AN OVERVIEW

5

E i s a rea l Banac h space , G i s a grou p of transformations o f E int o E, an d

I G C

l

(E, R). W e say I i s invariant unde r G i f I(gu) = I(u) fo r all g G G an d

u G E. A s a first example, consider (1.5). I t i s invariant unde r G = {id , -id},

where i d denote s th e identit y ma p o n E. Not e tha t w e can identif y G wit h

Z2. Mor e generall y i f p(x , £) i s continuou s o n [0,?r ] x R , i s od d i n £ , an d

P(x, £) = f£ p(x, t) dt, then

(1.9) I(u) = r$\uf - P(x,u)\ dx

Jo

is invariant unde r G . As another exampl e consider (1.7), recalling for this case

that function s i n E ar e 2n periodic. Le t 0 G [0,27r), Z = (p,q) G E, (g$z)(t) =

z(t + 0), and G = {g$\6 G [0,27r)}. The n i t i s easy t o see that T is invariant

under G. Moreover G ca n be identified wit h S

1.

The above examples show that functionals invarian t under a group of symme-

tries arise in a natural fashion. I t is often th e case that such functionals posses s

multiple critical points. Indee d results of this type are among the most fascinat -

ing in minimax theory. Th e first exampl e of such a theorem goes back to early

work o f Ljusternik an d Schnirelma n [LLS] . They studie d a constraine d varia -

tional problem, i.e . I restricte d t o a manifold (whic h must b e invariant unde r

G) an d proved

THEOREM. If I G

C1 (Rn,R)

and is even, then I\sn~i possesses at least n

distinct pairs of critical points.

Subsequently other researchers extended this result to an infinite dimensional

setting.

Another multiplicity result is provided by the following Z2 Symmetrie version

of the Mountain Pass Theorem [AR, R2]:

THEOREM. Lei E be a real Banach space and I G

C1(£?,R)

with I even.

Suppose 1(0) = 0 and I satisfies (PS), (h), and

(I2) for all finite dimensional subspaces E C E, there is an R = R(E) such

thatI(u) 0foru€E\BR{&).

Then I possesses an unbounded sequence of critical values.

In order to exploit symmetrie s of I, on e needs a tool to measure the size of

Symmetrie sets, i.e. subsets of E invarian t unde r G . Suc h a tool is provided by

the notion of an index theory. Wit h the aid of such theories, minimax charac-

terizations can be given for the critical points obtained in the two theorems just

cited. Inde x theories will be cüscussed in Chapter 7 and applied to constrained

and unconstrained variationa l problems in Chapters 8-9 . I n particular we will

see that generalize d version s of (1.5) satisf y th e hypothese s o f the Symmetri e

Mountain Pass Theorem and possess an unbounded sequence of critical points.

The nex t questio n w e will stud y i s what happen s t o a functiona l whic h i s

invariant unde r a group of symmetries when a perturbation i s made which de-

stroys the symmetry. N o general theor y ha s been develope d ye t t o treat suc h