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Topological Methods in Semilinear EllipticBoundary Value Problems
著者 TAIRA Kazuaki内容記述 発表会議: 第 541 回 応用解析研究会
開催場所: 早稲田大学理工学部講演日時: 平成24年5月12日(土) 15:30~17:00講演題目:「位相的手法による半線形楕円型境界値問題」
year 2012URL http://hdl.handle.net/2241/00142367
Kazuaki TAIRA
Topological Methods in Semilinear Elliptic
Boundary Value Problems
Purpose • The purpose of this talk is to study a class of
semilinear degenerate elliptic boundary value problems in the framework of Sobolev spaces
which include as particular cases the Dirichlet and Robin problems.
• The approach here is based on the following: (1) Minimax Method (2) Morse theory (3) Ljusternik-Schnirelmann theory.
Semilinear Elliptic Boundary
Value Problems Minimax Methods
Morse Theory
Ljusternik and Schnirelmann Theory
References (Monographs)
• Ambrosetti and Prodi: A Primer of Nonlinear Analysis, Cambridge University Press, 1993
• K.C. Chang: Methods in Nonlinear Analysis, Springer-Verlag, 2005
• Ambrosetti and Malchiodi: Nonlinear Analysis and Semilinear Elliptic Problems, , Cambridge University Press, 2007
References (Papers) • Ambrosetti and Lupo: On a class of nonlinear
Dirichlet problems with multiple solutions, Nonlinear Analysis, 8 (1984), 1145-1150
• Thews: Multiple solutions for elliptic boundary value problems with odd nonlinearities, Math. Z. 163 (1978), 163-175
• Amann and Zehnder: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa 7 (1980), 539-603
References (Papers) • Berger and Podolak: On the solutions of
a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1975), 837-846
• Ambrosetti and Mancini: Sharp nonuniqueness results for some nonlinear problems, Nonlinear Analysis, 3 (1979), 635-645
• Amann and Hess: A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh 84A (1979), 145-151
My Works
• Taira: Degenerate Elliptic Boundary Value Problems with Asymmetric Nonlinearity, Journal of the Mathematical Society of Japan, 62 (2010), 431-465
• Taira: Semilinear Degenerate Elliptic Boundary Value Problems at Resonance, Annali dell’Universit`a di Ferrara, 56 (2010), 369-392
My Works
• Taira : Multiple Solutions of Semilinear Degenerate Elliptic Boundary Value Problems, Mathematische Nachrichten, 284 (2011), 105-123
• Taira: Multiple Solutions of Semilinear Degenerate Elliptic Boundary Value Problems II, Mathematische Nachrichten, 284 (2011), 1554-1556
My Works
Taira : Degenerate Elliptic Boundary Value Problems with Asymptotically Linear Nonlinearity, Rendiconti del Circolo Matematico di Palermo, 60 (2011), 283-308 Taira: Multiple Solutions of Semilinear Elliptic
Problems with Degenerate Boundary Conditions, Mediterranean Journal of Mathematics, 10 (2013), 731-752
My Works
Taira: Semilinear Degenerate Elliptic Boundary Value Problems via Critical Point Theory, Tsukuba Journal of Mathematics, 36 (2012), 311-365 Taira: Semilinear Degenerate Elliptic
Boundary Value Problems via Morse Theory, Journal of the Mathematical Society of Japan, 67 (2015), 339-382
My Works
Taira: Semilinear Degenerate Elliptic Boundary Value Problems via Critical Point Theory, Tsukuba Journal of Mathematics, 36 (2012), 311-365 Taira: Semilinear Degenerate Elliptic
Boundary Value Problems via Morse Theory, Journal of the Mathematical Society of Japan, 67 (2015), 339-382
Bounded Domain
2,N N ≥R
∂Ω
Ω
Typical Example
( )
( ') ( ') (1 ( ')
in ,
on .
(H.1) .
) 0
(H. ( ') 1 on
0 ( ') 1 o
2)
n
.
f uuBu x a x a x u
a x
a x
u−∆ = Ω∂
= + − =
∂Ω
≤ ≤ ∂Ω ∂
≡ ∂Ω/
n
Absorption Phenomenon (Dirichlet Condition)
Reflection Phenomenon (Neumann Condition)
Associate Pseudo-Differential Operator (Fredholm Boundary Operator)
( )( ') 1 ( ')Laplace-Beltrami Operato
( )( ', ') ( ') ' 1 ( ')
0 ( ') 1 on .
r
T BP a x
T x a x
a
a
a x
x
x= + −
≤
= = −Λ + −
Λ =
≤ ∂Ω
σ ξ ξ
Non-Degenerate (Elliptic) Case
( )
( ): ' 0 on
1 (
: ' 0
')( ')(
on
)
( '
'
)
a x
T BP a x I Ia x
a xT BP a x Ia x
≡ ∂Ω
= = −Λ + =
> ∂Ω
−= = −Λ +
Robin
Dirichlet
e
Case
Cas
Elliptic Differential Operator
0
20
, 1
(1) ( ) ( ), ( ) ( )
for all and 0 such that
(2) ( ) ( ) and ( ) 0
( ) , , .
on .
Nij N
i ji j
ij ij jia x C a x a x
x a
c x C c
x a
x
a xξ ξ ξ ξ=
∞
∞
∈ Ω =
∈Ω ∃ >
∈ Ω ≥
≥ ∀ ∈Ω ∀ ∈
Ω
∑ R
1 1( ) ( )
N Nij
i ji j
uAu a x c x ux x= =
∂ ∂= − + ∂ ∂ ∑ ∑
Degenerate Robin Condition
(1) ( ') ( ) and ( ') 0 on .(2) ( '
( '
) ( ) and ( ') 0 o
) ( ') ( ') 0 on
n .
.uBu
a x C a xb x C b
x a b x u
x
x
∞
∞
∈ ∂Ω ≥
∂=
∂Ω
∈ ∂Ω ≥
+ = ∂Ω∂
∂Ω
ν
Conormal Derivative (1)
1 11.( ')
N N
i
Nij
i jii ij
a x nx x
νν = = =
∂ ∂ ∂= = ∂ ∂ ∂
∑∑ ∑
1 2 is the unit ex( , , , ) terior normal. Nn n n= ⋅⋅⋅n
Conormal Derivative (2)
∂Ω n
Ω ν
Degenerate Boundary Conditions
on .
(H.1) .(H
( ') ( ') ( ') 0
( ') 0 o( ') ( ') 0 on
. n . 2)a x
uBu x a x b x u
bb
xx
∂= + =
∂
≡ ∂Ω/
+ ∂Ω
∂
>
Ων
Semilinear Degenerate Elliptic Boundary Value Problems
For a given function ,find a function ( ) in such that
in ,0 o
)
(n
(
.)
uf t
fu u
x
ABu
Ω
= Ω = ∂Ω
Nonlinearities
1. Superlinear Nonlinearity 2. Odd Nonlinearity
Example of Superlinear Nonlinearity
1
0( )
0
1, 1Here
p
q
s sf s
s
p
s
q
s −
≥=
>
>
<
Example of Odd Nonlinearity
1( )e1
Her
pf
p
s s s −
>
=
Nonlinearity Conditions (1)
( ) ( ),f s s g sλ λ= − ∈R
Nonlinearity Conditions (2)
1(A) ( ), .
(B) The limits '( ) satisfiesthe conditio
(
(0) '(0) 0
)'( ) li
s
m
n
s
g sg
C
s
g gg
g
→±∞
= =∈
±
∞
∞
± = = +∞
R
Example 1
1
0( )
0
1, 1Here
p
q
s sg s
s
p
s
q
s −
≥=
>
>
<
Remarks on Nonlinearity Conditions
1
2
20
1
1
(0) '(0) 0
2( ) (1
(F.1) ( ),(F.2) 0 such that
(F.3) 0 , 0 such
), 12
1/ 2
0 ( ) ( ) ( )
t
,
hat
p
t
f f
n
f C
f t c t pn
F t f s ds t f t t
c
c
c
θ
θ
= =
+≤ + < <
∈∃ >
< ∃−
<
< = ≤
∃ >
≥∫
R
Linear Operator A
2
2 2,2
2
(a) The domain ( )
We define a linear operator
as follows:
is a , self-adjoint ope
is the set( ) ( ) ( ) : 0.
(b) , ( )
r
: ( )
a
)
or
.
t
(
DD u H W Bu
u
L
A u D
L
u= ∈ Ω = Ω ==
→ Ω
∀ ∈
Ω
⇒
positive definiteA
A
A
A A
A
1
1
1 1 1
1
(1) The first eigenvalue is and .(2) The corresponding eigenfunction may be chosen in :
(3) No other e , 2,igenvalue
( )
,( )
s ha
0
ve
in
j
x
j
x
λ
φ
φ λφφ
λ
Ω
=> Ω
≥
positivealgebraically simple
strictly positve
A
positive eigenfunctions.
Spectral Properties of A
Remark (Neumann Case)
( )
1
on ' 1
0
b x
λ
∂Ω≡
=⇒
Ne n)uman(
References • Taira: Boundary Value Problems and
Markov Processes, Second Edition, Lecture Notes in Mathematics, Springer-Verlag, 2009
• Taira: Degenerate Elliptic Eigenvalue Problems with Indefinite Weights, Mediterranean Journal of Mathematics, 5 (2008), 133-162
The Case
1λ λ>
Existence Theorem 1
1 2 1
The semilinear problemin ,
0 onhas at least solutions
0, 0 for eac
( )
h .
gAu uBu
u u
u
λ
λ
λ
= − Ω = ∂Ω
> ><
non - trivial two
Outline of ( ) ( )f s s g sλ= −
1λ λ
1u
2u
Comment 1
1'( )'( ) '(
'( )) :
0f sf f
g sλ
λλ< <∞ −∞ =
−==
1
at least one eigenval'( ) of
crosses goes froif m 0
u to
e.
fs
sλ ∞A
The Case
2λ λ>
Existence Theorem 2
2
The semilinear problemin ,
0 onhas at least
(
solutions
for each .
)
g uAu uBu
λ λ
λ
>
= − Ω = ∂Ω
non - trivial three
Outline of ( ) ( )f s s g sλ= −
1λ λ2λ
3u
2u
1u
Comment 2
1 2'( ) ''
(0'(
)( ) ) :f s s
fg
f λ λ λλ
∞ = − < =< <∞= −
1 2
at least two eigenvaluecrosses '( ) of if goes from
s 0 to, .
f ssλ λ ∞A
Nonlinearity Conditions (3) 1(A) ( ), .
(B) The limits '( ) satisfiesthe conditions
(0) '(0) 0
( )'( ) lim
(C) ( ) ( ), .
s
g g
g s
g s s
g
g C
ss
g
g
→±∞
= =
±∞ =
∈
−
±
= +∞
= − ∀
∞
∈
R
R
Example 2
1( )e1
Her
pg
p
s s s −
>
=
The Case
kλ λ>
Existence Theorem 3
The semilinear problemin ,
0 onhas at least of
solutions for each .
( )
-
k
gA uBu
u
k
u λ
λ λ
= − Ω
>
= ∂Ωpairs non - trivial
Outline of ( ) ( )f s s g sλ= −
1λ λkλ
1u
ku
Comment 3
1 2''( ) '( )
) (0):
( 'kfs
ff g s
λ λ λ λλ
∞ = −∞ =< < ≤ ⋅⋅⋅ ≤ <= −
1
at least eigenvalcrosses goes from 0
'( ) of
ue to
s, , f .ik
fs
s kλ λ⋅⋅⋅ ∞A
References
• Amann: Saddle points and multiple solutions of differential equations, Math. Z. 169 (1979), 127-166
• Thews: A resduction method for some nonlinear Dirichlet, J. Nonlinear Analysis 3 (1979), 795-813
• Amann and Zehnder: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa 7 (1980), 539-603
Semilinear Degenerate Elliptic Boundary Value Problems
For a given function ,find a function ( ) in such that
in ,0 o
)
(n
(
.)
uf t
fu u
x
ABu
Ω
= Ω = ∂Ω
Nonlinearity Conditions A
1(A) ( ).(B) The limit '( ) not an eigenvalue of
s
if C
f∈
∞R
A.
Example A
1 2
1 2
2( )
'(
2
2
1
)
1f s ss
f
λ λ
λ λ+
∞ =+
++
=
Existence Theorem A
The semilinear problemin ,
0 onhas at least solu
( )
tion.
f uAuBu
= Ω = ∂Ω
one
Nonlinearity Conditions B 1
not an eigenvalue of
(A) ( ),
(B) The limit '( ) is
(C) of s
(0)
uch that
o
'(0) '( )
0
'( ) '( )r 0
j
j
j
f f
f f
f C f
f
λ
λ
λ
< < ∞
∈
∞
<
=
∞ <
∃
R
A.
A
Example B
1 2
2
2
1
1
( )
'
12 2
2
2
( )
'(0)
f s
f
ss
s
f
λ λ
λ
λ λ
= +
=+
∞ =
+
Existence Theorem B
The semilinear problemin ,
0 onhas at least - one trivial solution and
soluti
(
on.
)f uAuBu
= Ω = ∂Ω
two solutions
non - triviao l ne
Gradient
( )
1Let be a and ( , ).The Frechet derivative ( ) can be
Hilbert space
( )
expressed as follo
(
ws:
( ) : the of at
) ( ), ,H
df u v f u
H f C Hdf u
f u H
v H
f
v
u
=
∇
∇ ∀ ∈
∈
∈ gra
R
dient
Palais-Smale Condition
1
'
Hilbert space
, ( ) ,
Let be a and ( , ).(1) satisfies if
(2) satisfiesif it satisfies f
(PS) condition
(PS)condition(PS)
( ) 0
is con
o
vergent
er y .v er
j j j
j
c
c
x H f x f x
H f C Hf
c
c
x
f
⊂ → ∇ →
⇒∃
∈
∈R
R
Minimizing Method for Minimum Points
1Let be a and ( , ).(1) ( ) is(2) ( ) sat
Hilbert space
(PS) condition withinf ( )
such that( ) i
i
n(
s
)
fi
0f ( )
es c
x H
x H
H f C Hf
c f x
x Hf x c f
f x
x
x
f x
∈
∗
∗
∗
∈
∈
⇒
=
=
∃ ∈
=∇
=
bounded from b wR
elo
Ekeland Variational Principle
Let ( , ) be a complete metric space
:(1) ( ) is and lower semi-continuous.(2) such that
such that
(a) ( ) ( ).(b) ( , ) 1.(c
0,( ) inf ( )
)
.x X
y X
f y
x
X df X
f x
f x
Xf x f
x yf
x
d
ε
ε ε
ε ε
ε
ε
εε
∈
→ ∪ +∞
∃ ∈
≤≤
∃ > ∃ ∈< +
⇒
bounded from R
below
( ) ( ) ( , ), .x f y d y x x yε ε εε> − ∀ ≠
Nonlinearity Approach Studied by •Asymptotically Linear Case
•Linking Theorem
Rabinowitz (1978)
•Superlinear Case
•Morse Theory Ambrosetti and Lupo (1984)
•Odd Nonlinearity Case
•Ljusternik and Schnirelmann Theory
Castro and Lazer (1979)
Existence of Saddle Points
Marston Morse
Marston Morse (1892-1977) American Mathematician
References (Papers)
• Palais: Morse theory on Hilbert manifolds, Topology 2 (1963), 299-340
• Palais and Smale: A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165-172
• Smale: An infinite-dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861-866
• Marino and Prodi: Metodi perturbativi nella teoria di Morse, Boll. Un. Mat. Ital. 11 (1975), 1-32
Regular and Critical Points
1Let be a and ( , )(1) is calle
Hilbert space
( ) 0
( )
d a of if (2) is called a
0of
if
f u
f
H f C Hu f
fu
u∇ ≠
∇ =
∈regular point
critical point
R
Hessian
( )
2
2
2
2 2
Let be a and ( , ).The Frechet derivative ( ) of ( )can be expressed as follows:
Hilbert space
( )( , ) , , ,
( ) : : the of at
( )H
d f u v w
H f C HD f u f u
D f u H H f
v w v w HD f u
u
= ∀ ∈
∈
∇
→ Hes ian
R
s
Non-Degeneracy of Critical Points
2
2
Let be a and ( , )A critical point of is called
if the Hessi
Hilbe
an( ) :
rt space
s a
.ha
H f C Hu f
D f u H H
∈
→
non
bounde
- deg
d in
en
ve
era
R
t
rse
e
Morse’s Lemma (1)
2
: compact manifold(
finite dimensiona, )
: f
l
o ,
Mf C Mp f∈
non - degenerate critical pointR
Morse’s Lemma (2)
1 2
2 2 22
2
2 2
1
21
( , , , )
( ) ( )
Then:a local coordinate system
near such that
Here is the of at
n
n
y y y
f y f p y y yy y y
p
f p
λ
λ λ
λ
+ +
− − − ⋅⋅⋅
⋅ ⋅ ⋅
=
+ + + ⋅⋅
∃
⋅
−
+
Morse index
2 2( , )f x y x y= + minimal point( )
0 : Morse Index
2 2( , )g x y x y= − saddle point( )
1: Morse Index
2 2( , )h x y x y= − − maximal point( )
2 : Morse Index
Morse Index
0
1
2
*( ,0)C f
0 0G⊕ ⊕
0 0G⊕ ⊕
0 0 G⊕ ⊕
Critical Groups and Morse Indices
Splitting Theorem (1)
2
2
: : convex neighborhood of 0 in
( , )Assume that:(1) 0 is the of
(2) (0) is with
Ker
Hilbert space
HU Hf C U
D
A
f
N
f
A
∈
=
=
only critical point
Fredholm
R
Splitting Theorem (2)
( ) ( )
1
Then:(1) an open ball about 0 in (2) : (3) : a mapsuch tha
( ) ( ( ))
t
1 ,2
,
Hf y f
B U HB B
h B N N C
y B N B N
y hA yϕ ξ ξ ξ
ϕ
ξ
⊥
⊥
∃ ⊂∃ →
∃ ∩ →
∀ ∈ ∩ ∈ ∩
+ = + +
homeomorphism
Betti Numbers
*
: Abelian group( , ) : pair of topological spaces with
( , ; ) :
the -th of ( , )
( , ) rank ( , ; ), 0,1, 2,q q
GX Y X
X Y H X
YH X Y G
q X Y
Y G qβ
⊂
= = ⋅⋅⋅
relative sing
Betti number
ular homology group
Euler-Poincare Characteristic
0( , ) ( 1) ( ,
( , ) r
the of (
ank ( , ; ), 0
)
,1, 2,
, )
q q
q
X Y
X Y
X
q
Y
H
X
Y
X Y G
χ
β
β∞
=
= = ⋅⋅⋅
= −∑
Euler - Poincare characteristic
Non-Trivial Interval Theorem (1)
( )
1
1
:
( , ] :
: Hilbert space(
)
, )
(
( ) 0
a
K
f f a x H f x
Hf C H
x H x
a
f
−
= ∈ ∇
= −∞ = ∈
=
∈
≤
R
Non-Trivial Interval Theorem (2)
( )1
( , ; ) 0,
[ , ]
b aqH
f
f b
K
G a
b
f
a−
≠ <
∩
⇒
≠∅
Deformation Retract (1)
: topological spaceA continuous map : [0,1]is called a of i
( ,0) identity f
on
XX X
XXη
η
⋅
×
=
→deformation
Deformation Retract (2)
( , ) : pair of topological spaces with (1) A continuous map : is ca
lled
a if
on on
(2) is called
identity iden
a
ti
of
ty r i
X Yr X Y
YXi
Y
r
Y X
X
=
⊂→
deformation retract
deformation retraction
Strong Deformation Retract
( , ) identity on for all [0,1]( ,1
A deformation retract : is called a if
a : [0,1] such th
on
at
) t Y t
i r X
r X Y
X Xη
ηη⋅ =⋅ =
→
∃ ×
∈
→strong deformation retract
deformation
Excision Property
( , ; ) 0, 0,1, 2,
( ; ) ( ; ), 0,1,
is a f
2,
o
q
q q
H X Y G q
H X G H
Y
Y G q
X
= = ⋅⋅⋅
= = ⋅ ⋅
⇒
⇒
⋅
strong deformation retraction
Non-Critical Interval Theorem (1)
( )1
1Let be a Hilbert space and ( , )(1) satisfies for (2) Let be the set of all
(PS) conditi
[ ,
of
is a
on [ , ]
of
]
a b
c
H f C Hf
K f
f
f
c a b
K
f
a b−
∈
⇒
∅
∀
∩
∈
=
critical points
strong deformation retraction
R
( ,
; ) 0, 0,1, 2,
( ; ) ( ; ), 0,1, 2,
is a of
q
q
a b
bq
b a
a
f f
f f
f
H G q
G H qfH G
= = ⋅⋅⋅
= = ⋅
⇒
⋅⋅
⇒
strong deformation retraction
Non-Critical Interval Theorem (2)
1
:( , )
: of a neighborhood of such
Hilber
that
t spaceHf C Hz fU
K zz
U =
∈
∩
isolated, critical pointR
Critical Group (1)
( )( )
( ) 1
( , ) is called
(
(
)( , ] : ( )
a of a
, ;
, )
t
\
q
q
c
c cq
c f zf f c x H f x
C f z H
C
f U f z
fc
z f
U G
z
−
= ∩ ∩
=
= −∞ = ∈ ≤
critical group
Critical Group (2)
Morse Index
0
1
2
*( ,0)C f
0 0G⊕ ⊕
0 0G⊕ ⊕
0 0 G⊕ ⊕
Critical Groups and Morse Indices
( )( ) ( )
1
: Hilbert space( , )
: o
( , ) , \ ;
;0 if
if
f
10
c c
q q
q
C f z H f U f z U G
Hf C Hz f
GH z G
= ∩ ∩
= = ≥
=
∈ Risolated, local minimum
Example 1 (Minimum Point)
y x
z
Minimum Point
2 2( , )f x y x y= +
-2 -1 1 2
-2
-1
1
2
x
y
1 2( , ) : ( , ) 1f x y f x y= ∈ ≤R
2B
( )( ) ( )
* *
1 1 2*
2
*
(1) ( ,0) 0 ; 0 0
(2) , ;
( , ) : ( ,
Her
0 0
)
e
;
a
C f H G
H f f
G
G
f x y f y
G G
x
H
a
B−
= = ⊕ ⊕
= =
= ∈
⊕ ⊕
≤R
Critical Groups and Homology Groups
: Morse Index0
( )( )( )
1
1
: Hilbert space( , )
:
( , ) , \ ;
,
of
;0 if
f
ij
c cq q
qj
Hf C Hz f
G q j
C f z H f U f z U G
H Gq
B Sj
− =
= ∩ ∩
= = ≠
∈
loR
isolated, cal maximum
Example 2 (Maximum Point)
y x
z
Maximum Point
2 2( , )g x y x y= − −
-2 -1 1 2
-2
-1
1
2
x
y
1 2( , ) : ( , ) 1g x y g x y= ∈ ≤R
2B
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
1 2( , ) : ( , ) 1g x y g x y− = ∈ ≤ −R
1S
( )( ) ( )
2 1* *
1 1 2 1* *
2
(1) ( ,0) , ; 0 0
(2) , ;
( , ) : ( ,
Her
, ; 0 0
e
)a
C g H B S G
H g g G
G
G
g x y g x
B
y
H
a
S G−
= ∈
= = ⊕ ⊕
= = ⊕ ⊕
≤R
Critical Groups and Homology Groups
:2 Morse Index
( )
2
1
: Hilbert space( , )
: critical point of Morse index
( , ) ,
with
;
0
iiff
j jq qC f
Hf C
z H B S G
qG
Hz f
j
j
q j
−
∈
=
=
= ≠
non - degenerateR
,
Example 3 (Saddle Point)
xyz
Saddle Point
2 2( , )h x y x y= −
1 2( , ) : ( , ) 1h x y h x y= ∈ ≤R
1B
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
1 2( , ) : ( , ) 1h x y h x y− = ∈ ≤ −R
0S
( )( ) ( )
1 0* *
1 1 1 0* *
2
(1) ( ,0) , ; 0 0
(2) , ;
( , ) : ( ,
Her
, ; 0 0
e
)a
C h H B S G
H h h G
G
G
h x y h x
B
y
H
a
S G−
= ∈
= = ⊕ ⊕
= = ⊕ ⊕
≤R
Critical Groups and Homology Groups
: Morse Index1
*( ,0) 0 0C f G= ⊕ ⊕
Minimum Point
: Morse Index0
*( ,0) 0 0C h G= ⊕ ⊕
Saddle Point
: Morse Index1
*( ,0) 0 0C g G= ⊕ ⊕
Maximum Point
:2 Morse Index
2
2 2
( , )0 : critical point of with Morse
1 1( )2 2
index
, dim
H Hf
j
H H H H j
x x
C
x
f Hf
+ − −
+ −
∈
⇒
=
= =
−
⊕
non - degeneR
rate,
Splitting Theorem
2 21 1( )2 2H H
f x x x+ −= −
jB
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
1jS −
( )1
Morse inde0 : critical point o
x
if ( , ) ,
f wi
t
;0 if
h
j jq q
j
G qC f z H B S G
q j
f
j− = = ≠
=
⇒
non - degen , erate
Critical Groups and Morse Indices
( )( )( ) ( )
1 0*
2 1*
1 11
(1) , ; 0
(2) , ; 0 0
(3
( 1)
( 2)
if
) , ; ;
0 if
( 3)
j j jq q
H B S G
H B S G
H B S G
G j
G j
G q j
H S G
q jj
− −−
= ⊕
= ⊕ ⊕
≅
≠
=
=
== ≥
Relative Homology Groups
2 1 0 1 2
11 2
1
... ...
( )
a Hilbert space( , )
of :
of :
, ,...,i
i i ii m
c c c c c
f c K
Hf C H
f
f
z z z
− −
−
∈
< < < < < <
∩ =
isolated, critical values
isolated, critical p s
R
oint
Morse Type Number (1)
1 1
( , ) rank ( , ; )
0
( , ) a of
of on
with
min ,
( , )
i i i i
i
c cq q
a c b
i i i i i
M a b H
a b fa
f a b
b
f f G
c c c c
ε ε
ε
+ −
< <
+ −
=
< < − −
<
∑
pair of regular values
Morse type number
Morse Type Number (2)
1
11
2
2
( , ) satisfies
( , ) a
( , ) rank
of
( , )
( ) , ,...,
i
i
i
mi
q q ja c b j
i i ii m
f C Hfa b f
M a b C f z
f c K z z z
< < =
−
=
∩ =
∈
∑ ∑
(PS)conditionpair of regular v
R
alue
s
Morse Type Numbers and Critical Groups
2
( , ) the number of critical po
(PS)condi
ints
of
tion(
in ( , ) with
, ) satisfies
qM a bf
f C
b q
Hf
a
∈
=
Morse index
R
Morse Type Numbers and Critical Points
2
: Hilbert space( , )
satisfiesAll critical points of are
Hf C Hf
f
∈(PS)condit
non - degeneio
Rn
rate
Fundamental Assumptions
( , ) rank ( ,
( , )
of ( ,
;
a of with
)
)b aq
b
q
a
a
f
b
a b fa
H f
b
G
f
fβ
<
=
pair of regular values
Betti number
Betti Numbers
( ) 1 ( , ] : ( )af f a x H f x a−= −∞ = ∈ ≤
2 ( , )(1) satisfies(2) is (3) has isolat
,ed
(4) has critical pointsof
f C Hffff
∈(PS)condition
bounded from belowloca
non - del minima
positive Morg
senerat
e i
R
endex
Morse Inequalities (1)
0
(Betti number)the number of isolated the number of ,
c
( ) rank (
ritical points
( )
of
;
i n
)
( )
bk k
mb
b H f G
Cf
C
m
bb
β==
=local minima
non - degenerateMorse index
Morse Inequalities (2)
0
1 0
0
0
1 0
0
( )
( ) ( )
(
( )
( ) ( )
( 1)1) ( ) ( )k
k mm
kk m
mmm
C b
C b C b
C
b
b
b b
b
β
β β
β−
=
−
=
≤
≤
≤
−
−−
−
∑ ∑
Morse Inequalities (3)
Four-Solution Theorem (1)
2
0
1
2 ( , )(1) satisfies(2) is (3) is a critical point
with
(4) has
2
0
,
f C Hf
q
u
f
f u
≥
∈
non
(PS)conditionbounde
- dd feg
R
tw
rom below
Morse index
local minima
enerate
o
Four-Solution Theorem (2)
3
has at least critical point
u
f another non - zero
Proof (1)
1 2
1 2 has only max ( ), ( ),
, 0(0)
,ff u f u f
u ub >
criReduction to Absurdity
tical points three
2( )f u
1( )f u
(0)f
: ( )bf x H f x b= ∈ ≤
Proof (2)
0
0
0 1
2 if 0
( )
1 if
0 if 1,
( ) 2, ( ) 0
2q
q
C b
q
q q q
b
q
C b C
=
=
= ≥⇒
≥ ≠
= =
Proof (3)
0 1
1 if
( ) rank ( ; )
ra0
( ) 1,
nk ( ; )
)
0 f
(
i 1
0
q
q
bq f
qH
b
b H G
H G
b
q
β
β
β
=
= =
=
≥=
⇒
=
Proof (4)
1 0
1 0( ) ( )( ) ( )
21
Cb
b C bbβ β
≤ −= −= −
−
Contradiction!
References (Papers)
• Schwartz: Generalizing the Lusternik-Schnirelman theory of critical points, Comm. Pure Appl. Math. 17 (1964), 307-315
• Palais: Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132
• Clark: A variant of Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972), 65-74
: real Hilbert space(1) A is said to be
with respesubset
ma
ct to 0 if
(2) A : is said to bep
( ) (
if
),
n
HH
Au A u A
f x f x x
A
A
f∈ ⇒ − ∈
− = − ∀
⊂
∈
→
symm
R
etric
odd
Symmetric Sets and Odd Maps
Krasnoselskii Genus (1)
( )the least integer such that( )
, \ 0n
A n
C A R
γ
φ ∈
=
∃ odd map
Krasnoselskii Genus (2)
( )the least integer such tha( )
,
( ) ,
t
0
n
A n
C H
x
R
x Aψ
γ
ψ
≠ ∀
∃
∈
∈
=
odd map
Tietze’s Extension Theorem
Let be a metric space and a closed subset.Let be a locally convex topologicallinear space and : a continuous map.Then there exists a
of .:
F X L
X AL
f A L
f
→
→
continuous extension map
Fundamental Properties
(1) ( ) ( )(2) ( ) ( ) ( )(3
( ) #( (
) 0,[ ] , ([ ]) 1
(
( ))4)
(5) ) 1n
A B A BA B A B
p
A
p p
m
S
m
p p
A
n
γ γ
γ γγ γ
γ
γ
γ⊂ ⇒ ≤∪ ≤ +
≠ =
=
= ⇒
⇒ =
+
≥
−
(Monotonicity)(Subadditivity)
(Normality)
Borsuk-Ulam Theorem
0
0
Let be a symmetric, bounded open subset of including the origin, with boundary .Let : be a continuous and
for .Then there exists a point such
( )
th
0
at
n
mg
m n
g
x
x
Ω
∂
<
Ω
∂Ω→
∈∂Ω
=
R odd mR
ap
Multiplicity Theorem 1
)
1
1
(
: real Hilbert space( , ),
(1)
( ) ( ),
( ) ( )
(PS)condit
(2)
( ) inf sup ( ), 1, 2,
( )
:
satisfies
, () 0
o
( )
i n
n
k k m
A n x A
c
c
f x f x x H
c c f c
c f f x n
K m
K x
f
f x
Hf C H
H fc
f
x
γ
γ
≥
+
∈
+
=
− = ∀ ∈
= = ⋅⋅
=
∈
⋅⋅⋅
≥
= ∈ ∇
⇒
⋅ = < ∞
= =
R
Multiplicity Theorem 2-1 (Analytic Version)
1Let be a , ( , )and (1) (0) and
(2) satis
( )
fie
(
Hilbert space
s
),
H f C
f x f
Ha b
f b
f
x x H
∈<>
− = ∀ ∈
(PS)condi
R
tion
Multiplicity Theorem 2-2 (Analytic Version)
(0)
Assume the following:
(i) , , 0 such that
(ii) , such that
(iii
s
) Then ( ) has at least of distinct critical
dim
dim
p
up ( )
inf ( )
( - ) points
ai.
rs
x E S
x F
E H
F H
f x
E m
f x b
f x a
m jm
F j
j
ρ
ρ
⊥
∈ ∩
∈
≤
>
>
∃ ⊂
=∃ ⊂
= ∃ >
Approach2L
Linear Boundary Value Problems
1 1( ) ( )
N Nij
i ji j
uAu a x c x ux x= =
∂ ∂= − + ∂ ∂ ∑ ∑
( ') ( ') ( ') 0 on .uBu x a x b x u∂= + =
∂∂Ω
ν
Linear Operator A
2
2 2,2
2
(a) The domain ( )
We define a linear operator
as follows:
is a , self-adjoint ope
is the set( ) ( ) ( ) : 0.
(b) , ( )
rato
: ( )
r
)
.
(
DD u H W Bu
u u u D
L L
= ∈
⇒
Ω = Ω == −∆ ∀ ∈
Ω → Ω
positive definite
A
A
A
A
A
A
1
1
1 1 1
1
(1) The first eigenvalue is positive and .(2) The corresponding eigenfunction may be chosen in :
(3) No other eigenva , 2,lues have
( )
,( ) 0 in
j
x
j
x
λ
φ
φ λφ
λ
φ=> Ω
≥
Ω
algebraically simple
strictly positve
A
positive eigenfunctions.
Spectral Properties of A
Ordered Vector Space
def
(i) ( , ) .(ii) (
(a) , , , .(b) , , ,
iii
0.
)
x y V x y x z y z z Vx y V x
V
VV
y x yα α α∈ ≤ ⇒ + ≤ + ∀ ∈∈ ≤ ⇒ ≤ ∀ ≥
⇔≤
≤
is an
is an ordered setis a real vector space.
linear
ordered vector spa
The ordering is
e
:
c
Ordered Banach Space
def
(i) (ii) ( , ) ( :iii) ,(a) , , , 0.(b) ( )
:
.
.
0
0P
E
EE
x y P x y PP P
x E xα β α β
⇔
≤
∈ ⇒ + ∈ ∀ ≥
∩ − =
= ∈ ≥
is an
is a Banach space.is an ordered vector space.
ispositive cone,
ordered Banach space
closed
Krein-Rutman Theorem (1)
( ) ,
:
E P
K E E→
Let be an ordered Banach space with non - empty interior
strongly p, and assume that
is andositive c ompact.
( ) ( )\ 0 IntK P P⊂
Krein-Rutman Theorem (2)
1/n(i) = lim 0, ( )
(ii)
:
n
nr K
r K
rr
E EK ∗
→∞
∗
>
→
positive eigenfunction
spectral radius
is a ofhaving a .
is .is also an algebraically sim
algebraically simple ple
eigenvalue of the adjoin
unique eigenv
t
alue
∗
with a positive eigenfunction.
Typical Example
( ') ( ') (1 ( ')) 0
( ') 1
in ,
on .
(H.1) .
(H.2
0 ( ') 1
) .o
on
n
uBu x a x a x u
a x
g
a x
u−∆ = Ω ∂
= + − =∂
≡ ∂Ω/
∂Ω≤ ≤ ∂Ω
n
( )( ') 1 ( ')
in ,
0 onux
u
a a x u
g−∆∂
+ − = ∂
= Ω
∂Ω
Consider the boundary value pro
n
blem
Reduction to the Boundary (1)
( )
( )
2,
,
1 ( ') 0
: !
( ),s p
s p
v a x v
v Gg
v H
W
g −
∂
−∆ = ∈ Ω
+ − =∂
= ∃ ∈ Ω
Solve the
W
Robin problem
e letn
Reduction to the Boundary (2)
( ')( 1 H.2) . a x on≡ ∂Ω/
Let:w u v u Gg= − = −
Reduction to the Boundary (3)
Reduction to the Boundary (4)
( )
( )
( )2
( ') 1 ( ')
1 ( ')
B Ggva x a x v
a x v
∂= + −
∂
= −
n
Gain of One Derivative
( )2
( )
1 ( ')
Bw Bu BvB Gg
a x v
= −= −
= − −
( )2
Then
,
1 (
0 in
in ,0
' n
on
) o
w u v
Bw a x
u gBu
v
⇔
=
−∆ = −∆
−∆
+ ∆ =
− − ∂Ω
= ΩΩ
Ω
= ∂
Reduction to the Boundary (5)
Reduction to the Boundary (7)
0 inw
w Pψ
−∆ = Ω
=⇔
(Poisson operator)
( )( )2
in ,
on
o
1 ( ')
0 n
BP
a
u gBu
v
B
x
wψ
⇔
=
= −
−∆ = Ω = ∂Ω
− ∂Ω
Reduction to the Boundary (8)
Fredholm Boundary Operator
( ) ( )( ) ( ') 1 ( ')BP a x P a xψ ψ ψ∂−
∂= +
n
Associate Pseudo-Differential Operator (Fredholm Boundary Operator)
( )( )(
( ') 1 ( '
.
', ') ( ') ' 1 ( '
)
0 ( ') 1 on
)T x
T BP a x a x
a
x a x
x
a=
= = −Λ + −
−
≤ ≤ ∂Ω
+σ ξ ξ
( )1,0
Let ( , ) be a properly supported pseudo-differential operator in the class
m
A p x D
L
=
Ω
Criteria for Parametrices (1)
( ) ( ) ( ) ( )
( ), ,
1
Assume that:
, , 1
, , , .
x K
K K
D D p x C p x
p x C x K C
α βα βξ α βξ ξ ξ
ξ ξ
− +
−
≤ ∃ +
≤ ∃ ∀ ∈ ⊂ Ω ∀ ≥
1/2
Criteria for Parametrices (2)
( )( )( )
1, such that
mod ,
mod .
B L
AB I L
BA I L
−∞
−∞
⇒
∃ ∈ Ω
≡ Ω
≡ Ω
01/2
Criteria for Parametrices (3)
1
1,1/
1,0
02
(
suc
') 1 ( ') ( )
( ) h
( )
that
mod
T a
S L
TS ST I L
x a x L
−∞
∃ ∈ ∂
= −Λ + − ∈ ∂
≡
Ω
⇒
Ω
≡ ∂Ω
Loss of One Derivative
Elementary Lemma
( )
( )
2( ) ,( ) 0 on ,
sup ''( )
'( ) 2 ( ) on .
x
f x Cf x
f x c
f x c f x
∈
∈
≥
≤ ∃
⇒
≤ 1/2
R
RR
R
Ordered Vector Space
def
(i) ( , ) .(ii) (
(a) , , , .(b) , , ,
iii
0.
)
x y V x y x z y z z Vx y V x
V
VV
y x yα α α∈ ≤ ⇒ + ≤ + ∀ ∈∈ ≤ ⇒ ≤ ∀ ≥
⇔≤
≤
is an
is an ordered setis a real vector space.
linear
ordered vector spa
The ordering is
e
:
c
Ordered Banach Space
def
(i) (ii) ( , ) ( :iii) ,(a) , , , 0.(b) ( )
:
.
.
0
0P
E
EE
x y P x y PP P
x E xα β α β
⇔
≤
∈ ⇒ + ∈ ∀ ≥
∩ − =
= ∈ ≥
is an
is a Banach space.is an ordered vector space.
ispositive cone,
ordered Banach space
closed
Example
def
( ),
( ) ( ),
Y C
u v u x v x x
= Ω
≤ ⇔ ≤ ∀ ∈Ω
Int(
( ) : ,
( ) : 0
0
)Y
Y
P
u C
u C
P
u
u= ∈ Ω
= ∈ Ω
≥ Ω
> Ω
on
on
( ) ( , ) ( )
( ) 0
( , )
) 0
0
(
k x y
Ku x k x y u y dy
u x Ku x
Ω=
⇔≥ ⇒
>
>
∫
strongly positive
Strong Positivity
( ) ( )\ 0 IntK P P⊂
Krein-Rutman Theorem (1)
( ) ,
:
E P
K E E→
Let be an ordered Banach space with non - empty interior
strongly p, and assume that
is andositive c ompact.
( ) ( )\ 0 IntK P P⊂
Krein-Rutman Theorem (2)
1/n(i) = lim 0, ( )
(ii)
:
n
nr K
r K
rr
E EK ∗
→∞
∗
>
→
positive eigenfunction
spectral radius
is a ofhaving a .
is .is also an algebraically sim
algebraically simple ple
eigenvalue of the adjoin
unique eigenv
t
alue
∗
with a positive eigenfunction.
( )( ', ') ( ') ' (( ') ( ')
Laplace-Beltram')
i OperatorT x a x b x
T BP a x b x= +
= = −Λ +
Λ =
σ ξ ξ
Associate Pseudo-Differential Operator (Fredholm Boundary Operator)
11
01,1
,
2
0
/
( ') ( ') ( )
( ') ( ') 0
( ) such t
on
hat
mod
( )
S L
TS ST I L
T a x b x L
a x b x
−∞
= −Λ + ∈ ∂Ω
+ > ∂Ω
∃ ∈ ∂Ω
≡ ≡ ∂Ω
⇒
, ,loc loc
, ,
1,
loc loc
Every properly supported operator( ), ,
extends to a continuous linear operator,1
and to a continuous: ( ) ( ),
linea
0 1
: ( ) ( ),r operator
,1
s p s m p
s p s
m
m pA B
A H H
A L
B
s p
s p
δ δ
−
−
Ω →
∈ Ω
∀ ∈ <∀ < ∞
∀
≤ <
Ω → Ω ∞
Ω
∈ ≤∀ ≤
R
R
Besov-Space Boundedness Theorem
Fractional Power of A
,= square roo oft C A A
1/2 1
0
1 ( )
( )
u s sI u ds
u Dπ
∞ − −= − +
∈
∫C A A
A
Function Space (1)
, of= square root C A A
the domain ( )with the inner product ( , ) ( , ).H
Du v u
Hv
==
CC C
Function Space (2)
, 1
0
the of the domain ( ) with respect to the
( , ) ( )
( ')( )( ')
Nij
Hi j i j
a
D
u vu v a x dxx xb xc x u vdx u v da x
H
σ
Ω=
Ω ≠
=
∂ ∂= ⋅
∂ ∂
+ ⋅ + ⋅
∑ ∫
∫ ∫
inner produc completion
t A
1
1
1
2
/
2
3 2
(
( ) ( )
( )
( )
( ) ( )
)C
C
C
D
D C HL
D C
D D
C D
−
−
−
Ω →
↑ ↑
→
↑ ↑
=
→
=
A A
A
Function Space (3)
( ) ( )1 1,2)( ) (D C HD H W⊂ Ω ==⊂ ΩA
Function Spaces (4)
( ) ( )( ) ( )
10
1
Observe the following fact (due to D. Fujiwara):
if on
if on
' 0
' 1
a xH
a xHH
Ω ∂Ω= Ω ∂Ω ≡
≡ ( ),
(Robin).
Dirichlet
Semilinear Elliptic Boundary Value Problems
For a given function ,find a function ( ) in such that
in ,0 o
)
(n
(
.)
up t
pu u
x
ABu
Ω
= Ω = ∂Ω
Definition of a Weak Solution
, 1
0
is a
( , )
( ) ( )
( ')( )
( )
( )'
0
H
Nij
i j i j
a
u v v dx
u va x dx c x u v
u H
p u
p u
dxx x
b xv dx u v da x
v H
σ
Ω
Ω Ω=
Ω ≠
⇔
−
∂ ∂= ⋅ + ⋅
∂ ∂
− + ⋅
= ∀ ∈
∈
∫
∑ ∫ ∫
∫ ∫
weak solution
Semilinear Elliptic Boundary Value Problems
For a given function ,find a function ( ) in such tha
( )
(
t
in ,0 n .
)o
u x
uBu
g t
u g uλ
Ω
−∆ = Ω = ∂Ω
−
Nonlinearity Conditions (1)
( ) ( ),f t t g tλ λ= − ∈R
Nonlinearity Conditions (2)
1(A) ( ), .
(B) The limits '( ) satisfiesthe conditio
(
(0) '(0) 0
)'( ) li
s
m
n
s
g sg
C
s
g gg
g
→±∞
= =∈
±
∞
∞
± = = +∞
R
The Case
1λ λ>
Nonlinearity Conditions (3)
0 s( )
uc( )
h that0
s ss s s g sg λλ
− +
+ + − −
∃ < <
−≤ ≤
∃
−
Outline of ( )p s
s+s−s
0
Example
1
11
( ) , 1p
p
g s ps
s
s
λ
−
± −
>
= ±
=
⇒
Nonlinearity Conditions (4)
( ),0
0,( )
, ss g s s s s
s
s
sp s λ
+
−
−
+
> <
< >
≤= ≤ −
( ) on
'( ) on
p s L
p s L
≤
≤
R
R
Modified Semilinear Problem
For a given function ,find a function ( ) in such that
in ,0 o
)
(n
(
.)
up t
pu u
x
ABu
Ω
= Ω = ∂Ω
( ) ( )
0
( ) ( )
( ) (
,
)
12 H
s
F u P u xu u dx
P s p t dt
Ω= −
=
∫
∫
Energy Functional
( ) ( )
( ) ( )( )
1
1
1
( )
( ( ))
, , ( )
,
( ( ))
( ) ( ( )
,
,
)
H H
H H
H
F u
u p u
u p u
F u u
v u v p u v dx
v
u
v
p
v
Ω
−
−
−
= −
= −
=
∇
−
∇ =
⇔
−
∫A
A
A
Gradient
1
( *) 0
* ( ( *))
* ( *)
F u
u p u
u p u
−
∇
⇔
=
=
=
⇔A
A
Critical Points (Euler-Lagrange)
* ( *) in ,* 0 o
* is a of
* is a of the prob
n .
em
l
F
Au p uB
u
u
u= Ω
= ∂Ω
⇔
critical point
weak solution
Critical Points and Weak Solutions
Regularity Theorem (1)
( )( )
( )
2,
, 1/
0
, 1
p
p
s p
s
s
u L
u
Au W
p
Bu
W
−
∈ Ω⇒ ∀∈ Ω >
=
+
∈ Ω
Regularity Theorem (2)
2
( )
0
( )
( )
pu L
u C
Au CBu
α
α+
∈ Ω=
∈ Ω
⇒
∈ Ω
( )( ', ') ( ') ' (( ') ( ')
Laplace-Beltram')
i OperatorT x a x b x
T BP a x b x= +
= = −Λ +
Λ =
σ ξ ξ
Associate Pseudo-Differential Operator (Fredholm Boundary Operator)
11
01,1
,
2
0
/
( ') ( ') ( )
( ') ( ') 0
( ) such t
on
hat
mod
( )
S L
TS ST I L
T a x b x L
a x b x
−∞
= −Λ + ∈ ∂Ω
+ > ∂Ω
∃ ∈ ∂Ω
≡ ≡ ∂Ω
⇒
, ,loc loc
, ,
1,
loc loc
Every properly supported operator
( ), ,extends to a continuous linear operator
,1and to a continuous
: ( ) ( ), linea
0 1
: ( ) ( ),r operator
,1
s p s m p
s p s
m
m pA B
A H H
A L
B
s p
s p
δ δ
−
−
Ω →
∈ Ω
∀ ∈ <∀ < ∞
∀
≤ <
Ω → Ω ∞
Ω
∈ ≤∀ ≤
R
R
Besov-Space Boundedness Theorem
Sobolev Imbedding Theorems
( )2, ( )1 if 2
if 22
q r
r q nnqr
Lq q n
n qW ∗Ω ⊂ Ω
∀ ≥ ≥ = = < −
Bootstrap Argument
Maximum Principle
( )
( ) in
( ) (
in ,0
)
on
( )
p u
s u x s
p u u g u f u
AuBu
λ
− +
= Ω = ∂Ω⇒
⇒
≤ ≤ Ω
= − =
Truncation
0,,
0( )
,( )
s s
s sp s s g s s s sλ
+
−
− +
= − ≤ ≤
>
> <
<
Outline of ( )p s
s+s−s
0
Original Semilinear Problem
in ,)on .
(0
AuB
u g uu
λ −= Ω = ∂Ω
Minimizing Method 1
*
( ) is
inf ( )
( , ).
(1)
(2) ( )
suc
sat
h that(
isfies
*) inf (
wit
0
h
( *))u H
u H
u HF u c
F C H
F u
F u
c u
u
F uF
F
∈
∈
∃ ∈
=
∇ ==
∈
⇒
=
c
bounded from below
(PS) condition
R
( ) ( )2
1
1( ) ,
,2
( )2 H
F u u u P u x dx
Lu H
λ
Ω
Ω≥ −
=
∀
−
∈
∫
Lower Bound for Energy Functional
( ) on
'( ) on
p s L
p s L
≤
≤
R
R
Remark (Neumann Case)
( )
1
on ' 1
0
a x
λ
∂Ω≡
=
⇒
Ne n)uman(
Palais-Smale Condition (1)
such that
( ) in
( ) n
0 i j
j
j
F u cF u
u H
H→
∇ →
⊂
R
* 21
in ( )
22
qju L
NqN
u
≤
→
∀ < =
∃
−
Ω
Palais-Smale Condition (2)
( ) ( )1 1,2( )D H WC H ⊂ Ω == Ω
Palais-Smale Condition (3)
2 /( 2) ( )2
(
,
(
)
(
( )
) ) N N
j H
j
H
HL
j H F uv v
C vp
v H
p
u
u
u
u + Ω
≤ ⋅
+∃ ⋅
∀
−
∈
− ∇
Palais-Smale Condition (4)
2 /( 2) ( )2( ) ( ) ( ) N N
j H
j jH LF u p u p u
u u
C + Ω
⇒
≤ +∇ −
−
Riesz Representation Theorem
: ( ) ( ( ))N u x p u x
Nemytskii Operator
* 21
in ( )
22
qju
Nu
qN
L
≤ ∀ =
→
<−
Ω
/
(i) ( , ) is for almost every (ii) x ( , ) is for
(
all s
(iii) ( , ) , ( , )
: ( , (
) ) )) (
q p
q p
s f x sx
f x s
f x s a b s x s
F u f xL Lu x
∈Ω∈
≤ ∃ + ∃ ∀ ∈Ω×
⇒
∈ Ω Ω∈
continuous
measurabl
conti
R
R
n ous
e
u
Continuity of Nemytskii Operator (1)
2 /( 2)( ) (: ) )(q N NL LN u P u +∈ ∈Ω Ωcontinuous
Continuity of Nemytskii Operator (2)
21 2*2
NqN
≤ ∀ < =−
Palais-Smale Condition (5)
( ) 0 in jF u H∇ →
2 /( 2) ( )) ( ) 0( N Nj L
N u N u + Ω− →
Palais-Smale Condition (6)
2 /( 2)2 ( )( ) ( ) (
0
) N Nj jH L
j H
F
u
u N u N u
u
C + Ω≤
−
∇ −
→
+
Existence Theorem (1)
1
The semilinear problemin ,
0 onhas at least
(
solutionfor each .
)Au uB
gu
u
λ λ
λ=
>
− Ω = ∂Ω
non - trivial one
Another Truncation
( ) ma
( ) ( )
x
(
(
)
),0
p t p t
p t p t
p t
+
− += −
=
( ) :p t± Lipschitz Continuous
Outline of ( )p t±
( )p t+
( )p t−
t0
Modified Semilinear Problems
For given functions ,find a function ( ) in such
(that
in ,0 on .
)
( )
u x
AuB
p uu
p t±
±
Ω
= Ω
= ∂Ω
( ) ( )
0
( ) ( )
( ) (
1 ,2
)s
HF u P u x dx
P s p t d
u u
t
± ±
Ω
± ±=
= − ∫
∫
New Energy Functionals
Existence Theorem (2)
11
The semilinear problem
in ,0 on
has at least solution for each .
( )A p uuBu
u λ λ
+ = Ω
= ∂Ω
> non - trivial one
Maximum Principle
1
1
1
1 1 1
1
1
( ) 0
0 ( ) in
( ) ( ) (
in ,0 n
)
oAuB
p u
u x s
p u u u
u
g u fλ
+
+
+
= Ω
= ∂Ω⇒
⇒
≥
≤ ≤ Ω
= − =
Outline of ( )p s
s+1u
s0
Outline of ( )p s
s−
2u
s0
Existence Theorem (3)
11
The semilinear problemin ,
0 onhas at leas
(
t solution
0 for each .
)Au uBu
u
u
g
λ λ
λ= − Ω = ∂Ω
> >
positive one
Existence Theorem (4)
12
The semilinear problemin ,
0 onhas at leas
(
t solution
0 for each .
)Au uBu
u
u
g
λ λ
λ= − Ω = ∂Ω
< >
negative one
Outline of ( ) ( )f s s g sλ= −
1λ λ
1u
2u
The Case
2 1λ λ λ> >
1 2 spa
'(
n , , ,
dim
) ,n
n
p s s
V
V n
φ φ
λ
φ
< ∃
⋅
∀
= ⋅⋅
∈
=
R
( ) on
'( ) on
p s L
p s L
≤
≤
R
R
1 2 span , , , nV φ φ φ= ⋅⋅⋅
V ⊥2 ( )L Ω
Orthogonal Decomposition (1)
0
( )1
,n
j jj
Qu u u φ φ=
= −∑
( )Y Cα= Ω
Orthogonal Decomposition (2) Y V ⊥∩
0 1 2 span , , , nV φ φ φ= ⋅⋅⋅
V
2
2
( )
( ) : 0B
B
X C
u C Bu
α
α
+
+
= Ω
= ∈ Ω =
Orthogonal Decomposition (3)
X V ⊥∩
0
V
2 ( )BX C α+= Ω
Orthogonal Decomposition (4)
X V ⊥∩
0
( ) ( )u v t w t= +
1( )
n
j jj
v t t φ=
=∑
X V ⊥∩
0
1( )
n
j jj
w t u t φ=
= −∑( ) ( )ttu v w= +
V
Orthogonal Decomposition (5)
( )( )
( ) in ,0 on
(
( )
( )(
)( ) ( )
( )
( ) ( )() ( ) )
w tw t w
Au p uBu
uA Q p
A
v tv t t
wI Qv v tt tp
= Ω= ∂Ω
⇔
= +
= +−= +
( )( ) ( ( ) ( ))
( ( ) ( )) (
( ) in ,
, (
) , 1
0 on
)
j j j
Aw t Q p v t w t w t
Au p u hB
p v t
u
X V
w t x dx t j nφ λ
⊥
Ω
= = Ω = ∂Ω
= + ∈ ∩
+ = ≤
⇔
≤∫
Lyapunov-Schmidt Procedure
( )( ) ( ( ) ( )) , ( )Aw t Q p v t w t w t X V ⊥= + ∈ ∩
Infinite-dimensional Equation (1)
( )
1
1 2
Her
( )
( )
(
: ( )
( , )
e
(
, , ,
)
)
n
n
j jj
nn
t v t
v t t V
t
A Q p
X V Y V
w w
t t t
w
φ=
⊥ ⊥
= ∈
= ⋅⋅⋅
Φ × ∩ → ∩
− +
∈
∑
R
R
Global Inversion Theorem metric space
metric space:
:
(1)( ) 2
MNF M N
M
F M N
→
⇒ →
arcwise connectedsimply connected
properlocally invertib
ho
le on all o
meomor
f
phism
( )
( )!
( ): ( )( , ) ( )
s( ) uch ( ) ( ( ))
t( )
tha
n
t v t
v t
X V Y VA Q p Qh
A Q p
w w w
w t Xw t t Q
Vw h
⊥
⊥ ⊥Φ
∃ ∈ ∩
− +
× ∩ → ∩
Φ = =
⇔
−
=
+
R
Infinite-dimensional Equation (2)
( )such th! (
0
( )
)
( ) ( ( )
t
)
aw t X V
wA Q p t
h
wv tt
⊥
⇒
∃
+
=
∈ ∩
=
Infinite-dimensional Equation (3)
Finite-dimensional Equation
( )
1
( )
( )
(1
( ) ()
( )
( ( ))
)
kjk
jj k
n
w t
w t
v tA I Q p v t
t
j n
p x dxtλ φ φΩ
=
= − +
⇔
= +
≤ ≤
∑∫
( )
( )2
1
1( ) ,2
1 (
( ) (
)2
)
( )
H
N
j jj
t
t P v t
t t
x
w w
w t dλΩ
=
Ψ =
+ − +∑ ∫
Energy Functional
( )( )
1 )
(
(
) ( )j j jj
w tt p v t x dx
jt
n
λ φΩ
≤
∂Ψ= +
∂
≤
− ∫
Gradient
( )
( )
( ) 0
( ) ( )
( ) (
( )
( ) ( ( )))
j j j
t
t p v t xw t
A I Q p w t
dx
v t v t
λ φΩ
∇Ψ =
= +
+
⇔
⇔
= −
∫
Critical Points
( )2
(
(1 , )
0) j iji j
i j nt t
λ λ δ∂ Ψ= −
∂ ∂
≤ ≤
Hessian (1)
2 1λ λ λ> >
, 3k kλ λ≠ ∀ ≥
Non-Degenerate Case
2 1λ λ λ> >
1
2
2
(0)
0 0 00 0 00 0 00 0 0
k
N
i jt t
λ λ
λ λλ λ
λ λ
−−
−−
∂ Ψ ∂ ∂ =
Hessian (2)
Hessian (3)
0
is a critical point
with
2
0
q ≥ Morse in
non - degener
x
te
de
a
Four-Solution Theorem (1)
1 2
0
2 ( , )(1) ( ) satisfies(2) ( ) is (3) is a critical point
with
(4) ( )
0
has
2
,
nC
t
t t
q
t
tΨ∈
ΨΨ
≥
Ψ
no
(PS)conditionbounden - de
d from below
Morse index
local minima
R R
tw
generate
o
Four-Solution Theorem (2)
3
( ) has at least critical point t
tΨ another non - zero
1 2 3 2
The semilinear problemin ,
0 onhas at least solutions
for eac
( )
, , h , , 3 k
g u
u u u k
Au uBu
λ
λ
λ λ λ
= − Ω
> ≠ ∀
= ∂Ω
≥
non -three trivial
Non-Degenerate Case
1 1 1
2 2 2
3 3 3
( ) ( )( ) ( )( ) ( )
u v t w tu v t w tu v t w t
= += += +
Outline of ( ) ( )f s s g sλ= −
1λ λ2λ
3u
2u
1u
, 3k kλ λ= ∃ ≥
Degenerate Case
2 1λ λ λ> >
1
2
2
(0)
0 0 00 0 00 0 00 0 0 N
i jt t
λ λλ
λ
λ
λ
−
−
∂ Ψ ∂ ∂ =
−0
Hessian (4)
Proof (1)
2
2
1
1, ,( ) has only max ( ), ( ), (0
0)
t tb
tt t>
Ψ
Ψ Ψ Ψ
crReduction to Absurdity
itical pointhree t s
Resolution of Critical Points (1)
0
2 ( , )(1) satisfies(2) is an isolated ( )critical p
oint
ffx
C H∈(PS)conditi
degenerate
Ron
Resolution of Critical Points (2) 2
0
0
2 2
0, such that(a) satisfies(b)(c) has a of critical points in
(d
g ( , )
( ) ( ) for
( ) () )
C Hgg x f x x xg
x x
D g x D f x
ε
ε
ε
ε
∈
= − ≥
− <
− <
∀ > ∃(PS)condition
R
non - degefinit nerae n e teumb r
( )tθ
02ε ε
t
1
Proof (1)
Proof (2)
( )
( )2 2 21 2
0
1
0
( )
( )
( )
( , )n
n j jj
g xx x y
x x x
f x
f x
x
x y
θ
θ=
= −
= − + ⋅⋅⋅
=
+ ∑
2
2
( ) 0,2
( ) :
( ) ,2
( ) :
: of in 2
g x x
D g x
f x y x
D f x
y f x
ε
ε
ε
∇
⇔
=
∇ = ≤
≤
⇔
≤
∇
singular
singular
critical value
Proof (3)
Proof (4)
( ) has a of
in the open b
ll ,
a
g
x
x
ε<
⇒
non - degenerate critical poin
Sard's Lemma
finite numbts
er
( )
( )2
1
1( ) ( ), ( )2
1 ( ) ( )2
H
N
j jj
t w t w t
t P v t w t dxλΩ
=
Ψ =
+ − +∑ ∫
Energy Functional
0
1 2
2 ( , )(1) ( ) satisfies(2) ( ) is (3) is a critical point
with
(4) ( )
0
,
2
has
nCt
t
q
t t
t
Ψ∈ΨΨ
Ψ
≥
(PS)conditionbounded from below
R R
two
Morse index
local minim
dege
a
nerate
2
1 2
1 2
such that:(1) satisfies(2) has only
in the closed set
(3) has
( , )(
only a finite number of critical p
) ( )
(
,
ts
,
in)
,
o
nCtt
t
t t
t
β β
ε
β
Ψ∈
Ψ
Ψ
Ψ
⋅
∃
≥
⋅⋅
R R
two critical points
non - degenerate
(PS)condition
2with
in the open s
et t ε
≥
<
Morse index
2( )tΨ
1( )tΨ
( )jβΨ
: ( )b nt t bΨ = ∈ Ψ ≤R
Proof (5)
1 0
1 0( ) ( )( ) ( )
21
Cb
b C bbβ β
≤ −= −= −
−
Contradiction!
Proof (6)
3
( ) has at least critical point t
tΨ another non - zero
2
2
1
1, ,( ) has only max ( ), ( ), (0
0)
t tb
tt t>
Ψ
Ψ Ψ Ψ
crReduction to Absurdity
itical pointhree t s
Degenerate Case
1 2 3 2
The semilinear problemin ,
0 onhas at leas
( )
,
t solutions
for each ,
g u
u u u
Au uBu
λ λ
λ= − Ω Ω
>
= ∂
non - trivial three
1 1 1
2 2 2
3 3 3
( ) ( )( ) ( )( ) ( )
u v t w tu v t w tu v t w t
= += += +
Odd Nonlinearity Conditions 1(A) ( ), (0) '(0) 0 .
(B) The limits '( ) satisfiesthe conditions
(C) ( ) (
( )'( l
, ) .
) ims
g C
g sgs
g g
g
g s g s s
→±∞±∞ =
∈
= +∞
= =
−
∞
= − ∀
±
∈
R
R
Outline of ( )p s
s+
s s− += −s
0
( ) ( )
0
( ) ( )
( ) (
,
)
12 H
s
F u P u xu u dx
P s p t dt
Ω= −
=
∫
∫
Energy Functional
( ) ( )
( )2
2
( )
0
1
1( ) , ( )21 ( )2
,2
H
u x
H
F u u u P u x dx
u p t dt d
Lu H
x
λ
Ω
Ω
Ω≥ −
=
∈
−
= −
∀
∫
∫ ∫
Lower Bound for Energy Functional
Multiplicity Theorem (1)
1( , )
(1)
(2) satis
(
fies
) ( ),
F C H
F
F u F u u H∀
∈
− = ∈
(PS)con
R
dition
Multiplicity Theorem (2)
2
(0)
Assume the following:(i) , 0 such that
(ii) ( ) is Then ( ) has at least
di
o
1sup ( )
m
f.
14
-
u V S k
V k
F u
k pairu sF u
F
ρ
λ
ρ
ρλ∈ ∩
≤ −
∃ >=
bounded from belo
distinct critical points
w
( )
( )
2
2
2 2
( )
0
( )
0
12
121 ( )2 2
( )
(
(
( )
)
)
u x
H
u x
H
H
P u x dx
t g t d
u
u
u u x
t dx
g t dtdx
F u
dxλ
λ
Ω
Ω
Ω Ω− +
−
−
=
=
=
−
∫
∫ ∫
∫ ∫∫
Behavior of Energy Functional (1)
Finite-Dimensional Linear Space
1 2 k span , , ,
dim
k
V
V k
φ φ
λ λ
φ= ⋅ ⋅
⇒
⋅
=
>
on the a e .rV
finite - dimenAll nor sionamsequiva
ll
spa eent
c
Behavior of Energy Functional (2)
Nonlinearity Conditions 1(A) ( ), .
(B) The limits '( ) satisfiesthe conditions
(0) '(0) 0
( )'( ) lim
(C) ( ) ( ), .
s
g g
g s
g s s
g
g C
ss
g
g
→±∞
= =
±∞ =
∈
−
±
= +∞
= − ∀
∞
∈
R
R
( ) 2
0( ) ( )
as and 0
( ) ( ) as 0
u x
H
g t dt dx o
u V
g t o t t
u
ρ
ρΩ
=
∈ →
= →
=
⇒
∫ ∫
Behavior of Energy Functional (3)
( )
( )
2
2 2
1( )2
1 12
( )
as and 0
H
k
H
P u x dx
V u
F
o
u u
u
λ ρ ρλ
ρ
Ω= −
≤ − +
∈ =
→
∫
Behavior of Energy Functional (4)
Behavior of Energy Functional (5)
2
(0)
, 0 such tha
su
dim
14
p ) 1
t
(u V S k
u
k
F
V
ρ
ρ
λ ρλ∈ ∩
≤ −
=
∃ >
Outline of ( ) ( )f s s g sλ= −
1λ
λ
kλ