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:
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.
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
Au = /(x), 2 0 ,
u = 0, x G du,
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