where here and in future example s Q denotes a bounded domai n i n R
whos e
boundary, 50 , i s a smoot h manifold . Suppos e / G C(ll). A funetion u i s a
elassical Solution of (1.1) if u G
0 C(fl). Fo r such a Solution, multiplying
(1.1) by p G C§° (Q) yields
(1.2) f{Vu-
Vp-fp)dx = 0
after a n Integratio n b y parts. Le t
denot e the elosure of C§°(Q) with
respeet t o
If u G WQ
an d satisfies (1.2) for all p G Q)°(n), then it is said to be a weak
Solution of (1.1). By our above remarks, any elassical Solution of (1.1) is a weak
Solution. Unde r slightly stronger hypotheses on / (e.g . / Holde r continuous) the
converse is also true. Choosin g E = Wo'
(1.3) I(u)= t {\\Vu\* - fu)dx.
It is not difficult t o verify tha t I i s Frechet differentiable o n E an d
(1.4) I'(u)p= f (Vu-Vp-fp)dx
for p G E. Thu s u is a critical point of I i f and only if u is a weak Solution of
As was noted earlier, when E = R
th e most familiär sort s of critical points
obtained ar e maxim a o r minima . I n thes e lecture s w e will be dealing mainl y
with functionals whic h may not b e bounded from above or below even modulo
finite dimensional subspac^ or submanifolds. Suc h "indefinite " functionals may
not posses s any local maxima or minim a other tha n trivia l ones . Fo r example
let 0 = (0 , TT) C R, E = W^
TT]), and
(1.5) I(u)= r ( | K |
- | u
) d x ,
where ' = d/dx . I t i s not difficult t o show that / i s differentiable o n E an d has
u = 0 as a local minimum. Fo r any other u G E an d a G R,
I(au) = J*(^\ur-^u*)dx
-0 0
as |a | o o so i" is no t bounde d from below . Furthermor e fo r eac h k G N,
sin kx G U, and
- j oo
as k —• oo so I is not bounded from above. Thus it is not obvious that I possesses
any critical points other than the trivial one u = 0. Nevertheless we will see later
as an application of the Mountain Pass Theorem thati* possesses positive critical
values and the same thing is true for higher dimensional versions of (1.5).
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