A bifurcation theorem for critical points of variational problems

Nonlinear Analysis, Theory. Methodr & Applicoriom. Printed in Great Britain.

Vol. 12, NO. 1. pp. 51-61.

19% 01986

0362~546&‘88 $3.00 + .@I Pergamon Journals Ltd.

A BIFURCATION THEOREM FOR CRITICAL POINTS OF VARIATIONAL PROBLEMS SHUI-NEE CHOW* Department

of Mathematics,

Michigan State University, Wells Hall, East Lansing, MI 48824, U.S.A.

and REINER LAUTERBACH~ Institut fur Mathematik,

Universitlt

Augsburg, Memminger Strage 6, D-8900 Augsburg, West Germany

(Received 3 July 1986; received for publication 27 October 1986)

1. INTRODUCTION IN THIS

AND MAIN

THEOREM

note we consider the problem zn;l,n@, A)

where E is a real Hilbert space, il E R and F:ExR+R is a C*-functional.

(1.1)

Let f(x, A) = D,&

A)

(1.2)

be the gradient of F, which by the Riesz representation theorem may be identified with an element of E. Let (. , . )E denote the inner product on E, and 11.IIE = ((. , . )E)‘/2. An element (x0, A,,) E E x R such that f(xo9 Ao) = 0

(1.3)

is called a critical point of F. We assume that a C*-smooth curve of critical points of F is given, and that it may be parametrized x = x(A).

(1.4)

Without loss of generality, we may assume that x(n) = 0, i.e. (0, A) is a critical point of F for all A E R. We want to investigate the branching of further critical points from this curve, which we shall call the trivial solution of (1.3)). Let D,f(O, A) = A(1) and assume 0 < n = dim ker A(0) < m.

* Research partially supported by National Science Foundation Grant DMS 8401719. t Supported by Deutsche Forschungsgemeinschaft LA 525/1-l. 51

(1.5)

SHUI-NEE CHOW and R. LAUTERBACH

52

Moreover, assume that 0 is isolated in the spectrum a(A(0)). Then, for small \A/# 0, there exist n eigenvalues near zero, we assume that none of these is zero, i.e. there exists a number so > 0, such that kerA(ii)=(O) forO0

(1.7)

r,(d) = Fi r(A(A)) d
(1.8)

and

exist. Then we have the following. MAIN THEOREM. Let F : E x Iw-+ 07pbe C* and f(x, 3L)= D,F(x, il). Suppose that f(0, ;1) = 0 for all I E R and 0 is an eigenvalue of DJ(0, 0) = A(0). If

‘i(A) - T&d) f 0, then (0,O) is a bifurcation

point of the equation f(x, A) = 0.

Remark 1. In applications, we consider partial differential equations expressed in terms of integral equations. For example, many boundary value problems may be written as an operator equation (AI - L)x + N(x, n> = 0

where 1 is the identity, L : E + E is linear, compact and selfadjoint, and N = o(]xl) as Ix/ -+ 0 uniformly for bounded A. Thus, our theorem is applicable if f(x, 3L)= (AZ- L)x + N(x, A). Remark 2. This theorem

generalizes earlier results by Rabinowitz 114, 151, Clark [(il. Fade11 and Rabinowitz [S], Biihme [3], Marino [ll], Berger [2] and Takens [16]. The major difference between these earlier results and ours is that we allow an arbitrary dependence of F on A, while the other authors required that F depends linearly on h. Remark 3. The major difference

that he uses in using the not possess the method

between our proof and the proof given by Rabinow~tz [15] is the Ljapunov-Schmidt method, while we use center manifoId theory. The difficulty Ljapunov-Schmidt method arises from the fact that the bifurcation equation might a variational structure; compare also with [5]. In a recent paper by Kielhiifer [18] of Ljapunov-Schmidt is used to prove a theorem similar to ours.

Remark 4. Below we shall indicate a slight generalization

admit a bounded linear map

K:H-+E

to those real Banach spaces E which (1.9)

53

A bifurcation theorem for critical points of variational problems

with range R(K) dense in E, where His a real Hilbert space. Let J : H-, given by the Riesz representation theorem, if K : H-t E then

H” be the isomorphism

J

EtH-+H*+--E*

(1.10)

3ie. *

K =: KJ-‘iy*

is a bounded linear operator,

(1.11)

which in fact is injective, since the range of K is dense in E.

Remark 5. The construction

(1.9)-( 1.10) applies to all Sobolev spaces W”‘.p(R ) for bounded domainsBCRNand1 2 then

(1.12)

lV*P(Q) K

ww’qs2)

where m’ satisfies

1

-+

1 --

W~.P(Q)

m’--m

P=?

N

.

Remark 6. Problems in nonlinear elasticity may lead to variational problems in W’,J’(Sl) for 52 C RN, N = 1,2,3. Compare with [12f. Our more general formulation given below applies

to the problems in hyperelasticity

2. THE

which have a convex strain energy function.

CENTER

MANIFOLD

THEOREM

In this section we are going to describe the center manifold theorem to use it. Moreover we shall provide a proof of the crossing number relation between crossing numbers in the full space and crossing manifold. Let E be a Hilbert space, A : E -+ E be a bounded linear operator. A by a(A) and define q,(A)

in the way we are going theorem which gives a numbers on the center Denote the spectrum of

= a(A) fl @r/rE iw).

We assume a&l) # 0 and go(A) as well as its complement a(A)\a,(A) in a(A) are closed subsets of a(A). If this assumption is satisfied, we say o,(A) is a spectral set 19, p. 301. If 5,(A) is a spectral set it can be separated from a(A)\a,(A) by a closed curve r : R --, C. Here we use that a*(A) is compact. We assume that r‘ winds once around a*(A) in the mathematically positive direction. Let

(2.1) where R(z, A) is the resolvent of A at z E Cc. PO is a projection of E, the restriction A0 = A/El, maps E. into itself and 5(Ao)

=

onto a subspace E.

50(A),

[lo, p_ 1781. Let E, = (I - P&Y be a closed complement following.

see

operator

of E. in E. Then we have the

SHUI-NEE

54

C~o~and R.LAUTERBACH

CENTER MANIFOLD THEOREM. Let A be a bounded linear operator g(0) = 0, Dg(O) = 0. We consider the differential equation dx z

=

on E, g E C’(E, E) with

Ax + g(x).

(2.2)

We assume (a) %(A) f azI, (b) a,,(A) is a spectral set. U of 0 in E, there exists a Cl-mapping

Then, for a sufficiently small neighborhood

G:UnE,-+E,

(2.3)

such that the manifold M = {x + G(x)[x E U, n E}

(2.4)

is a Cl-center manifold for (2.2), i.e. (i) M is a Cl-manifold, which is locally invariant under the flow $(t, X) which is generated by (2.2); (ii) all solutions x(t) of (2.2) which remain in U for all t E R are on M; (iii) G(0) = 0, D,G(O) = 0, i.e. M is tangent to E. at the origin; (iv) the (local) flow on M in Cl-conjugate to the flow on U fl E. which is generated by the differential equation dx z = A ,,x + P,g(x + G(x)),

(2.5)

where x E E. fl U. Remark 1. Since M is an invariant

manifold, we know that the map x--t Ax + g(x),

x E M

(2.6)

maps M-,

TM

the tangent bundle on M, i.e. (2.6) defines a vector field on M. The projection &JIM :M-+EonU is a diffeomorphism

with inverse x-+x+

G(x),

XE Un Eo.

The right-hand side of (2.5) gives a vector field on U fl Eo, which is the push-forward with the diffeomorphism (2.7). This gives the Cl-conjugation in (iv). Remark 2. Different

(2.7)

(2.8)

of (2.6)

versions of the center manifold theorem are contained in Sections 6.1, 6.2 in Henry [9]. His results are more general than what we need at this point: he allows closed linear operators A which are sectorial, and nonlinearities which are defined on fractional power

A bifurcation theorem for critical points of variational problems

55

spaces E” = D(A”). However, he does not give complete and detailed proofs. In our case A is bounded and therefore it generates a flow (in contrast to the situation of an unbounded sectorial operator, then (-A) generates a semiflow). For this reason all the proofs given in the ODE case carry over to our situation. [4, 5, 131 give proofs for the existence of Lipschitz continuous manifolds satisfying (i)-(iv). However, it is difficult to follow the smoothness proof in [4,5]. Vanderbauwhede and van Gils [17] provide an existence proof and a transparent proof of the smoothness of G using functional analytic methods. A new geometric approach [l] proves the existence of a Lipschitz continuous center manifold for some infinite dimensional problems. It seems to be well known that the right hand side of (2.5) is CK whenever (2.2) is CK. The difficulty with the proofs in [4,5] is the use of the implicit function theorem in spaces of functions which are CK. We should note that c” or real analyticity of g do not imply that G is C” or real analytic. Moreover local center manifolds M are not unique. A further discussion of these matters may be found in [4, 5, 91. Next we would like to discuss the crossing of eigenvalues for parameter dependent equations. We are especially interested in whether the crossing of eigenvalues can be seen on the center manifold. We consider the equation dx z

= Ax + B(A)x + g(A, x),

AE A c R

x E E,

(2.9)

where A satisfies the hypotheses of the center manifold theorem, B(k) is a family of bounded linear operators depending differentiably on A E A, A is an open interval containing 0, and Z?(O)= 0. We assume that g is twice continuously differentiable on W X A, where W is an open neighborhood of 0 in E, moreover we assume g(0, A) = 0

for all A E A

(2.10)

and D,g(O, A) = 0.

(2.11)

In order to apply the center manifold theorem to this equation we write (2.9) as a differential equation on E’ = E x [w

$ =Ax dA dt-

+ B@)x + g(L, x)

(2.12) - 0.

We write x’ for the pair x’ = (x, A) E E’. Let A’ be the linear operator on E’ defined by A’x’ = (Ax, O),andg’(x’) = (B(A)x + g(A,x),0).Theng’(0) = g(O,O) = 0, let h’ = (h,, h,) E E’, then D,,g’(O)h’

=

=

D,g’(O,0) D,g’(O, 0) h, 0

0

B(O) + Dxg(O, 0) 0

= 0.

)( hA>

56

SHUI-NEE CHOW and R. LAUTERBACH

The spectrum a(A')is given by

a(A')= o(A)U (0). In particular we have

is a spectral set. Therefore

%(A’) = o,(A) U (01

(2.13)

the center manifold theorem

can be applied to (2.12). Define

Eb,Pb ,EA , U' as before with the obvious modifications. The center manifold theorem guarantees the existence of a map

G':E;,nU'+E:

(2.14)

such that the set M’ = {x’ + G’(x’)Jx’ E U’ n E;} is invariant under the flow corresponding In particular the set

to (2.12).

MA = {x+ G(x,il)l(x,L)E v'} is invariant, where G(x, ,%)is the first component of G’(x, A) in E' = E X R. Again we can push the vector field on MA forward on E, and obtain

$ =Aox + P,B(A)(x+ G(x, A)) + g(k

x + Gb

A))

= C(A)x + g(A, x),

(2.15)

This last equation defines C(k) to be the linearization of the right hand side of (2.15) along the stationary solution x = 0, A.E A (by the construction of the center manifold this is a stationary solution) and g contains the nonlinear terms, g is of class C’. We write A(A)= A + B(h).We add one more hypothesis in order to define crossing numbers for A(A),C(A) and to study their relation:

o,(A)is finite and consists of eigenvalues of finite multiplicity.

(2.16)

Since q,(A)is isolated in a(A),there exists a number a > 0, such that

a(A)rl{rE cl0-c(Rezj
0.

By the continuity of isolated eigenvalues, the upper continuity of the spectrum [lo, p. 2081 and (2.16) there exists a number E > 0, such that for IA]< E the intersection 1

z E @)Oc /Re

consists only of finitely many eigenvalues zero as h+ 0. For fixed A, (AJ< E, let eig#(A))

~1 -c ;)

rl

a(A(A))

of finite multiplicity,

whose real parts converge to

= ( z E @I(Re zj < i) n u(A(A)).

A bifurcation theorem for critical points of variational problems

57

Let us assume for I # 0, IA)< E we have 0 65{Re P/,UE eigo (A(A))}. As before write r(A(A)) and

(2.17)

for the number of elements in eigO(A(A)) having negative real parts, +)

= pn~ +(A)) I>0

rA(q = yz @(A)) a<0 Let c’A(n)= r&) - ~2~~).Similarly we define crccA).Then we have the following. CROSSINGNUMBERTHEOREM. CrA(L)

Proof. In order to prove the theorem

=

C’C(A).

it suffices to show that for ]A)sufficiently small

+W)

(2.18)

= @(A)).

This follows from the following lemma. LEMMA. If the neighborhood U in the above construction there exists a number a > 0 such that

is chosen sufficiently small then

(a) if 1< E then each stationary solution f E U of (2.9) is of the form X=X0 + G(Zo,J.),

forsomeZo

EE,

n U;

(b) the linearization L of (2.9) with respect to x at 2 leaves the tangent space T,MI, of MA at 2 invariant; (c) the intersection of the spectrum of L with a strip of width 2a a(L) rl {z E C) [Re z ( c a} is exactly the set of eigenvalues of the linearization t of (2.15) at X0 E E. fl U; and the multiplicities of these eigenvalues are the same. Proof of the lemma.

(a) is a rewording of property (ii) of the center manifold theorem; (b) follows from [19, p. 259, definition 4.3.1. and exercise 4.3 A (iii)]; (c) since the vector field in (2.15) is the push forward of the vector field on the invariant manifold M,, it is clear that (6) Therefore

= @IT&).

it suffices to show that ~(L(T~~~) = a(L) fl {z E C( IRe z/ < a}.

(2.19)

Since T,M, is an invariant subspace for L it is clear that @I T&i) c o(L)*

(2.20)

SHUI-NEECHOWand R. LAUTERBACH

58

We shall show the reverse inclusion in (2.20) using a multiplicity argument. Observe that A0 =

AlToMo. Since q,(A)is a spectral set o(A)\oo(A) has a positive distance from the imaginary axis (cJ(A)&,(A) is compact). a(A) fl

Corresponding

{zE Q=jO
to a we can find a neighborhood

Choose a such that

2a}= 0.

V such that for any stationary solution X E V

dist(44Tf&,4Ao)>
z mk0) ~O~a(LITfMI) ZOEZla)m(zo)*

of finite sets of eigenvalues of finite multiplicity,

2

this last quantity equals

+0>.

zoEo(L)n{rEC(~Rez(
This proves part (c) of our lemma. 3. PROOF

OF THE MAIN

THEOREM

We assume the notation of the first part and the hypotheses f(x, A) = &F(x, A) an d consider the differential equation

of the main theorem.

Let

(3.1) dA z=

0.

(3.2)

By (1.5), the fact that A(0)= DJ(0, 0) is a bounded and symmetric linear operator assumption that 0 is isolated in a(A(O)), we have

and the

(0) = a(A(0)) 17{zE @IRe z = 0) and (01, o(A(O))\{OI are spectral sets. Since f is of class C’ the hypotheses of the center manifold theorem are satisfied for (3.1) and (3.2). Therefore we have an (n + 1)-dimensional manifold M' C E x R which (i) is locally invariant; (ii) contains all invariant sets which are contained (0,O) E E x R; (iii) is tangent to kerA(0) x R at (0,O); and

in a sufficiently small neighborhood

of

A bifurcation theorem for critical points of variational problems

(iv) the flows on the shces iVlkare C’conjugate dV -

at

=

&,f(v + G(u,

59

to

A), A), u E E,

where PO is the eigenprojector onto E0 = kerA(O), which is constructed ’ + EC is the map obtained by the center manifold theorem. G : (E. x KS)f~ U We rewrite (3.3) as dv - = -B(A)v + g(v, n) dt

(3.3)

as in (2.1),

(3.4)

where g(v, A) = O(llv flE) uni formly for /A/sufficiently small. From the crossing number theorem it follows that r(A(L))

and

= r(-B(1)) cr(A(A))

for 111sufficiently small = cr( - B(I)).

(3.5) (3.6)

In particular we see cr( -B(1)) # 0. We want to obtain a contradiction to the assumption that there is no bifurcation using Conley’s generalization of the Morse index. Suppose there is no bifurcation. The first step is to show that for A = 0 the set (0) is an isolated invariant set for equation (3.3). Let il = 0, and N be an open neighborhood of 0 in EO, with fi compact (and sufficiently small). Let S be the maximal invariant set in N. Then the set S, = {v + G(o, 0)ju E S} is an invariant set on MO. Let x(t) be a trajectory in S,, i.e. a solution curve to (3.1) for A = 0. Then its rr - (w-) limit sets are both contained in S,. Since the flow on M,, is the restriction of a gradient flow these sets consist of critical points. The assumption that there is no bifurcation of critical points means that 0 is an isolated critical point in E, i.e. the limit sets must be the set {O}.Again from the fact thatfdefines a gradient flow we get x(t) = 0, i.e. S, = (0). Therefore S = (0) and N is an isolating neighborhood. This isolating neighborhood continues for small A # 0. By an argument which is similar to the one given above, we show that the maximal invariant set S, C N for small A # 0 is given by S, = (0). (We use again our assumption that there is no bifurcation of critical points). By our assumption (1.6) and the lemma used to prove the crossing number theorem, we see that 0 is a hyperbolic singular point of equation (3.3) for I # 0, llz] small. The homotopy index [7, p. 511 of this isolated invariant set (v = 0) is the pointed sphere whose dimension equals the dimension of the unstable manifold of 0 with respect to (3.3). The crossing number theorem and our assumption of nonzero crossing number imply that the dimension of the unstable manifold for nonzero 3L1,A2 with sgn il, = -sgn A2 is different. By 17, theorem 1.4, p. 671 the homotopy index is invariant under continuation, yielding a contradiction to the assumption that there is no bifurcation.

4. A GENERALIZATION

There are two directions in which one could try to generalize our theorem: (a) more general spaces; (b) less assumptions on the smoothness of F.

SHUI-NEE CHOW and R. LAUTERBACH

60

For example, if one wants to prove a similar theorem which is applicable to problems in elasticity one needs a generalization in both directions. The simplest problems in hyperelasticity as for example the buckling of a rod lead to variational problems as

I

min

1

UE wm+y[o,l]) o

IV%?,

(4-l)

A) ch

where Wm,J’([O,11) is the Sobolev space of functions in Lp([O, 11) having weak derivatives of mth order in D’([O, 11). We note that “real problems” are formulated over domains & C R3. We want to assume p > 1 although in applications this is not always the case. The functional 1 F=

i0

W(4 9A) dx

(4.2)

isnotC*andifp#2, W= Wyd$ is not a Hilbert space. If p < 2, and S2 C RN is bounded, then the construction given in remark 4 above applies with H = Wm,‘(Q) ifp > 2, take m’ satisfying 1 j=?

1 -- m’-m N

and H = W’“‘~*(Q). In these cases we look at the differential i = -Kf(x,

equation

A).

(4.3)

If F is differentiable (in some sense) we get for a solution of (3.3) ((. , .)cE.,EJ refers to the pairing E*, E, while (. , . )H denote the inner product in Z-Q

By the injectivity of K we conclude that any point x0 in the (strong) w-limit set of x(t) is an equilibrium for (4.2) and a critical point of F. The main ingredient in our proof is the center manifold theorem, which holds under more general hypotheses, than we needed above [9]. In elasticity one looks at strain energy function W(u,, A) where u is the deformation, U, the deformation gradient and the problem is to minimize the function 1 F=

I0

over Wm,P([O,l]), wh ere b incorporates

W(U,) A’) + b(u, x) dx

external forces. We assume

(i) WE C*(R, R) (ii) 5

W(z, A) > 0, i.e. W(. , A) is convex for each A.

(4.4)

A bifurcation theorem for critical points of variational problems

The Euler Lagrange equation of (4.4) is quasilinear, W”(Ux7lb)u,

61

namely

+ b(u, x) = 0.

(43)

We look at al.4

(Wux, ~)u, + b(u,4).

at=

(4.6)

Changing the time variable as in [9, p. 591, we obtain the semilinear parabolic equation

at4 $

1 = uxx

+ W”(J$) A)

b(u, x)

which may be treated by our theory. Acknowledgments--The second author wishes to thank John Maddocks for mentioning the problem. He also would like to express his gratitude to the IMA at Minneapolis for the hospitality which he received during the 1984-1985 academic year. We would like to thank Paul Rabinowitz for his comments which clarified the exposition as well as our ideas on the subject. REFERENCES 1. BATESP. W. & JONESC. K. R. T., The center manifold theorem with applications (preprint, 1985). 2. BERGERM. S., Bifurcation theory and the type number of Marston Morse, Proc. Natn. Acad. Sci. 69, 1737-1738 (1972). 3. B~HME R., Die Losung der Verzweigungsgleichung fiir nichtlineare Eigenwertprobleme, Marh. Z. 127, 105-126 (1972). 4. ~ARR'J., Applications of Centre Manifold Theory. Springer, New York (1981). 5. CHOW S.-N. & HALE J. K., Methods in Bifurcation Theory. Grundl. der wathe. Wss. 250, Springer, Berlin (1983). 6. CLARK D. C., Eigenvalue perturbation for odd, gradient operators, Rocky Mt. 1. Math. 5, 317-336 (1975). 7. CONLEY C., Isolated invariant sets and the Morse index, Regional Conference Series in Mathematics 38, Conference Board of the Mathematics Society (1976). 8. FADELL E. R. & RABINOWTZ P. M., Bifurcation for odd potential operators and an alternative index, J. funct. Analysis 26, 48-67 (1977). 9. HENRYD., Geometric theory of semilinear parabolic equations; Lecture Notes in Mathematics 840, Springer, New

York (1982). 10. KATO T., Perturbation theory for linear operators, Grundf. der mathe. Wiss 132, Springer, Berlin, corrected printing to 2nd edition (1980). 11. MARINOA., La biforcaxione nel.caso variationale, Conf. Sem. Mat., Bari (1973). 12. MARSDENJ. E. & HUGHEST. J. R., Mathematical Foundations of Elasticity. Prentice Hall, Englewood (1983). 13. PALMER K., Qualitative behaviour of a system of ODE near an equilibrium point-a generalization of the Hartman-Grobman theorem (preprint, 1980). 14. RABINOWITZP. H., Variational methods for non-linear eigenvalue problems, C.I.M.E., Varenna (1974). 15. RABINOWITZ P. H., A bifurcation theorem for potential operators, J. funct. Analysk 25, 412-424 (1977). 16. TAKENSF., Some remarks on the Bohme-Berger bifurcation theorem, Math. Z. 129, 359-364 (1972). 17. VANDERBAUWHEDE A. & VON GILT S. A., Center manifolds and contractions on a scale of Banach spaces (preprint, 1985). 18. KIELHOFERH., A bifurcation theorem for potential operators (preprint, 1985). 19. ABRAHAMR., MARSDENJ. E. & RAT~UT., Manifolds, Tensor Analysis, and Applications. Addison Wesley, London (1983).