Table Of ContentLectures on
The Ekeland Variational Principle
with Applications and Detours
By
D. G.DeFigueiredo
TataInstituteofFundamentalResearch, Bombay
1989
Author
D.G. DeFigueiredo
DepartmentodeMathematica
Universidade deBrasilia
70.910–Brasilia-DF
BRAZIL
(cid:13)c TataInstituteofFundamentalResearch,1989
ISBN3-540-51179-2-Springer-Verlag,Berlin,Heidelberg.NewYork.Tokyo
ISBN0-387-51179-2-Springer-Verlag,NewYork.Heidelberg.Berlin. Tokyo
No part of this book may be reproduced in any
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out written permission from the Tata Institute of
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PrintedbyINSDOCRegionalCentre,Indian
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andpublished byH.Goetze,Springer-Verlag,
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PRINTEDININDIA
Preface
Sinceitsappearancein1972thevariationalprincipleofEkelandhas
found many applications in different fields in Analysis. The best refer-
ences for those are by Ekeland himself: his survey article [23] and his
book with J.-P. Aubin [2]. Not all material presented here appears in
those places. Some are scattered around and there lies my motivation
in writing these notes. Since they are intended to students I included
a lot of related material. Those are the detours. A chapter on Nemyt-
skii mappings may sound strange. However I believe it is useful, since
their properties so often used are seldom proved. We always say to
the students: go and look in Krasnoselskii or Vainberg! I think some
of the proofs presented here are more straightforward. There are two
chapters on applications to PDE. However I limited myself to semi-
linear elliptic. The central chapter is on Bre´zis proof of the minimax
theorems of Ambrosetti and Rabinowitz. To be self contained I had to
develop someconvex analysis, whichwaslater usedtogiveacomplete
treatment of the duality mapping so popular in my childhood days! I
wrote these notes as a tourist on vacations. Although the main road
is smooth, the scenery is so beautiful that one cannot resist to go into
the side roads. That is why I discussed some of the geometry of Ba-
nach spaces. Some of the material presented here was part of a course
delivered at the Tata Institute of Fundamental Research in Bangalore,
India during the months of January and February 1987. Some prelimi-
nary drafts were written by Subhasree Gadam, to whom I express may
gratitude. I would like to thank my colleagues at UNICAMP for their
hospitalityandEldaMortariforherpatienceandcheerfulwillingnessin
texingthesenotes.
Campinas,October1987
Contents
1 MinimizationofLowerSemicontinuousFunctionals 1
2 NemytskiiMappings 9
3 SemilinearEllipticEquationsI 23
4 EkelandVariationalPrinciple 31
5 VariationalTheoremsofMin-MaxType 39
6 SemilinearEllipticEquationsII 55
7 SupportPointsandSuportFunctionals 73
8 ConvexLowerSemicontinuousFunctionals 81
9 NormalSolvability 97
v
Chapter 1
Minimization of Lower
Semicontinuous Functionals
LetXbeaHausdorfftopologicalspace. AfunctionalΦ : X → R∪{+∞} 1
is said to be lower semicontinuous if for every a ∈ R the set {x ∈ X :
Φ(x) > a}isopen. Weusetheterminologyfunctionaltodesignateareal
valued function. A Hausdorff topological space X is compact if every
coveringofX byopensetscontains afinitesubcovering. Thefollowing
basic theorem implies most of the results used in the minimization of
functionals.
Theorem 1.1. Let X be a compact topological space and Φ : X →
R ∪ {+∞} a lower semicontinuous functional. Then (a) Φ is bounded
below,and(b)theinfimumofΦisachievedatapoint x ∈ X.
0
Proof. The open sets A = {x ∈ X : Φ(x) > −n}, for n ∈ N, constitute
n
anopencoveringofX. Bycompactness thereexistsan ∈ Nsuchthat
0
n0
A = X.
j
[j=1
SoΦ(x) > n forall x∈ X.
0
(b) Now let ℓ = InfΦ, ℓ > −∞. Assume by contradiction that ℓ is
1
2 1.MinimizationofLowerSemicontinuousFunctionals
notachieved. Thismeansthat
∞
1
x ∈ X :Φ(x) > ℓ+ = X.
( n)
[n=1
2 Bycompactness againitfollowsthatthereexistan1 ∈ Nsuchthat
n1 1
x ∈ X :Φ(x) > ℓ+ = X.
( n)
[n=1
But this implies that Φ(x) > ℓ+ 1 for all x ∈ X, which contradicts
thefactthatℓistheinfimumofΦ. n1 (cid:3)
Inmanycasesitissimplertoworkwithanotion oflowersemicon-
tinuity given in terms of sequences. A function Φ : X → R∪{+∞} is
said to besequentially lowersemicontinuous iffor every sequence (x )
n
withlimx = x ,itfollowsthatΦ(x ) ≤ liminfΦ(x ). Therelationship
n 0 0 n
between the two notions of lower semicontinuity is expounded in the
followingproposition.
Proposition 1.2. (a) Every lower semicontinuous function Φ : X →
R ∪ {+∞} is sequentially lower semicontinuous. (b) If X satisfies the
FirstAxiomofCountability, theneverysequentially lowersemicontinu-
ousfunction islowersemicontinuous.
Proof. (a) Let x → x in X. SupposefirstthatΦ(x ) < ∞. Foreach
n 0 0
ǫ > 0considertheopensetA = {x ∈ X :Φ(x) > Φ(x )−ǫ}. Since
0
x ∈ A,itfollowsthatthereexistsn = n (ǫ)suchthat x ∈ Afor
0 0 0 n
all n ≥ n . For such n’s, Φ(x ) > Φ(x )−ǫ, which implies that
0 n 0
liminfΦ(x ) ≥ Φ(x )−ǫ. Sinceǫ > 0isarbitrary itfollows that
n 0
liminfΦ(x ) ≥ Φ(x ). If Φ(x ) = +∞ take A = {x ∈ X : Φ(x) >
n 0 0
M}forarbitrary M > 0andproceedinsimilarway.
(b) Conversely weclaimthatforeachrealnumberatheset F = {x ∈
Ω : Φ(x) ≤ a}isclosed. Suppose bycontradiction thatthis isnot
the case, that is, there exists x ∈ F\F, and so Φ(x ) > a. On
0 0
theotherhand,letO beacountablebasisofopenneighborhoods
n
3
of x . For each n ∈ N there exists x ∈ F ∩O . Thus x → x .
0 n n n 0
Using the fact that Φ is sequentially lower semicontinuous and
Φ(x )≤ aweobtainthatΦ(x )≤ a,whichisimpossible.
n 0
(cid:3)
Corollary 1.3. If X is a metric space, then the notions of lower semi-
continuity andsequentially lowersemicontinuity coincide.
Semicontinuity at a Point. The notion of lower semicontinuity can be
localizedasfollows. LetΦ : X → R∪{+∞}beafunctionaland x ∈ X.
0
We say that Φ is lower semicontinuous at x if for all a < Φ(x ) there
0 0
exists an open neighborhood V of x0 such that a < Φ(x) for all x ∈ V. 3
It is easy to see that a lower semicontinuous functional is lower semi-
continuous at all points x ∈ X. And conversely a functional which is
lowersemicontinuous atallpoints islowersemicontinuous. Thereader
canprovidesimilardefinitionsandstatementsforsequentiallowersemi-
continuity.
SomeExamplesWhenX = R.LetΦ: R → R∪{+∞}. ItisclearthatΦ
islowersemicontinuousatallpointsofcontinuity. If x isapointwhere
0
thereisajumpdiscontinuity andΦislowersemicontinuous there, then
Φ(x ) = min{Φ(x −0),Φ(x +0)}. If limΦ(x) = +∞ as x → x then
0 0 0 0
Φ(x ) = +∞ if Φ is to be lower semicontinuous there. If Φ is lower
0
semicontinuous the set {x ∈ R : Φ(x) = +∞} is not necessarity closed.
Example:Φ(x) = 0if0≤ x≤ 1andΦ(x) = +∞elsewhere.
Functionals Defined in Banach Spaces. In the case when X is a Ba-
nach space there are two topologies which are very useful. Namely
thenorm topology τ(also called thestrong topology) whichisametric
topology and the weak topology τω which is not metric in general. We
recallthattheweaktopologyisdefinedbygivingabasisofopensetsas
follows. Foreachǫ > 0andeachfinitesetofboundedlinearfunctionals
ℓ ,...,ℓ ∈ X∗, X∗ is the dual space of X, we define the (weak) open
1 n
set {x ∈ X : |ℓ (x)| < ǫ,...,|ℓ (x)| < ǫ}. It follows easily that τ is a
1 n
finer topology than τω, i.e. given a weak open set there exists a strong
opensetcontainedinit. Theconverseisnottrueingeneral. [Weremark
that finitedimensionality of X implies that these twotopologies are the
4 1.MinimizationofLowerSemicontinuousFunctionals
same]. It follows then that a weakly lower semicontinuous functional
Φ : X → R∪{+∞},X aBanachspace,is(strongly) lowersemicontinu-
ous. A similar statement holds for the sequential lower semicontinuity,
sinceeverystronglyconvergent sequenceisweaklyconvergent. Ingen-
eral, a (strongly) lower semicontinuous functional is not weakly lower
semicontinuous. Howeverthefollowingresultholds.
Theorem 1.4. Let X be a Banach space, and Φ : X → R ∪ {+∞} a
convex function. Then the notions of (strong) lower semicontinuity and
weaklowersemicontinuity coincide.
Proof. (i) Case of sequential lower semicontinuity. Suppose x ⇀
n
x (the half arrow ⇀ denotes weak convergence). We claim that
0
4 the hypothesis ofΦbeing (strong) lower semicontinuous implies
that
Φ(x )≤ liminfΦ(x ).
0 n
Let ℓ = liminfΦ(x ), and passing to a subsequence (call it x
n n
again) we may assume that ℓ = limΦ(x ). If ℓ = +∞ there is
n
nothing to prove. If −∞ < ℓ < ∞, we proceed as folows. Given
ǫ > 0thereisn = n (ǫ)suchthatΦ(x ) ≤ ℓ+ǫ foralln ≥ n (ǫ).
0 0 n 0
Renaming the sequence we may assume that Φ(x ) ≤ ℓ + ǫ for
n
all n. Since x is the weak limit of (x ) it follows from Mazur’s
n
theorem [which is essentially the fact that the convex hull co(x )
n
of the sequence (x ) has weak closure coinciding with its strong
n
closure] thatthereexistsasequence
kN kN
yN = αNx , αN = 1, αN ≥ 0,
j j j j
Xj=1 Xj=1
suchthatyN → x asN → ∞. Byconvexity
0
kN
Φ(yN) ≤ αNΦ(x ) ≤ ℓ+ǫ
j j
Xj=1
and by the (strong) lower semicontinuity Φ(x ) ≤ ℓ + ǫ. Since
0
ǫ > 0 is arbitrary we get Φ(x ) ≤ ℓ. If ℓ = −∞, we proceed
0
