1. An Overview

A focus of these lectures is the existence of critical points of real valued func-

tionals. Th e most familiär exampl e occurs when we have a continuously differ -

entiable map g:

Rn—

• R. A critical point o f g is a point £ at whic h ^(f)» the

Frechet derivative of?, vanishes. The simplest sort of critical points of g are its

global or local maxima or minima.

g(x)

The setting in which we will study critica l point theor y i s an infinite dimen-

sional generalization o f the above. Le t E b e a real Banach space. A mapping

I o f E t o R wil l be called a functional. T o make precise what w e mean by a

critical point of I, recal l that I i s Rrechet differentiable a t u G E i f there exists

a continuous linear map L = L(u): E — R satisfying : fo r any e 0, there is a

6 = Sie, u) 0 such that \I(u + v) - I(u) - Lv\ e\\v\\ fo r al l |jv| | 6. Th e

mapping L is usually denoted by I'(u). Not e that I'(u) G E*, the dual space of

E. A critical point u of / i s a point at which V{u) = 0, i.e.

I'(u)p = 0

for all p G E. Th e value of I a t u is then called a critical value of I.

In application s t o differentia l equations , critica l point s correspon d t o weak

Solutions of the equation. Indee d this fact makes critical point theory an impor-

tant existenc e tool i n studying differentia l equations . A s an exampl e consider

the linear elliptic boundary value problem

(1.1)

Au = /(x), 2 € 0 ,

u = 0, x G du,

l

http://dx.doi.org/10.1090/cbms/065/01