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, £) = 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
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