Topological Methods in Semilinear Elliptic Boundary Value Problems · 2017-10-14 · Dirichlet...

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

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


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


0 onhas at least

(

solutions

for each .

)

g uAu uBu

λ λ

λ

>

= − Ω = ∂Ω

non - trivial three


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


0 onhas at least of

solutions for each .

( )

-

k

gA uBu

u

k

u λ

λ λ

= − Ω

>

= ∂Ωpairs non - trivial


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


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


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


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

qq

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


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

= = ⋅⋅⋅

= = ⋅

⇒

⋅⋅

⇒


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⊕ ⊕


( )( ) ( )

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

qq

= ∩ ∩

= = ≥

=

∈ 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


:

( , ) , \ ;

,

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


:2 Morse Index

( )

2

1


: 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


: 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+ −= −

-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


( )( )( ) ( )

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


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


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−

=

−

=

≤

≤

≤

−

−−

−

∑ ∑


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


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⊂


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


( ')( 1 H.2) . a x on≡ ∂Ω/

Let:w u v u Gg= − = −



( )

( )

( )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

= ∂



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


Fredholm Boundary Operator

( ) ( )( ) ( ') 1 ( ')BP a x P a xψ ψ ψ∂−

∂= +

n


( )( )(

( ') 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


( )( )( )

1, such that

mod ,

mod .

B L

AB I L

BA I L

−∞

−∞

⇒

∃ ∈ Ω

≡ Ω

≡ Ω

01/2


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⊂


( ) ,

:

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⊂


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


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


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

−


( ) ( ),f t t g tλ λ= − ∈R


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>


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

−

± −

>

= ±

=

⇒


( ),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


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


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


( ) ( )1 1,2( )D H WC H ⊂ Ω == Ω


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

≤ ⋅

+∃ ⋅

∀

−

∈

− ∇


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

≤ ∀ < =−


( ) 0 in jF u H∇ →

2 /( 2) ( )) ( ) 0( N Nj L

N u N u + Ω− →


2 /( 2)2 ( )( ) ( ) (

0

) N Nj jH L

j H

F

u

u N u N u

u

C + Ω≤

−

∇ −

→

+

Existence Theorem (1)

1


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


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


11


0 onhas at leas

(

t solution

0 for each .

)Au uBu

u

u

g

λ λ

λ= − Ω = ∂Ω

> >

positive one


12


0 onhas at leas

(

t solution

0 for each .

)Au uBu

u

u

g

λ λ

λ= − Ω = ∂Ω

< >

negative one


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 =


X V ⊥∩

0

V

2 ( )BX C α+= Ω


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


( )( )

( ) 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


( )such th! (

0

( )

)

( ) ( ( )

t

)

aw t X V

wA Q p t

h

wv tt

⊥

⇒

∃

+

=

∈ ∩

=


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


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


3

( ) has at least critical point t

tΨ another non - zero

1 2 3 2


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

= += += +


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


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 .


(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


Nonlinearity Conditions 1(A) ( ), .


(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

=

∈ →

= →

=

⇒

∫ ∫


( )

( )

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= −

≤ − +

∈ =

→

∫



2

(0)

, 0 such tha

su

dim

14

p ) 1

t

(u V S k

u

k

F

V

ρ

ρ

λ ρλ∈ ∩

≤ −

=

∃ >


1λ

λ

kλ

Topological Methods in Semilinear Elliptic Boundary Value Problems · 2017-10-14 · Dirichlet...

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Transcript of Topological Methods in Semilinear Elliptic Boundary Value Problems · 2017-10-14 · Dirichlet...