Local bifurcation and symmetry

E

c~(n) •

(6)

Analogously we find from equation (2) : (F w - F w)n + (F w - F w)n xyy yyx x xyx xxy y

+·In pn

Vn E c~(n)

, (7)

where, as before, F = FO+f. 2 2 Furthermore, we know that f and w belong to WO ' (n) (see lemma 3.21); consequently, the relations (6) and (7) remain meaningfull for all

(8)

with similar inequalities for the other terms. This suggests the following definition. 2.4.6. Definition. A gen~alized ~otution of the equations (1)-(3) is a pair of functions f and w belonging to w~,2(n), and such that the relations (6) and (7) are satisfied for each

2 2 ' (n). We have, for each = WO

00

CO(n) (9) 49

By theorem 3.19 the norm 1.l z Z is equivalent to the usual norm in 'H. Since C~(Q) is dense in H, (9)show~ that we can use the following inner product and associated norm in H : (u,v)

fQ

=

(~u)(~v)

Vu,v E H ,

(10)

Vu

(11)

and lIuli

= (fQ (~u)Z)l/Z

E H •

Let now FO E WZ,Z(Q) and p E LZ(Q) be fixed. For each f and w in H, the right-hand sides of (6) and (7) define bounded linear functionals of ¢ and Using the Riesz representation theorem this allows us to define elements B(f,w), Aw and p in H such that: (B(f,w) ,¢)

=

fQ (fxywy - fyywx)¢x

+

fQ (fxywx - fxxwyHy , V¢

E H

(13) V<jJ E H •

(14)

Z.4.9. Theorem. The generalized solutions of (1)-(3) coincide with the solutions (f,w) E HxH of the equations w = Aw

+

B(f,w)

+

p ( 15)

1

f =-2 B(w,w) which are equivalent to f

=

-21 B(w,w) ,

w - Aw

+

C(w)

=

(16 )

p ,

where C : H + H is given by

so

(17)

C(w)

1 = ZB(B(w,w),w).

o

(18)

2.4.10. Lennna. 'The operator A : H

-+

H is a bOllilded, selfadjoint and compact

linear operator. proof. Estimates similar to (8) show that

which implies the boundedness and compactness of A, since H is compactly imbedded in W6,4(rl). The selfadjointness follows from (Aw,v) is easily verified for v,w E C~(~).

=

(Av,w), which

0

2.4.11. Lennna. The operator B : H x H -+ H is bounded, bilinear and compact. Moreover, the mapping (u,v,w)

+>

(B(u,v),w) defines a synnnetric multilinear

fonn over H : (B(u,v),W)

=

= B(u,w) ,v)

(B(v,u),w)

2.4.12. Lemma. The map C

Vu,v,w E H

(19)

H -+ H is compact and cubic Va

Em.

and is the gradient of the CCO -fllilctional (J (J(w)

o

1 = "4(C(w),w)

Vw

E H •

Vw H -+

m.

E H

(20)

defined by

o

(21)

The proofs are similar to the proof of lemma 10. Remarkt that the synnne-

try relation (19) implies that (C(w),w) = IIB(w,w) 112

Vw

E

H .

(22)

We also have :

DwC(w) . u =

1 -2B (B(w, w) ,u)

+ B (B(u,w) ,w)

Vu,w E H .

(23)

51

2.4.13. Example. Consider a circular plate subjected to a uniform compressive pressure at its boundary. We can aSSlnne that the radius is equal to 1, so we have :

We will assume that the compressive pressure is proportional to a real parameter A ;;. 0; then we can take 1

2

2

FO(x,y) = -ZA(X +y )

(24)

(If A < 0, this represents a uniform tension along the boundary). The equations take the form : 2

1

f'. . f "'-jw,wj (25)

2

f'. . w = [w, f 1 - Af'...W + P together with the clamped plate boundary conditions f

Y

o=

w '" w = w along 3Q . x Y

(26)

The operator A is defined by

fQ

=

-fQ

-fQ

cjJM

(Aw)f'...2cjJ

wf'...cjJ

E Ii

VcjJ

E

C~(Q)

\lcjJ.\lcjJ ;;. 0

VcjJ

E

C~W)

Vw

(27)

It follows that

(AcjJ,cjJ) =

fQ

SO A is a positive operator, i.e. (Aw,w) ;;. 0

Vw

E

H .

(28)

Since A is compact and self-adjoint, its spectYllffi consists of real eigenvalues converging to zero. So the characteristic values of A, i.e. those A E 1R for which the equation 52

w - AAwO

(29)

has a nontrivial solution w E H, form a sequence A1 < A2 < ... < An < ... converging to +00. Each of these characteristic values has a finite lTD.lltiplicity. We now determine some of these characteristic values. 2.4.14. Characteristic values corresponding to radially symmetric eigenfunctions. Since Q is a smooth domain, solutions of (29) correspond to classical solutions of the following boundary value problem 2

in Q

6. w + A6.W = 0

w=wx =wy

0

in

em

(30)

Since both the domain Q and the equation (30) are rotationally invariant, it follows from the representation theory for the rotation group as given in section 6 that eigenfunctions of (30) either are radially symmetric, or are linear combinations of functions of the form ak(r)coske and ak(r)sinke, for SOIre k = 1 ,2, . .. (Here (r, e) are the polar coord ina tes of the point x E Q). In the first case the corresponding characteristic value is generically simple, in the second case its lTD.lltiplicity is at least two, generically it is equal to two. In order to determine the characteristic values of (30) correspondong to radially symmetric eigenfunctions, let w = w(r) be a radially symmetric solution of (30). Then it is easily seen that ~(r) = ~;(r) satisfies 2 d

~ +.l~ + (A-2-2)~ = 0

dr 2

~(O)

r dr

=0

r

~(1)

(31) =

0 .

By the positiveness of A we only have to consider the case A > o. Putting ~(r)

= ~(~r), the equation (31) and the first boundary condition ~(O) = 0 show that ~(x) = J 1 (x), the first order Bessel function. Then the second boundary condition ~(1) = 0 takes the form

o,

(32) S3

and gives us the characteristic values of A corresponding to radially syrrnne-':, tric solutions. These characteristic values are simple and form a sequence \1 < \2 < ... < \n < ... tending to +00; \1 corresponds to an eigenfunction which satisfies w(x) > 0, Vx E Q. All other characteristic values of (30) are at least double; one can prove that they are all strictly greater than

2.4.1S. The simply supported plate. Consider again the von Karman equations for the buckling of a plate, bllt let us assume now that the plate is simply supported instead of being clamped. The equations (1) and (2) remain, but there are different boundary conditions; the general theory for this case has been described in Berger & Fife [18]. Here we will only consider the partiClllar case of a rectangtllar plate, for which the theory can be considerably simplified. Let the domain occupied by the plate in its undeformed state be {(x,y)

Q

I -t<x< t,

-1
We assume that a compressive thTllst T is applied to the edges x = ±t, while also a normal load acts on the plate. After some rescalings, the von Karman equations for the equilibrium state of the plate take the form (see Bauer & Reiss [ 14 ]) 6, 6,

2

1 f =-'i£w,w]

2

w = [w,f] -

(33) AWXX +

P

together with the boundary conditions f

M

o

w

6,w

0 along aQ •

(34)

Here A is proportional to the thTllst T, and p is proportional to the normal load; the other entities have the same meaning as in the case of the clamped plate. 2.4.16. Definition. A

of the problem (33)-(34) is a pair of functions (f,w) belonging to c4(Q)nc2(~) and satisfying (33) and (34) S4

cta6~~cal ~of~on

pointwise. Before defining generalized solutions and putting the problem in an operator form, let us remark that the domain ~ has a smooth boundary, except at the four corners; even at these corners the singularity is not too bad, since 3~ (corners included) satisfies a Lipschitz condition, that is, 3Q is of class CO,1. Under this condition the imbedding and compactness theorem 4 remains valid if we replace W~,p(Q) by wm,p(Q) (see Adams [11). Also wm,p(Q) is the completion of Coo(~) under the wm,p(Q)-norm. 2.4.17. A function space. Let us define now a Hilbert space which will be the underlying function space for the generalized form of the problem (33)(34). Consider the space of all functions u E Coo(~) satisfying u(x) = 0 for x E 3~. Let H be the completion of this space under the W2 ,2(Q)-norm. It is clear that

Moreover, since W2 , 2 (Q) is continuously imbedded in cO ,ct(~) for any 0.;;; ct < 1 , H will consist of continuous functions which are zero along the boundary 3Q. In fact : 2 2 H = {uEW ' (Q)

I u(x)

=0, ¥xE 3Q}

(35)

2.4.18. Lenma. The inner product (u,v)H = fQ (6U)(6V)

Vu,v E H

(36)

Vu

(37)

and the corresponding norm E

H

2 2 determine a topology in H equivalent to the W ' (Q)-topology. Proof. It is immediate that lIuli H .;;; Cllull 2 2 for some constant C. In order , to prove the coercivity of II.II H, remark first that

f

Q

(6U)2 =

I Ictl =2

r IDctul 2

VuEH.

(38)

.Q

55

Indeed, (38) can be explicitly verified for u E COO (~) n H; the boundary terms appearing after integration by parts cancel out because of the special geometry of the domain ~ and the fact that u = 0 along 3~. It remains to show that there is a constant C > 0 such that

I lal=2

II ~

Dau 12

~

(f u 2

C

I

+

~

letl=l

f IDau I2)

Vu

E H ,

~

(39)

(i.e. lult2 ~ CIIUIlL2)' If no such constant exists, then there is a sequence {u I nElN} in H such that lIu 111 2 = 1, Vn E IN, and lu 17 ') -+ 0 as n n, 2 2 7 2n ~,'" n -+ This implies that {u } is bounded in W ' ; since W~' is compactly . b e dd e d 'ln W1 , 2 , we may assume n . W1 ' 2 • Slnce . 1m t h at {u } converges to some u ln Iun 12 , 2 -+ 0, {un } also converges in W2 , ~, and its limit u has vanishing second derivatives. So u must be linear, but then u E H implies that u(x) = 0, Vx E ~. This, however, contradicts the fact that lIun 111 ,~7 = 1 for all nElN. 0 From now on we equip H with the topology given by the lemma. 00.

2.4.19. Generalized solutions. Let (f,w) be a classical solution of the problem (33)-(34). 'Then f and w both belong to H. Let $ and n be Coo(fi) functions vanishing along d~. If we multiply (33) by $, and (34) by n, and integrate over ~, then we find : w -w w) $ _.1. f (w w -w w) $ f~ (M)(M) = _.1.2 I~ (wxyy yyx x 2 ~ xyx xxy y

f~ (6w) (6n)

=

(40)

f (fxyy w -f w)n + f (f w -f w )n yyx x xyx xxy y ~

+

~

AI ~wxnx

+

f pn . ~

(41)

Using the compact imbedding of W2,2(~) into W1 ,4(s"2) , we can, as in tion 7, prove that the relations (40) and (41) remain valid for each $ in H. This suggests the following definition : 2.4.20. Definition. A g~n~a£iz~d botution of the problem (33)-(34) is a of functions (f,w) , both belonging to H, such that (40) and (41) are satisfied for each $ and n in H. 2.4.21. Theorem. Every classical solution of (33)-(34) is also a solution. 56

Conversely, every generalized solution is a classical solution in :JUd':J, where d':J is equal to d:J with the corners deleted. 0 The main (and most difficult) part of the proof consists in showing that each generalized solution is of class C4 in:J U d':J. The proof uses a socalled bootstrapping argument, and can be found in Knightly & Sather [124] for the particular case under consideration here, in Berger & Fife [18] for the general case. 2.4.22. Theorem. The generalized solutions of (33)-(34) coincide with the solutions f E H and w E H of the pair of equations : w - "AAw + C(w) f

=

=

(42)

P

1

( 43)

-Z-B(w,w)

Here A is a compact selfadjoint positive linear operator of H in itself, B is a compact, continuous bilinear operator mapping H x H into H, and finally C is a compact, cubic COO map of H into itself. 0 The proof, and the definition of the operators are analogous to the case of the clamped plate. So we have (Aw,ep)

=

f wxepx

(44)

Yep E H ;

:J

(B(u,v) ,ep) C(u)

=

In

f (u v -u v)cp + (UxyVx-Uxxvy)epy :JXYYYYxx"

1

= ~B(B(u,u),u)

Yep E H

(45) (46)

t..

Yep E H .

(47)

2.4.23. The linearized problem. The linear problem w-"AAw

0

(48)

has, by the compactness, selfadjointness and positivity of A, only nontrivial solutions wE H for a sequence of positive values for "A, tending to +00. Let us detennine these characteristic values. By theorem 21, equation (48) is equivalent with the classical boundary value problem: 57

Indeed, (38) can be explicitly verified for u E eco(~)nH;the boundary terms appearing after integration by parts cancel out because of the special geometry of the domain ~ and the fact that u = 0 along d~. It remains to show that there is a constant e > 0 such that

I lal=Z

I IDau IZ ;;. e (J ~

u Z+

~

I

J IDau IZ)

10,1=1

Vu

E

H ,

~

(39)

(i.e. lul~ , z;;' ellull~ , Z)· If no such constant exists, then there is a sequence {u I nE:lN} in H such that lIu 111 Z = 1, Vn E:lN, and lu I? Z -+ 0 as "1"1es t hat {un }"1S ·b oun n d'd" " 1 Z, Zn" ~, n -+ co. Th ~1S llnp e 1n h.T Z' Z; Slnce vV 1S compact1y imbedded in W1 ,Z, we may assume that {u } converges to some u in W1 ,Z. Since lun Iz, Z -+ 0, {un } also converges in WZ,~, and its limit u. has vanishing second derivatives. So u must be linear, but then u E H implies that u(x) = 0, Vx E ~. This, however, contradicts the fact that !lun Ill? = 1 for all nE:lN. 0 ,~ From now on we equip H with the topology given by the lemma. Z.4.19. Generalized solutions. Let (f,w) be a classical solution of the problem (33)-(34). 'Then f and w both belong to H. Let ¢ and T) be eCOc?}) functions vanishing along d~. If we multiply (33) by cP, and (34) by n, and integrate over 0" then we find : w -w w) cP _.l J (w w -w w) cP J~(M)(M) = _.lZ f~ (wxyy yyx x Z ~ xyx xxy y

I~ (lIw) (LIT))

= J~ (fxyy w -fyyx w)T) x + +

AI ~wx T)x

+

f

~

(40)

(fxyx w -fxxy w)n y

I~PT) .

(41 )

Using the compact imbedding of WZ,Z(0,) into W1 ,4(Sl), we can, as in tion 7, prove that the relations (40) and (41) remain valid for each cP and n in H. This suggests the following definition : Z.4.Z0. Definition. A gen~alized ~ofution of the problem (33)-(34) is a of functions (f,w), both belonging to H, such that (40) and (41) are satisfied for each ¢ and T) in H. Z.4.Z1. Theorem. Every classical solution of (33)-(34) is also a solution. 56

Conversely, every generalized solution is a classical solution in S1 U d' S1, where 3'S1 is equal to 3S1 with the corners deleted. 0 The main (and most difficult) part of the proof consists in showing that each generalized solution is of class C4 in S1 U d' S1. The proof uses a socalled bootstrapping argument, and can be found in Knightly & Sather [124] for the particular case under consideration here, in Berger & Fife [18] for the general case. 2.4.22. Theorem. 'The generalized solutions of (33)-(34) coincide with the solutions f E H and w E H of the pair of equations : w - "AAw + C(w) f

=

=

P

(42) ( 43)

--±B(W,w)

Here A is a compact selfadjoint positive linear operator of H in itself, B is a compact, continuous bilinear operator mapping H x H into H, and finally C is a compact, cubic COO map of H into itself. 0 The proof, and the definition of the operators are analogous to the case of the clamped plate. So we have : (Aw, ¢)

=

JS1wx¢x

(B(u,v) ,¢) C(u)

=

=

IS1 (uA)'y v -u v)¢ yyx

¥¢

X

+

(44)

E H ;

In (UxyVx· -UxxVyHy

¥¢

E H

( 45)

"

-±B(B(U,u),u)

(46) ¥¢

E H •

(47)

2.4.23. The linearized problem. The linear problem w-"AAw

0

(48)

has, by the compactness, selfadjointness and positivity of A, only nontrivial Solutions wE H for a sequence of positive values for "A, tending to +00. Let us determine these characteristic values. By theorem 21, equation (48) is equivalent with the classical boundary value problem: 57

f:, 2w + AWXX

in Q

0

in

w = f:,w = 0

(49)

em

This problem can easily be solved by using a Fourier expansion for w 00

I

w(x,y) =

m,n=1

amn si~(X+£) si~(y+1) .

(SO)

Bringing this in (49) one observes that (49) has only the trivial solution w = 0 except when (51)

The characteristic value Amn corresponds to the eigenfunction (52)

where the constant emn can be chosen in such a way that the set {
(53)

It is clear that each characteristic value Amn has finite multiplicity if we fix some (m,n), then the set cr(m,n)

=

{(m' ,n')

I Am'n'

=

Amn}

(54)

is finite. If we define : (55)

I - AmnA then ker Lmn and 58

=

span{
I (m' ,n') E cr(m,n)}

(56)

R(L

mn

)

(ker Lmn) {WE

HI

1

(w'¢m'n')

=

0, V(m' ,n')

E

o(m,n)} .

(57)

2.4.24. First characteristic value. Now we prove the following result TIle lowest characteristic value for C49) is simple, except when i 2 = m(m+1) for some m = 1,2, ... In that case the lowest characteristic value has llRlltiplicity two. Pro a f. It is immediate from (51) that Amn> Am1 if n> 1, and that for each m. Now Am1

=

02 i2 2 -2(m +-) 4i m

2

The fWlCtion x

+>-

x +~ attains its minirum in x> 0 at the point x x

L So

we can expect the lowest characteristic value AO to be equal to Am,l for m = [ i 1 or m = [i+ 1 ]. If i = m, then AO = A l' and AO is simple. If m, 2 . m < .~ < m+l, then Am,l = Am+1, -I if and only if i = mCm+1). In that case AO has multiplicity two. In all other cases AO is simple. The special importance of the first characteristic value follows from the next resul ts . 2.4.25. Lemma. Let H = W~,2(~), respectively as in 2.4.17. Let C : H 7 H be as in (18), respectively (46). Let w E H be a solution of (29), respectively (48), for some A. E R. Suppose also that (C(w),w) = O. 'Then w = o. Pro a f. We have in both cases that (C(w),w) = II B(w,w) II 2 (efr. (22)), so (C(w),w) = 0 implies that B(w,w) = 0, or, by the definition of B and since w is sufficiently smooth :

f [W,W]¢=o ~

Consequently : w w

xx yy

w(x)

=

-w2

xy

0

=

0

in

~

Vx

E

d~

•

59

Considering the surface w = w(x,y), this implies that the Gaussian curvature of this surface is identically zero. This means that the surface is developable and contains a family of straight lines. Because of the boundary condition on w this is only possible for w = o. 0 2.4.26. Theorem. Let 1..0 be the lowest characteristic value for the problem (29), respectively (48). Then the equation w - I..PJN + C(w)

o

(58)

has, for A < 1.. 0 ' only the solution w

=

O.

Proof. 1..0 has the following variational characterization (see Courant & Hilbert [48], Weinberger [ 234]) :

-1 = sup (Aw,w) 1..0 \\EH (w,w)

(59)

wrO

Let w t 0 be a solution of (58), for some 1..< 1.. 0 . Then (22) implies that (w- Aw,w) < O. By (59) this is only possible if A = 1..0 and w E ker LAO. But then (58) gives (C(w),w) = 0, which, by lemma 25 gives w = 0, a contradiction. 0 2.4.27. The cylindrical plate. As a last example we consider a cy~indrical plate which is simply supported, axially compressed, and subject to a radial loading. The von Karman-Donnel equations which govern the equilibrium state of such system reduce after some rescaling to : 2

1

t, f = - -2I w, w 1 + awxx ' t, 2w = [w,f] - AWxx - afxx

(60) +

p .

These equations hold in the domain the boundary condition w

=

t,w

=

0

f

t,w

and the periodicity condition 60

n = {(x,y) I -TIl<x< TIl},

0 along 8Q

together with

w(x,y) = w(x,y+2n) , f(x,y) = f(x,y+2n)

, V(x,y)

E

rt .

(62)

The constant a is proportional to the curvature of the cylinder. In our setting x measures distances parallel to the axis, while y measures distances along the circumference of the cylinder; this explains the periodicity condition (62). One can use the same approach to these equations as we did before for flat plates. The basic Hilbert space will be the closure in the W2 ,2(rt O)norm of the Coo(~)-functions which are zero along art and 2n-periodic in y. (Here rto = {(x,y) I-nt<x
P

(63)

Elimination of f gives (w - !cAw +

i A2w)

+ aQ (w) + C(w)

=

p ,

where the quadratic and compact operator Q

(64 ) H + H is given by

1 Q(w) = B(w,Aw) + ZAB(w,w) .

(65)

2.4.28. Characteristic values. The characteristic values for the cylindrical plate problem are those. A E lR for which the linearized equation (66) has a nontrivial solution. We have 1

.rz-:-:i

lJ±(A) = Z(A± VA -4a ) ,

(67)

which implies that (68)

Consequently, A E

lR

is a characteristic value if and only if lJ+ (A) or lJ_ (A) 61

are a characteristic value of A, that is, if and only if A = ~ + a2/~ for some characteristic value~ of A. The characteristic values ~ of A are detennined by the bOLmdary value problem : I'!,zw + w

~wxx

0

= 6w = 0

in rI in

(69)

,m ,

which we have to solve for solutions w = w(x,y) which are 2'IT-periodic in y. Using Fourier expansions one finds : m=1,2, ... The characteristic values ~m, 0 for m to the eigenfunction

0,1,2, ...

n

(70)

1,2, ... are simple, &id correspond

(71 ) For n > 0, ~mn is a characteristic value of ponding to the eigenfunctions :

A

of multiplicity two, corres-

(72)

and m,n sin 2rr;,7v (X+1TJI,) sin ny .

C

(73) .

The constants Care nOllllalization constants. m,n 2.4.29. Bibliographical notes. There exists an extensive literature on the buckling problem for plates. An account of the physical motivation for the von Karman equations can be found in Landau and Lifschitz [139J and Leipholz [ 142 ]; the basic ideas leading to these equations were introduced by von Karman [ 107 ]. A general theory of the von Karman equations can be found in Berger and Fife [ 17] ,[ 18]. Further references are Ambrosetti [6], Antman [ 8 I, Bauer, Keller and Reiss [ 12],[ 14], Berger [20],[ 21], Dickey [ 58], Fife [62], 62

Friedrichs and Stoker [68], Keener [109], Keener and Keller [110], Keller, Keller and Reiss [113], Knightly [121],[122], Knightly and Sather [123],[124], [125],[ 126],[ 127], Koiter [131], Magnus and Poston [152], Matkowsky and Putnick [158], Reiss [181], Sather [190 ],[ 191], Shearer [206], Wolkowisky [236]. Recently a very nice discussion of the von Kannan equations has been given in the lecture notes of Ciarlet and Rabier [246]. A somewhat different approach can be found in Duvaut and Lions [60]. 2.5. GROUP REPRESENTATIONS AND EQUIVARI~"IT PROJECTIONS This section contains some general results on representations of groups over Banach spaces. Most of these results are taken from Rudin [184] (in particular sections 5.11 to 5.18), where full proofs can be found. Finite-dimensional representations will be considered in the next section. 2.5.1. Definition. A topofogi~al g~oup is a group G on which a topology is defined such that : (i) every point of G fonns a closed set; (ii) the map cjl

GxG+G, (s,t) +>-cjl(s,t)

st -1

(1)

is continuous. The topology of G is a Hausdorff topology, and is completely determined by any neighbourhood base of the identity element e E G. 2.5.2. Definition. Let X be a topological vector space, and G a topological group. A ~ep~~entation 06 G ove~ X is a map r : G + L(X) such that : (i)

r is a group homomorphism : r(e)

I

rest)

res) ret)

Vs, t

E

G

(2)

(ii) r is strongly continuous lim r(t)x = x

Vx EX.

(3)

t+e

63

2.5.3. Lemma. Suppose that X is a Frechet-space, G a locally compact topological group, and r : G +. LeX) a representation of Gover X. Then the map (s,x) ++ r(s)x from GxX into X is continuous. Proof. For each x E X the map s ++ r(s)x from G into X is continuous, by the definition of a representation. Fixing a compact neighbourhood U of So E G, this implies that {r(s)x I s E U} is bounded in X, for each x E X. It follows from the Banach-Steinhaus theorem that {res) I s E U} forms an equicontinuous set of bounded linear operators. This proves the lemma. 0 2.5.4. The groups used in this book will in general be ~ompact topological groups. On such groups it is possible to define an invariant measure; in order to introduce this me.asure, we need some concepts from integration theory. If X is any set, then a o-atgebna over X is a collection L of subsets of X, such that (i)

X·E L

(ii)

A E L ~ AC E L ;

(iii) An

E L ,

Vn

E N

~

u

nEN

A

n

E L

A set X together with a o-algebra L over X is called a me~unable ~pa~e (X,L). In particular, if X is a compact or locally compact Hausdorff space, then the smallest o-algebra LB containing all open subsets of X, is called /

the o-algebra of BOfl.u ~ w A po~-i;ttve me~uJte over

in X. (X,L)

is a map].l

L + [0,00]

which is cruntably

additive : A.

J

E L ,

A. nA. 1

J

= ¢ ,

i 'f j

~

].l( u

j EN

A.) J

Such positive measure is &~nite if ].l(X) < 00. In case ].l(X) = 1, ].l is called a p~obab~y m~uJte. In case X is a locally compact Hausdorff space, a po~-i;ttve Bo~u m~uJte is a positive measure over (X,LB)' Such Borel measure is

~e.gulM

if

sup{].l(K) I KcE compact} in£{ ].l(G) lEe G open} 64

2.5.5. Definition. Let \1 be a positive measure over a measure space (Q,L), X a locally convex topological vector space, and f : Q -+ X a mapping such that x*of E L'(\1), for each x* E X*. If there exists a vector x E X such that x* (x)

f

=

Vx* E X* .,

(x* of)d 11

(4)

Q

then x is called the PeA.:U.J.,-iVl:t!2gJt.af on f ovM Q wi:t:h ItVlP!2C.:t :to :t:h!2 m!2aJ.>Wl.!2 11 ; we use the notation : x = f

(5)

fd\1. Q

2.5.6. Theorem. Suppose X is a Frechet space, and \1 is a Borel probability measure on a compact Hausdorff space Q. Tnen the integral JQ fd\1 exists for every continuous f : Q -+ X. Moreover,

JQ fdll E co

f(Q) ,

the closed convex hull of f(Q) in X.

0

2.5.7. Let G be a compact topological group. Then we denote by C(G) the Banach space of all complex-valued continuous functions on G, with the supremum norm. For given s E G, we then define L : C(G) -+ C(G) and s R C(G) -+ C(G) by : s

(1, f) (t)

s

= fest)

(R f) (t) = f(ts) s

VtEG , Vf E C(G) .

(6)

2.5.8. Theorem. On every compact topological group G there exists a unique Borel probability measure m which is left-invariant, in the sense that :

J

JG fdm = G(Ls f)dm

Vs E G , Vf E C(G) .

(7)

This measure is also right-invariant

f G fdm =

JG(Rs f)dm

Vs E G , Vf E C(G) ,

(8)

and it satisfies the relation 65

fG f(t)dm(t)

=

f

f(t- 1)dm(t) G .

Vf E C(G) •

This measure is called the HaaJl me.a6UJLe. on G.

(9)

0

2.5.9. Theorem. Let X be a Frechet space, and Y a closed subspace of X, having a topological complement. Let G be a compact topological group, and r : G ~ L(X) a representation of G over X, such that r(s)(Y) c Y

Vs E G •

Then there exists a continuous projection Q of X onto Y which is e.quA-vaJuaVLt with respect to r r(s)Q = Qr(s)

Vs E G •

(10)

When P E L(X) is a continuous projection with R(P) = Y, then one can construct an equivariant projection Q from P by

f

Qx = G res

-1

)pr(s)xdm(s) ,

Vx EX,

(11 )

where m is the Haar measure of G, and the integral is defined in the sense of definition 5. 0 2.5.10. Example. Let G be a finite group G

{s. l

I i = 1 , ..• ,N}

( 12)

•

Using the discrete topology, G becomes a compact topological group. The corresponding Haar measure is given by m(E) = ~ x (number of elements in E)

VE E G •

( 13)

Under the hypotheses of theorem 9, the formula (11) for an equivariant projection takes the form : 1

Q =N 66

N

I

i=1

1 r(s~ )pr(s.) l

l

(14)

For this particular case the proof is much easier than for the general situation of theorem 9 (see Vanderbauwhede [225]). ·2.5.11. Example. Let X be a Hilbert space, G and r G + L(X) a unitany representation : Vs

E

a compact topological group,

G.

(15)

If Y is a closed subspace of X which is invariant under the representation (i.e. such that r(s)(Y) c Y, Vs E G), then the orthogonal projection Q of X onto Y is equivariant with respect to r. Proof. Let y

E

Y,

Z E

Y1 and s

E

G. Then

= = = 0 , -1

1

since res )y E Y. This shows that also Y is invariant under r. For each x E X we have: r(s)x = r(s)Qx

+

r(s) (I-Q)x .

Since, by the foregoing, we have r(s)Qx that Qr(s)x = r(s)Qx. 0

E

Y and r(s)(I-Q)x

E

Y

it follows

2.5.12. Example. In the subsequent chapters we will frequently meet the following situation, which is somewhat similar to the preceding example. Let X be a (real) Banach space, and <.,.> : XxX -+:ffi. a continuous, synunetric and positive definite bilinear form over X. Let G be a compact topological group, and r a representation of G over X, such that = <x,y>

Vs

E

G , VX,y

E X

(16)

Let Y be a finite~imensional subspace of X, which is invariant under r. Let {e i I i = 1, .•. ,n} be a basis for Y, which we can assume to be orthonormal with respect to the bilinear form <.,.>. Define a projection Q of X onto Y by

67

n

Qx =

I

i=1

<x,ei>e i

Vx EX.

(17)

Defining y1= {xEXi <x,y>=O, ¥xEY} one easily sees that Qx = 0 is equivalent to x E y1, i.e. ker Q = y1. Since, for each s E G, res) is an automorphism of X, the restriction of res) to Y is an isomorphism of Y onto r(s)(Y) C Y; we conclude that r(s)Y = Y, since Y is finite-dimensional. Then the same argument as in example 11 shows that r(s)(y1) C y1, and that the projection Q given by (17) is equivariant. We conclude this section with the following theorem, whose proof is very similar to the proof of theorem 9, as given in Rudin [ 184]. 2.5.13. Theorem. Let X be a Frechet space, G a compact topological group, and r : G + L(X) a representation of Gover X. Let Xo = {xEX i r(s)x=x, VSEG} .

( 18)

Then Xo has a topological complement in X, and Po E LeX), defined by (19)

is a continuous projection of X onto XO. Moreover Vs

E

G •

(20)

Proof. It is clear that Xo is closed. By theorem 2.1.2 it is sufficient to prove that PO' as defined by (29), is a continuous projection, and that R(P O) = XO· First, the integral in (19) exists since the map s r(s)x from G into X is continuous, for each x E X. It is clear that Po is linear. Since G is compact, the set {r(s)x i sE G} is canpact, for each x E X. By the BanachSteinhaus theorem (see Rudin [184], theorem 2.6) it follows that {r(s)isEG} is an equicontinuous family of linear operators in X. Let V be any neighbourhood of 0 in X. Choose a convex neighbourhood W of o such that WC V. There exists a neighbourhood U of 0 such that ;+

r(s)(U) C W 68

Vs

E

G.

it follows from theorem 6, the definition of Po and the convexity of W,

Po(U) ewe V This shows that Po is continuous. It is clear that POx = x for x E XO' Now remark that it follows from the . definition of the Pettis-integral that the integral commutes with bounded linear operators. Using the left invariance of the Haar measure, we have Vt

E

G , Vx EX.

This shows that R(P O) C XO; since PO(XO) = XO' we have in fact R(P O) = XO' Finally, it follows from the right invariance of the Haar measure that por(t)x =

fG

r(s)r(t)xdm(s)

This proves (20).

Pef

r(t)Pox

,Vt E G , Vx EX.

D

2.6. IRREDUCIBLE GROUP REPRESENTATIONS In this section we give an introduction to the theory of irreducible group representations, and then determine the irreducible representations of the groups SO(2), 0(2), SO(3) and 0(3). These groups will be frequently encountered in the subsequent chapters. Let us remark that we will concentrate on real representations, in contrast to the complex representations considered almost exclusively in classical handbooks on group theory (see e. g. Hamermesh [ 92], Miller [ 166], Husain [99]). Our approach to the representations of the group SO(3) is an adaption to real representations of the approach in Miller [ 166]. A nice account of the theory of spherical harmonics can be found in MUller [ 168]. We consider representations of the form

r

n G -+ LOR) ,

where G is a topological group. The natural number n is called the of the representation.

dimen6~on

69

2.6.1. Lennna. Let f be an n-dimensional representation of G, and let T E L(lRn) be non-singular. Then n

f1 : G -+ L(IR) , s

+>-

-1

f 1 (S) = T f(s)T

also defines a representation of G.

(1)

0

2.6.2. Definition. Two n-dimensional representations f and f1 of a topological group G are J.JJJYlilaJL or equivalent if there exists some non-singular T E L(lRn) such that Vs

E

G.

(2)

Similarity defines an equivalence relation among the representations of a group G. This reduces the study of all representations of G to the study of equivalence classes of representations. For compact topological groups, each such equivalence class contains an o~hogonal ~ep~eJ.Jentation, that is, a representation such that f(S) is orthogonal for each s E G. 2.6.3. Theorem. Each representation of a compact group is equivalent with an orthogonal representation. Proof. Denote by <.,.> the inner product inlRn , and by m the Haar measure of the group G. Define a positive definite, synnnetric bilinear form B : lRn xlRn -+ lR by : B(x,y) = fGdm(S)

Vx,y

E

lRn .

(3)

It follows from the invariance of the Haar measure that B(f(t)x,f(t)y) = B(x,y)

Vt

E

G , Vx,y

E

lRn

(4)

We have B(x,y) = , for some synnnetric posltlve definite A E L(lRn). Also, there is a synnnetric positive definite T E L(lRn) such that T2 = A. Then B(x,y) = , and (4) can be written in the form : -1

-1

= <x,y> 70

Vt

E

G , Vx,y

E

lRn

(5)

thiS shows that Tr(t)T-1 is orthogonal, for each t E G.

0

2.6.4. Definition. A set M = {Mk I kE K} C LORn) of linear operators over:nf is said to be fLeduc.ible if there exists a subspace V of]Rn such that (i) 0 < dim V < n ; (ii) Mk(V) C V , Vk E K If M is not reducible, we say that M is If JIf =

N

I

ffi

j=l

(i)

(ii)

0<

v., such that for each

j = 1, •.. ,N :

J

dim V . ..;; n

J M.(V.)CV. • Ok J J

(iii) Mj

~edueible.

= {~Iv.

VkEK

I kEK} is irreducible

J

then we say that M is c.omple:tely fLeduc.ible. 2.6.5. Definition. A representation r of a group G is called fLeduc.ible (respectively ~educ.ible, c.ompldely fLeduc.iblel when the set of linear operators {res) I sE G} is reducible (respectively irreducible, completely reducible) .

2.6.6. Lemma. Each orthogonal representation of a group G is completely reducible. Proof. If V is a proper subspace oflRn which is invariant under r, then the argument used in example 2.5.11 shows that also VI is invariant under r. Applying this argument a finite number of times one obtains a complete reduction of r. 0 The foregoing reduces the study of all representations of a compact topological group to the study of all irreducible orthogonal representations. The following result is fundamental for the study of such representations. 2.6.7. Theorem (Schur's lemma). Suppose that MC L~) and N C LORn) are irreducible, and that A E LORn ,1!f1) is such that for each HEM there is some N EN, and for each N E N there is some M E M such that 71

MA=AN. Then either A = 0, or n = m and A is non-singular. Proof. Let V = ker A C JRn . By the hypothesis, we can find for each N E N some M E Msuch that

ANx

0

MAx

Vx E V .

This shows that N(V) C V, \IN E N. Since N js irreducible, we conclude that V = {O} or V = JRn . In the first case n < m, and A is injective; in the second case A = O. Similarly, if V' = RCA) C M E M and x E JRn , then we have

r,

MAx

A\!x E V' .

So M(V') C V' for each M E M. Then the irreducibility of M implies that V' = {O} or V' =r. In the first case A = 0, in the second case n > m and A is surjective. Combining both results proves the theorem.

and r Z be two irreducible representations of a group = m and dim r Z = n. Let A E LORnJRm) be such that

Z.6.8. Corollary. Let G, with dim

r,

D

r,

(6)

Vs E G •

Then either A = 0, or n = m and A is non-singular. In this last case

r Z are equivalent.

r,

and

D

2.6.9. Corollary. Let r, and r Z be two n-dimensional representations of a group G, and assume r Z is irreducible. If there exists a non-zero A E L~) such that (6) holds, then A is non-singular, and is an irreducible representation equivalent to r Z. D

r,

Z.6.'0. Corollary. Let r : G + LORn) be an irreducible representation, and let A E LORn) be such that Ar(s)

72

r(s)A

Vs E G •

(7)

(C:IIDPC)Se also that A has at least one real eigeIwalue c E JR. Then A = cI. proof. It is sufficient to apply corollary 8 to A - cI .

0

2.6.11. Lemma. Let r : G + LORn) be an irreducible· representation. Then either n = 1 and r(s)x = x, Vx E JR, Vs E G (Le. r is the trivial representation), or {XEJRn

I r(s)x=x,

VsEG} = {a} .

(8)

Proof. The set at the left-hand side of (8) is an invariant subspace for r, and so equals JRn or {a}. This proves the lemma. 0 2.6.12. Definition. For each n = 1,2, ... we denote by O(n) the group of orthogonal linear operators on nf : (9)

By 9O(n) we denote the subgroup of O(n) containing those R E O(n) which are orientation preserving, i.e. which satisfy det R = 1. Both O(n) and SO(n) are compact topological groups if we give them the topology induced by the usual topology on LORn). It follows from lemma 2.5.3 that if r : O(n) + LQRU) is a representation of O(n) over~, then r is continuous. The same holds for representations of SO(n). We will study in particular the representations of 0(2), SO(2), 0(3) and 90(3).

2.6.13. Representations of 0(2) and SO(2). Starting with SO(2), consider the following mapping : sinal . eosa

(10)

This is a Coo-mapping, satisfying
I

¢(a+S) = ¢(a)¢(S)

, Va,S

Em .

(11)

73

Also, cp(ffi.) = SO(2), and cp-'(cpCa)) = {a+2nk I kELZ}. So we can consider cp as homeomorphism betweenJR (mod 2n), with the quotient topology, and SO(2). In fact, cp can be used to give SO(2) a differential structure. If now r : SO(2) + LORn) is a representation, then r = r o ¢ : JR + LORn) is a continuous mapping such that : (i)

r(a+2n) = rea)

(ii)

reO) = I ;

Va E JR

(iii) r(a+S) = r(a)r(S) ,

( 12)

Va,S E JR •

Conversely, given a continuous mapping r : JR + LORn) satisfying (i)-(iii), then there exists a unique representation r : SO(2) + LORn) such that r = ro¢. Therefore, r is uniquely determined by r0ci>, which we will itself denote by r, and also call a representation of SO(2). So, a representation r of SO(2) is a continuous map r : JR + LORn) satisfying (i)-(iii) above. As for 0(2), define 0 E 0(2) by : ( 13)

Then 0(2) 02

ci>OR)u(oo¢)(IR). Also I

and

0 o¢(a) =

¢(-a)oo

Va E JR •

( 14)

A representation of 0(2) is uniquely determined by a representation r : JR + L(IRn) of SO(2), together with an operator r = reo) E L(IRn) such that o

r2 o

I

and

r .r(a) 0

=

r(-a).r

0

Va E JR •

( 15)

2.6.14. Lemma. Let r : JR + LORn) be a representation of SO(2). Then r is continuously differentiable. If we define A

= dr CO ) da

then we have :

74

'

(16 )

dr .

(i)

-(a) = Ar(a) dO',

(ii)

r is irreducible if and only if lRn is irreducible for A.

Va

E

lR ;

Moreover, if r is an orthogonal representation, then we also have: """) AT = -A . (111

(17)

proof. If we can show that r is differentiable at a = 0, then the differentiability everywhere and (i) follow immediately from (12). In order to show the differentiablity at a = 0, define B : lR -+ LORn) by B(S) =

f: r(a)da = sf~ r(sS)ds = SB(S)

VS

E

lR .

Then B : lR -+ LORn) is continuous, and B(O) = I. Consequently B(S) is nonsingular for lsi sufficiently small. Using (12), it is easily verified that (r(a)-I)B(S) = Cr(S)-I)B(a)

Va,SElR.

Dividing by as, we find, for a f 0, S f 0 and lsi sufficiently small (18)

Since the right-hand side of (18) has a limit for a -+ 0, the same holds for the left-hand side, proving the existence of ~(O). It follows from (i) and reO) = I that rea) = exp(Aa), Va ElR. Since a subspace V of lRn is invariant under exp(Aa) if and only i f V is invariant under A, we have (ii). Finally, if r is orthogonal then rT(a) = r- 1 (a) = rC-a). Differentiation at a = 0 gives (iii). 0 2.6.15. Theorem. Let r : lR tion of SO(2). Then either

(i)

n

=

rCa)

-+

L~) be an irreducible orthogonal representa-

1, and =

r CO ) (a)

def

Va

E

lR ,

(19)

75

or (ii) n = Z, and there exists some k

r

rea)

(k)

(a)

def

=

[COSka -sinka

E ~\{O}

such that

Sinka] coska

Va

E

lR .

Proof. Define A E L(lRn) by (16). Since r is irreducible, lRn is irreducible for A. It follows from (17) that AZ is a symmetric operator, and consequently has real eigenvalues. Since AZ commutes with A, Schur'S lemma that AZ = AI, for some A E lR. Let ~ E ~ be an eigenvalue of A; then exp(~a) is an eigenvalue of rea) exp(Aa). Since r(Z1T) = I, it follows that exp(Z1T~) = 1, i.e. ~ = ik k E ~. We conclude that AZ = -kZI for some k E I. First suppose that 0 is an eigenvalue of A, i.e. there is some Uo E lRn, U o t- 0 such that AuO = O. Then V = span{u O} is invariant under A. By the irreducibility hypothesis it follows that V =lRn, n = 1 and A = O. This proves (19). Next suppose that AZ = -kZI for some k E 1\{0}. Take any Uo E~, Uo t- 0, and let vo = Au O' Then vo is linearly independent from uo' since otherwise would have a real eigenvalue, which can only be zero, and we would have AZ = O. Since AvO = AZu O = -kZuo it follows that V = span{uO'v O} is invariant under A. Since lRn is irreducible for A, we have V =:nf and n = Z. Let {e 1,e Z} be the canonical basis of lRZ. Then <ei,ATe j > for i,j = 1,Z. It follows that

for some A E lR. The condition A2 = -kZI then implies that A = k (possibly after replacing k by -k). Since rea) = exp(Aa) this gives r(a)e 1

=

coska.e 1 - sinka.e Z '

r(a)e Z = sinka.e 1 This proves (ZO).

76

0

+

coska.e Z '

Va

E

R .

2.6.16. Lemma. For each k f 0 the representations r Ck ) and r C- k ) given by theorem 15 are equivalent. proof. The foregoing proof shows that we obtain r( -k) from r Ck ) by interchanging the role of e 1 and e Z ' More formally, if we define T E L(IR2) by _ _ -1 lk) 0 Tel - e 2 , Te 2 - e 1 , then T r (a)T = r(-k)(a), Va E R.

2.6.17. Theorem. Let r : R -~ L(E) he an irreducible representation of SO(2) over a finite-dimensional vector space E. Then one has either dim E = 1 and rCa)u = u, Va E R, Vu E E, or dim E = 2, and E has a basis {u 1 ,u2} such that sinka.u 2

for some k E]N\{O}.

(21)

0

2.6.18. Lemma. A finite-dimensional representation r of 0(2) is irreducible if and only if the restriction of r to SO(2) is irreducible. Proof. r is determined by a representation r : R + L(IRn) of SO(2) , and by r E L(IRn) such that (15) is satisfied. By lemma 14, r is differentiable. 0" If we define A by (16), then r is irreducible if and only if Rn is irreducible for {A, r }. We have to show that:uf is irreducible for {A, r } if and 0" 0 only if Rn is irreducible for A. It is clear that if Rn is irreducible for A, then Rn is also irreducible for {A,r o }' Conversely, suppose Rn is irreducible' for {A,r o }' Differentiation of (15) at a = 0 gives

r A o

=

-Ar

0

(22)

This implies that A2 commutes with A and r . Then, as before, it follows o 2 from Schur's lemma and r(2n) = I that A2 = -k I for some k E &. Moreover, since r2 = I, r must have an eigenvalue equal to +1 or -1. Let Uo ERn, o 0 Uo f 0 be such that rouO = ±uo. Define Vo = AuO and V = span {uO,vO}. We have Av = A2u = _k 2u and r v = r Au = -Ar u = +Au = +v . This o 0 000 0 0 0 0 0 0 shows that V is invariant under A and r ; consequently V = Rn. Now there are o two possibilities. First, assume that Vo is linearly dependent of u O; then A has a real eigenvalue, which can only be zero, with eigenvector UO. Then 77

n = 1, since dim V = 1, and V is irreducible for A. If Uo and v0 are ly independent, then n = 2. and the relations vo = AuO' AvO = -k 2u O (k f 0) show that V =R2 is irreducible for A. This proves the lemma. 0 2.6.19. Theorem. Let r : 0(2) -+- L(E) be an irreducible representation of 0(2) over a finite-dimensional vectorspace E. Then we have the following possibili ties n = 1 , r(a)u = u, r 0 u = u Va ER Vu E E (i) (ii)

n '" 1 , r(a)u = u, r 0 u = -u

Va ER

Vu E

E

(iii) n = 2, and E has a basis {u 1 ,u 2} such that coska.u 1

sinka.u 2

sinka.u 1 + coska.u 2

rou 2 = - u 2

Va ER

'

for some k E N\{O}. Proof. The proof of the foregoing lemma shows that either n = 1 or n = 2. In case n = 1, then A = 0 and r 0 = ±1, which gives the possibilities (i) and 22 (ii). In case n = 2, both +1 and -1 are eigenvalues of ro; also A = -k I for some k E N\{O}. Let u l E E, u l f 0 be such that roul = u l . Define D u 2 E E such that AU l = -ku 2 ; then Au 2 = kUl' giving us the case (iii). Next we turn to the representations of the groups 0(3) and SO(3). 2.6.20. Lemma. Let r : 0(3) -+- LORn) be a representation of the group 0(3). Then r is irreducible if and only if its restriction to the subgroup SO(3) is irreducible. If this is the case, then r

r(-I)

=

±I .

Proof. If the restriction of r to SO(3) is irreducible, then also r is irreducible. As for the converse, suppose that r is irreducible. Then r r(-I) commutes with r(R), for each R E 0(3). Moreover, r: = I, and cons~­ quently r_ has real eigenvalues ±l. By corollary 10, it follows that r ±I. Since 0(3) = SO(3) U {-R I RE SO(3)} it is then clear that the irreducibi78

lityof r implies the irreducibility of the restriction of r to SO(3). This lemma allows us to restrict attention to the subgroup SO(3).

0

·2.6.Z1. The Lie group SO(3). We will show now that SO(3) forms a 3-dimensional smooth submanifold of the 9-dimensional space LOR3). (For the general theory of differentiable manifolds, see e.g. Lang [141]). Using the group structure of SO(3) , it is sufficient to show that there is a neighbourhood U of I in LOR3) such that SO(3) n U is a submanifold. Since the determinant function is continuous on LOR3), we can take U sufficiently small such that 50(3) n U = 0(3) n U. Let Ls OR3) and La OR3) be the subspaces of symmetric, respectively antisymmetric operators :

nCR)

=

1

T

1

T

Z(R R-I) +Z(R -R)

Then n(I) = 0, and Dn(I) is the identity in LOR3). Consequently, the restriction of n to an appropriate open neighbourhood U of I in LOR3) is a Coo _ diffeomorphism from U onto an open neighbourhood n(U) of 0 in LOR3). Moreover, 0(3) n U = n -1 (La OR3) n n (U)). This shows that 0(3) n U is a three-dimensional smooth submanifold of LOR3). The tangent space to SO(3) in the point I is given by Dn -1 (0) (La(lRh) = La OR3). Let V be an open neighbourhood of I in LOR3) such that {A.B I AE V,B E V} C U. Then it is clear that the mapping

is a Coo-mapping from n(V)xn(V) into n(U). So also its restriction to (n(V) n La(lR3)) x (n(V) n La OR3)) is a Coo-mapping into n(U) n La OR3). This shows that the mapping (R1 ,RZ) * R1 from SO(3)xSO(3) into SO(3) is smooth. Then an easy application of the implicit function theorem shows that the mapping R * R- 1 from SO(3) into itself is also smooth. This proves that

.Rz

79

SO(3) is a 3-dimensional Lie group. 2.6.22. Lemma. Let f : SO(3) ~ L~) be a finite-dimensional representation of the Lie group SO(3). Then f is a Coo-mapping. Proof. Using the fact that f is a group representation, it is sufficient to proof that the restriction of r to an appropriate neighbourhood U of I SO(3) is smooth. For that purpose, we will use a special system of local coordinates, which we introduce now. Let {e 1 ,e 2 ,e 3 } ~e the canonical basis oflR3 , and define a basis {J 1 ,J 2 ,J 3 } of LaOR ) by J 1 ·e 1

0

J 1 ·e 2

-e 3

J 1 ·e 3

=

e2

J 2 ·e 1

-e".)

JZ-e 2

0

JZ-e 3

==

e1

J 3 ·e 1

-e 2

J 3 ·e 2 = e 1

For i = 1,2,3, define ¢i : JR

~

(23)

J 3 ·e 3 = 0

SO(3) by Va E lR .

(24 )

Finally, define W : lR3 ~ SO(3) by Va i E JR, i == 1,2,3 .

(25)

w is a Cco-mapping, with weO) = I, an~ Th)J(O)'(;:1';:2';:3) = ;:1 J -\+;:7J 2+;:3J 3' So DW(O) is an isomorphism between JR and the tangent space La QR3) to SO(3) at I; consequently W defines local coordinates near I on SO(3). It remains to show that foljJ : lR3 ~ LORn) is C. We have :

Now the argument used in the proof of lemma 14 shows that for each i = 1,2,3, the mapping a-

#

f(¢. (a.)) fromlR into LORn) is differentiable. This implies

I I

1

that also foW is differentiable.

80

0

La (IR3).

.23. The Lie algebra so(3)

Fix A,B

E

La (IR3)

and a

E

lR. Then the

--------------------~~--

mapping 6 * exp(Aa)exp(B6)exp(-Aa) defines a 1-parameter subgroup of SO(3); differentiating at 6 = 0 we obtain a smooth mapping from lR into La (IR3) given by 0.* exp(Aa)B.exp(-Aa) . Differentiating again at a = 0 we find an element of L (IR3), which we denote a 3 3 by [A,B], and call the Lie bracket of A and B. So [ • ,.] : La (IR ) xLa (IR ) -7 L (IR3) is defined by a

d

d

[A,B] = da(d6 exp (Aa)exp(B6)exp(-Aa) '6=0) 10.=0 ' and is explicitly given by

[A,B] = AB - BA .

(26)

An explicit calculation shows that for the J i (i=1,2,3) defined by (23) we have: (27) La (IR3), equipped with the Lie bracket, forms the Lie algebra of the

group SO(3); as such, it is sometimes denoted by so(3). 2.6.24. Lemma. Let r : SO(3) define L. E LORn) by

-7

LORn) be a representation. For i

1,2,3

1

d L.1 = -:orua r(
(28)

Then (29) 81

Proof. By the differentiability of r we have for each smooth curve I;; : lR + SO(3) such that 1;;(0) = I : d Ia=O = Dr(I)·da(O) dl;; dar(l;;(a)) ,

L.1 = Dr(I).J.1 It

i

1,2,3.

(30)

follows that d

d

[L 1 ,L 2 ] = aa(QEr(¢1(a))r(¢2(s))r(¢1(-a)) IS=O)!a=O d d

= aa(QEr(¢1 (a)¢2(S)¢1 (-a)) IS=o) la=O

2.6.25. Lemma. Let r : SO(3) + LOR~ be a representation. Then r is irreducible if and only iflRn is irreducible under-{L 1 ,L 2 ,L 3}. Proof. Each R E SO(3) can be written as an appropriate compositon of operators of the form ¢i(ai ) (i=1,2,3). Since ¢i(a+S) = ¢i(a)¢i(S) it follows that Va E lR , i = 1,2,3 , and consequently r(¢i(a)) = exp(Lia). It follows that r is irreducible if and only if1!f is irreducible under {r(¢i(a)) I i=1,2,3, aElR}, which in tUrn holds if and only if~ is irreducible under {L i ! i=1,2,3}. 0 2.6.26. The Casimir operator. Let now r : SO(3) + LORn) be an orthogonal irreducible representation. Then the Li (i=1,2,3) are anti-symmetric. Consider the restriction of r to {¢3(a) I aElR}, which is a subgrcup of SO(3) isomorphic to SO(2). So this restriction of r forns a representation of SO(2). It follows from the theory of such representations that we can wri te lRn as a direct sum of one- and two-dimensional subspaces, on which the 82

representation of SO(2) is irreducible. The one-dimensional subspaces are spanned by vectors u such that L3u = 0, the two-dimensional ones by vectors u, v such that L3u = -qv, L3v = qu, for some q = 1,2,3, ... Consider such nonzero vectors u, v for which L3u = -qv, L3v quo Define + + U , v , u and v by (31)

and (32) Then it follows from (29) that + L3 u

and L3u

-

+ -(q+1)v -(q-1)v

-

L3v L3v

+ -

+ (q+1)u

(33)

(q-1)u

(34)

Now we define the so-called Casimir operator (35)

C

It is a symmetric operator, and consequently its eigenvalues are real. MOreover, it is easily seen from (29) that C commutes with each of the Li . SincelRn is irreducible under {Li I i=1,2,3}, it follows from corollary 10 tha t C AI for some A E lR. Let t be the maximal value of q appearing in the decomposition oflRn discussed above, and let u, v be a corresponding pair of nonzero vectors such that L3u = -tv, L3v = tu. Then (33) implies that L2u-L 1v = 0 and L1u+L 2v = O. But then

t(t+1)u We conclude that

C

t(t+1)I.

83

2.6.27. Lemma. Let r : SO(3) ~ LORn) be an orthogonal irreducible representation. Then there exists some

U

o~

°such that L3u O = o.

Proof. Let s be the minimal value of q appearing in the decomposition ofRn discussed above, where s = 0 if there is some u ~ 0 such that L3u = O. We have 0 < s < £, and we must prove that s O. Suppose that 0 < s < £. Then there are nonzero vectors u, v such that L3u = -sv, L3v = su, while the definition of s and (34) imply that -L 2u - L1v = 0 and L1u - L2v = 0. But then 2

Cu

L 2 ( -L 2u-L 1v) - L1 (L 1u-L 2v) - L3v - L3u

s(s-1)u . Since UfO we must have s(s-1)

o<

s < £.

£(£+1), which is impossible for

0

2.6.28. Theorem. Let r : SO(3) ~ LORn) be an irreducible representation. TIlen n = ned by L

2S',+1

for some integer £. Up to equivalence, r is uniquely determi-

Proof. We may asswne that r is orthogonal. By lemma 27, we can find some U

o ERn, U o ~

0 such that L3uO

= O. Define u 1 = L2u O' v 1 = L1u O' and more

generally : Vq EN.

(36)

Vq EN

(37)

Then -qv

q

L3vq = quq

If (uq,V q ) ~ (0,0), then the vectors {u O,u 1 ,v" ... ,uq ,vq } are linearly independent. Consequently, there exists some £ EN such that (u£,v£) f (0,0) and (u£+1,v£+1) = (0,0). Let U = span{u O'u 1 ,v 1 , ... ,u£,v£}. Then U is a (2£+1)dimensional subspace ofRn, which is clearly invariant under L3 . Also, the same argument as used before shows that C = £(£+1)1. Moreover, L 1u O E U, L2u O E U, and we have for 0 < q < £ :

84

2

-L 2uq -L 1vq = Cuq- 1 +L3u q- 1- L3vq- 1 = [.\',(.\',+l)-q(q-1)]

u q- 1 '

(38)

L1uq -L 2vq = [.\',(.\',+1)-q(q-1)]vq- 1. Together with (36) this shows that U is invariant under L1 and L2 . By lennna 25, this implies that U =Rn, and n = 2.\',+1. Since the action of L1, L2 and L3 on the basis {u O,u 1,v 1 , ... ,u.\'"v.\',} is uniquely detennined by the foregoing relations, also the second part of the theorem is proved. 0 2.6.29. Corollary. A representation r : SO(3) ~ L~) is irreducible if and only if 1 •

(39)

Proof. This follows from lemma 27 and the proof of theorem 28. Remark also that

2.6.30. Spherical hannonics. For each.\', EN one can realize a (2R-+1)-dimensional representation of SO(3) by using spherical harmonics, which we introduce now. Let H.\', be the space of homogeneous polynomials of degree.\', in three scalar variables H.\', : R3 ~ R, which are also hannonic : (40)

Let S2 be the unit sphere in R3 : (41)

Finally, let (42)

85

element Y£ of U£ is called a spherical harmonic of order £ in 3 it is the restriction to the unit sphere of a harmonic homogeneous of degree £. We can define a representation r(£) SO(3) + L(U£) of SO(3) over the finite-dimensional space U£ as follows

An

V8

E

S2, VR

E

SO(3), VY £

E

U£ .

(43)

We have the following result. Z.6.31. Theorem. We have for each £ E W

(i)

dim U == 2£+1 . £

'

(ii) the representation r(£) irreducible.

SO(3)

L(U£) defined by (43) is

+

Proof. It is clear that dim U£ == dim H£. Writing H£

E

H9~ in the form

£

I

j ==0

(44)

x3jA£_J-Cxl'XZ)

where Ai (x 1,x Z) is a homogeneous polynomial of degree i in xl and x z , we find £

6H£C x) == ()Z

I

j==2

£-2 jU-l)X1-2A£_J-(xl'XZ) + (12_

where 6 2 == --2 + --2 ()X 1

lS

I

j==O

x3j6zA£_J-Cx1,x2) ,

the Laplacian in 2 variables. Bringing this in (40)

aX 1

and equating coefficients of equal powers of x 3 ' we find

62AnJ<.,-]- == -(j+2)(j+l)An",-]-L - ~

j

0, ... ,£-2 .

( 45)

This shows that there are no conditions on the coefficients appearing in A£ and A£_l' while A£_2"" ,AO are uniquely deternlined by A£ and A£_l' Since A£ and A£_l contain respectively £+1 and £ coefficients, we conclude that dimH£==2£+l. 86

Now consider the subspace

(46)

Ai (x1coscx + x 2 sina, -x 1sina + x 2cosa) = A (x ,x ), i 1 2 Vi = 0, ... ,£ , Va

E JR

(47)

This implies that Ai (x 1 ,x 2 )

2 2 i/2 a i (x 1 +x2 )

if

i = even

0

if

i = odd

Since the A.l also have to satisfy (45), this shows that dim V follows from Corollary 29 that r(Q,) is irreducible. 0

=

1. Then it

2.6.32. Remark. The representation r(£) of SO(3) defined by (43) can be extended to a representation of 0(3), by using (43) for all R E 0(3). Then r (Q,) ( - I) = (-1) £1.

87

3 Symmetry and the Liapunov-Schmidt method The main objective of this chapter is to introduce the Liapunov-Schmidt reduction method, and to show how this reduction can be performed in a way which is compatible with the symmetries which may be present in the problem under consideration. The Liapunov-Schmidt- method results in the so-called bifurcation equations, which form a finite set of equations, equivalent to the original problem. Our main result shows that if the reduction is done properly, then the bifurcation equations inherit the symmetry properties of the original problem. In section 1 we describe the Liapunov-Schmidt method under very general hypotheses. In section 2 we introduce the concept of an equivariant mapping; the definition of such mappings contains all the ingredients of what we will consider as a mapping with symmetry properties. Section 3 contains the main result referred to above. In section 4 we restrict attention to symmetryinvariant solutions, and we show that for such solutions there is a further reduction of the bifurcation equations. In section 5 we use our formalism to describe some properties of reversible systems of ordinary differential tions (also called "systems with property E"); these results were found lier by Hale [77]. Some aspects of the approach for such systeJll5- can be put in a more abstract form, as is done some further examples : subharmonic solutions of periodic differential tions, oscillations in conservative systems, Dirichlet bOlll1dary value problems, and the von Karman equations for the buckling problem of rectangular and cylindrical plates. 3.1. 1HE LIAPUNOV-SrnMIDT METIIOD

In this section we describe the Liapunov-Scnmidt method. Application of method allows in most cases to reduce an infinite-dimensional problem to a finite-dimensional one. Sometimes this approach is referred to as the native method, although this last method is in fact somewhat more general than the method described here (see e.g. Cesari[31], Hale [78]).

88

1.1. The hypotheses. Let X, Z and r be real Banach spaces, S"l an open neighbourhood of the origin in X, and w an open neighbourhood of the origin in 11.. Let M : S"lxwcXxll. ->- Z be a C1-map, such that M(O,O) = 0. We want to . study the solution set of the equation M(X,A) i~

=

°

(1)

a neighbourhood of (0,0) in XxII.. Define (2)

Our basic assumption will be the following : (C) ker L has a topological complement in X, while R(L) is closed and has

a topological complement in Z. By definition 2.1.5 and theorem 2.1.3 (C) will be verified under the following hypothesis : (F) L is a Fredholm operator. Not only does (F) imply (C), but (F) also has an important advantage on the general hypothesis (C) : under the hypothesis (F) the result of the LiapunovSchmidt reduction will be a finite nwnber of scalar equations in a finite number of scalar unknowns, and depending on the parameter A. In all applications considered further on (F) will be satisfied. 3.1.2. Under the hypothesis (C) it follows from theorem 2.1.2 that there exist continuous projections P E L(X) and Q E LeZ) such that ker L = R(P)

ReL) = ker Q .

(3)

in the form x = u+v, where u = Px E ker L = X and p v" (I-P)x E ker P = XI_po (In general, if P E LeX) is a projection, we write Xp for the image Rep) of P). Then we can rewrite the equation (1) X E S"l

89

in the fonn

(I-Q)M(u+v, A) QM(u+v, A)

o o

(4. a)

(4.b)

3.1.3. Lemma. Assume (C). Then there exist a neighbourhood U of the origin in Xp ' a neighbourhood V of the origin in XI _P ' a neighbourhood Wo of the origin in fl, and a continuously differentiable mapping v* : UxwO -+ V such that : (i)

UxV

C

:\2

(ii) for each (U,V,A) E UxVxwO C4a) is satisfied if and only if v = V*CU,A) . Moreover v*(O,O) = 0

(5)

and

Proof. The map t/l origin by

XpxXI _pXfl

-+

RCL), defined in a neighbourhood of the

t/l(U,V,A) = (I-Q)M(u+v,A) is continuously differentiable, t/l(O,O,O) = 0 and DvIHO,O,O) = Ll ker P' which is an isomorphism between key P and R(L). Then the result follows from the implicit function theorem. 0 3.1.4. Remark. Define N : :\2Xw

-+

Z by

N(X,A) = Lx - M(X,A)

(7)

Then N is a C1-map, with NCO ,0)

90

o

o,

(8)

(1) can be rewritten in the form Lx '" N(X,A) .

(9)

By the same argument as used in the proof of theorem 2.1.6, the hypothesis (C) implies that the restTiction of L to keT P has a bounded inverse K : ReL)

->-

keT P, satisfying relations such as (2.1.7). It is then easily

seen that (9) is equivalent to the equations : v = K(I-Q)N(u+v,A)

(10.a)

o = Q NCU+V,A)

(10.b)

.

The Tesult of the pTeceding lemma then also holds for the equation (10. a) ; in fact, the two solutions V*(U,A) coincide.

Sometimes it is interesting to write the equations in the form (10). For example, if the original problem can be WTitten in the form (9), without N being necessarily of class C1 it may still be possible to prove that the right-hand side of (IO.a) defines a contraction in a sufficiently small neighbouThood of the oTigin. Then (10.a) can still be solved in a unique way for v as a function of u and A. Also, in case one would like to geneTalize the Liapunov-Schmidt method to more general topological vector spaces X and

Z, it may in certain cases be possible to prove the existence of a pseudoinverse K for L such that (9) is equivalent to (10). Then (lO.a) may be solved by a fixed point theorem. For more details on generalized inverses, one can consult Z. Nashed [ 262].

3.1.5. The result of 1enma 3 allows us to define Cl-mappings X" and F : UxwO -)- R(Q) by :

x *(U,A)

=

u + v *(u,'\)

( 11)

and

F(u,A) = QMCu+v*(u,'\) ,A) = M(u+v* (U,A) ,I,) .

(12)

Then x'CO,O)

o

Du x'CO,O)

(13) 91

and F(O ,0) = 0

DF(O,O)

u

=0

•

(14 )

The following theorem summarizes the essential result of the LiapunovSchmidt reduction. 3.1.6. Theorem. Assume (C). Let U, V and Wo be the neighbourhoods given by lemma 3. Let x* and F be the mappings defined by (11) and (12). Then for all u E U, X E UxV C X and A E Wo the following statements are equivalent (i)

Px = u

(ii) x = X*(U,A)

F(U,A)

M(X,A)

and

=0

0

and .

o

( 15)

3.1.7. Conclusion. Theorem 6 gives us a one-to·-one relation between the solution set of (1) and the solution set of (15), both restricted to appropriate neighbourhoods of the origin. This reduces the bifurcation problem for (1) to the bifurcation problem for (15). Since u E ker Land F(U,A) E R(Q) this is a considerable reduction, both of the domain and the range of the nonlinear operators. In case (F) is satisfied (15) is a finite dimensional problem: for each A E Wo (15) is equivalent to m scalar equations in n scalar unknowns, where n = dim ker L and m = codim R(L). Equation (15) is called the b~6un~~on eq~~on cOITesponding to (1). Remark that by (14) the b-t6UJLC.~(m 6un~Uon F is completely degenerated as far as a direct application of the implicit function theorem is concerned. In the sequel we will consider several examples for which we can analyse the structure of the bifurcation equation and give methods to solve the equation. For a survey of some simple cases we refer to Hale [80], Lichnewsky (Expose nr. 2 in [ 204]) and Vainberg and Trenogin [ 224]. 3.1.8. Bibliographical notes. Although the Liapunov-Schmidt method is explained in almost any text on bifurcation theory (see the references given in chapter 1) we can refer more in particular to Hale [ 78], Cesari [ 31], Vainberg and Trenogin [ 223] ,[ 224]. The method itself o"{iginated from the work 92

of Liapunov [143] and Schmidt [202] on nonlinear integral equations. 3.2. EQUIVARIANT MAPPINGS 3.2.1. An example. Consider the following simple example of a problem showing some symmetry invariance : find the 2n-periodic solutions of the autonomous ordinary differential equation

.

x

=

f(x,A)

(1)

where f : lRn x1\. + lRn is, for example, a C'-function. Since (1) is autonomous, each 2n-periodic solution x : lR40lRn of (1) will generate a whole family of such solutions, obtained from x by a phase shift over an arbitrary e E lR : (2)

However, since x is 2n-periodic, e and e + 2kn will give the same result. In order to put this in an abstract form, we introduce the spaces Z = {z : lR + lRn and X =

M

Iz

is contimous and 2n-periodic}

{x E Z I x is C'}, and the opera tor Xx1\.

+

Z

x(.) ~ M(X,A)(.)

= x(.) -

f(x(.)'A)

(3)

Then our problem can be written in the form M(X,A)

o.

(4)

Moreover, for each e ElR we can define a bounded linear operator ree) : Z + Z by r(e)z(t) = z(t+e)

Vz E Z , lit E

lR .

(5)

It is clear that r(e+2kn) = r(e); so r is in fact a representation over Z of the rotation group SO(2) introduced in section 2.6. If (X,A) is a solution 93

of (4), then the same holds for (r(e)x,A), for each e E JR. Also, the operator M defined by (3) satisfies M(r(e)x,A) = r(e)M(x,A)

ve E JR , V(x,A) E Xxfl .

(6)

This example motivates the following definition. 3.2.2. Definition. Let X, Z and fl. be real Banach spaces,

c Xxfl open and Z. Let G be a compact topological group. Let r : G + L(X) and ~lxw

M : ~xw + f : G + L(Z) be representations of G over X, respectively Z. Then we say that M is l'-qu)V{,(/L{an): wah Jte!.lpec;t ;to (G,r,f) if : (i)

res) (~) =

(ii) M(r(s)x,A)

~

f(s)M(x,A)

Vs

E

G

Vs

E

G

V(x,A) E

~xw

.

(7)

3.2.3. Remark. Although in many applications X will be a subspace of Z, we

did not make such an assumption here. This forces us to consider two different representations (r and f) of G. Again, in many applications r will be obtained from f by an appropriate restrictioil. However, our formalism allows different representations, even if X c Z. As we will see in section 5, this is especially important for the treatment of reversible systems. 3.2.4. Some irmnediate properties. Let M be equivarjant with respe<2t to

(G,r,f). Then we have the following; (i)

if (X,A)E\txw is a solution of M(x,A) allsEG;

(ii)

if G1 is a closed subgroup of G, and

0, then so is (r(s)x,A), for

then M is also equivariant with respect to (G 1 ,r 1 ,f1) (iii) i f M is of class C1 , and if we define L as in 3.1.2, then Lr(s) = f(s)L 94

Vs

E

G,

(8)

i.e. : L is equivariant with respect to (G,r,f) under the conditions of (iii), we also have r(s) (ker L) = ker L

Vs

E

G,

(9)

Vs

E

G

(10)

and

f(s)(R(L)) = R(L) (v)

define Xo = {xEX I rcs)x=x, VsEG}

(11)

and Zo = {zEZ Ir(s)z=z, VsEG}

(12)

then, for each (X,A) E (nnxO)xw

M(X,A) E Zo .

(13)

3.2.5. Remarks. (i) The relations (9) and (10), in combination with theorem 2.5.9, will be at the basis of the theory in the next section. (ii) If we consider only solutions property (v) above, in a reduction the symmetry-invariant part of the a similar situation arises for the Liapunov-Schmidt reduction.

x E Xo of M(X,A) = 0, this results, by of the equation; we only have to consider equation. In section 4 we will show that bifurcation equation resulting from a

(iii) As an example of such a reduction, consider the problem introduced at the beginning of this section. In this case Xo consists of all constant

functions, and the reduced problem if that of finding the singular points of (1). The reduced equation is f(x,A)

o•

(14)

95

3.2.6. Bibliographical note. Hypotheses similar to (7) were used by Loginov and Trenogin [ 148],[ 149], Sattinger [195],[ 247], and Dancer [57]; in these references one has X c Z, and r is taken equal to The definition as given here (using different representations) was first introduced in [225], for the case of finite groups.

r.

3.3. THE LIAPUNOV-SCHMIDT METHOD FOR EQUIVARIANT EQUATIONS 3.3.1. The hypotheses. In this section we reconsider the Liapunov-Schmidt reduction for the equation

M(X,A) = 0 ,

(1)

as given in section 3.1, under the supplementary condition that !vI is equivariant with respect to some triple (G,r,r). We resume the hypotheses on M (H) (i)

M

r2xw c Xx1\

M(O,O)

-+

Z is of class C1 , and

=0

ex,

Z and 1\ are real Banach spaces, r2 is a neighbourhood of the origin in X, and lei is a neighbourhood of the origin in 1\) . (ii)

The linear operator L

= DxM(O ,0' )

satisfies the hypothesis (C) of section 3.1; this implies that we can apply the Liapunov-Schmidt reduction. (iii) M is equivariant with respect to some (G,r,r), with G compact. 3.3.2. Theorem. Let (H) be satisfied. Then the projections P and Q satisfying (3.1.3) can be chosen in such a way that also:

r(s)p = pres)

Vs E G

r(s)Q

Vs E G

and 96

=

Qr(s)

proof. This follows from (2.9), (2.10) and theorem 2.5.9.

0

3.3.3. Important remark. From now on we will always asS1.ID1e that, under the

. hypotheses (H), the projections P and Q are chosen such that also (4) and (5) are satisfied. TiU.6 will Ylo-t be men.:t[oYlYled expUc-ifty. Remark also that the pseudo-inverse K : R(L) ~ ker P of L depends on the choice of P. In case P satisfies (4) we can prove the following. 3.3.4. Lemma. Ass1.ID1e (H), and let P E L(X) be a projection on ker L, such

that (4) is satisfied. Then we have on R(L) : Kr(s) = r(s)K

Vs

G

E

proof. Let z = Lx E R(L) , then, since KL Kr(s)z

(6)

I-P

Kr(s)Lx = KLr(s)x = (I-p)r(s)x

= r(s) (I-P)x = r(s)KLx = r(s)Kz This proves (6).

0

3.3.5. Lemma. Ass1.ID1e (H). Then the neighbourhoods Uand V given by lemma 1.3 can be chosen in such a way that res) (U)

=U

r(s)(V)

Then the mapping v* : UxwO v*(r(s)u,>..)

=

~

=V

Vs

E

G.

(7)

V given by the same lemma satisfies

r(s)v*(u,>..)

Vs E G , V(u,>..) E UxwO •

(8)

Proof. Let U, V, Wo and v*(u,>..) be as in lemma 1.3. First, we can shrink U and Wo in such a way that r(s)v*(u,>") E V for each (u,>..) E UxwO and for each s E G. This follows from the fact that r(s)v*(O,O) = 0 for all s E G, the compactness of G, the continuity of V* and lemma 2.5.3. The same lemma also implies that we can find a neighbourhood U' c U of 0 in ker L such that r(s)(U') C U for all s E G. Then let 97

U,

=

u sEG

r(s)(u')

C

u

v,

=

u sEG

r(s) (V)

.

= U1 and r(s)(V1 ) = VI for each s E G. Moreover, u rcs)(UxV)cQ. Let us show that for (u 1 ,v 1 ,A) E U1xV1xwO the SEG tion (1.4a) is satisfied if and only if v 1 = v*(u l ,I\). Let (u l ,v 1 ,1\) E U1xV1xwO be a solution of (1.4a). Then v 1 = r(s)v for

It is clear that r(s)(U 1)

U1xV1

C

some s

E

G and some v

E

V. It follows from the equivariance of M and Q that

also (r-1(s)u 1 ,v,I\) E UxVxwO is a solutionof (1.4a). 111en lemma 1.3 implies -1 -1-1 that v = v*(r (s)u 1 ,1\) and v 1 = r(s)v*(t (s)u 1 ,:\). Since r (s)u 1 E U and 1\ E wo' it follows that vI E V. Then the result follows from lemma 1.3.

Moreover, (8) follows from the foregoing and the fact that (r(s)u 1 ,r(s)v*(u 1 ,1\),),)

E

01xV1xwO is a solution of (1.4a) for each s

E

G,

and each (u l ' 1\) E 01 xwO' lJ Prom now on we will assume implicitly that the neighbourhoods U and V satisfy (7). 3.3.6. Theorem. Assume (H). Then the mappings x * : UX()I O -7- X and F : UxwO -7R(Q), defined by (1.11) and (1.12), are equivariant, i.e., we have for all s E G and all (u,/\) E UxwO : x *(r(s)u,/\)

r(s)x *(u,/\)

(9)

and F(r(s)u,/\) = ~(s)F(u,l\) Pro (5).

0

(10)

£. (9) follows from (8), while (10) follows from (8), (2.7) and 0

3.3.7. Conclusion. As a result of the foregoing we may conclude that the bifurcation equation HU,I\)

=

0

obtained from (1) by the Liapunov-Schmidt reduction inherites the symmetry properties of the original equation (1), provided the projection operators used for the reduction are equivariant with respect to the symmetry

98

3.3.8. Bibliographical note. The Liapunov-Schmidt reduction for symmetric equations as given in this section is a generalization of the treatment in Vanderbauwhede [ 225], where attention was restricted to finite groups. Some. what similar results were obtained by Loginov and Trenogin[ 148], who introduced the commutation relations (4) and (5) as a supplementary hypothesis, and by Sattinger [195], who used particular projections P and Q, defined in tenns of dual bases. The possibility to choose P and Q such that (4) and (5) ate satisfied was recognized by Sattinger in [ 197], where also the formula (2.5.11) is given. 3.4. SYMMETRIC SOLUTIONS 3.4.1. In this section we show that there is a one-to-one relation between the solutions of M(x,>.)

=0

(1)

which are invariant under the symmetry-operators {r (s) try-invariant solutions of the bifurcation equation

I s E G},

and the symme-

(2)

F(u,>.) = 0 •

We assume the hypotheses (H) of the preceding section, and use also the notation of that section. Let Xo and Zo be the subspaces of X, respectively Z, given by (2. 11) and (2.12). By theorem 2.5.13 we can define projections Po and QO on Xo and ZO' respectively, by : Vx E X

(3)

Vz E Z •

(4)

and

Q z = J f(s)zdm(s)

o

G

3.4.2. Lemma. We have (5)

99

and (6)

~ = GoQ •

Proof. Let x E X. Then

PPox = pfGr(S)Xdm(S)

fGPr(S)Xdm(S)

fGr(S)Pxdm(S)

= POPx . An analogous proof holds for (6).

3.4.3. Lennna. Let u

E

0

unXO and A E wOo Then (7)

and (8)

F(U,A) = GoF(U,A) •

Proof. We have, for u X*

E

(U,A) =x* (r(s)U,A)

un XO' A E

Wo

and

S E

G

r(s)X* (U,A)

and F(U,A)

= F(r(s)U,A)

This implies (7) and (8).

r(s)F(u,A)

0

3.4.4. Theorem. Assume (H). Let U, V, wO' X* and F be as in lemma 3.5 and theorem 3.6. Then the following statements are equivalent for each x E UxV, U E

U and A E x

E

Xo

(ii) u

E

Xo

(i)

Wo :

Px = u and M(X,A) = 0 '

X =

x *(U,A)

and (9)

100

pr 0 0 f. Let (i) be satisfied. TIlen POu = POPx = PPOx = Px = u; so UElhXO' Further, we have from theorem "1.6 that x x*(u,\) and F(u,\) = O. This implies (ii). Conversely, let (ii) be satisfied. Then, using the preceding lemma x

x *(u,\)

and

F(u,\) By

QOF(u,\) = 0

theorem 1.6 this implies (i).

0

3.4.5. Corollary. Under the hypotheses of theorem 4, let x E UxV, \ E wO' U = Px and M(x,\) = O. Then x E Xo if and only if u E XO' 0 3.4.6. Conclusion. It follows from theorem 4 and corollary 5 that, as long as we restrict to a sufficiently small neighbourhood of the origin, there is a one-to-one relationship between the solutions x E Xo of (1), and the solutions u E ker L n Xo of (2). When one wants to construct the bifurcation equations for such solutions two approaches are possible : (i)

For x E Xo we can replace (1) by (10) (see remark 2.5(ii)). The left-hand side of (10) is a map from (Si n XO) Xw into ZO' Then we can apply the Liapunov-Schmidt method to this reduced equation, using the projections P1 = PIXo and Q1zo' By lemma 2 these projections have the desired properties.

(ii) We can also construct the bifurcation equation (2) for the equation (1); when restricting attention to u E Un Xo in (2), we can replace (2) by : QOF(U,A)

=

0 ,

(11)

where the left-hand side of (11) is considered as a map from (U n XO) into R(Q) n ZO' 101

By the theory above both approaches give exactly the same bifurcation equations for the reduced problem. 3.3.7. Bibliographical note. Stokes [210] has given a somewhat different approach to the problem of the reduction of the bifurcation equations for solutions in certain invariant subspaces. A discussion of ~le relation between both methods is given in [226]. The same paper also describes a set of hypotheses which are intermediary between (H) and the basic hypotheses of [210]; starting from these intermediary hypotheses one can recover most of the results of this chapter. 3.5. APPLICATION : REVERSIBLE SYSTEMS In this section we show how some results of Hale [77] for reversible systems (also called systems with "property E") can be derived from the results of the preceding sections. 3.5.1. The problem. Consider the ordinary differential equation

x = A(t)x

+

f(t,x,A)

(1)

where ~

(i)

x

(ii)

A: R ~ L~) is continuous and 2n-periodic;

E

(iii) f : R xRn x w ~ ~ is continuous, and C1 in (x, A); here w is a neighbourhood of 0 in some Banach space A ; (iv)

f(t,x,A) is 2n-periodic in t;

(v)

f(t,x,O) = 0

V(t,x) EJRxRn.

We will be looking for 2n-periodic solutions of (1). 3.5.2. The functional formulation. Let us introduce the Banach spaces (of 2n-periodic functions) X and Z, and the linear operator L : X~ Z as in section 2.2. Defining the nonlinear operator N : Xxw ~ Z by 102

N(X,A)(.) = f(.,xC.),A)

(2)

we can reformulate the problem as follows Find (X,A) E Xxw such that Lx = NCX,A)

(3)

It is easy to see that N is once continuously Frechet-differentiable, and

N(x,O)

Dx N(x,O) = 0

°

Vx EX.

(4)

We use the results of section 2.2. Let P and Q be the projections defined by (2.2.19) and (2.2.20). Also, let K : R(L) ~ ker P be the pseudo-inverse of L; i.e., for each z E R(L), Kz is the unique 2n-periodic solution of

:ic = A(t)x

+

z(t)

(5)

also satisfying Px = O. The operator K(I-Q) is then a bounded linear operator from Z into X. Let us define the following isomorphisms betweenJRP (p = dim ker L) and ker L, respectively RCQ) : JRP ~ ker L

a ...,. a

l a.¢. l a.l/!. i=l i=l

IjI

: mP

~ R(Q)

a ...,. ljIa

l

l

l

l

(6)

.

Using an adapted version of the Liapunov-Schmidt method, we can prove the following result. 3.5.3. Ineorem. For each R> 0, there exists a AO = AO(R) > 0 such that, for each (a,A) E JRPXA with I all .;;;; Rand IlAIl .;;;; AO(R) , the equation x = a

+

K(I-Q)N(x,A)

(7)

has a unique solution satisfying I (I-P)xll .;;;; R. This solution x *(a,A) is a C1-function of (a,A) and x*(a,O) = a. 103

~reover, if for I all " R and 11 All " AO (R) we define G(a, A) E lRP by

G(a,A) = ,¥-l[QN(x*(a,A),A) 1

(8)

then, for each (a,x,A) E lRPxXxA satisfying I all "R, I (I-P)xll " R and IIAII " AO(R) , the following are equivalent (i)

Px

= ~a

,

and

(ii) x = x*(a,A) G(a,A)

=

Lx

= N(X,A)

and

0 •

Proof. As in the Liapunov-Schmidt reduction one proves that equation (3) is equivalent to v = K(I-Q)N(u+v,A)

(10

o = QN(U+V,A)

(10

•

Fix some R> 0, and let A(R) A E w consider the map : F(a,A) : A(R)

+

{vE X1 _P IIIVII';;;R}. Also,

X1 _P , v

#

F(a,A)v = K(I-Q)N(~a+v,A)

Choose AOCR) > 0 sufficiently small, such that : sup{lIf(t,x, A) I I tElR, I xII "(l+II¢II)R, !lA1l "AO(R)} " RIIK(I-Q) 11- 1 and sup{lI af ax (t,x, )11 I tE:ffi., IIxll 1

" ZIlK(I-Q) I

"(l+II~II)R, nAil

-1

"AO(R)}

•

This is possible because of (4). Then it is easy to see that for each (a,A) E :ffi.Pxw satisfying lIall " R and !lA1l " AO(R) we have 104

F(a,A)

A(R)

-)0

A(R)

while F(a,A) is a contraction, with a contraction constant (= 1/2) independent of (a,A) in the domain considered (i.e. we have a uni60JUn contraction). It follows that the map F(a,A) has a unique fixed point v*(a,A) E A(R) , for each (a,A) such that lIall ,,;;; R, HII ,,;;; AO(R). It is then clear that x*(a,A) = ~a+v*(a,A) will satisfy the requirements of the first part of the theorem. The second part follows then as in the Liapunov-Schmidt approach, reminding that '¥ is an isomorphism between m.P and R(Q). 0 For each A, the bifurcation equations (8) form a set of p scalar equations in the p unknowns a.1 (i = 1, ..• ,p). Also G(a,O)

o

Va

E

m.P

(11 )

while G(a,A) is a C1-function of its arguments (where defined). Now we introduce some symmetry in the problem. 3.5.4. Definition. Let g : m.xm.nxw -)om.n , and consider the first order

differential equation x

=

g(t,x,A) .

(12)

Then we say that (1) is ~ev~ib{e if there exists a symmetric S E L~) such that S2 = I and Sg(-t,SX,A) = -get,x,A)

V(t,X,A) Em. xnf x w .

One also says that system (12) has the "pMpeJL:ty E wU;h (see Hale [77]).

( 13) ~e.ope& :to

S"

3.5.5. Reversible periodic systems. Assume that equation (1) is reversible

SAC -t)

-A(t)S

Vt Em.

(14)

and

105

Sf(-t,Sx,A) = -f(t,x,A) , Vt

E

n· lR , Vx

n lR , VA

E

w •

E

( 15)

In order to introduce the fonnalism of the preceding sections, consider a two-element group G = {e,i}, where e is the identity, and i 2 = e. We define representations rand r of G by x

r(i)x(t)

Sx( -t)

Vx

E

X, Vt E lR

(16 )

f(e)z = z

rei) z(t)

-Sz(-t)

Vz

E

Z, Vt

(17)

r(e)x

=

E

lR

It is easily verified from (14) and (15) that

r(s)L

Vs

Lf(s)

E

G

and N(r(s)x,A)

Vx

r(s)N(x,A)

E

X, VA

E

w, Vs

E

G.

That means: (3) is equivariant with respect to (G,f,r). Remark that rand do not coincide on X. According to the example 2.5.12 the projections P and Q satisfy the commutation relations (3.4) and (3.5). So ','le are in a position to apply the results of the preceding sections. 3.5.6. Since r(i)(kerL) = ker Land f(i)(R(Q)) singular (pxp)-matrices Band 13 such that :

r

R(Q) there exist non-

p S<jJ.(-t) = J

I

Bk·~(t) J

j

1, ... ,p , Vt

E

lR

( 18)

I l\'1f\(t) J

j

1, ... ,p , Vt

E

lR •

( 19)

k=l

and p -Slj,I. (-t) J

k=l

This implies f(i)cPa

106


r(i)'¥a

=

'!13a

Va

E

lRP .

3.5.7. Theorem. Suppose that equation (1) is reversible. Let x*(a,A) and G(a,A) be the functions given by theorem 3. Then, i f nil ';;;Ao(lIall) : x*(Ba,A)(t)

=

Vt

Sx*(a,A) (-t)

E

JR

(21)

and G(Ba,A) = BG(a,A)

(22)

D

3.5.8. Theorem. Under the conditions of theorem 7 let aEJRP be such that a = Ba .

(23)

Then x *(a,A)(t)

=

Vt

Sx *(a,A)( -t)

E

JR

(24)

and

G(a,A) = BG(a,A)

(25)

D

Theorems 7 and 8 contain precisely the results given by Hale in [77], p.267269. The following result (Meire and Vanderbauwhede [ 163]) may help to determine the elements of ker L. Consider the linear 2n-periodic equation

:ic = A(t)x

(26)

and asswne that (26) is reversible, i.e. we have (14) for some S. Let T(t)

T(t,O) be the transition matrix of (26), and C = T(2n) the corresponding monodromy matrix. We have T(t+2n) = T(t)C

Vt

E

JR .

(27)

3.5.9. Theorem. Let A(t) satisfy (14). Then T(-t) = ST(t)S

Vt

E

JR

(28) 107

and C = ST

-1

(n)ST(n) .

Proof. It is immediate to see that T1 (t) = ST(-t)S will be matrix for (26). Since T1 (O) = I, we have T1 (t) = T(t) for all t proves (28). Putting t = -n in (27) and using (28) gives (29). 0

E

R. This

3.5.10. Corollary. Let A(t) in (26) satisfy A( -t)

=

-A(t)

Vt ER .

(30)

Then all solutions of (26) are even and 2n-periodic. Proof. We can take S I in theorem 9. Then C = I. The result follows from (28), (29), and the fact that all solutions of (26) have the form x(t) T(t)xO for some Xo E Rn . 0

~

3.5.11. Example. Suppose that the right-hand side of (1) is odd in t; i.e., (30) holds while also f(-t,x,:\)

=

-f(t,x,:\) .

(31)

Then equation (1) is reversible, using S = 1. It follows from cOrollary 10 that dim ker L = n, while the elements of ker L are even functions of t. So B = 1. Similarly, the elements of ker L* are even functions of t, and B = -1. We conclude from theorem 8 that : x*(a,:\)(-t) = x*(a,:\)(t)

Vt ER

and

G(a,:\)

=

-G(a,:\) = 0 .

So, for each (a,:\) ERnxw satisfying

(33)

nil ,;;;; :\o(lIall), x*(a,:\)(t) is an even

function of t, and is a 2n-periodic solution of (1). Or : for !lA1l ly small, all solutions of (1) are even and 2n-periodic. 108

The foregoing forms a local version of a global result which can easily be obtained using the approach of "autosynartetic solutions" of differential equations, as developed by D.C. Lewis ([258],[ 259]). In fact, the argument . of the proof of theorem 9 is a special case of the arguments used by Lewis. The result is that all solutions of

x = f(t,x)

(34)

are even and 2IT-periodic , as soon as f is of class C1, odd and 2IT-periodic in t. another application of theorem 8 we derive the frequency response curves for a class of oscillation equations with a small nonlinearity and a As

small forcing term. 3.5.12. Example. Consider the following scalar equation Z d x + x + g(x) dT2

where g : JR

-+

h(wT) ,

=

(35)

JR is a C1-function, h : JR

-+

JR is ZIT-periodic and even, while

w is a real number near 1. We also assume that g and h are small, in a sense to be made precise. Making the time rescale WT t, we can rewrite (35) as a first order system : x =y y

=

-x -

(w -Z -l)x + W-Z [-g(x)+h(t)] .

(36)

We are looking for ZIT-periodic solutions of (36); these will correspond to 2n/w-periodic solutions of (35). Equation (36) has the form (1), with:

A

f 1 (t,x,y;w,g,h) = 0 , (37)

fZ(t,x,y;w,g,h)

=

-(w

-Z

-Z

-l)x + w [-g(x)+h(t)] .

This means that we consider Ie = (w,g,h) as the parameter; we let wE JR+ and 109

hE {z :lR->-lRl z is continuous, even and 21T-periodic}, with the usual supremum-norm. The space of the functions g is given a Banach space structure as follows. We choose in advance a value of R> as appearing in theorem 3. All solutions in the bounded set described by theorem 3 will satisfy Ix(t) I ~ 2R, Vt E lR. We take

°

g E {g: [-2R,2Rl +lR

I g is

of class C'}

this last space is equiped with the C'-norm. Let us now look for the symmetries of (36), and apply theorem 8. 3.5.13. The system (36) is reversible, using

We can take : [

cost

I

-sintJ

sint] . [ cost

It follows that

By theorem 8 one of the two bifurcation equations will be identically satisfied if we take a = (r,O)T, for some Irl ~ R. The remaining bifurcation equa tion takes, after multiplication by w2 , the form :

°

(w 2-l)r + TI1 J21T[-g(reost +vi(r,O,w,g,h) (t)) + h(t) leost dt

°.

(38)

Let (w,g,h) be given, with Iw-ll, II gil and IIhII sufficiently small, in accordance with theorem 3. Then to each solution r of (38), with Irl ~ R, there corresponds a 21T-periodic solution of (36); by theorem 8, this solution will also be an even function of t. Usually (38) is approximated by (w 2-1)r +-1 f21T[ -g(rcost) + h(t)l cost dt = 1T

°

110

°.

(39)

and h (sufficiently small), this is a relation between the frequency w and the "ampli tude" r of the corresponding solution.of (36). The curves in (w,r)-plane given by (39) (or (38)) are referred to as frequency response curves. For a discussion of this curves, for example in the case of the Duffing equation, see Hale [77]. 3.5.14. In [ 160], Mawhin defined an extension of the synnnetry property E, as follows. Let S be a constant symmetric matrix satisfying S2 = I, and let £ and , be real numbers such that £2 = 1 and (1+£), = 2mm, for some mEl. Assume that the function g in (12) is 2n-periodic with respect to t. Then we say that equation (12) has the property E with respect to (S,£,,) when : Sg(£t+"Sx,A) = £g(t,x,A)

Vt E JR, Vx E nf-, VA E w .

(40)

For £ = -1 and , = 0 this reduces to definition 4. We can repeat the foregoing approach for this case; (16) and (17) .are replaced by : r(i)x(t) = Sx(£t+,)

(41)

and

(42) Similar changes have to be made in the definition of B and

B,

and in (21)

and (24).

3.6. FURTHER RESULTS AND APPLICATIONS When further hypotheses are imposed on the restrictions of the synnnetry operators to ker L and R(Q), then more specific conclusions can be obtained from the results of sections 3 and 4. The next chapters contain several such results; here we give a few of the most direct examples, together with a mmilier of applications. 3.6.1. Theorem. Suppose that (H) is satisfied, while ker L

C

Xo •

(1)

111

Then we can find a neighbourhood W of 0 in X, and a neighbourhood in A, such that all solutions (x,>..) E WxwO of the equation

Wo

of 0

M(x,>") = 0

satisfy : r(s)x = x

Vs

E

G

(3)

that is : x E XO' Proof. This follows from theorem 1.6 and theorem 4.4.

0

As a consequence all sufficiently small solutions of (2) can be found by solving the reduced bifurcation equation (4.11). Using property 2.4(ii) we have the following corollary which formulates a practical way to apply theorem 1. 3.6.2. Corollary. Assume (H), and let GL = {SEG I r(s)u=u, VUEker L}

(4)

Then there exist a neighbourhood W of 0 in X, and a neighbourhood Wo of 0 in A, such that all solutions (x,>..) E WxwO of equation (2) satisfy r(s)x = x

o

(5)

3.6.3. Theorem. Assume (H) and (6)

Zo C R(L) ,

i.e. (7)

Let U be the neighbourhood of the origin in ker L given by theorem 4.4. Then (x*(u,>..),>..) is a solution of (2), for each Cu,>..) E (unXO) xwO; also, each of these solutions belongs to XO' 112 , I

proof. It follows from (7) that, for z E R(Q), QOz tion equation (4.11) is trivially satisfied. D

O. So the bifurca-

3.6.4. Example. The foregoing theorems are abstractions of the results in example 5.11. Indeed, in this example the restriction of res) to ker L is the identity, for each s E G; also, the restriction of rCs) to R(Q) equals minus the identity. For this example the conditions (1) and (6) are satisfied. 3.6.5. In the next theorem we will make a more detailed hypothesis on the symmetry properties of the elements of ker L. In order to formulate the hypothesis, consider a closed subgroup G' of G, together with the corresponding invariant subspace

Xb

= {xEX I r(s)x=x, YsEG'}

(8)

and projection operator (9)

(m' is the Haar measure of G'). We will assume that ker L n Xb generates ker L by the action of the symmetry operators; more precisely : (R) For each u E ker L one can find s E G such that

r(s)u E ker LnXb .

(10)

~otheses

similar to (R) will play an important role at several places in the following chapters, and more in particular in the main result of chapter 4.

3.6.6. Theorem. Assume (H) and (R). Let U, V and Wo be the neighbourhoods given by lemma 3.5. Let (X,A) E (UxV)xwO be a solution of (2). Then there exists some s E G such that r(s)x E

Xb .

(11)

113

Proof. Let u = Px; then x = X*(U,A) and F(u,A) = O. By (R) we can find s E G such that f(s)u E ker L n XO. Then f(s)x

=

f(s)x *(U,A)

=

x *(f(S)U,A) ,

and, for each t E G' : f(t)f(s)x = x*(f(t)f(s)U,A) = X*CfCS)U,A) This proves the theorem.

f(s)x .

0

3.6.7. Corollary. Under the hypotheses of theorem 6, all solutions

(X,A) E (UxV)xwO of (2) have the form x = rcs)xO '

with s E G, and with (XO,A) a solution of (2) such that Xo E XO. Conversely, of (XO,A) is such a solution, then (f(S)XO,A) is also a solution of (2), for each s E G. 0 This corollary allows us to restrict attention to solutions of (2) belonging to the subspace XO. By theorem 4.4 this gives a reduction of the bifurcation equation. 3.6.8. Example : subharmonic solutions. Consider again the 2n-peri~dic equation

x = A(t)x

+

f(t,x,A) ;

( 12)

making the same assumptions as in subsection 3.5.1, we will this time however look for ~ubhanmonLe solutions of (12). These are solutions having a least period which is an entire l1Ultiple of the basic period 2n of (12). Using the adapted Liapunov-Schmidt reduction for (12), (as in section 5) am theorem 1, we obtain the following negative result: Let m EN\{O} and suppose that each 2nm-periodic solution of

:ic 114

=

A(t)x

(1

is in fact 21T-periodic. Then each 2mn-periodic solution of (12) with sufficiently small, will also be 21T-periodic. For the proof, we replace 21T by 2nrn in the treatment of section 5. For example, we consider the spaces x(m) and Z (m) of continuously differentiable, respectively continuous, 2nrn-periodic functions. Consider also the group G = l (mod m) and its representation (f(p)z)(t) = z(t+21Tp)

Yt E JR , Yz E Z(m)

(14)

p = 0,1, ... ,m-1 . Defining L(m) : x(m) + z(m) formally as before, the hypothesis shows that r(p)u = u for each u E ker L(m) and each p = 0,1, ... ,m-1. The result follows from theorem 1. The region where the result is valid is the same as the one given by theorem 5.3 for the validity of the Liapunov-Schrnidt reduction. 3.6.9. Example: conservative oscillation equation. Consider the problem of finding 21T-periodic solutions of the autonomous scalar equation :

x + x = g(x,A)

(15)

where g : JRxJ\ +JR is a C1-function, satisfying g(O,O) = ~~(O,O) = o. Of course there is no reason whe we should only consider 21T-periodic solutions of (15). However, if we want to find periodic solutions of (15) having a period near to 2, then by a time rescale (as in example 5.12) the problem can be reduced to that of finding 21T-periodic solutions of another equation, which is of the form (15) if we include the (unknown) rescaling parameter in the new parameter A.

Let

z = {z :JR+JRI z X = {x E Z I x is L : X

+

is continuous and 21T-periodic} of class C2}

x-»Lx=x+x

Z

and

N

XxJ\

+

Z

(x,A) -» N(x,A)(t)

g(xCt) ,A) .

115

Our problem can be written in the form

= N(X,A)

Lx

We have ker L

span{cos(.),sin(.)}

and R(L) = {z E Z

2n

I f0

J2n

z(t)cost dt = 0 z(t)sint dt = OJ .

( 18)

Let G = 0(2), and define a representation of Gover Z as follows (f(a) z)(t)

z(t+a)

Va EJR, Vz E Z, Vt EJR,

( 19)

Vz E Z, Vt E JR •

(20)

and (f z)(t) = z(-t) a

(See subsection 2.6.13 for the notations f(a) and fa)' The restrictions of f(a) and fa to X define a representation of 0(2) over X, which we also denote as f. The mappings Land N are equivariant with respect to (G,f); the hypothesis (R) is satisfied for G' = {<jJ(O)=I,a}, and Zo is the space of continuous, 2n-periodic and even functions. An application of the6rem 6 and corollary 7 gives the following conclusion Each solution (X,A) E XxA of (15), with IIxll and I All sufficiently small, becomes an even function of t after an appropriate phase shift; i.e. xC -t+a)

x(t+a)

Vt

E

JR ,

for some a EJR. When solving (15) for small 2n-periodic solutions it is sufficient to consider even 2n-periodic solutions. The blfurcation equations for (16) form a two-dimensional problem for each fixed A; by our result, this reduces to a one-dimensional problem.

116

3.6.10. Example: Dirichlet boundary value problem. Let n be a bounded C2 ,a_

domain inJK1 (n>2, a E ]O,l[), and consider the following Dirichlet problem: ,0,u+jJ.u ]

u

=

in n ,

f(x,u,>..)

=

°

(21)

em ,

in

where p. is an eigenvalue for the problem ]

°

,0,u + jJU u

=

in n , (22)

°

in eln .

We suppose that f : fi xJR x/\. ... JR is of class C1 , and satisfies f(x,O,O) =

°

L. : X

N:

Vx

°

E

n

(23)

{U E C2 ,a(fi) iu(x)=O, VxEeln} and Z

Let X

]

elf elu(x,O,O)

-+

u

Z

XX/\' -+

Z

+>

L.u = ,0,u+ p.u

(u,>..)

]

]

+>

N(u,>..) (x)

f(x,u(x),>..).

Our problem takes the fOTIn (24)

We know from the results of section 2.3 that L. is a Fredholm operator with J zero index. n

Let G be the group of transformations S : JR

-+

n JR , which are compositions

of translations, rotations and reflexions, and which leave the domain n invariant: Sen) = n. Define a representation of Gover CO(fi) (and so also over X and Z) as follows (r(S)u)(x)

= ueS -1 x)

°-

_.

Vu E C (n), Vx E n, VS E

G.

(25)

AsslUJle also that

117

V(x,u, A) E ~ xJR x fe, VS E G .

f(Sx,u,A) = f(x,u,A)

This condition is in particular satisfied when f(x,u,A) = f(u,A) does not depend explicitly on x. Application of corollary 2 gives the following result Let f in (21) satisfy (23) and (26), and let Gj = {SEG

I r(S)u=u,

VUEker Lj } .

Then each solution (U,A) of (21), with lIul1 2 ,a and II Ali sufficiently small, satisfies u(Sx) = u(x)

Vx

i1

E

VS

E

G. J

A more precise result can be given for bifurcation from the lowest value wl of (22). 3.6.11. Theorem. Consider the boundary value problem

L'lu + wlu u

= f(X,U,A)

in rI

=0

in art

where f satisfies (23) and (26), and Wl is the lowest eigenvalue .£or (22). Then each solution of (28) with lIull 2 and IIAIl sufficiently small satisfies: ,a u(Sx) = u(x)

Vx E ~ , VS E G .

Proof. From the foregoing it follows that it is sufficient to show that G1

=

G. By theorem 2.3.31 we have dim kel' L1

=

1; let ker L1

= {au,!

I aEJR}.

By the same theorem, u 1 can be chosen such that u 1 ex) > 0, Vx E rI. Since r (S)(kel' L1) = ker L1 , for each S E G, we have Vs

G,

E

for some a(S) E JR \{O}. Suppose

Ia(S) I < 2

3

1; since G is compact (rl is bounded)

we may suppose that the sequence S,S ,S , ... converges to some So E G (in the operator topology). But then 118

r(So)u 1 = lim r(Sn)u 1 = lim (a(S))nu1 = n->= n->=

°,

contradicting the fact that r(SO) has an inverse. In case la(S) I> 1, then . laC S- 1) [ = Ia(S) 1- 1 < 1, again leading to a contradiction. So Ia(S) I = 1. Finally, also a(S) = -1 is impossible, since (r(S)u 1)(x)

= u,(S -1 x) >

We conclude that f(S)u 1

=

°

Vx E

~

•

u 1 ' for each S E G. This proves the theorem.

3.6.12. Example. As a particular example of problem (21), let

~ =

B

0

=

{XElR2 [ II xII 2 < 1} be the lmi t ball in JR2 • The eigenvalues can easily be found

by using polar coordinates and a Fourier expansion in the angle variable. One finds that each eigenvalue ]J must satisfy :

(30)

for some kEN, J k (x) being the k-th order Bessel function. For each k E :N, equation (30) has a sequence of solutions {]Jkm I mEN}, tending to +00 as m-+ oo • The eigenvalues ]JOm (mE N) are simple, and correspond to radially symmetric eigenfunctions JOC/POmr); ]JOO is the lowest eigenvalue. For k

~

1, ]Jkm is a

double eigenvalue, the corresponding eigenfunctions being linear combinations of

Assume now that f is radially symmetric; that means, we consider the Dirichlet problem : L1u + llkmU =

u =

fer ,u,;\)

in B (31)

a

in dB

with

f(r,O,O)

= df du(r,O,O)

lir E [O,l] •

(32)

119

The synnnetry group leaving B invariant is the group 0(2). Next theorem follows then from corollaries 2 and 7. 3.6.13. Theorem. Under the foregoing conditions, let (u,A) be a solution of (31) , with II ull 2 (i)

u

and I All sufficiently small. Then :

,a

= u(r)

( 1"1" ) u ((211)) ¢ l(P x

1n case k

= u (x )

=0

VX E B- ,p

" = 0 , 1 ,"', k - 1,1n

case k

~

1 .

Also, for each such solution there is a S E]R such that the function vex) u(¢(S)x) is symmetric around the xl-axis:

v(ax) = vex)

or

v(r,-¢)

= v(r,¢)

o

(33)

So, when solving (31) for small solutions, one can restrict attention to solutions satisfying (33). In a later chapter we will discuss a 3-dimensional version of problem (31). 3.6.14. Syrmnetry of the von Karmtm equations. Another class of problems for which similar resul ts can be proved are the buckling problems for plates considered in section 2.4, and described by the von Karman equations. We want first to investigate the synnnetry properties of the operators appearing in the abstract formulation of the problem. The approach is the same "for each of the different cases considered in section 2.4 (clamped plate, simply supported rect@lgular plate and simply supported cylindrical plate); so we will not make the distinction here. Let Sl

C

JR2 be the basic domain of the plate problem, and let S :]R2

be @lY distance preserving transformation leaving

~

invariant :

S(~)

JR2

+

= ~.

Since the elements of H, the basic Hilbert space, are continuous functions, we can define the representation (r(S)u)(x)

= u(S -1 x)

Vu E H, Vx E

Let u,v,¢ E H be sufficiently smooth, and w

120

?i

B(u,v), then

(34)

This implies :

J~[ r(s)u,r(S)vl¢

=

f~[U'Vl(S-lX)¢(X)dX J~[ u,v j(x)¢(Sx)dx

f~I'lW(X).I'l¢(SX)dX J~ I'lw(S -1 x) .I'l¢(x)dx (r(S)w,¢) .

We conclude that B(r(S)u,r(s)v)

r(S)B(u,v)

Vu,vEH.

(35)

Vw E H .

(36)

It follows also that ccr(S)w) = r(S)C(w)

If the function FO appearing in the definition (2.4.13) of the operator A satisfies : Vx E

~

(37)

(that means: the external forces have the same symmetry as the domain ~), then also: Ar(S)

r(S)A .

(38)

Finally, also the operator Q appearing in the equation for a cylindrical plate satisfies Q(r(s)w) = r(S)Q(w)

Vw E H •

(39)

121

3.6.15. The clamped plate. Consider the buckling problem for a clamped plate, which is also subject to a normal load; this load is supposed to 'be proportional to a small parameter v. The equation takes the form : (I - '\A)w

+

C(w) = vp

(40)

2 2 where p is a fixed element of H = WO ' (rn. We let G be the group of transformations S considered in the preceding subsection; suppose that r(s)p

=p

VS

E

G.

(41 )

Then the solutions of (40) have s)~etry properties similar to those of the solutions of the Dirichlet problem considered in example 10. More precisely: Let .\. be a characteristic value of A, and L. I-A.A. Let J

Gj

J

J

{SE G I u(Sx) =u(x), VxErl, VuEker Lj }

Then for IX-'\j I and Ivl sufficiently small, each sufficiently small solution w of (40) satisfies w(Sx) = w(x)

Vx E rl , VS E G. . J

(42)

For example, in the case of a circular plate we have essentially tl;e same result as in theorem 13. In particular, when solving (40) for a circular plate, we can restrict attention to solutions which are symmetric around the Xl-axis. 3.6.16. The rectangular plate. In case p = 0 (or v = 0) in (40), this equation has an additional synnnetry : all operators corrrrnute with -I; if (w,.\) is a solution, then the same is true for (-w,.\). This will double the number of symmetry operators, as compared with the group G cons ide red in the preceding subsection. As an example, consider the simply supported rectangular plate. Denoting the elements of the group G by their representation over H, we have for this case:

122

(43)

where

rxw(x,y) = we -x,y) r yw(x,y) = w(x,-y)

(44)

rxy = r xr y . The characteristic values Amn (m,n E N\{O}) are given by (2.4.51); they correspond to the eigenfunctions ¢mn given by (2.4.52). Denoting by Gmn the subgroup of G which leaves ¢mn invariant, we have : G

mn

{I,-rx ,-ry ,r xy } if m = even and n

even ;

{I ,-rx,r y,-r xy}

if m = even and n

odd

{I,rx,-ry,-rxy}

if m = odd and n

even

{I,rx,ry,rxy }

if m = odd and n

odd

Solutions (W,A) with Ilwll H and IA-Amnlsufficiently small will be iIwariant under the opera tors of Gmn ,under the condition that Amn is a simple characteristic value. 3.6.17. The cylindrical plate. In the case of a simply supported plate the equation is no longer invariant under the operator -I, the normal load is zero; this is a consequence of the appearmlce dratic operator Q in the equation. Suppose that the normal load has the symmetry of the cylinder then we can take G

{rea) ,r(a)rx ,r(a)r y ,r(a)rxy I aEJR}

cylindrical even when of the qua(see (41));

(45)

where r x ' ry and r xy are as before, and 123

crCa)w)(x,y) = wCx,y+a)

Va E]R •

The subgroups Gmn corresponding to the characteristic values Amn given in subsection 2.4.28 are: G

G

{rea) ,r(a)r x I aE]R}

Gm,n

21T ,rC-p)r 2IT \ -} { rc-p) n n x p = 0,1, ... ,n-l

if m = odd and n

Gm,n

{rc~TIp) \ p = 0,1, ... ,n-l}

even and n

m,O

if

m = odd ;

Gm,O

if m

=

even;

if m

~

~

1

1 .

Solutions CW,A,V) in a neighbourhood of (0,Anu1,0) will be invariant under the operators of G . Also, when n > -I, one can restrict attention to solumn tions satisfying r w = w; this follows from corollary 7. Y

3.6.18. Bibliographicalp.ote. For other applications of symmetry argmnents in bifurcation theory (e.g. for the Benard problem), one can see the recent lecture notes by Sattinger [265].

124

Perturbations of symmetric nonlinear equations 4.1. INTRODUCTION

Consider the problem of finding 21T-periodic solutions of the ordinary differential equation : ~ +

x =

g(x,~} +

where g(O,O) =

Wp(t) ,

~~(O,O) = 0, p :

(1)

lR

-r

lR is continuous and 21T-periodic, and W

is a small parameter. In In any particular examples of such problem the function pet) is also even: p(-t)

=

pet). Under such condition one usually re-

stricts attention to 21T-periodic solutions which are also even in t. (See e.g. Hale [77], Hale and Rodrigues [ 88]). There is a very practical reason for this restriction: using the results of section 3.4 it is easily seen that the bifurcation equations for this problem reduce from two scalar equations in the general case to only one scalar equation if one restricts to even solu hons . In case W = 0 the restriction to even solutions can be justified by corol-

lary 3.6.7

each solution of (1) with

after an appropriate phase shift. When applicable; indeed, equation (1) is for

w

0 becomes an even function of t

t

lJ lJ

0 the corollary is no longer

t

0 not invariant under arbitrary

phase shifts. Although the restriction to even hlnctions gives us information

on Mme, solutions, the question remains whether or not there are any other solutions. In this chapter we describe conditions which ensure that all sufficiently small solutions of (1) are even when

lJ

to.

In the particular case of the Duffing equation, where g(x,~)

=

AX - x

3

(2)

such result was proved by Hale and Rodrigues [88 J. The main property of (1) ~ed

in their proof can best be eA~ressed by the hypothesis (R) used in

section 3.6. Let G

=

0(2), with a representation over the space of continuous

Zn-periodic functions as given by (3.6.19) and (3.6.20); let G'

=

{¢(O),o}. 125

°

°

For ~ = the equation (1) is equivariant with respect to (G,r); for ~ f the equation is only equivariant with respect to the subgroup ot. This means that the perturbation destroys the symmetry of the unperturbed equation. For

°

~ = (1) reduces to the equation studied in subsection 3.6.9; it follows that (1) satisfies the hypothesis (R) of section 3.6. The difference with the situation of section 3.6 is that (1) is no longer equivariant with respect to G = 0(2) when jJ f 0. In section 2 we prove some abstract theorems which generalize the result of Hale and Rodrigues to a large class of nonlinear equations having symmetry properties similar to those of (1). Our presentation is an adaption of some earlier joint work of H.M. Rodrigues and the author [182]. In section 3 we discuss the particular case where the unperturbed equation has an 0(2)symmetry; we also consider applications to periodic solutions of forced conservative oscillatory equations such as (1), elliptic boundary value problems in the unit circle, and some buckling problems having rotational symmetry. In section 4 we briefly discuss the situation when the unperturbed equation has an 0(3)-symmetry.

4. 2. THE ABSTRACT RESULTS

4.2.1. 1ne hypotheses. Let X, Z, A and L be real Banach spaces, rl a neighbourhood of the origin in XxAxl: and M : rl ->- Z; we will consider the equation

M(X,A,O) =

°.

(1)

Remark that we have replaced the parameter space A used in the previous

°

ter by a product space Ax=' When ° = we call (1) the Uf1pe~tLULbed equation, v.hile the p~LULbed eQuctUon. corresponds to f 0. We make the following assumptions about M. (Hl) M is of class C2 , M(O,O,O) thesis (C) of section 3.1.

°and

°

L

DI1(O,O,0) satisfies the hypo-

(H2) There is a compact topological group G, a closed subgroup GO' and sentations r : G -r LCX) and f : G ->- L(Z), such that M is equivariant with respect to (G,r,f) when ° = 0, and with respect to (Go,r,f) when

°f 126

°:

Vs E G, V(X,A,O) E n

M(r(S)X,A,O) = r(S)M(X,A,O)

(2)

and M(r(s)x,A,a) = r(S)M(X,A,a)

Vs EGO' V(X,A,a) En.

(3)

As in chapter 3 it follows from (Hl) and (H2) that L is equivariant with

respect to (G,r,r), and that we can find equivariant projections P E L(X) and Q E L(Z) such that R(P) = ker Land ker Q = R(L). We w~{,use the follo(J wing subspaces of X and Z : XI = {xEX I r(s)x=x, VSEG} , Xo = {xEX I r(s)x=x, VSEGO} , (4)

ZI = {ZEZ

I r(s)z=z,

VsEG} , Zo = {ZEZ I r(s)z=z, VSEGO} ,

together with the corresponding projections : PIX =

fG r(s)xdm(s) , Poz = f

r(s)xdmo(s)

Vx EX

GO Ql z =

fG r(s)zdm(s) , Qoz = f

(5)

r(s)zdmo(s)

Vz

E

Z

GO In (5) m and mO are the Haar measures of G and GO respectively; one has

We will also use the mappings y : G + 1R and y .

G + 1R defined by

1

yes) = sup{llCI-po)r(s- )ullluEker Lnxo' lIuli =1} and

(7)

yes) = inf{U (I-QO)f(S)wlI

I WE R(Q) n ZO'

IIwll = 1}

Now we can formulate the remaining hypotheses : (H3) For each u E ker L there is some s E G such that r(s)u E ker Ln XO;

i.e. :

127

(H4) There is as>

° such that

Sy(s) ,,;; yes)

Vs E G •

Hypothesis (H3) is in fact the hypothesis (R) used in section 3.6; (H4) a technical hypothesis, which can be verified as soon as ker Land R(Q) determined. At the end of this section We will briefly return on these hypotheses.

lS

4.2.2. Theorem. Assume (H1), (H2) and (H3). Then there is a neighbourhood W of (0,0) in XxI\. such that, for each (X,A) E W,

M(X,A,O)

=

°

(10)

implies the existence of some s E G such that f(s)x

E

Xo

'

i.e. f(t}f(s)x

f(s}x

Vt

E

GO •

Proof. This is just a restatement of theorem 3.6.6.

0

The next theorem forms the main result of this chapter; it describes conditions which ensure that the solutions of the perturbed equation (1) have the same symmetry as the equation itself. 4.2.3. Theorem. Assume (H1), (H2), (I-l3) and (H4). Let S c Z\{O} be a compact subset such that inf lIoll- 1l1QD M(O,O,O}oll _ o(S} > oES

Let

128

0

°.

be the cone generated by S. Then one can find a neighbourhood W of (0,0) in Xxf\, and a neighbourhood

w of 0 in L:, such that for each solution (X,A,O) r5

I 0, we have x

E

0

Wx(w n Cs ) of (1), with

XO' i. e . :

r(s)x = x p.r

E

Vs

E

(14) .

GO .

a f. Let (x, A,O) be a solution of (1), sufficiently near to the origin,

such that the Liapunov-Schmidt reduction holds. Let u x = x *(U,A,O)

and

F (u , A,0)

=

Px; then we have :

o.

(15)

Applying the results of chapter 3 separately for the cases

0 and

0=

0

I 0,

we see that x * and F have the following symmetry properties : x*(r(S)U,A,O)

r(s)x *(U,A,O) , Vs

F(r(S)U,A,O)

E

(16)

GO if

0

f 0, Vs

E

G if

0

0

r(s)F(U,A,O) Vs

By (H3) we can find some s

E

(17) E

GO if

0

f 0, Vs

E

G if (}

Gand some Uo E ker L n

r(s)u O' where, for simplicity of notation, we put

=

0 .

Xo such that u =

5 = s -1 for each s

E

G.

Then (15) implies that

o. The mapping G is defined for all s

E

( 18)

G and for (UO,A,O) in a sufficiently

small neighbourhood of the origin in (ker L n XO) x f\ x L:. It follows from (17)

that

o

( 19)

o

(20)

and

129

Let us analyse G in somewhat more detail. Using (19) we can write .

1

G(S,UO,A,O) = (I-QO)r(s) foDoFCr(S)Uo,A,to).Odt 1

(I-Qo)f(s)IoDoF(Por(~)uo,A,tO).Odt

1 +

1

(I-Qo)f(s)fodtfodt'

DuDaF(POr(~)uO + t' (I-PO)r(s)uO,A,to). (0, (I-PO)r(s)u O)

We have (21 ) where C2 (S,U O,A,0) remains bounded for s E G and (uO,A,a) in a neighbourhood of the origin; indeed G is compact and Du DaF(u,A,a) is continuous. Since F(POU,A,O) E R(Q) n Zo for each (U,A,O), it follows that Di'(POU,A,O) .0' E R(Q) n ZO' Then the definition of yes) implies that (22) where, for

° i=

0 : (23)

Since DoF(u,A,O) is a continuous function of its arguments, and

we have by (12) and the compactness of S

for all s E G and all (UO,A,O) sufficiently near to the origin, with and a E CS . Now suppose the theorem is false. Then we can find 130

° =I

0

nEN}, converging to (0,0,0) as n

+

each term being a solution of (1),

00,

°

n E CS\{O} and xn ¢ XU' Let u n = Pxn ; then x n = x'(un ,An ,0n ) for n large enough. Since u n E Xo implies xn E XU' by (16), we have u ¢ XU' Let s E G and u o n n ,n E ker LnxO be such that u = res )u o ; then it follows from the definition of yes) and from n n ,n f for all n EN. Moreover, our foregoing analysis (I-POJun f 0 that y(s) n and such that for each n EN we have

°

shows that (y(sn )110n II)

-1

G(s n ,u O,n ,A n ,0) n

Taking the limit for n

+

00

°

=

Vn EN .

(25)

and using the estimates (21), (22) and the hypo-

thesis (I-l4) , we conclude that: lim C (s ,u ,A,O) = n+oo 1 n O,n n n

°.

(26)

o

This, however, contradicts (24).

4.2.4. Corollary. Assume (H1)-(I-l4). Suppose that dim L

operator QDoM(O,O,O) : L

+

<

00

and that the

R(Q) is injective. Then there is a neighbourhood

Wof (0,0) in Xxii. and a neighbourhood w of 0 in L such that (X,A,O) E Wx(w \{ O}) and M(X,A,O) =

° imply x

E

XU'

Proof. We apply theorem 3 with S = {o E L:

I 11011

= 1}.

S is compact since

dim L: < co; together with the injectivity of QD M(O,O,O) this also implies

°

the condition (12) of the theorem. As for the conclusion, remark that Cs

=

L.

0

4.2.5. _Corollary. Let

Q

be a neighbourhood of the origin in Xxii., and

° E IR.

Consider the equation M(X,A)

= OMO(X,A)

(27)

Assume : Z is of class C2 , M(O,O,O) = 0, L = Di1(0,0) satisfies (C)

(i)

M:

(ii)

and M is equivariant with respect to some (G,r,r), with G compact; MO : Q + Z is of class C2 and equivariant with respect to some closed subgroup GO of G;

Q +

131

(iii) (H3) and (H4) are satisfied; (iv)

QMO(O,O) f o.

Then there is a neighbourhood W of (0,0) in XxA and a a O > 0 such that for each solution of (27) with (x,A) E Wand 0 < lal < a O we have x E XO' Proof. The hypotheses of theorem 3 are easily verified by taking and S = {'}. 0

~

4.2.6. Corollary. Let n be a neighbourhood of the origin in XXA, M and p E Z. Consider the equation

n+ Z

M(x,A) = p .

=R

(28)

Assume: (i)

M is of class C', M(O,O) = 0, L = DxM(O,O) satisfies (C) and M is equivariant with respect to some (G,f,r) (G compact);

(ii)

(H3) and (H4) are satisfied for some closed subgroup GO of G;

(iii) PO E Zo and Qpo f o. Then there is a neighbourhood W of the origin in XxA, a neighbourhood w of Po in Zo \{O} and a number aO > 0 such that for each p E Cw = {lJP 1 pE w,aER} with 0 < IIpll ,.;;; aO' each solution (x,A) E. W of (28) is such that x E XO' Proof. Let ~ = Zo and w a neighbourhood of Po in Zo on which I QPII remains bounded away from zero; such neighbourhood exists because of condition (iii). If dim ~ < 00 we can innnediately apply theorem 3, with S = w. In the general case one has to reconsider the last part of the proof of theorem 3; from the expression for C,(s,uO,A,a) it is easily seen that (26) gives a contradiction. Also, a careful examination of the proof shows that it is sufficient for M to be of class C'. 0 4.2.7. Corollary. Under the conditions of corollary 6, consider the equation M(x,A) = 0p0 ' wi th a E R. Then there is a neighbourhood W of the origin in XxA and a 132

(29)

°0 > 0 such that, that x

E

XO.

for 0 < D

lal

~ aO' each solution (x,A) E W of (29) is such

. 4.2.8. Remark. Let the hypotheses (H3) and (H4) be satisfied for some closed subgroup GO of G, and let sl E G. Consider the subgroup : (30)

We claim that (H3) and (H4) are also satisfied for the subgroup Gl • Indeed, we have : (31)

This shows that x E Xl if and only if x = r(sl)P Or(sl)x. So the corresponding projection on Xl is given by Pl = r(sl)P Or(sl). Now we have: ker L = {r(s)uO I sEG, uOEker LnxO} =

{r(s.sl)r(sl)uO I SEG, uOEker LnXO}

= {r(s)u l

I sEG,

u l Eker Lnx l }

This proves (H3) for the subgroup Gl • As for (H4) , we have for each s E G

and similarly

It follows that Sl Yl(s) ~ Yl(s) for each s E G, with S1

=

-

~

~

-

S[ I r (s 1) I I r (s 1) 1111 r (s 1) 1111 r (s 1) II]

-1

. 133

Using this remark one can easily prove some variants of the foregoing results. For example, one obtains the following modification of corollary 7. 4.Z.9. Corollary. In the hypotheses of corollary 7, replace the condition Po E Zo by the condition f(sl)PO E ZO' for some sl E G. Then the solution of (Z9) considered in the conclusion of corollary 7 will satisfy :

o

(31 )

4.Z.10. We conclude this section with a remark on the hypotheses (Hl)-(H4). The following situation frequently appears in applications : X is continuously imbedded and dense in Z, while Z = ker L ffi R(L), and dim ker L < Let Q be a projection in Z onto ker L, and such that ker Q = -R(L); then we can take P = Qi x ' Assume also that res) = r(s)i x ' for each s E G; we win denote both representations by f. The claim is that under such conditions the hypotheses (H1) - (H4) imply that the restriction of f to ker L is irreducible over ker L, except when ker L C XO' This last case is uninteresting as far as theorem 3 is concerned, since the conclusion of theorem 3 is in that case an immediate consequence of the results of chapter 3. In order to prove the claim, suppose that ker L Ul ffi UZ' with Ul and Uz nontrivial subspaces, such that 00.

Vs E G •

By (H3) we may assume that Ui nxO f {a}, for i = 1,Z. Excluding the case ker L C XO' we may also suppose that e. g. Uz\XO f if>. Take u l E Ul n XO' u l f 0, and Uz E Uz\ XO' By (H3) there is some s E G such that r(s)(u l +u Z) E XO' i.e. such that r(s)u l

E

Ul n Xo

and

Since u l E Xo and r(s)u l E Xl it follows that yes) and

134

O. Also, r(s)u z E Xo

imply that yes) > 0. This however contradicts (H4). 4.3. PERTURBATIONS OF EQUATIONS WITH 0(2)-SYMMETRY

In this section we apply the preceding results to the particular case where the unperturbed equation is equivariant with respect to the symmetry group G = 0(2). We will denote the elements of 0(2) by cp(a) and Tocp(a), where aEJRand: cp(a)

=[

cosa -S1-na

Sinal

(1)

cosa

LORn) is a representation, then we denote rCa) = r(cp(a)) and = r(T)r(cp(a)). We refer to section 2.6 for the irreducible T T representations of 0(2) . If r : 0(2)

r

(a) =

+

r .r(a)

4.3.1. The hypotheses. We will consider equations of the form

M(X,A,O)

=

°

(2)

with hypotheses on M similar to the ones described in subsection 4.2.10. ~lore precisely, we assume : (a) M : XxI\xL: + Z is of class C2 and M(O,O,O) imbedded and dense in Z; (b) if L = DxM(O,O,O), then dim ker L < (c) there is a representation r : 0(2)

M(r(R)x,A,O) = r(R)M(x,A,O)

00

+

0, where X is continuously

and Z = ker L ffi R(L); L(Z) such that

\IR E 0(2), V(X,A) E XxI\.

(3)

(d) the restriction of r to ker L is irreducible. By the theory of section 2.6 the hypothesis (d) implies that dim ker L = 1

or 2. We will consider separately the three possibilities given by theorem 2.6.19. 4.3.2. Theorem. Assume (a)-(d), dim ker L

1 and r u = u for each u E ker L. T

Suppose also that

135

,

,

M(r x,A,o) = r M(x,A,o)

V(x,A,a) .

(4)

Then each solution (x,A,o) of (2), sufficiently near to the origin, satisfies :

,

r x =x .

(5)

Proof. This follows from theorem 3.6.1, using the basic group GO {¢CO),,}.

0

4.3.3. Theorem. Assume (a)-(d), dim ker L Suppose (4) and

M(-x,A,o) = -M(x,A,o)

,

1 and f u

-u for each uE ker L.

V(x,A,o) .

(6)

Then each solution (x,A,o) of (2), sufficiently near to the origin, satisfies :

r ,x

-x.

Proof. Define

rea)

=

rca)

(7)

r

0(2) ~ L(Z) by

r, (a)

= -r , (a)

Va

Em..

(8)

r

It is immediate that defines a new representation of 0(2), while (3), (4) and (6) imply that the conditions of theorem 2 are satisfied, if we replace r by The result follows then from theorem 2. 0

r.

4.3.4. Lemma. Assume (a)-(d) and dim ker L = 2. Let GO = {m(O),,}. Then the hypotheses (H3) and (H4) of section 2 are satisfied. Proof. By theorem 2.6.19 there is a basis {u 1 ,u 2} of ker L such that f(a)u 1 = coska.u 1 - sinka.u 2

r(a)u 2 = sinka.u 1

136

+

coska.u 2 '

Va

Em.

(9)

for sane k E N\{O}. It follows that ker L () Xo = span{u 1}. Also, for each a,b E lR we can find some p ;;;. 0 and some a E lR such that a

= pcoska

= -psinka •

b

'Iben we have :

which proves (H3). 1 As for (H4), we have Po = QO = 2(I+r,). It follows that

'Ibis shows that: yea)

=

,

y (a)

=

yea)

=

Y, (a)

=

Isinkal

,

Va

E lR ,

and that the hypothesis (H4) is satisfied. 0 Using remark 2.9 we see that (H3) and (H4) will also be satisfied if we take GO = {~(0)"o~(2aO)}' for some fixed a O E lR. Then Xo consists of those x E X such that (10)

4.3.5. Theorem. Assume (a)-(d), dim ker L = 2 and : (i) Mis equivariant with respect to the subgroup GO (4) holds; (ii) for a compact S

C ~\{O}

{HO),,}, i.e.

we have

o(S) :: inf lIoll- 1l1QD M(O,O,O)oll aES a

> 0

(11)

Then there is a neighbourhood W of (0,0) in XXA, and a neighbourhood w of 0 in ~, such that each solution (x, A,o) E Wx (CS () w) of (2), with a f 0, satisfies (5). 137

Proof. This is an immediate consequence of theorem 2.3.

0

4.3.6. Corollary. If in theorem 2 and theorem 5 the condition (4) is replaced by r T (2a O)M(x,A,a)

(12)

V(x,A,a) E XXAXL for some a O E R, then the theorems remain valid, if we replace the conclusion (5) by (10). Proof. This follows from the remark before theorem 5, using GO

{¢(0),T o ¢(2aO)}'

0

4.3.7. Application: Periodic perturbations of autonomous oscillation equations. Consider the problem of finding 21T-periodic solutions of the scalar equation

x + x = g(X,A)

+ h(t,x,A,a)

(13)

We assume the following : (i)

g: RxA

-+

g(O,O)

=

R is of class C2 , with

o

0

(ii) h : R xR x A XL

-+

R is of class C2 , 21T-periodic in t, and

h(t,x,A,O) = 0

V(t,X,A) •

Using the formalism of subsection 3.6.9, this problem can be brought in the form (2). Using the representation (r(a)x)(t) = x(t+a)

(r T (a)x)(t) = xC-t+a) ,

the hypotheses (a)-Cd) are easily verified. We have ker L = span{cos(.),sin(.)} , 138

which transforms under r according to (9), with k we can take: 1

(Qz)(t) = -eost IT

f2IT

°

1

z(s)cossds +-sint IT

f2IT

°

1. For the projection

z(s)sinsds,

.

Q

(16)

Vz E Z

Application of theorem 5 gives the following results. 4.3.8. Theorem. Suppose that

h(-t,x,A,a)

= h(t,x,A,a)

Let S be a compact subset of 8(S)

=

(17)

~\{o}

such that

inf lIall- 1 1 f 2ITCOStD h(t,O,O,O)adtl >

°

aES

()

°.

(18)

Then for nil and II all sufficiently small, a E Cs and a f 0, each sufficiently small 2IT-periodic solution of (13) will be an even function of t : x(-t) = x(t)

Vt E lR •

(19)

Proof. (17) and (19) correspond to the conditions (i) and (ii) of theorem 5. 0 4.3.9. Particular case. Let

~

= lR and

h(t,x,A,a) = ap(t) ,

(20)

where pet) is continuous, 2IT-periodic and even. If

f

2IT

°

cost p(t)dt f

°,

(21)

then, for II All and 1a 1 sufficiently small, and for a f 0, each sufficiently small 2IT-periodic solution of

x+

x

= g(X,A)

+

ap(t)

(22) 139

will be an even function of t. For the case of the Duffing equation this sul t was proved by Hale and Rodrigues in [88]. Remark that for this case is sufficient for g(X,A) to be of class C1 (see corollary 2.6). 4.3.10. Theorem. Suppose that g( -X,A)

-g(X,A)

V(X,:A)

and h(-t,-x,A,o) = -h(t,x,A,o) ,

V(t,X,A,O)

(24)

Let S be a compact subset of Z\{o} such that 6(S) = inf DES

~o~-1\J2nBintD h(t,O,O,O)odt\ ° °

>

°.

(25)

Then, for IIA~ and Ilo~ sufficiently small, with ° E Cs and ° f 0, each sufficiently small 2n-periodic solution of (13) will be odd in t : X( -t)

=

VtEill..

-x(t)

(26)

Proof. This result is proved in a similar way as theorem 8, this time using the following representation of 0(2) over Z, the space of continuous 2n-periodic functions :

(fCex)z)(t)

z(t+ex)

(f

T

(ex) z)(t)

-z(-t+ex) .

o

(27)

4.3.11. Application: an elliptic boundary value problem. Let B = {XEill.2 \ ~x~ < 1} be the unit sphere in ill. 2 . (11.11 is the Euclidean nonn). We will denote by (r,e) the polar coordinates of a point x E B ; correspondingly, the value of a function u at the point x will be denoted by u(x) or u(r,e). Let f B xill. x l\ -+ ill. and h B xill. x l\ -+ ill. be C2-functions; we will assume that f does not depend on e :

f(x,u,A) = f(r,u,A) , 140

(28)

af f(x,O,O) = au(x,O,O) =

°

Vx E

B

(29)

Consider the following boundary value problem !'Iu + flkmU u(x) =

f(x,u,A) + oh(x,u,A)

xEB

°

X E

(30)

aB

Here 0 E R is a small parameter, and flkm is an eigenvalue for the Dirichlet problem for the Laplacian in B !'Iu + flU

=

u(x) =

°

°

xEB, X E

(31)

aB .

We have seen in subsection 3.6.10 how the problem (30) can be brought into 2 a (B) - and Z = C ' a (B) - for some a E ] 0, 1[ • the abstract fonn (2). We have X = CO' The operator L : X + Z defined by

°

(32) is a Fredholm operator with zero index (see section 2.3). The representation r of 0(2) over CO(B), as introduced in section 3.6, takes the form : (r(a)u)(r,8) = u(r,8+a)

(r (a)u)(r,8) = u(r,-8+a) . T

.

(33)

The discussion of the solutions of (31), as given in section 2.3 and subsection 3.6.12, shows that the hypothesis (a)-(d) of the foregoing general theory are satisfied. We will denote the eigenfunctions, given in 3.6.12, as follows :

Xo ,m(r,8)

=

°

J (lflOmr)

k

=

Xk ,m (r,8) = Jk(lflkmr)cosk8

k;;;.

1:k ,m(r,8) = Jk (lflkmr )sink8

k;;;.

° (34)

141

For the projection Q we take : (Qu)(x)

=

~Xkm(x)

fB u(y)Xkm(y)dy

+ '1ooskm(x) fB u(y) skm(y)dy Here if k

'100 is a normalisation constant, such that Q is indeed a projection; = 0, then the second term in (35) does not appear.

4.3.12. Theorem. Let k h(r,-6,u,A)

=

=

° in (30). Assume that

h(r,e,u,A)

V(r,e,u,A)

Then each solution of (30), with lIull 2,a ,II All and satisfies : u(r,-e) = u(r,e) Proof. If k

=

lal

(36) sufficiently small,

VCr ,e) .

0, then dim ker L

4.3.13. Theorem. Let k

~

=

(37)

1, and theorem 2 applies.

0

1 in (30). Assume (36) and

fB h(x,O,O)Xkm(x)dx f

°

Then each solution of (30), with a f small, satisfies (37).

(38)

° and I ull 2,a ,II All

Proof. By application of theorem 5.

and Ia I sufficiently

0

4.3.14. Corollary. If in theorem 12 or theorem 13 the condition (36) is replaced by h(r,eo-e,u,A)

=

h(r,eO+e,u,A)

V(r,e,u,A)

(39)

for some eO E lR, then the theorems remain valid, i f in the conclusion (37) is replaced by 142

VCr ,e)

(40)

In the case of theorem 13, the condition (38) must also be replaced by (41)

pro a f. By application of corollary 6. 4.3.15. Theorem. Let k

~

0

1 in (30), and assume that

(i)

f(r,-u,A) = -f(r,u,A)

V(r,u,A) ;

(42)

(ii)

h(r,-e,-u,A) = -h(r,e,u,A)

V(r,e,u,A)

(43)

(iii)

IB

h(x,O,O)skm(x)dx

Then each solution of (30), with and a 0, will be such that

r

u(r,-e) = -u(r,e)

r °.

(44)

lIull 2,a ' HAll and icri sufficiently small,

(45)

V(r,e)

Proof. The proof is similar to that of the foregoing theorems, by using this time the following representation of 0(2) over COCB) (r(a)u) (r,e) = u(r,e+a)

(r (a)u) = -u(r,-e+a) T

o

(46)

4.3.16. Remark. Theorem 15 has a corollary analogous to the corollary 14 of theorem 13. Also, one can combine several of the foregoing results, on condition that the function h has enough symmetry. The main point to be careful about is that the symmetry imposed on h does not contradict the condition (41). We give a few examples. 4.3.17. Corollary. Let k be even in (30). Suppose that: (i)

h(-x 1,x 2 ,u,A) = h(x1 ,-X 2 ,U,A) = h(x 1 ,x 2 ,u,A) ,

(47)

for all values of the arguments (ii) (38) holds, in case k

r O. 143

Then each solution of (30), with lIull 2 ,11'\11 and ,a with 0 f 0, will be such that :

101

sufficiently small,

Proof. By combination of theorem 12 (if k = 0) or theorem 13 (if k f 0) with corollary 14, taking 80 = TI/2. One could also use theorem 2.3 directly, by taking GO = {4J(o),4J(TI),LO¢(O), LO¢(TI)}. If k is odd, then the condition (47) on h implies that both integrals in (41) vanish; so the conclusion only holds for k even. 4.3.18.

Coroll~ry.

0

Let k be odd in (30). Assume (42), (44) and ( 49)

Then each solution of (30), with lIuli Z ,11'\11 and ,a wi th 0 f 0, will be such that

101

sufficiently small, and

(50) Proof. By combination of theorem 15 with corollary 14, in which one takes 80 = TI/2. Again, under the sY1lllnetry condition (49) for h, (41) can only be satisfied if k is odd.

0

4.3.19. Remark. For k > 1, the elements of ker L remain invariant under the symmetry operators :

{r(~j)

1

j

= 0,1, ... ,k-1}

It follows from the theory of chapter 3 that, if

her,8 + iTIj,u,'\) = h(r,8,u,;.)

j

0,1, ... ,k-1 ,

j

O,l, ... ,k-l ,

then also 2TI .) =ur,8 ( ) ( ur,8+TJ 144

for sufficiently small solutions of (30). In case k group of 0(2) to get a similar result.

0 one can use any sub-

4.3.20. Application: the circular and cylindrical plate. The buckling problems for a clamped circular plate and for a simply supported cylindrical plate have, in the absence of normals loads, also an 0(2)-symmetry; in the case of the cylindrical plate the symmetry group is even larger than 0(2) . A physically interesting perturbation is obtained by the introduction of a normal load. The problem dealt with in this chapter reduces for these examples to the question under wllat conditions the symmetry of the normal load determines the symmetry of the corresponding solution. Our abstract results can be applied in a way similar as for the problem (30). Here we will simply state a few of the results which one can obtain. Using the appropriate Hilbert space H (see section 2.4) the buckling problem for the clamped circular plate is described by the equation : (I-AA)w + C(w) = vp

(53)

here v is a small parameter, controlling the amplitude of the normal load. For the simply supported cylindrical plate we have the equation : 2 2 (I-AA+a A )w + aQ(w) + C(w) = vp •

(54)

For problem (53) we denote by A (m = 1,2, ... , n = 0,1,2, ... ) the charactenm ristic values of A; for n = 0, AmO corresponds to a radially symmetric eigenfunction: (55) for n ;;;, 1, Anm corresponds to the eigenfunctions a

nm

(r)cosne

(56)

(r)sinne.

(57)

and a

nm

145

In the case of the cylindrical plate Amn will denote the characteristic values given in subsection 2.4.28, with corresponding eigenfunctions ¢ and mn Wmn , as given in (2.8.71)-(2.8.73). If we assume that all characteristic values are distinct, then the hypotheses (a)-(d) of this section can easily be verified, using the symmetry operators introduced in section 3.6, and using the fact that A is self-adjoint. 4.3.21. Theorem. Assume that

V(r,e) for some

eO

E JR. Let mE

IN\{O}

(58)

and n E IN; in case n;;'

1,

assume that

((p'¢mn)'(p,Wmn )) f (0,0) .

(59)

Then each solution (W,A,V) of (53), with IIwll, [A-Amn [ and [v[ sufficiently small, and with v f 0 in case n;;' 1, will be such that VCr,e).

o

(60)

4.3.22. Theorem. If the condition (58) of theorem 21 is replaced by

V(r,e) ,

(61 )

then in the conclusion (60) should be replaced by V(r,e) . 4.3.23. Theorem. Consider (54) and

aSSlli~e

o

(62)

that (63)

V(x,y) ,

for some yo E JR. Let mE IN\{O} and n E IN. In case n;;' 1, let (59) be satisfied. Then each solution of (54), with IIwll, [A-Amn [ and [v[ sufficiently small, and wi th v f 0 if n ;;. 1, will be such that V(x,y) . 146

o

(64)

4.3.24. Remark. ·For the cylindrical plate there is no analogue of theorem~ 22; the reason for this is the presence of the quadratic term Q(~). 4.4. AXISYMMETRIC PER1URBATIONS OF A PROBLEM WIlli 0(3)-SYMMETRY In this section we briefly discuss an application 6f theorem 2.3 and its corollaries to perturbations of a boundary value problem with 0(3)-symmetry. The results which we can obtain are far less general than for perturbations of problems with 0(2)-symmetries. This is mainly due to the fact that the irreducible representations of 0(3) (and, more generally of O(n) with n ~ 3) are much more complicated than those of 0(2). The main difficulty is in finding a representation of 0(3) and a subgroup GO for which (H3) and (H4) are satisfied. We will only give one particular example. 4.4.1. The problem. In dary value problem : ~u + ~£jU

u(x)

=

B

= {XER3 I IIxll < 1} we consider the following boun-

= f(x,u,A)

+

xEB

crh(x,u,A)

(1)

°,

X E

Here f : B xR x A -+ R and h f(Rx,u,A)

3B

B xR x A -+ R are of class C2 , and such that

f(x,u,A)

VR

E

0(3)

(2)

Vx

E

B

(3)

and

3f f(x,O,O) = 3u (x,O,O) =

°

further cr E R is a small parameter, while Dirichlet problem ~u + ~u

u =

°

° in

~£j

is an eigenvalue for the

B

(4)

on 3B

(See further on for the notation ~£.). We bring this problem in the fo~ (2.1) by defining X = C~,a(B), Z = CO,a(B) and

147

M(U,A,O) (x)

llU(x) + ]J9, j u(x) - f(x,u(x) ,A) - oh(x,u(x) ,A) , Vx E

B.

(5)

The operator L = Du M(O,O,O) is given by : Lu(x) = L'lu(x) + )J9, j u(x)

Yu EX, Vx E :8

(6)

We know from the theory of section 2.3 that L is a Fredholm with index zero. On CO (:8) we can define the following representation of 0(3) f(R)u(x) = U(R- 1x)

Vx E :8 , VR E 0(3) .

(7)

Then it is clear from (2) that M(U,A,O) is equivariant with respect to (0(3),r) . 4.4.2. The eigenvalue problem (4). In order to obtain some more information on ker L, let us consider the eigenvalue problem (4). It appears that because of the spherical symmetry of this problem, it can be handled in a sui table way by using the spherical harmonics introduced in subsection 2.6.30. (See e.g. Courant and Hilbert [48]). The space U9, of spherical harmonics of order 9, has dimension 2S',+1; let {Y9,m I m= 1,2, ... ,2t+1} be a basis of US'" which is orthonormal with respect to the L,(S2)-inner product. One can show that ~ 2 {Y S', EN, m= 1 ,2, ... ,29,+1} fonus a complete orthonormal subset o,f L2 (S ). Then we can solve (4) by expanding u in spherical harmonics; one writes

J

u(r,8)

(8)

(where (r,8) are polar coordinates for x E:8) and brings (8) into (4). It follows that n9,mCr) must be a solution of DS',(]J)n = 0 , nCr) regular at r =

° and n(1) = ° ,

(9)

where D9,(]J) is a linear second order ordinary differential operator, singular at r = 0, and depends on 9, and ]J, but not on m. For fixed 9, problem (9) has nontrivial solutions if and only if ]J belongs to an infinite {]J 9,j I j EN} of eigenvalues; if ]J = ]J 9,j then (9) has a one-dimensional space 148

of solutions, spanned by a fllilction nQ,j (r). The eigenvalues llQ,j are strictly positive, llQ,j ~ 00 as j ~ 00, and llQ,j f llQ,fjf if Q, f Q,f. We conclude that (4) has nontrivial solutions if and only if II = llQ,j for some (£,j) E ~xN; to each eigenvalue ll£j there corresponds a (2Q,+1)-dimensional eigenspace, spanned by the functions m = 1,2, ... ,2£+1 . Under rotations the eigenspace transforms according to the irreducible representation r(£) of 0(3), introduced in subsection 2.6.30. So the index £ in iJ£j refers to the dimension 2Q,+1 of the corresponding eigenspace and to the way the eigenvectors transform llilder rotations. 4.4.3. The case Q, = O. First suppose that £ = 0 in (1). This means that dim ker L = 1 and that the elements of ker L are spherically symmetric. Using the theory of chapter 3 one then immediately proves the following : if GO is any closed subgroup of 0(3), and h(Rx,u,iI)

h(x,u,iI)

(9)

then each sufficiently small solution of (1) will satisfy u(Rx)

u(x)

( 10)

In particular an analogue of theorem 3.12 holds.

For a particular subgroup, namely GO = {I ,-I}, this result can be extended for all £ EN. Indeed, for general ,~ E ~ one has : Yu E ker L ,

(11 )

(see remark 2.6.32). Then the theory of chapter 3 gives us the following result. 4.4.4. Theorem. Suppose that £ is even in (1), and that h(-x,u,iI) = h(x,u,iI)

Y(x,u,iI) •

(12) 149

Then each sufficiently small solution (u,A,o) of (1) will be even u( -x)

=

u(x)

Vx E

Ii

( 13)

In case £ is odd, f(x, -u, A)

-f(x,u,A)

V(x,U, A)

(14 )

V(X,U,A) ,

(15)

and h(-x,-u,A) = -h(x,u,A)

then each sufficiently small solution will be odd u( -x)

-u(x)

Vx E

Ii

o

(16)

4.4.5. The case £ = 1. Let now £ = 1 in (1). It is immediately seen from the defini tion that to each spherical harmonic Y1 E U1 there corresponds a unique a E 1R3 such that

a.e

ve

E

S2· .

( 17)

0(3) ,

( 18)

Then

(r(R)Y 1)(e) = (Ra).e

VR E

and for each Y1 E U1 we can find some R E 0(3) such that (19)

where cElli and e 3 is the unit vector along the x3-axis. It follows that the hypothesis em) of section 2 is satisfied if we take (20) As for (H4) , we are in a si tua tion as described in subsection 2. 10. As a norm for elements in ker L = R(Q) we can take 1I all, the Euclidean norm of the

150

vector a ER3 which determines Y1 via (17). The restriction of ~ to ker L = R(Q) corresponds to projection of a onto the x3-axis. Using this correspondence it is easy to see that y(R) = y(R) for all R E 0(3). So also (H4) is satisfied, and we can apply the results of section 2.

4.4.6. Theorem. Let £ =

in (1). Suppose that

h(Rx,u,A) = h(x,u,A)

VR E GO '

(21)

and

J1

°

r2dr

J 2 h(r,6,0,0)n 1 · (r)6 3d6 f S

°.

Then, for 1 All and ICJ I sufficiently small and solution of (1) will satisfy : u(Rx) = u(x)

(22)

J CJ

f 0, each sufficiently small

VR E GO .

(23)

Proof. The result follows from corollary 2.5.The condition (22) implies Qh(.,O,O) f 0. Remark that, because of (21), the integral in (22) becomes zero if we replace 63 by 61 or 62 , 0

4.4.7. Remark. The vector en in the definition of GO can of course be replaced by any e E S2. We conclude that in general an axisymmetric perturbation will lead to axisymmetric solutions with the same axis, if £ = 1.

151

5 Generic bifurcation and symmetry

5.1. INTRODUCTION In this chapter we start the study of the bifurcation equation and the bifurcation set for the important case that dim ker L = codim RCL) = 1. It appears that the behaviour of the bifurcation set is mainly determined by the degree of the first nonvanishing parameter-independent term in the Taylor expansion of the bifurcation function. We will discuss the cases where the dominant term is either quadratic or cubic. Our presentation combines elements from Chow, Hale and Mallet-Paret [39] with the approach used in [229]. We will show that in both cases considered in this chapter it is possible to describe the bifurcation set as a finite union of submanifolds in the par~ meter space; each of these submanifolds has a finite codimension. For each value of the parameter in the same connected component of the complement of the bifurcation set, the bifurcation problem has the same number of solutions; This number of solutions can only change when A crosses the bifurcation set, i.e. when a bifurcation takes place. In section 2 we discuss the case where the dominant term is quadratic, while the case of a cubic dominant term is studied in sections 3 and 4. It appears that under a certain "generic condition" the bifurcation second case is cusp-shaped. In section 5 we analyse the situation when the equivariance of the equation under a symmetry group prevents the generic condition from being satisfied. An example of such a nongeneric situation can be obtained by considering the buckling problem for rectangular plates, subjected to symmetric normal loads; in section 6 we study how different types of symmetry for the normal load affect the corresponding bifurcation set. A number of results in this chapter show a close relationship to some results from singularity theory. We will briefly discuss this relationship. In this connection let us remark that we obtain our results by "classical" methods (rescaling techniques and the implicit function theorem), while singularity theory uses as one of its most important more difficult preparation theorem of Malgrange and Mather (see e.g. [26], [74]).

152

BIFURCATION PROBLEMS WITH A QUADRATIC DOMINANT TERM 5.2.1. The problem. Let X, Z and A be real Banach spaces, ~ C X and we A open subsets, and M : ~xw -+ Z a Cr-function, with r:;;' 1. We want to study the solution set of the equation

M(X,A)

=

0

(1)

under the following hypothesis (H1) For some (XO,A O) E ~ x w, we have M(XO,A O)

=

0, while L

DxM(xO,A O) is

a Fredholm operator, with dim ker L = codim R(L) = 1. More in particular, since simple examples (see chapter 1) show that (1) may have a different number of solutions for different values of the parameter, we would like to solve the following problem: determine a neighbourhood ~1 of Xo in X, a neighbourhood w1 of AO in A, and a partition of w1 ' such that for A varying within each of the subregions determined by the partition, the number of solutions x E ~1 of (1) remains constant. This will also determine the bifurcation set B for (1); this is the set of all A E w1 which are a bifurcation point for (1) at a solution x E ~1 (see the definition in chapter 1). Without loss of generality, we may assume that (0,0) E ~xw and (XO,A O)

=

(0,0).

5.2.2. TIle generic problem. A particular problem of the form just described is the following. Let X, Z and ~ be as before; let Cr(~;Z) denote the Banach space of all r-times continuously differentiable functions m : ~ -+ Z satisfying

Imlr

=

sup{llm(x)1I

+

I Drn(x) I

+ ••• +

IIDrm(x)ll I XE~} <

00

•

Let Xo E~, mO E Cr(~;Z) and mO(xO) = o. Suppose also that L = DIDa(x O) is a Fredholm operator with dim ker L = codim R(L) = 1. We want to determine, for each m E Cr(~;Z) with Im-molr sufficiently small, the number of solutions of the equation 153

m(x)

=

a

(2)

belonging to a sufficiently small neighbourhood of xO. This problem has the form (1) by taking A = Cr(~;Z) and M(x,m)= m(x)

Vx E

~ ,

r

Vm E C

(~;Z)

•

(3)

It is clear that this "generic problem" contains each of the problems (1); once we have solved the generic problem,- then it is sufficient to restrict attention to L = {M(. ,A) I AE w} C Cr(~;Z) to find the solution for (1). We will nevertheless give the analysis for the equation (1). 5.2.3. Reduction of the equation. We can apply the Liapunov-Schmidt reduction to equation (1). Let P E L(X) and Q E L(Z) be projections, such that ker L = R(P) and ker Q = R(L). Let ker L = span{uO} and R(Q) = span{zO}. Define the linear functional ~ E Z· by Vz E Z •

(4)

Let v·(u,A) be the unique solution of the auxiliary equation (3.1.4a). It is a Cr-function, defined for (u,A) in a neighbourhood of the origin in ker LxA, and taking values in ker P. Also v* (0,0) = a and Duv* (0,0) = o. Equation (1) reduces then to the following bifurcation equation (5)

F(p,A) is a real valued Cr-function, defined for (p,A) in a neighbourhood of the origin in RxA, and satisfying: F(O,O) = a

DpF(O,O)

O.

(6)

5.2.4. Now we supplement (H1) with the following hypothesis

Here, as before, 154

U

o is

a vector generating ker L. By (HZ) we are allowed to

(7)

in (4). Assuming again (XO,A O) = (0,0) we find then (8)

Remark that the hypotheses (H1) and (HZ) are in fact hypotheses on the function mO : Q ~ Z defined by mO(x) = M(X,A O), Vx E Q. 5.2.5. Theorem. Assume (H1) and (HZ). Then there is a neighbourhood Q1xw1 of (XO,AO) in XxA and a C1-functional a w1 ~ 1R such that for A E w1 the number of solutions of (1) in Q1 equals (i) zero (ii) one (iii) two

if a(A) > 0 if a(t..) = 0 i f a(A) < 0

Proof. By (8) and the continuity of DZF(p,A) we can find p neighbourhood w1 of the origin in A such that :

and

DZ pF(p,A) > 0 D F(o,A) > 0 P

Vp

F(o,A) > 0

[-0,0]

VA

E

w, ,

D F(-o,A) < 0 , P

VA

E

w, ,

F(-o,A) > 0

VA

E

w1

E

°> 0 and a

Let A E w1• Then Dl(p,A) is a strictly increasing function of p E [-0,0], and has exactly one zero in the interior of this interval; let PO(A) be this zero. The function A * PO(A) is continuously differentiable, as follows from an application of the implicit function theorem on the equation DpF(p,A) = 0

(9)

Also POCO) = O. It follows that the function p * F(p,A) has a strict minimum at the point 155

P = PO(A). Let a(A) be the corresponding minimal value (10)

TI1e equation (5) has no solution P E [-8,8] if a(A) > O. TI1ere is exactly one such solution if a(A) = 0, namely p = PO(A). Finally, (5) has two different solutions P E [-8,8] in case a(A) < 0. TI1is proves the theorem. 0

5.2.6. TI1eorem. Suppose (Hl) and (H2). TI1en there is a 0 > 0, a neighbourhood w1 of AO in A, mld a constant C > such that each solution (p,A) E [-8,o]xw1 of (5) satisfies

°

(11)

Proof. If the theorem is not true, then there is a sequence {(p ,A ) I n n nEN}, converging to (0,0), such that F(p,A) = 0, P f PO(A), while also -2 n n n n a(An).(Pn-PO(An )) converges to zero. By the definition of PO(A) and a(A) we have:

Dividing F(pn ,An ) = find

°by (pn-PO(An

TI1is, however, contradicts (8).

))2 and taking the limit for n

+

00, we

0

5.2.7. Corollary. Suppose (Hl) and (H2). Suppose also that there is a ~ E A such that (12)

TI1en A E w1 is a bifurcation point for (1) corresponding to a solution x E Ql' if and only if a(A) = 0. TI1e bifurcation set

156

{A E W1 I a(A) =o}

(13)

contains AO' and is a submanifold of A, with codimension equal to 1. proof. To see that (13) is a submanifold of codimension 1, we remark that DI, a(O).~ = QOD\M(O,O).~ !\.

VA EA.

Because of (12) we can find some '~ E A for which this expression is different from zero. So a : w1 -'>- IR is a submersion in a neighbourhood of AO; this proves the las t part of the theorem (see Lang [ 141 ]) . It is clear that bifurcation points of (1) corresponds to bifurcation points of (5), at least if we restrict to a small neighbourhood of the origin. When a(A) > 0 then (5) has no solution, and A cannot be a bifurcation point. When a(A) < 0 and F(p,A) = 0, then it follows from the proof of theorem 5 that DpF(p,A) f 0; an application of the implicit function theorem shows that A is not a bifurcation point. Let finally A E w1 be such that a(A) = O. It follows from (12) that each neighbourhood of A contains parameter values A' for which aCA') < 0; for such A' the equation F(p,A') = 0 has two different solutions. When A' -'>- ,\ in the region {A' E w1 I a(A') < O}, then the two conesponding solutions will converge to PO(A), as follows from theorem 6; also PO(A) is the unique solution of F(p,A) = O. 1bis shows that A is a bifurcation point. D

S.2.S. Remarks. TIle submanifold (13) divides the neighbourhood in two connected compon~nts w~+) = {A E w1 I a(A) >O} and w~-) = {A E w1 I a(A)
( 14)

This does not belong to R(L) for an appropriate mE Cr(r/;Z). A reformulation of the foregoing for this generic problem gives precisely the results contained in Chow, Hale and Mallet-Paret [39]. 157

J:vIore generally, the first part of the proof of corollary 7 shows that condition (12) means that the inclusion map A * M(.,A) from A into Cr(Q,Z) is transversal to the submanifold B = {mECr(Q;Z) I a(m) =O} in the point mO = M(.,A O); B is the bifurcation set for the generic problem (2). For this reason we can call (12) a transversality condition. This condition will "generically" (i.e. almost always) be satisfied, as soon as dim A ;;'1. So we may also refer to (12) as a generic condition. The hypotheses (H1) and (H2) imply that the bifurcation function F(p,>.) is such that FO(p) = F(p,O) = p2 + higher order terms. We may consider F(p,>.) as an unfolding of the function FO. Since DAF(O,O).\ = QODAM(O,O).\, it follows from singularity theory (see e.g. Brocker [26], GOllIDitsky and Guillemin [72], ~nrtinet [261]) that (12) is precisely the necessary and sufficient condition for F(p,>.) to be a univ~al unno!d{ng of the function FO(p)· For >. E w1 we define : ( 15) 'The following theorems show that when a(A) 0, i.e. when A is a bifurcation point, then the hypotheses (H1) and (H2) are also satisfied at the point (xO(>.) ,>.) . 5.2.9. Theorem. Suppose (H1) is satisfied at the origin. Then there is a neighbourhood Q1xw1 of the origin in XxA and a 6 > 0 such that the only points (X,A) E Q1xw1 at which (Hl) is satisfied have the form (puO+ V' (puO' A) ,>.) for some (p, >.) E ] -6,0[ x w1 such that : F(p,A) = 0 and Dp F(p,A) = 0

(16 )

For all other (X,A) E Q1xw1 we have either M(X,A) f 0 or dim ker DxM(x,A) codim R(DxM(x,A)) = o. Proof. Since (H1) includes the condition M(x,>.) = 0, it is clear that (x,>.) should have the given form, with (p,>.) a solution of F(p,A) = 0; this is a consequence of the Liapunov-Schmidt reduction. For each (p,>.) near the origin we define

158

we have L(O,O) = L, and we want to determine ker L(p,A) and R(L(p,A)). To do . so, we consider the equation (17)

wi.th z E Z given. Let into two equations :

x

=

puO+v, with P EJR and

vE

ker P; then (17) splits

( 18)

(I -Q) z

and

(19) Moreover, differentiating the equation defining

V*

(pUO,A) , we find (20)

Multiplying this relation by

p,

and subtracting from (18), we obtain (21)

Since (I-Q)L(O,O) = L is an isomorphism between ker P and R(L) = ker Q, the same holds for (I-Q)L(p,A); denote by K(p,A) the inverse of this isomorphism. Then (21) gives : (22)

Substitution of (22) into (19) gives : (23)

This equation has a unique solution p for each z E Z if and only if DpF(p,A) f 0; then ker L(p,A) = {O} and R(L(p,A)) = Z. If DpF(p,A) = 0, then (24)

159

to see this, it is sufficient to put z : 0 in the preceding calculations. Under the senne condition (23) has a solution if and only if the right-hand side of (Z3) is zero. For (p,A) : (0,0), this right-hand side reduces to QO(z); since QO is a nontrivial functional, we conclude that codim R(L(p,A)) : 1 if D F(p,A) = O. This proves the theorem. 0 p

5.Z.10. Theorem. Suppose (Hl) and (HZ) are satisfied at the point (O,O)EXxA. Then the set of points (X,A) E S61xw1 at which (Hl) is satisfied is given by (Z5) At these points, also (HZ) is satisfied. Proof. The first part follows immediately from theorem 9 and the definition of alA). Let L1 (A) = L(PO(A),A), K1 (A): K(PO(A),A),

(see (Z4)), and

If alA) : 0, then Z1(A)

~

R(L 1 (A)) if and only if (Z6)

The left-hand side of this inequality is contimlous in A, and reduces for A = 0 to QOD;M(O,O)(uo'u O), which is different from zero by assumption. We conclude that (Z6) will be satisfied for all A in a sufficiently small neighbourhood of the origin in ~. In particular, (HZ) will be satisfied at the points of (Z5). 0 5.3. BIFURCATION PROBLEMS WITH A CUBIC DOMINANT TERM In this section we discuss the number of solutions of the bifurcation equation (Z.5) in the case where DZF(O,O) ~ 0 and D3F(0,0) t O. This case has alp p ready been treated in the basic paper of Chow, Hale and Mallet-Paret [39] , 160

also in Vanderbauwhede [ 229] . In this last paper a rescaling technique used. The presentation here keeps somewhat the middle between these contributions. Our results allow in a nwnher of cases an easy calculation of . the approximate form of the bifurcation set. The approach seems to be particularly useful when many parameters appear in the problem; we will illustrate this by an example in section 6. 5•.3.1. The hypothesis. In this section we replace the hypotheses (H2) of the

preceding section by : (H3) r

~

3, and the bifurcation function F(p,A) given by (2.5) satisfies:

o

(1)

and (2)

The condition (1) can be reformulated as (3)

(see the previous section). As for (2), we obtain from a direct calculation the following expression for D3F(0,0) : p

D~F(O,O)

=

QOD~M(O,O).(uo'uo'uo) 2

2

- 3QODxM(0, 0) . (uO' K(I -Q) DxM(O ,0) . (uO,u O)) . We remark that again (H3) is a condition on the function mO(x)

(4)

=

M(X,A O),

and does not involve parameter values A f AO . Let (5)

We may suppose that a O > 0; in case a O < 0, it suffices to replace Zo by -zO in the definition of QO' The next lemma's describe the solutions of the 161

bifurcation equation F(p,>")

=

0 .

(6)

°

5.3.2. Lemma. Suppose (H1) and (H3). Then there is a > 0, a neighbourhood w1 of the origin in 1\, and a C1-map y 1 : w1 ->- ffi such that for each A E w1 satisfying Y1(>");;' 0 the equation (1) has a unique solution p E ]-rS,o[. Moreover, Y1(0) = O. Proof. Since F(p,O) = aop3 + 0(p3) it follows from continuity arguments that we can find rS > 0 and a neighbourhood w1 of the origin in 1\ such that 3 p

D F(p,>..)

> 0

Vp

D2F(8,A)

>0

D2F(-rS,>..)

D F(8,>..)

>0

D F(-o,>..)

p

P

E

p

P

V>"

E

w1 '

(7)

< 0

V>"

E

w1

,

(8)

> 0

V>..

E

lU

,

(9)

VA

E

w1

[-0,0]

1

and FCo,>..) > 0

F(-o,>..) < 0

(10)

From (7) and (8) it follows that, for each A E w1 , the equation

D2pF(p,>..) = 0

(11 )

has a unique solution p = PO(>..) E 1-0,0[. The implicit function theorem shows that Po : w1 ->- ]-0 ,or is continuously differentiable. Let now (12) Application of theorem 2.5 on the equation D F(p,>..) = 0 p

( 13)

shows that (13) has no solution p E j-o,8[ when Y1(>") > 0, just one solution p = PO(>..) when Y1(>") = 0, and two solutions in case y,(A) < O. Fix some 162

AE w1 • If Y1(A) > 0, then the map p * F(p,A) is strictly increasing; because of (10) the equation (13) has exactly one solution in ] -0,0[. If Y1 (A) =0 then F(p,A) has a strictly positive derivative, except at the point p = PO(A). thiS implies that F(p,A) is strictly increasing, and again has a unique solution p E ]-0,0[. This proves the lemma. 0 5.3.3. Lemma. Using the notation of lemma 2, define, for A E w1 ( 14)

If A E w1 and Y1(A) ~ 0, then A can only be a bifurcation point for (6) at a solution p E ] -o,o[ if YO(A) = Y1 (A) = O. proof. It follows from the implicit function theorem that A E w1 can only be a bifurcation point for (6) at a solution p if F(p,A) = 0 and D F(p,A) = o. We see from the proof of lemma 2 that the equation D F(p,A) p p = 0 has no solution if Y1(A) > o. If Y1(A) = 0 then the equation has exactly one solution, namely p = PO(A). This will also be a solution of F(p,A) = 0 if and only if YO(A) = o. This proves the lemma. 0 Remark that for all A E w1 satisfying YO(A) = Y1(A) = 0 we have (15)

5.3.4. Lemma. Suppose (H1) and (H3). For A E w1 , let (16)

Then there exist functions s+(A,n) and s_(A,n), defined and continuous for AE w1 and Inl sufficiently small, and continuously differentiable for n f 0, such that: (17)

(ii) if Y1(A)';;;; 0, then the solutionsof(13) in ]-o,o[ are given by : (18)

163

where n(A)

=

(-Y (A)) 1/2

( 19)

1

Proof. First, let us study the equation

o.

H(p,n,A)

(20)

By the argument of theorem 5.2.6 it is easily shown that solutions of (20) in a neighbourhood of the origin will satisfy Ipi ,,;;; Clnl for some constant C> o. Therefore we put = ns in (20), which gives us the equation:

p

(21)

The function Hl is of class C1 in all variables, and of class C2 in nand s; moreover H1(0,s,A) = DnH(O'S,A) = 0 for all (S,A). So we can write: (22) where HZ(n,s,A) = f01 sds

f1 ds'D 3F(PO(A) + SS'nS,A)S 2 - 1 .

o

P

(Z3)

" It follows from (ZZ) that in the domain n f 0 the function HZ has the same smoothness properties as the function H1. Further, H2 is continuous everywhere, and

H2 (0,s,A) = 3a(A)S Z - 1 ,

(Z4)

which shows that HZ(O,s,A) is continuously differentiable In s. Finally -7

lim n -D H1(n,s,A) n-+O s

=

6a(A)s ,

so that we can conclude that HZ has a continuous partial derivative in the variable s. Now Hz(0,±(3a(A))-1/ Z,A) 164

application of the implicit function theorem gives us the existence of functions s±(A,n) having the smoothness properties given in the statement of the lemma, satisfying (17) and such that for (n,A) near the origin, s = ,+(A,n) are the only solutions of H2 (n,s,A) = 0 in [-C,C]. Consequently the o~IY solutions of (20) in a neighbourhood of the origin are given by p = n,±(A,n). Part (ii) of the statement of the lemma then follows from the observation that (20) reduces to (13) if we put p = p-PO(A) and 0 n ~ (-Y1(A))1/2. Remark. It is possible to define the functions P+(A), as given by (18), for ~in a neighbourhood of the origin by taking-n(A) = 1Y1(A)11/2. The functions ob tained in this way are continuous, and of class C1 in {A E w1 I Y1 (A) f O}. MJreover, P± (A) are of class C2 in w = {A E w1 I Y1(A) < O}, since for A E w P = PiCA) solve (13), while F is of class C3 and D~F(P±(A)'A)

1

1,

f O. If we define yeA) for A E w1 by (25) then yeA) has the same smoothness properties as the functions PiCA), in particular, yeA) is of class C2 in w1 . 5.3.5. Lemma. Assume (H1) and CH3), and define yeA) by (25). Then we have the following for each A E W

1

(i)

if yeA) > 0, then (6) has exactly one simple solution pE]-o,o[;

(ii)

if yCA) = 0, then (6) has one simple and one double solution in ]-o,o[ ;

(iii) if yCA) < 0, then (6) has three simple solutions in ] -0, o[ . Proof. If Y1(A) < 0, then it is easily verified that F(p,A) has a local maximum at P = p_(A) and a local minimum at P = p+(A). It follows from (10) that (6) has one simple solution if the corresponding values of F(p,A) have the same sign, and three simple solutions when these values have an opposite sign. When one of these values is zero, then there is one simple and one 165

double solution. The result follows since yeA) is precisely the product of these maximum and minimum values of F(p,A). 0 5.3.6. 1nere is another way of characterizing the set {A E W1 I Y1(A) < 0 and yeA) =O}. It follows from the definition of YO(A) and Yl(,\) that (26) where R(p, A) is a continuous function which can be defined by 3 R(p,A) = f 0l s 2ds fl0 s'ds' fl0 ds"Dl(po(A) +sS'S"p,A)

(27)

It is then easily seen that, if Yl(A) < 0, then (28)

where n(A) = (-Yl(A)) 1/2 and (29)

The functions 0±(A,n) are continuous, and 0±(A,0) = ±2(27a(A))-1/2 .

(30)

The following theorem summarizes our results up to now. 5.3.7. Theorem. Assume (Hl) and (H3). Then there exist a 0 > 0, a neighbourhood w1 of the origin in fl., C1-functionals YO: w1 .. lR and Y1 : w1 .. lR, and continuous functionals 0+(A,n) and 0_(\,n), defined in a neighbourhood of the origin and satisfying (30), such that for A E w1 the following holds (i)

if Yl(\) > 0, then (6) has one simple solution p

E ]

-o,o[

(ii) if Yl(A) < 0, and if n(\) = (-Yl(A)) 1/2, then (6) has the following solutions in ] -o,o[ : (a) one simple solution if

166

(b) three simple solutions if

(c) one simple and one double solution if y,(A) < 0 and

(d) one triple solution if YO(A) = y,(A) = 0 .

5.3.8. Theorem. Assume (Hl) and (H3). Then

D

°> 0 and the neighbourhood w"

appearing in the statement of theorem 7, can be chosen sufficiently small such that each solution (p,A) E j-o,o[ x of (6) will satisfy:

w,

(3') for some constant C. Proof. If not, then we can find a sequence of solutions (Cpn,An ) I nEN} of (6), such that pn f PO(An ) for each n EN, and

as n + 00. Using the expression (26) for F(p,A), and dividing F(pn ,An ) = 0 by IPn-PO(An) 1 3 , we find in the limit for n + 00 : R(O,O) = 0 However, R(O,O) = aO' and so (32) contradicts (H3).

(32) D

We conclude this section by giving an alternative formulation of the hypothesis (H3). Since the definition of the function F(p,A) contains the projections P and Q, and the vectors uo and zO' we can ask whether (H3) will still be satisfied for another choice of these projections and vectors. Next '67

theorem shows that this is indeed the case. 5.3.9. Theorem. Suppose that mO(x) and Zo be as before, and define :

=

M(X,A O) satisfies (H1). Let P, Q, uo

(33) Suppose that r (i)

k

~

~

2. Then the following statements are equivalent

D~FO(O) = D~FO(O) = and

= Dkp- 1 F0 (0) = 0

,

D~FO(O) f 0 ;

(ii) there is a Ck-map X'" : j-o,o[ D X'" (0) f 0, such that

+

X, with X" (0).

=

Xo and

p

Vp E ] -0,0 [

(34)

for some zl ¢ R(DxMO(xO)) and some function a(p) satisfying a(O)

k = Dpa(O) = ... = Dk-1 p a(O) = 0 , Dpa(O). f 0 .

(35)

Proof. We remark first that (H1) implies that FO(O) = DpFO(O) = o. Assume (i), and let

Then (32) and (33) are satisfied, if we take zl = Zo and a(p) = FO(p). Conversely, suppose (ii) is satisfied. Let P(X'" (p)-x O) = S(p)u O; then S(O) = 0 and

(36)

for some function ~ : j-o,o[ (I -Q)mO(X" (p))

168

=

+

ker P. From (35),

a(p)(I -Q) zl '

and the equation defining V*(S(p)UO,A O) it follows that:

yeO) = a

DpyeO)

=

a ,

Dk - 1y(0) = p

a•

(37)

(38)

and (39) From (36) and (37) it follows that :

Indeed, the only term in D~F1(0) containing D~Y(O) has the form

which is zero by the definition of QO' Now QO(zl) f 0, since zl ~ R(DxmO(x O))' From F1(p) = a(p)~(zl) and (35) it follows that (40)

Finally, we compare the definition (39) of FZ(p) with the definition of FO(p). We obtain: FZCO) = FOCO) DpFZ(O) DZFZ(O) p

= =

DpFO.D pS(O) , DZFOCO). CD p(3(O))Z + Dp FZ(O) , p ' .DZS(O) p

DkFZ(O) = rfFOCO).CD p p pSCO))k

+

terms containing -1<--1

CFOCO),DpFO(O), ... u~ FOCO)). 169

Since D X" (0) f 0 it follows from (36) that D S(O) f o. Then (i) follows P P from (40). 0 In the formulation of the next theorem we use the following notation BO = ((PO(A),A)

I AE w1 ,

YO(A) =Y1(A) =O} ,

(41)

B+ = {(P+(A),A)

I AE w1 ,

Y1(A)<0, YO(A) = 0+(A,n(A))(n(A))3},

(42)

and

Remember also the notation : (44) 5.3.10. Theorem. Suppose that (H1) and (H3) are satisfied at (0,0). Then there is a neighbourhood ~lxwl of the origin in XxA such that for (X,A) E ~lxw1 the hypothesis (H1) is satisfied at (X,A) if and only if x = X* (p,A) with (p,A) E B=BOUB+UB_. If (p,A) E BO then also (H3) is satisfied at the point (X* (p,A),A). If (p,A) E B+ UB_, then (H2) is satisfied at (x· (p,A) ,A). Proof. The first part follows from theorem 2.9 and the fact that F(p,A) = DpF(p,A) = 0 if and only if (p,A) E B. The second part follows from theo/ rem 9, as follows: Let (p,A) E B, and define :

Then: M(X(p) ,A) = F(p + p,A)ZO = a(p)zO. For (p,A) E BO we have a(O) = D-a(O) 2 3 P = D-a(O) = 0 and D-a(O) f o. If (p,A) E B+UB - , then a(O) = D-a(O) = 0 and , p p p ~a(O) f o. This proves the theorem. 0 p

170

5.4. GENERIC AND NON-GENERIC BIFURCATION 5.4.1. Introduction. Consider again the equation M(X,A)

=

0

(1)

and suppose that the hypotheses (Hl) a~d (H3) from the preceding sections are satisfied. For a sufficiently small neighbourhood :it 1XeD1 of the origin in Xx~, we define 6

=

{:\Ew 1

1:\ is a bifurcation point for (1) at a solution xE:it 1} (2)

From theorem 3.10 we see that 6c60U6+U6_, where

6+

(3)

However, we have not necessarily that 6 = 60 U 6+ U 6_. For example, assume that M(X,A) is such that Yl(:\) = 0 for all A E w1 ; then the equation (1) will have exactly one solution in rll for each :\ E w1 , and the bifurcation set 6 will be empty. In order to determine 6 we have to study the following subsets : (4)

and (5)

5.4.2 ..Theorem. Suppose that (H1) and (H3) are satisfied at (0,0) for the equation (1). Define a map ~ w1 ~R2 by : (6)

171

Suppose that the linear map DA'¥(O) E L(A,]l2) is surjective. Then MO and M, are C'-submanifolds of A, with codimension equal to 1, and transversal at the origin A = O. ~reover, under these conditions we have 6 = 60 u 6+ u 6 _. Proof. The condition implies in particular that the linear functionals DAyO(O) E L(A;.lR) and DAY1(0) E L(A;.lR) are surjective. This shows that MO and M, are C1-submanifolds with codimension equal to 1. Let now ~1 E A and ~2 E A be such that DA,¥(O)'~l and DA,¥(0)'~2 are linearly independent. We may suppose that DAYO(O)'~l f o. Then:

This implies that ~2 etA 1 + ~2' for some et E ill. and ~2 Eker DAy O(0). The vectors DA,¥(O)'~l and DA,¥(0)'~2 = (0,DAY1(0)'~2) are linearly independent; so DAY1(0)'~2 f O. It follows that

Since ~2 E ker DAyO(O) we conclude that A is spanned by ker DAyO(O) and ker DAY1(0); i.e., MO and M1 are transversal at the point A = O. This also implies that 60 = MO n M1 is a C1-submanifold with codimension equal to 2. Each point AO E 60 is a bifurcation point. Indeed, each neighbourhood of AO contains A at which Y1(A) < 0, and, for example, YO(A) = 0; for such A the equation (1) has three different solutions in ~1; it follows frdm theorem 3.8 that these solutions converge to the unlque solution of M(X,A O) = a ~1 as A -+ AO' For A+ E 6+ equation (1) has one single and one double solution; let x+ be this double solution. It follows from theorem 3.10 that the conditions of corollary 2.7 are satisfied at the point (X+,A+). So A+ is a bifurcation point for (1); the bifurcation takes place from the double solution. A similar argument shows that A_ E 6_ is a bifurcation point. 0 The following figures show the bifurcation set, which forms a cusp along 60 , and the corresponding solutions of the bifurcation equation F(p,A) = O. The bifurcation set 6 divides w1 into two subregions; for A in region I, the equation (1) has a unique solution in ~1' for A belonging to region II there are three such solutions.

in

172

.}("

II

(a)

Y1 (A)

(b)

P-P. (A)

o

(c)

Fig. 5. (a) The bifurcation set (b) The solution set along YO(A) = constant> 0 (c) The solution set along YO(A) = 0

173

5.4.3. Remark. The condition of theorem 2 can be refonnulated as follows there exist ~1 E A and ~2 E. A such that

(7)

The derivatives appearing in (7) can be calculated as follows. It follows from the definition of YO(A) and Y1(A) (see (3.12) and (3.14)), from POCO) = 0 and from (H3) that VA E A

(8)

and (9)

Using the definition of F(p,A) one finds (10) To calculate DAY1(0).~ from (9), one needs DAv* (0,0); this can be obtained by differentiating the relation defining v*(U,A); one finds :

Finally one obtains

5.4.4. CorOllary. Consider the generic problem described in subsection 2.2. Suppose that mO E C3(~;Z) satisfies (H1) and (H3) at xO' Then the bifurcation set in a sufficiently small neighbourhood w1 of ~ in C3(~;Z) is given by :

174

pro a f. We only have to show that the condition (7) is satisfied (with A = C3 (Q;Z)). From (10) and (11) we find for this case

and

DmYl(~)·m = QODxm(xO)·u o - OoD~~(XO)·(UO,K(I-Q)m(Xo)) Vm

E

C3(Q;Z)

Now make the following choice for m1 and ~ : Vx E Q

and

The matrix in (7) becomes the identity matrix. This proves the corollary.

0

5.4.5. We can use corollary 4 to reformulate the condition of theorem 2. We suppose that not only M(x,A), but also DAM(x,A) is of class C3. We define the mapping, : we 11. -+ C3 (Q;Z) by ,(A)

=

M(.,A)

VA E w •

(13)

Let ~ = ,(A O)' and let;;; be a neighbourhood of ~ in C3 (Q;Z) such that for each mE;;; the map w(m) = (YO(m)'Yl(m)) is well-defined. We put So = {mE;;; I w(m) = o}; So is a C1-subrnanifold of C3 (Q;Z) with codimension equal to two.

Using this framework, the condition of theorem 2 can be restated as saying that the mapping Wo,: w1 = ,-1(;;;) -+:JR2 is submersive at the point AO• The proof of theorem 2 shows that this means that

where Tmo So is the tangent space to So a t ~. So the condition says nothing else than that the map, is transversal to So at A = AO. It is clear that 175

this can only be satisfied if dim A ? 2, since So has codimension 2. if dim A ? 2, then the transversality condition will be satisfied "generi-

cally', i.e. for almost all T : A ~ C3(~;Z). For this reason the transversality condition will also be called the geJ'lvUcc CCOYldi:UOYl, and the corresponding bifurcation set a geYlvUcc b~6LlJLc-atiOVl f.,e;t. 5.4.6. Sjmilarly as it was the case with the results of section 2, also the resul ts of sections 3 and 4 show a clear relationship with some resul ts from singularity theory; this relationship was already commented by Chow, Hale and Mallet-Paret in [39] and [40]. Suppose that M(X,A) is of class Coo, that dim A = P < and that M satisfies (H1) and (H3), say at (0,0). Then also the bifurcation function F(p,A) is of class Coo, and can be considered as an unfolding 'of the function FO(p) = F(p,O), which has the form FO(p) = aop3 + higher order terms. The transversality condition can be written as : 00

rank(a .. ) = 2

(i=1, ... ,p

1J

j=1,2),

(14 )

where, for i = 1, ... ,p :

of oYo ail = OA, (0,0) = OA. (0,0) 1

1

and

(15)

Assume that (14) is satisfied, then we may suppose, without loss of generality, that

This implies that the map (p,A 1 , ... ,Ap ) ~ (P,111, ... ,l1p ) given by p

=

p - Po (A)

111 = YO(A) 112 = Y1 (A) 11 1. A.1 176

(16) i

3, ... ,p

is a COO-diffeomorphism in a neighbourhood of the origin in lR xlRP • Then (3.26) shows that in the new variables (p,~) the bifurcation function takes the

( 17) with R(O,O) = aO. Comparing with the basic unfolding result of singularity theory this shows that the transversality condition (14) is the necessary and sufficient condition for F(p,A) to be a universal unfolding of FO(p). When this is the case, then there is a smooth change of coordinates near the origin of the form V·

1

=¢.(A) 1

i

1, ... ,p

P = r;(p,A)

such that in the variables (p,v), F(p,A) takes the normal form (18) The genericity of the condition (14) means the following. Consider the space of all Coo-mapping G : lR xlRP -+ lR. This space can be given the structure of a Baire topological space by introducing the so-called Whitney topology. Consider the subset U of all G(p,A) such that G(p,O) = FO(p); U is a linear submanifold. Then it follows from Thom's transversality theorem that the subset of all G E U which satisfy (14) is generic in U, i.e. forms an open and dense subset of U, equiped with the Whitney topology. (For the results on singulari ty theory, quoted in this subsection, we refer to Golubi tsky and Guillemin [72], Brocker [26] and Martinet [261 ]). 5.4.7. It is interesting to compare the bifurcation sets for the equations

° ° is given by

F1(P'~) = and F2 (p,v) = 0, with Fl and F2 given respectively by (17) and (18). It follows from our foregoing results that the bifurcation set for

F1(P'~)

(19)

177

with 0±(0,0) = ±2(27ao)-1/2. The bifurcation set for F2 (p,v) 62

=

{vE1RP

I v 2 ";; 0, 27V~ + 4v~ = O}

o is

given

,

as can easily be verified. By a rescaling of p one can achieve that a O = 1. 'Then one obtains (20) from (19) by replacing 0±(~) by its value at ~ = 0; similarly, one obtains (18) from (17) by replacing the remainder term R(p,~) by its value at (0,0). 5.4.8. We have obtained our bifurcation results for the equation (1) in a general setting, under relatively weak smoothness conditions, and using only classical tools such as rescaling techniques and the implicit function theorem. To the contrary, the approach via singularity theory requires that the equation is COO and that the nwnber of parameters is finite, while the proof uses the less elementary preparation theorem. (In a recent contribution, Michor [ 164] has extended the preparation theorem to the case where the parameter belongs to a Banach space). However, in more complicated situations the unified treatment via singularity theory seems to have some advantage; indeed, once the general theory is established, the application of the results of singularity theory to particular examples reduces to merely algebraic manipulations. For example, one could try to use our method to study bifurcation in the general case that the binlrcation function satisfies : F(O,O) = DpF(O,O) = ... = D~-1F(0,0) = 0 However, the treatment becomes very complicated for k > 3. Using singularity theory, the problem reduces to the study of the solutions of a polynomial equa tion of the form k

p

+

o,

with ai(A O) = O. For a somewhat different approach to bifurcation theory (in particular imperfect bifurcation) via singularity theory, see the basic papers by Golubitsky and Schaeffer ([ 74] and [ 75]).

178

.4.9. Non-generic bifurcation. We conclude this section with the description of a particular situation where the transversality condition (7) is not satisfied. This case is of some interest when the equation has additional symmetry, as we will see in the next section. The hypothesis is that yoO,) = a for all A E w1 ; Then the bifurcation equation has the solution P = PO(A) for each A E w1 . All other solutions near (0,0) can be written in the form (p,A) = (PO(A)+p,A), where Cp,A) has to be a solution of the equation 1

F1 CP,A)

=f

o

D F(pO(A)+sp,A)ds P

(21)

O.

Irrieed, F(PO(A)+p,A) pF 1 (p,A), and (21) is equivalent with F(PO(A)+p,A) = 0 if p f O. From (21) we see that : F1 (0,A) = Y1 CA) 1 2

a , (22)

LTDpF1(0,A) = a(A) ,

We can apply theorem 2.5 to equation (21); remembering that the bifurcation equation has always the solution P = PO(A), while (21) gives us the solutions p of PO(A), we obtain the following result. 5.4.10. Theorem. Suppose that the equation (1) satisfies the hypotheses (Hl) and (H3) at (0,0), while also YOCA) = a for all A E w1 . Assume that we can find some>: E A such that (23) Then the bifurcation set for Cl) is given by So = {A E w1 I YO(A) =Y1CA) =O} = {A E w1 I Yl(A) =O}, which is a C1-submanifold of A, with codimension 1. If Y1(A) > a then (1) has one simple solution in a sufficiently small neighOCurhood ~1 of the origin in X; this unique solution corresponds to the solution P = PO(A) of the bifurcation equation. If Y1(A) = a (i.e. A E SO) then the solution of (1) corresponding to P = POCA) remains the unique solution in ~1' but is now a triple solution. If Y1 CA) < 0, then (1) has three simple Solutions in ~1' one of which corresponds to POCA). 0 179

The following figures give a sketch of the bifurcation set and the solutions.

(a)

(b)

Fig. 6. (a) Bifurcation set (b) Solution set.

5.S. GENERIC CONDITIONS

h~

SYMMETRY

5.5.1. In this section we study how the symmetry of the equation M(X,A) = 0

(1)

can affect the bifurcation set. Our hypotheses are the same as in the preceding sections: we assume that M satisfies (H1) and (m) at (XO,A O) = (0,0). We will also assume that M is equivariant with respect to some (G,r,r), where G is a compact group. The projections P and Q used in the LiapunovSchmidt reduction of (1) are chosen such that they commute with the symmetry operators (see chapter 3). The bifurcation function

H(u,A) _ QM(U+V*(U,A) ,A) satisfies

180

U E

ker L,

AE

W

(2)

H(r(s)U,A) = r(S)H(U,A)

Vs E G , V(U,A)

(3)

The relation between H(U,A) and F(p,A) is given by V(p,A) .

(4)

5.5.2. The one-dimensional subspaces ker L = R(P) and R(Q) remain invariant under the operators res), respectively res). It follows that there exist scalar functions a : G -+ R and ~ : G -+ R such that : Vs

E

G

(5)

It is clear that both a and ~ are one-dimensional representations of G. Con-

sequently, {a(s) [SEG} is a compact subset ofR\{O}, which is closed under multiplication and inversion; the same holds for {~(s) [ s E G}. We conclude that

Vs

[a(s) [

E

G •

(6)

It follows from (3), (4) and (5) that FCa(s)p,A)ZO = H(a(s)puO,A) = H(r(S)pUO,A)

= r(s)H(PUO,A) = ~(S)F(p'A)ZO ' i.e.

F(a(s)p,A)

=

~(S)F(p,A)

Vs E G , V(p,A) .

(7)

5.5.3. Lemma. Let M be equivariant with respect to (G,r,r), and satisfy (H1) and (H3) at ( 0 , 0). Then :

a(s) = ~(s)

Vs

E

G •

(8)

Proof. It follows from (7) that 3 3

(a(s)) DpF(O,O)

=

~

3

a(s)D pF(O,O)

Vs

E

G •

181

Since Dp3p(0,0) f 0, we conclude that (a(s))3 = ;;:(s) , which together with (6) implies (8). 0 5.5.4. Define GO={SEG! a(s) =1} and assume that GO r(s)u = u

=

(9)

G, i.e. Vs E G , Vu E ker L .

(10)

Then it follows from theorem 3.6.1 that all solutions (x,A) of (1) in a neighbourhood of the origin in XxA will satisfy : r(s)x = x

Vs

G •

E

( 11)

The bifurcation problem can then be reformulated and discussed using the spaces Xo and Zo instead of X and Z (see chapter 3). In this reduced formulation symmetry plays no further role, and the general discussion of the preceding section applies. 5.5.5. Now suppose that GO is a proper subgroup of G, i.e. a(s) = -1 for some s E G. Then (6) implies that GO will be a no~al subgroup of G, i.e. s -1 DGODS = GO for all s E G. Application of theorem 3.6.1 shows that' all sufficiently small solutions (x,A) of (1) will satisfy r(s)x = x

Vs

E

GO .

( 12)

This allows us to restrict our attention to solutions (x,A) of (1), with x E XO' i.e. we can consider M as a mapping of the type M : XOXA + ZO' where

Xa

= {XEX! r(s)x=x, VSEGO} ,ZO = {ZEZ !r(s)z=z, VSEGO}.

(13)

The action of the group G on Xo and Zo reduces to the action of a two-element group G= {e,i}, with i 2 = e; the group Gis isomorphic to the quotient group G/GO. So the action of Gon Xo and Zo is given by : 182

r(e)x=x.

r(i)x

r(s)x

(14)

f(e)z = z

f(i)z = f(s)z

(15)

where s is an arbitrary element of G\G O' We have (16 )

and (7) shows that F(p,A) is odd in p : F(-p,A) = -F(p,A)

V(p,A)

( 17)

It follows that

(18)

So we have a case of non-generic bifurcation, as described at the end of the preceding section. 5.5.6. Theorem. Assume M satisfies (H1) and (H3), and is equivariant with respect to some (G,r,f), where G is a compact group. Define the subgroup GO by (9), and suppose the following: (i)

GO f G ;

(ii) there is some },

E 11.

such that

Then the bifurcation set for (1) is given by (19)

For all A E w1 the equation (1) has a solution V*(O,A) E XO' corresponding to the zero solution of the bifurcation equation. If y 1 (A) < 0 there are also two other solutions, which are invariant under the action of the subgroup GO' 183

and which can be obtained one from the other by application of res), sEG\GO these two solutions correspond to nontrivial solutions ±p of the bifurcation equation. 0 5.5.7. The proof of theorem 6 follows immediately from the foregoing remarks and theorem 4. 10. The condition (ii) corresponds to the transversali ty condition (4.23) of theorem 4.10. The following more direct proof, which uses the oddness of the bifurcation function F(p,A), was suggested to us by J.K. Hale. We have, from (17) : F1 (p,A) =

fo1 D F(op,A)do p

.

(20)

F1 (p,A) is of class C2 , and even in p. For p t- 0 the bifurcation equation F(p,A) = 0 is equivalent to :

o.

(21 )

Define G(p,A) by (22)

Since F1 (p,A) = Yl(A)+p2 R(p,A), with R(O,O) = aO' it is easily seen that G is of class C1 • Moreover, G(O,O) = 0 and D~G(O,O) = a O > O. Now F(p,A) = G(p2,A), and (p,A) is a soiution of (21) if and only if (p2,A) is a solution of G(p,A) = o. So we look for solutions of :

o

p

>

0 .

(23)

We can use the implicit function theorem to solve the equation in (23) : for each A near zero the equation G(p,A) = 0 has a unique solution p = peA). It is easily seen that this solution has the form : (24) where ~(A) is a CO-function, with ~(O) a~l (~ is of class C1 in the region Yl(A) f 0). It follows that ~(A) > 0 for all A near zero. 184

Now bifurcation occurs because of the condition

P~

0 in (23). We have

p(>J > 0 if Yl (A) < 0, and peA) < 0 if Yl (A» O. It follows that (21) has two nonzero solutions if Yl(A) < 0, and no such solutions if Yl(A) ~ O. The bifurcation points are given by Y1 (A) at the zero solution.

=

0, and the bifurcation takes place

5.5.8. Remark. There are more general situations than the one described in this section where the transversality condition of section 4 is not satisfied because of some symmetry properties of the operator M. To give an example, let mO = M(.,O), and suppose that mO is equivariant with respect to some (G,r,r). Under the hypotheses (Hl) and (H3) we can still determine the subgroup GO; suppose that GO f G. Suppose that A =JRP (as in section 4), and define mi = :~ (.,0). Finally, 1 suppose that for each i E {1,2, ... ,p} there is some si E G\G O such that:

res. )m. (x) 1

(25)

1

Then, using (4.15) and (4.10) we find

i

We conclude that ail = 0 for all i = 1, ... ,p,

1, ..• ,p .

the generic condition (4.14) will not be satisfied, although we do not necessarily have that YO(A) = 0 for all A. and

5.6. APPLICATION: 1HE VON KARMAN EQUATIONS FOR A RECTANGULAR PLATE 5.6.1. The problem. In this section we consider the buckling problem for a simply supported rectangular plate, subjected to a compressive thrust at its short edges and to a normal load, proportional to a small parameter v. In the appropriate Hilbert space H (see section 2.4 for the details) the equation for this problem takes the form : (I-AA)w

+

C(w) = vp ,

(1) 185

where p is some fixed element of H; the parameter A measures the magnitude of the compressive thrust. We want to Stlrly the bifurcation of solutions of (1) for (W,A,V) in a neighbourhood of (O,AO'O), where AO is a characteristic value of the operator A. These characteristic values were detennined in section 2.4, where we found

,i (m2+n ZQh 2

m,n

4.Q.2m2

1,2, . .. ,

(2)

wi th corresponding eigenflllctions

4.Q.3/2 . mn 7 2 2 slllzo (x+.Q.) n (m- +n .Q. )

¢mn = 2

nn

sin~

(y+1)

m,n

1 ,2, ...

(3)

Yv

Fix some (m,n) such that Am,n fA" m ,n for all (m' ,n') f (m,n). For notational convenience we denote Amn and ¢mn by AO and ¢O. Let (4)

Then ker L = span{¢O} and R(L) = {wEH I (w,¢O) = A}. For the projections P and Q used in the Liapunov-Schmidt reduction we take the orthogonal projection onto ker L : Vw

E

H .

(5)

5.6.2. Equivariance group when p = O. In subsection 3.6.16 we discussed already the symmetry of (1) when there is no normal load, i. e. when p = O. Each of the operators A, C and P COllllTlute with the operators of the finite group : (6)

where, as in (3.6.44) : (TXW)(X,y) = w(-x,y) Let

186

(TyW)(X,y) = w(x,-y)

Txy

T T

xy

.

(7)

(8)

this is precisely the subgroup GO defined by (5.9), here denoted by G1 for "notational convenience further on. This subgroup depends on our choice of (m,n). Since : Tefl x mn

=(-1)

m+l

ymn =(-1)

efl mn

Tefl

n+l

efl mn

(9)

it follows that (10)

where T1 = I , TZ = (-1)

m+l

(11)

TX' T3

Using this notation we have (lZ)

T·

J

j

1,Z,3,4

(13)

and TZT4 = T3 •

(14)

For general p E H equation (1) will only be equivariant with respect to the isotropy subgroup of p, that is the subgroup Gp

=

hE G I TP = p} .

We want to study how the bifurcation properties of (1) depend on the symmetry of p, i.e. how they depend on Gp . First we consider the irreducible representations of G1• 5.6.3. Irreducible representations of G1 • If T E G, then also -T E G; consequently, the action of G on an element wE H is uniquely determined by the 187

action of the subgroup Gl on w. For this reason we restrict our attention the subgroup Gl . Let r : Gl ~ LORn) be an irreducible representation of Gl . It follows (13) that for each, E Gl , r('l) has an eigenvalue equal to ±1. MOreover, the group Gl is commutative. Then corollary 2.6.10 implies that reT) = ±I each, E Gl . Since the representation is irreducible, it follows that n = Such an irreducible representation r : Gl ~JR = LOR) is uniquely determined by the real numbers (1., = r(,,) J

j = 1,2,3,4

J

moreover, (1.j = ±1 for each j, and (1.2(1.3 = (1.4' (1.3(1.4 = (1.2 and (1.2(1.4 = (1.3' These relations are sufficient to show explicitly that there are only four different irreducible representations of G, denoted by r i (i=1,2,3,4), and determined by the numbers

r, (,,) J

1

(1." 1J

E

i,j

JR

1,2,3,4

(15)

given in the following table

'1

'2

'3

'4

rl

1

1

1

1

r2

1

1

-1

-1

r3

1

-1

1

-1

r4

1

-1

-1

1

5.6.4. Remark 1

-4 1

It follows from the table above that

4

L (1.1'k(1.J'k

k=l

1J

4

4 i~l (1.ij(1.ik 188

<5, ,

<5 jk

1 -4

4

I

i=l

(18)

a .. lJ

The following approach shows that these relations are not just a coincidence, but follow from more general relations that have to be satisfied by the matrix elements appearing in irreducible representations of finite groups. One

can see Miller [166] for the general theory. Let r : G1 ~R and

r : G1 ~ill be

two irreducible representations of G1 ;

let a· = r(T.) and ~. = rCT.). Define

J

J

_ 1

a - 4

J

J

4

I

rCTk)r(T

k=l

1

1

4

k ) ="4 k~1

~

~~

Then a·a J

1

= -4

~

4

I

-1

1

r(TJ.Tk)r(T k ) = -4

k=1

4

I

k=1

aa.

~

r(T.Tk)rC(T.T k) J

j

J

=

-1 ~

J

)rCT.) J

1,2,3,4 .

It follows that a = 0 if r f r, i.e. if a· f~. for sane j. In case r = r, J

then:

J

since r (T"I) = 1 for each representation. We conclude that ( 19)

where 6 ~ = 1 if r =

r,r

r,

and = 0 if r f

r.

Now let X be the space of all functions x : G1

~

ill. Under pointwise

addition and multiplication with a scalar, X is a 4-dimensional vectorspace, which becomes a Hilbert space if we use the inner product <x,y>

1

=

-4

4

I . 1

Y

XCTJ·)Y(T J.)

x,y EX.

(20)

Now we define a representation r 0 : G1 ~ L(X) as follows 189

Vx EX, Vj,k = 1,2,3,4 .

(21)

It is clear that fO is orthogonal with respect to the inner product (20).

If f : G1 ~~ is an irremlcible representation, then we can consider f as an element of X, and Vj ,k

1,2,3,4 .

This means that (22) i.e. span{f} is an invariant subspace of X, whjch transforms tmder fO according to the irreducible representation f. Since by (19) different irreducible representations are orthogonal when considered as elements of X, it follows that G1 has at most four different irreducible representations. Let V be the subspace of X spanned by all the irreducible representations of G1 ; the foregoing shows that V is invariant under the representation fO' If V X, then also V 1 is invariant uIlder f O,and Vi contains a one-dimensional subspace span{x}, which is invariant under f 0 and transforms according to some irreducible representation f. It follows that

r

j,k

1,2,3,4 .

Taking Tk = T1 we see that j

1,2,3,4 ,

i.e. x = X(T 1)f, and consequently x E V. Since also x E Vi, it follows that x = 0, which contradicts dim span{x} = 1. We conclude that V = X; consequently G1 has precisely four different irreducible representations, which we denote by f·1 (i=1,2,3,4). Then (19) takes the form (16) . .Another way of writing (16) is in the form AAT = I, where A is the 4x4-matrix 1/2(a .. ). 1J This implies ATA = I, i.e. we also have (17). Denote by {x j I j = 1,2,3,4} the canonical basis of X : xj(Tk) = 0jk' Using this basis it is easily seen that 190

j

= 1,2,3,4 .

(23)

Expressing f a(T j ) with respect to the basis {f iii = 1,2 ,3,4} of X, and using (22), we find 4

trace fa ( T .) = I a.. . J i=l 1J

(24)

A.combination of (23) and (24) gives (18).

5.6.5. For each i 1

1,2,3,4 we define a synnnetric operator Pi E L(H) by

4

P. = -4 I a1·kTk · 1 k=l

(25)

It follows from (18) that

4

1 4 4 4 P. = -4 I I a·kTk = I 0lkTk 1 i=l k=l 1 k=l

I i=l

= Tl

=I .

(26)

We also have 1

4

1

4

TjPi = 4 k~l aikTjTk = 4 k~l aij(aijaik)(TjTk) 1

= a ij

4

4 ~~1 ai~T~ = aijPi

(27)

This implies 1

PkP. = -4 1

4

I j=l

1

~ .T.P. = -4

KJ J

1

4

I j=l

~J.al·J·Pl· = 0k1· Pl· .

K

(28)

We conclude from (26) and (28) that the Pi are orthogonal projections on mutually orthogonal subspaces Hi = Pi(H), which together span H : H = H1 ffiH 2 ffiH 3 ffiH 4 . If w E H is such that TjW = aijw for each j, then (25) and (16) show that Piw = w; this, together with (27) proves that

{wEH I TW=W, VwEG.} , 1

(29)

191

where the subgroup G.1 of G is defined by G. 1

=

{a. ·T·

lJ J

I j = 1,2,3,4}

.

So the elements of Hi are precisely those w E H which transform according to the representation r.; H.1 also consists of those wE H which remain invariant 1 under the subgroup Gi of G. 5.6.6. Remark. We will see in the remainder of this section that the introduction of the irreducible representations r.l facilitates very much the cal. culation of certain terms in the bifUrcation equation. This is mainly due to the simple form of the representations, the splitting of H into orthogonal subspaces associated with each r i , and also to the fact that the nonlinear term in the equation (1) is homogeneous. From a general point of view one may ask the question how useful group representation theory really is when dealing with nonlinear bifurcation problems with synnnetry. At first sight there seem to be a few severe handicaps: group representation theory is a linear theory, which can only be fully developed when considering complex representations. From the other side bifurcation theory is clearly nonlinear, and usually deals with problems in real spaces. The foregoing chapters seem to indicate that only concepts such as "equivariance with respect to some subgroup" or "invariant subspace" will be relevant. (See also Ruelle [185], Poenaru [174] , Golubitsky and Schaeffer " [ 75]) • However, once the problem has been pushed down to that of solving a finite-dimensional bifurcation equation, group theory can actually be of great help in two respects. First, it can sometimes be used to obtain a further reduction of the equations; in the next chapter we will give an example where the specific form of the representation is used to reduce a 5-dimensional problem to a 2-dimensional one. Second, the fact that the bifurcation equations remain equivariant may result in a serious restriction on the terms which can appear in the Taylor development of the bifurcation fUnction; here group representation theory is the appropriate tool when one has to find out which terms will vanish and which terms remain. This has been beautifully illustrated by the work of D.H. Sattinger ([195-197],[265] ). Also the remainder of this section illustrates this point. 192

5.6.7. From (26.) it follows that each p E H can be written in the form 4

I

p

i=l

p.

p.

1

1

=

P.p E H. • 1

(31)

1

We will use this form of p when dealing with the equation (1). Since P is an orthogonal projection on span ¢o c H 1 , it follows that (32)

Vp E H •

We put w

=

P

P¢o + v

E

lR ,

V E

ker P

(33)

and

in (1), and project on R(L), using I-P. We find Lv - AO~V + (I -P)C(p¢O+v) = v( (I -P)P1 +

4

I

i=2

(34)

p.) 1

For (p,~,v) in a neighbourhood of (0,0,0) the equation (34) has a unique solution v = v*(p,~,v) near v = O. 5.6.8. Lemma. There is a constant c > 0 such that for ciently small neighbourhood of the origin we have IIv*(p,~,v)1I

(p,~,v)

in a suffi-

3

.:;; c(lpl +Ivl) .

(35)

Proof. If no such constant c exists, then we can find a sequence {(vn ' Pn' ~n' vn ) I nEN}, converging to (0,0,0,0) as n -+ each n EN, (vn ,pn ,~n ,vn ) solves (24), IIvn I f 0 and

00,

and such that for

3

IPn l liVT -+ 0

asn-+

oo

•

n

Expressing that (vn,Pn'~n,vn) is a solution of (34), dividing by IIvnll and taking the limit for n -+ gives 00

193

vn

lim L IIvnll

n-+oo

0.

Since L has a bounded pseudo-inverse K v lim _n_ n-+oo II Vnll

=

R(L)

~

ker P, it follows that

°,

which is impossible.

0

5.6.9. The bifurcation equatio~. Using (32) and the solution v*(p,lJ,v) of (34) gives us the bifurcation equation for (1) in the form : (36)

From lemna 6 and the fact that C(w) is homogeneous of degree three it follows easily that F(O,O,O)

D F(O,O,O)

°

P

and

(37)

(38) (See lemna 2.4.25). We conclude that equation (1) satisfies the hypotheses (H1) and (H3) of this chapter. From (36) and (4.8)-(4.9) we see that DlJyoCO,O)

o

DvYO(O,O)

Dj) Y1 (0,0)

-1

DvY1(0,0)

(39)

The transversality condition (4.7) will be satisfied if (p,¢O) = (P1'¢O)/O. If this is the case, then, in a (j),v)-plane, the bifurcation set for (1) will have the generic cusp-form which we found in subsection 4.2. A necessary condition to have the generic situation is that Pl = P1p I O. This implies that when we consider normal loads having a particular symnetry (e.g. p E H2) then it may very well be that this symnetry prevents the generic condition to be satisfied. So, for certain symnetries of the normal load we expect to have non-generic bifurcation sets. In the remainder of this

194

section we will study how the bifurcation set depends on the components Pi (i = 1, Z, 3,4) of p; more in particular, we are interested in the non-generic case that p E HZ ffi H3 ffi H4 (i.e. P1 = 0), and in the transition between the ge·neric and the non-generic case (i.e. P1 small). In order to do so we will suppose that in equation (1) p E H has the form 4

P = EP1

+

2

p.

p.

i=Z

E

1

1

H. (i=1,Z,3,4) , E EJR. 1

We consider € as an additional parameter; the elements p.1 and we suppose that P1 is such that

E

(40)

H.1 are fixed,

5.6.10. Let us return to the auxiliary equation (34). If we use the form (40) for p (i.e. replace P1 in (34) by EP1)' put ~ = € = 0 and neglect the cubic term, then the remaining equation has the solution v* = \l o

4

2

i=Z

Kp. ,

(4Z)

1

where K is the pseudo-inverse of L. Therefore we put 4

v

= \l

I

i=Z

Kp.

+

~

~ E ker P

(43)

1

in (34); we obtain the following equation for ~ 4

I

L~ - AO]JA~ + (I-P)C(p¢O + \l

i=Z

Kp.

+ v)

1

4

= E\l(I-P)P1 + AO~\l

2

i=Z

(44)

AKp . . 1

We denote by v*(p,~,\l,€) the unique solution of (44); such solution exists for (p,~,\l) small and € bounded. A proof similar to the one of lemma 8 shows that we have the following estimate for ~*. 5.6.11. Lemma. There is a constant c > 0 such that for Ciently small neighbourhood of the origin we have

(p,~,\l,E)

in a suffi-

195

The bifurcation function

F(p,~,V,E)

F(p,~,V,E) = -~p+ (C(P¢o+v

4

I

takes the form Kp- +v*(P,~,V,E)),¢O) -Evk

i=2

1

°

5.6.12. Lemma. There is a constant c > such that all sufficiently small solutions (p,~,V,E) of the bifurcation equation F(p,~,V,E) = satisfy

°

(47)

Proof. If no such constant c can be found, then there is a sequence {(pn ,~n ,vn ,En ) I nEN}, converging to (0,0,0,0) as n -+ such that, for each n EN, we have F(pn ,~n ,vn ,En ) = 0, pn t and 00

°

I~ n I --2-+ 0 , IPnl

IEn II vn I 3 -+0, IPn l

°

Ivn I ---+0 Ipnl

asn-+

oo

•

3

Dividing F(pn'~n,Vn,En) = by IPn l , and taking the limit for n -+ us (C(¢O)'¢O) = 0, which contradicts (38). 0

00

gives

5.6.13. Let 4

FO(P,~,V,E) = -~p+ (C(p¢o+v

I

i=2

KPi)'¢O) -Evk .

( 48)

Since C(w) = iB(w,B(W,w)), where B(.,.) is a bounded bilinear operator on H, it is easily seen that there is a constant c > 0 such that ( 49)

Using (49) and the estimate (45) for v*, it is straightforward to show that IF(p,~,V,E)-FO(p,~,V,E)

I

~ c[(lpl+lvl)5 + (1~1+IEI)(lpl+lv\)31 , for some c > 196

° and for all

(p,~,V,E)

in a neighbourhood of the origin.

(50)

We now determine a more explicit form of the function (2.4.18) and (2.4.19) we find 332

FO(P,~,V,E) =aOp -~p-Evk+2pv +

4

I

i=2

FO(P,~,V,E).

Using

(B(KPi,B(¢O'¢O))'¢o)

4 4 pv 2 il2 jl2 [(B(KPi,B(KPj'¢O))'¢O) +

4

4

+~v3 I

~(B(¢O,B(KPi,KPj))'¢O)]

4

I

I

i=2 j=2 k=2

(B(Kp.,B(Kp.,KPk))'¢O)

(51)

J

l

As a consequence of symmetry considerations several terms under the summation signs in (51) will vanish. In order to study this in detail we will use the irreducible representations of the group G1 • 5.6.14. Let w. E H. n R(L); then TKw. = KTW. = Kw. for each T E G.• This shows l l l l l l that K maps H.l n R(L) into Hl.• Next, we have for each T E G : Vu,v,w E H •

TB(u,B(v,w)) = B(TU,B(TV,TW)) This implies that for w.

H., w.

E

l

l

J

E

H. and wk

J

E

(52)

Hk we have

(53) From this relation we see that B(w. ,B(w.,wk )) transforms under the operators l J T,Q, E G1 according to the representation f : G1 -+]{ given by ai,Q,aj,Q,ak,Q,

fi(T,Q,)fj(T,Q,)fk(T,Q,) ,Q, = 1,2,3,4 ;

(54)

this represenation is denoted by f· 19 f. 19 f k . The next table gives the prol J duct representations f. 19 f. as a function of f. and f. l

J

l

J

197

@

f1

f2

f3

f4

f1

f1

f2

f3

f4

f2

f2

f1

f4

f3

f3

f3

f4

f1

f2

f4

f4

f3

f2

f1

5.6.15. Let us now consider the different terms appearing in (51). First, since ¢O E H1, the table above shows that B(KPi,B(¢O'¢O)) E Hi for i=1,2,3. Since the subspaces Hi are mutually orthogonal, we have (B(KPi,B(¢O'¢O))'¢O) = O. Next, BCKPi,B(KPj'¢O)) transforms according to f i @f j , which equals f1 if and only if i = j. It follows that

the same conclusion holds for (B(¢O,B(KPi,KPj))'¢o). Finally, B(KPi,B(KPj,KI\:)) transforms according to fi@fj@f k ; f6r i,j,k E {2,3,4} this equals f1 if and only if (i,j,k) forms a permutation of (2,3,4). Taking all this together, we find the following expression for FO(p,~,v,E) (55) where b

and

198

4

I [IIB(Kp·'¢O)1I i=2 1

2

1

+-2(B(Kp.,Kp.),B(¢O'¢O))]' 1

1

(56)

c

=

(B(KPZ, KP3) ,B(KP4'