
(1.3) θ φ
a = v ω+v Ω.
θ φ
(cid:26)
From (1.1) that implies
(1.4) X = (rǫ ) , X =(R+rcosθ)ǫ , X X = 0,
θ r θ φ 2 θ φ
·
and (1.2)(1.3) and a routine computation we get
(1.5) v(θ,φ) = ω(rǫ ) +Ω(R+rcosθ)ǫ
r θ 2
= ω(r ǫ +rǫ )+Ω(R+rcosθ)ǫ ,
′ r θ 2
and
a(θ,φ) = [(r r)ω2+rω ω (R+rcosθ)cosθΩ2]ǫ
′′ ′ ′ r
− −
+ [((R+rcosθ)Ω)ω+(rcosθ)ωΩ]ǫ
′ ′ 2
(1.6) + [2r ω2+rω ω+(R+rcosθ)sinθΩ2]ǫ ,
′ ′ θ
where ǫ is defined in (1.1) and ǫ = sinθǫ + cosθe ,ǫ = sinφe + cosφe . Let
r θ 1 3 2 1 2
− −
n = n(θ,φ) = Xφ∧Xθ the exterior unit normal to 0 at X(θ,φ) and δ = δ(θ) the
thickness of the s|Xtrφa∧tXuθm| along n that is δ = sn ǫ . TThRe balance between inertial forces
r
·
and newtonian self-attraction at the typical point X(φ,θ) of 0 reads
TR
X(α,β) X(θ,φ)
∗
(1.7) a(φ,θ) = Gµ − X X (α,β)δ(α)dαdβ,
ZS1×S1 |X(α,β)−X(θ,φ)|3| φ ∧ θ|
where the integral in the r.h.s. is to be intended in the sense of Cauchy principal value.
That is
∗
= lim
ZS1×S1 l→0+Z(S1×S1)\Bl
with B a ball or radius l centered at X(φ,θ).
l
Explicit expressions of the components of the newtonian force and of their dependence
on R will bepresented in Section 3. Here we only observe that, due to the axial symmetry
of the mass distribution in the component on ǫ of the r.h.s. of (1.7) vanishes and
R 2
T
therefore (1.6) and (1.7) imply the first integral
(1.8) (R+rcosθ)2Ω = J(R),
thatexpressestheconservation of momentumof momentumwith respecttothesymmetry
axis of . Equation (1.7) must be complemented with the continuity equation that
R
T
expresses the constance of the flux through the section S of obtained by cutting
θ R R
T T
at right angle with respect to 0 along the line θ = const :
TR
(1.9) S v ǫ = C(R),
θ t
| | ·
3
where S is the measure of S and ǫ = n ǫ is a unit vector tangent at X(θ,φ) to the
θ θ t 2
| | ∧
line φ = const on 0. Observing that
TR
(rǫ ) r ǫ +rǫ
r θ ′ r θ
ǫ = = ,
t
(rǫr)θ r 2+r2
 | | ′

 n = rǫr −r′ǫθ ,p

(1.10)  r′2+r2


 p sr
δ = ,
 r 2+r2
 ′


 p
 |Sθ| = 2π(R+rcosθ)δ,



where denotes differentiation with respect to θ, we can rewrite (1.9) in the explicit form
′
(1.11) (R+rcosθ)srω = C(R).
By means of the first integral (1.8) and the continuity equation (1.11) we can determine
s and Ω once r and ω are known. This allows for transforming system (1.7), (1.11) into
an equivalent system, see (2.7) below, for the unknowns r and ω. We let r and ω be the
0 0
averages of r and ω and we represent the unknowns r and ω in the form
(1.12) r = r (1+ερ),
0
ω = ω (1+εw),
0
where ε := r0 is regarded as a small parameter and ρ and w are 2π periodic functions
R −
with zero average. We observe that the representation of in (1.1) is not unique. Indeed
R
T
aslightchangeofRcanbeexactly compensatedbyacorrespondingchangeofthefunction
θ r(θ). To make the representation (1.1) of unique we impose on the unknown ρ
R
→ T
the conditions
ρcosθ = 0,
(1.13) S1
ρsinθ = 0.
(cid:26) RS1
Our main result is the following R
Theorem1.1. Givenr > 0and ω > 0, there existsε > 0suchthat foreach ε = r0 < ε
0 0 0 R 0
system (2.7), has a 2π periodic solution r = r (1+ερ), ω = ω (1+εw) such that:
0 0
−
(i) The maps ρand w are of class C2,γ and C1,γ respectively for some γ (0,1). Moreover
∈
ρ and w have zero average and ρ satisfies (1.13).
(ii) ρ and w satisfy the estimates
lim ρ = 0,
ε 0+k kW2,2
→
lim w w¯ = 0,
ε 0+k − kW1,2
→
where w¯ := 1cosθ.
−4
4
(iii) The solution is unique in the set of maps that satisfy:
ρ + w 2 w¯ .
k kW2,2 k kW1,2 ≤ k kW1,2
(iv) The function s is of class C1,γ and the function Ω is of class C2,γ. Moreover
s s¯ = o(ε),
k − kW1,2
Ω Ω = o(ε),
− W2,2
where s¯= ω02r0 (1 ε3 cosθ) (cid:13)(cid:13)and Ω =(cid:13)(cid:13) ω0 εν(ε) with
2πµG − 4 2√π
1
ν(ε) = O( log ).
ε
r
Theorem1.1, in the limit case of infinitesimal thickness, yields an example of a station-
ary motion of a self-gravitating fluid which can not be reduced to relative equilibrium. It
is natural to expect that many other mass distribution of a self-gravitating fluid can allow
for a similar situation. For instance several toroidal strata like one inside the other
R
T
with suitable choice of the functions r,s,ω,Ω for each stratum should do.
The paper is organized as follows. In sec.2 we reduce system (1.7), (1.11) to a system,
see (2.7), of two scalar equation for the unknowns r,ω after eliminating s and Ω via the
continuity equation (1.11) and the first integral (1.8). In Sec.3 we analyze in detail the
newtonian forces and study their dependence on the parameter R. Using the analysis in
sec.2 we can write the system for r,ω as a weak nonlinear equation which, for R >> 1,
can be solved leading to the proof of Theorem 1.1.
j
We denote by φ , φ , j = 1,2, n = 1, the Fourier coefficient of a 2π periodic
0 n
··· −
integrable function φ.
2 The system for r and ω
In the following, if h : is a 2π-periodic function, we set
R → R
(2.1) h = h(θ)dθ.
h i ZS1
Let F the integral term on the r.h.s. of (1.7), that is the force of newtonian interaction.
Define
(2.2) fr = F ǫ , fθ = F ǫ , f = F ǫ = frcosθ fθsinθ.
r θ 1
· · · −
The axial symmetry of the problem implies that, as can be also verified by inspecting the
expression of F, the components fr, fθ, f depend only on the variable θ. Rewrite (1.7)
in the form
(2.3) v ω+v Ω = F.
θ φ
From the kinematic identity
(2.4) v ǫ = (v ǫ ) (v ǫ ) = 0,
θ 1 1 θ 1 θ
· · ⇒ h · i
5
and (2.3) it follows
Ω Ω2 f
(2.5) v ǫ = (R+rcos()) = ,
φ 1
h · ωi −h · ω i hωi
where we have also used (1.5) that implies v ǫ = (R+rcosθ)Ω. From (1.8) and (2.5)
φ 1
· −
we obtain
f
J2(R) = hωi ,
− 1
 hω(R+rcos())3i
(2.6)  ·


 1 f
Ω2(θ) = hωi .
−(R+rcosθ)4 1
 hω(R+rcos(·))3i



By means of this expressions of Ω2 and (2.2), (1.6), from (1.7), after dividing by ω, it
follows
cos() f fr
(r r)ω+r ω = · hωi +
′′− ′ ′ −ω(R+rcos())3 1 ω
 · hω(R+rcos())3i
(2.7)  ·


 sin() f fθ
2r ω+rω = · hωi + .
′ ′ ω(R+rcos())3 1 ω
 · hω(R+rcos(·))3i



From (1.4), (1.10) we obtain
rs
(2.8) δ X X = r 2+r2(R+rcos())
θ φ
| ∧ | √r 2+r2 ′ ·
′ p 1
= rs(R+rcos()) = C(R) .
· ω
We note that from (2.8), we have
h
(2.9) hδ X X dθdφ = C(R) dφ,
θ φ
ZS1×S1 | ∧ | ZS1hωi
for each function h : S1 S1 . As observed fr,fθ,f depend only on the variable θ,
× → R
and therefore it suffices to compute these components at φ = 0. From this observation
and (2.8) it follows
GµC(R) 1 Nr(α,β,θ)
fr = dαdβ,
R2 ω(α)D3(α,β,θ)
 ZS1×S1
(2.10)


 GµC(R) 1 Nθ(α,β,θ)
 fθ = dαdβ,
R2 ω(α)D3(α,β,θ)
 ZS1×S1


where 
RNr = (X(α,β) X(θ,0)) ǫ (θ,0)
r
− ·
= (1 cosβ)(R+r(α)cosα)cosθ+r(α)cos(α θ) r(θ),
(2.11) − − − −
RNθ = (X(α,β) X(θ,0)) ǫ (θ,0)
 θ
 = (1 cosβ−)(R+r(α)·cosα)sinθ+r(α)sin(α θ),
− −



6
and
(2.12)R2D2 = X(α,β) X(θ,0)2
| − |
= ((R+r(α)cosα)cosβ (R+r(θ)cosθ))2
−
+(R+r(α)cosα)2sin2β+(r(α)sinα r(θ)sinθ)2.
−
= 2(1 cosβ)(R2 +R(r(α)cosα+r(θ)cosθ)+r(α)r(θ)cosαcosθ)
−
+2(1 cos(α θ))r(α)r(θ)+(r(α) r(θ))2.
− − −
In the following we denote by ar, aθ and by r, θ the expressions on l.h.s. and on the
ω ω F F
r.h.s of (2.7) respectively. For later reference we list the identities
ar aθ
(2.13) cos() sin() = 0= rcos() θsin()
hω · − ω · i hF · −F · i
ar aθ
(2.14) sin()+ cos() = 0 = rsin()+ θcos() .
hω · ω · i hF · F · i
Thefirsttwoarejustarewritingof(2.4)and(2.5),thethirdisequivalentto ((rsin())ω) =
′ ′
h · i
0 and says that the component of the acceleration of the center of mass of on the axis
R
T
of is zero. The fourth identity is equivalent to F e3 = 0 which is a consequence of the
TR h ω· i
fact that there are no exterior forces acting on . In the following we will also use the
R
T
identities
(2.15) ω(r r +r θ) = 0,
′
h F F i
r ar +raθ = 0.
′
h i
The identity (2.15) says that the power p of the conservative field of force Φ sum of the
1
newtonian and centrifugal forces on the stationary motion of the torus is zero. To derive
(2.15) we note that (2.9) implies:
1
Φ v
(2.16) 0= p = Φ vδ X X dθdφ= C(R) · dφ
θ φ
ZS1×S1 · | ∧ | ZS1h ω i
= 2πC(R) ω(r r +r θ) ,
′
h F F i
where we have also used the definition of r, θ and (1.5) that implies Φv = ω(r r +
F F ω· ′F
r θ). The kinematic identity (2.15) follows from
2
F
(2.17) r ar +raθ = ω[r ((r r)ω+(r2ω)]
′ ′ ′′ ′
h i h − i
ω
= ω[(r 2ω) (r2+r 2) +(r2ω)]
′ ′ ′ ′ ′
h − 2 i
(ω2)
= (r 2ω)ω + ′(r2+r 2) (r2ω)ω
′ ′ ′ ′
h− 2 − i
= 0.
We can replace (2.7) with the equivalent system
ar = r,
(2.18) ω F
aθ = θ + r′( r ar).
( ω F r F − ω
The advantage of (2.18) with respect to the original system (2.7) is that (2.15) imply
that, in the analysis of (2.18), we don’t need to consider the projection of (2.18) on
2
the subspace of constants functions. This is a consequence of the following lemma for
f = ω(r r +r θ) (r ar +raθ) and h = 1 .
′F F − ′ rω
7
Lemma 2.1. Let f, h be 2π periodic L2( π,π) functions. Assume that c h C for
− − ≤ ≤
some constants c, C > 0. Let g = hf, k = 1. Then a necessary a sufficient condition in
h
j
order that g = 0 is that f = 0 and g = 0, n = 1, ; j = 1,2.
0 n
···
Proof. A standard computation reveals that
(2.19) f = k g + (kig1 +k2g2).
0 0 0 n n n n
n 1
X≥
The lemma follows from this and from the assumptions on h that imply k > 0.
0
3 The Newtonian Forces
In this section we assume that the unknowns ρ and w in (1.12) are 2π periodic functions
−
such that
(3.1) ρ C1,γ, w C0,γ, γ (0,1)
∈ ∈ ∈
ρ C, w C.
C1,γ C0,γ
|| || ≤ || || ≤
We analyze the smoothness and the dependence on ε = r0 of the forcing terms r, θ
R F F
on the r.h.s. of (2.7). This analysis involves the study of certain integral operators with
singular kernels which, even though are of the type considered in the literature [12], are
extended to a manifold which depends on the singular parameter ε. For this reason, in
order to estimate the dependence on ε <<1 of the norms of these operators we develop a
direct analysis.
By using (1.12), after setting
(3.2) z(t) = 2(1 cost),
−
we rewrite (2.11) as
Nr = 1( z(β)cosθ+ε( z(β)cosαcosθ z(α θ))
2 − − − −
+ε2( z(β)ρ(α)cosαcosθ z(α θ)ρ(α)+2(ρ(α) ρ(θ)))),
(3.3)  − − − −
Nθ = 1(z(β)sinθ+ε(z(β)cosαsinθ+2sin(α θ))
 +2 ε2(z(β)ρ(α)cosαsinθ+2ρ(α)sin(α θ−))).
−


A similar computation leads to
(3.4) D2 = z(β)[1+ε(cosα+cosθ)
+ε2(cosαcosθ+ρ(α)cosα+ρ(θ)cosθ)
+ε3(ρ(α)+ρ(θ))cosαcosθ+ε4ρ(α)ρ(θ)cosαcosθ]
+ε2z(α θ)[1+ε(ρ(α)+ρ(θ))+ε2(ρ(α)ρ(θ)
−
(ρ(α) ρ(θ))2
+ − )].
z(α θ)
−
Lemma 3.1. The components fr and fθ given by (2.10) can be expressed in the form:
fr = GµC(R) 3 Kr(β,α θ,ε)(1+Sr(ρ,w;α,β,θ,ε))dαdβ,
(3.5) 2ω0R2 h=1 S1 S1 h − h
( fθ = G2µωC0R(R2) P2h=1RS1×S1Khθ(β,α−θ,ε)(1+Shθ(ρ,w;α,β,θ,ε))dαdβ,
×
P R
8
where
2ε2(ρ(α) ρ(θ))
(3.6) Kr = − ,
1 3
(z(β)+ε2z(α θ))2
−
1 εz(α θ)
Kr = K = − ,
2 ε 2 −(z(β)+ε2z(α θ))32
−
z(β)cosθ
Kr = cos(θ)K = ,
3 − 3 −(z(β)+ε2z(α θ))32
−
1 2εsin(α θ)
Kθ = K = − ,
1 ε 1 (z(β)+ε2z(α θ))32
−
z(β)sinθ
Kθ = sin(θ)K = ,
2 3 3
(z(β)+ε2z(α θ))2
−
and, under the assumption (3.1), Sr(ρ,w) and Sθ(ρ,w) are C0,γ functions such that
h h
(3.7) Sr(ρ,w) , Sθ(ρ,w) Cε,
|| h ||C0,γ || h ||C0,γ ≤
Sr(ρ ,w ) Sr(ρ ,w ) Cε (ρ ,w ) (ρ ,w ) ,
h 1 1 h 2 2 C0,γ 1 1 2 2
|| − || ≤ || − ||
Sθ(ρ ,w ) Sθ(ρ ,w ) Cε (ρ ,w ) (ρ ,w ) ,
h 1 1 h 2 2 C0,γ 1 1 2 2
|| − || ≤ || − ||
where (ρ,w) := ρ + w .
C1,γ C0,γ
|| || || || || ||
Proof. If we factor out D2(β,α θ) = z(β)+ε2z(α θ) from the expression of D2
0 − −
given by (3.4) we get D2 = D2(1+d(ρ;α,β,θ,ε)) with
0
z(β)
(3.8) d(ρ;α,β,θ,ε) = (ε(cosα+cosθ)
D2(β,α θ)
0 −
+ε2(cosαcosθ+ρ(α)cosα+ρ(θ)cosθ)
+ε3(ρ(α)+ρ(θ))cosαcosθ+ε4ρ(α)ρ(θ)cosαcosθ)
ε2z(α θ)
+ − (ε(ρ(α)+ρ(θ))+ǫ2(ρ(α)ρ(θ)
D2(β,α θ)
0 −
(ρ(α) ρ(θ))2
+ − )).
z(α θ)
−
From (3.1) and (3.8) it follows
(3.9) d(ρ) Cε(1+ ρ ),
C0,γ C1,γ
k k ≤ || ||
d(ρ ) d(ρ ) Cε ρ ρ .
1 2 C0,γ 1 2 C1,γ
k − k ≤ || − ||
and therefore we have
1 1 ∞ 3
(3.10) = 1+ −2 d(ρ;α,β,θ,ε)n ,
D3 D3(β,α θ) n
0 − n=1(cid:18) (cid:19) !
X
where, if ε (0,ε ] for some ε > 0, the series on the r.h.s converges absolutely and
0 o
∈
uniformly on S1 S1 S1.
× ×
9
Similarly
1 1 ∞ 1 1 ∞
(3.11) = 1+ εm − wm = 1+ ( 1)mεmwm .
ω ω m ω −
0 m=1 (cid:18) (cid:19) ! 0 m=1 !
X X
In conclusion we can rewrite (2.10) in the form
GµC(R) 2Nr
(3.12) fr = (1+S(ρ,w))dαdβ,
2ω R2 D3
0 ZS1×S1 0
GµC(R) 2Nθ
fθ = (1+S(ρ,w))dαdβ.
2ω R2 D3
0 ZS1×S1 0
where
n 3
∞
(3.13) S(ρ,w) = ( 1)mεm −2 wmd(ρ)n−m
− n m
n=1m=0 (cid:18) − (cid:19)
X X
From (3.9) and the inequalities
(3.14) Πnu (n+1)Πn u
|| 1 j||C0,γ ≤ 1|| j||C0,γ
Πnu Πnv
|| 1 j − 1 j||C0,γ ≤
n
n Πh 1 v Πn u u v
0− || j||C0,γ h+1|| j||C0,γ|| h− h||C0,γ
h=1
X
valid for u ,v C0,γ,j = 1,...,n, u = u = v = v = 1 we see that
j j 0 n+1 0 n+1
∈
n 3
∞
(3.15) ||S(ρ,w)||C0,γ ≤ εn(n+1)Cn n−2m
n=1 m=0(cid:12)(cid:18) − (cid:19)(cid:12)
X X (cid:12) (cid:12)
(cid:12) (cid:12)
∞ εnCn (cid:12) (cid:12)
≤ 1
n=1
X
n 3
∞
||S(ρ1,w1)−S(ρ2,w2)||C0,γ ≤ εnn2Cn−1 n−2m ||(ρ1,w1)−(ρ2,w2)||
n=1 m=0(cid:12)(cid:18) − (cid:19)(cid:12)
X X (cid:12) (cid:12)
(cid:12) (cid:12)
∞
εnCn (ρ ,w )(cid:12) (ρ ,w ) ,(cid:12)
≤ 1|| 1 1 − 2 2 ||
n=1
X
where C and C are constants independent on n. From this and the expressions (3.3) of
1
Nr and Nθ that we rewrite as
2Nr = z(β)cosθ(1+εcosα+ε2ρ(α)cosα)
−
εz(α θ)(1+ερ(α))+2ε2(ρ(α) ρ(θ)),
(3.16)  − − −
2Nθ = z(β)sinθ(1+εcosα+ε2ρ(α)cosα)


 +2εsin(α θ)(1+ερ(α)),
−


the lemma follows.
10