Download Chapter 7 : Existence Theorems

Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Although the discussion focuses
on the Dirichlet problem , the
idea are widely applicable to the
study of PDE in general
 u  0 for 
u  f for 
2
Vector Space
S = set of objects , closed under addition and scalar multiplication
for u, v  S, a, b  R
Example
then au  bv  S
  x

S  R 2     : x, y  R 
  y


the vector space of real - valued
S  C ([ a, b])  continuous
functions defined on [a, b]

Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Inner Product
This operation associates with each pair of objects x and y
In the vector space, a real number denoted <x,y>. The inner
product is assumed to satisfy:
1) commutativ e :  x,y    y,x 
2) linear :  x  y, z     x,z     y,z 
3) For any nonzero vector x,  x,x  is positive
4) if 0 denotes the zero vector :
Example
x  x 
In R 2 ,   x, y     1 ,  2  
 y1   y2 
 0,0  0
 x1 x2  y1 y2
b
In, C ([a, b])  f,g   f(x)g(x)dx
a
f  x, g  1  x.a  0, b  1
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Orthogonal:
f and g are orthogonal if  f,g   0
Example
In R 2 ,
x  x 
 x, y     1  ,  2  
 y1   y2 
b
In, C ([a, b])  f,g   f(x)g(x)dx
a
 x1x2  y1 y2
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Norm
The norm of a vector x is a real number x
following properties:
with the
1) x  0 for all x  S
2) x  0 if and only if x is the zero vector.
3) cx  c x for any vector x and scalar c.
4) x  y  x  y
Example
In R 2 ,
In, C ([ a, b])
x  x12  x22
1
2
b
f    [ f(x)]2 dx 
a

In, C ()
1
2

f    [ f(x)]2 


Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Cauchy-Schwarz Inequality
 u,v 

u v
Example
In R 2 ,
In, C ([ a, b])
x  x12  x22
1
2
b
f    [ f(x)]2 dx 
a

In, C ()
1
2

f    [ f(x)]2 


Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Length of a vector
2
In R ,
x 
x  ( a, b)
x12

x
x22
Distance between 2 vectors
x y
x  y  (a1  a2 ) 2  (b1  b2 ) 2
y  (a2 , b2 )
x  (a1 , b1 )
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Norm induced by an inner product
x   x, x 
2
In R ,
In R 2 ,
x   x, x 
 x12  x22
f   f,f 
In C ([a, b]),

b
  f ( x) dx
a
2
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Convergent sequence
v1 , v2 , v3 ,V converges to v V If
We say the sequence :
lim vn  v  0
n
Example
In R 2 ,
x  x12  x22
In, C ([ a, b])
1
2

f    [ f(x)]2 dx 
a

b
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Cauchy sequence
We say the sequence :
v1, v2 , v3 ,V is cauchy If
lim vn  vm  0
n ,m
Example:
In R 2 ,
x  x12  x22
In, C ([ a, b])
1
2

f    [ f(x)]2 dx 
a

b
Remark:
Every convergent sequence in V is cauchy. (proof)
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Complete Space
A normed space is complete if every cauchy sequence in V
is convergent.
Hilbert Space
A compete inner product space ( with respect the norm
induced by the inner product) is called a Hilbert Space.
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Hilbert Space
A compete inner product space ( with respect the norm
induced by the inner product) is called a Hilbert Space.
Example
R n is a Hilbert space, because any cauchy sequence
of n - vectors converges to a vector in R n .
C ([0,1])
??? Hilbert space ???
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Hilbert Space
A compete inner product space ( with respect the norm
induced by the inner product) is called a Hilbert Space.
Example
L2 ([0,1])  the space of real - valued functions defined
1
on the interval [0,1] and such that

f 2 
0
1
Is a Hilbert space with the inner product .  f , g   fg
0
and norm
1
2
1 2
f    f 
0 
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Hilbert Space
A compete inner product space ( with respect the norm
induced by the inner product) is called a Hilbert Space.
Example
 compact set on R 2
L2 ()  the space of real - valued functions defined
on  and such that  f 2  
Is a Hilbert space with the inner product .  f , g   fg
and norm
f 


f

1
2 2
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Hilbert Space
A compete inner product space ( with respect the norm
induced by the inner product) is called a Hilbert Space.
Example
 compact set on R 2
L2 ()  the space of real - valued functions defined
on  and such that  f 2  
 f , g   fg
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
The completion of V
If V is an inner product space that is not complete, then V can
be embedded as a dense subset of a Hilbert space H.
This means that there is a Hilbert space such that V inside H.
We call such H the completion of V.
Example
  x

Q 2     : x, y are rational numbers 
  y

It is an inner product space with standard inner product.
It is not complete ( give an example)
The completion of
Q2
is
R2
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
The completion of V
If V is an inner product space that is not complete, then V can
be embedded as a dense subset of a Hilbert space H.
This means that there is a Hilbert space such that V inside H.
We call such H the completion of V.
Example
consisting all functions that are cont.,  f , g   fg  f x g x  f y g y
C ()  

with
cont.
first
partial
derivative
s


1


C01 ()  f : f  C1 (), f  0 on 
 f , g   fg  f x g x  f y g y
It is an inner product space. But It is not complete
The completion of
H 01 ()

 f :

C01 ()
is
H 01 ()

f f
2
f , ,  L (), and f  0 on 
x y

Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Linear functional
A linear functional on an inner product space V is a realvalued function  : V  R satisfying the linearity condition
 (x  y)  ( x)   ( y)
1
In L ([0,1]) ,  ( f )  0 f ( x) sin( x)dx
2
Example
Bounded Linear functional
A linear functional 
is bounded if there exist a positive M, such that
 ( x)  M x
Example
x V
1
In L2 ([0,1]) ,  ( f )  0 f ( x) sin( x)dx
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Bounded Linear functional and inner product
Bounded linear functionals on an inner product space V are
intimately tied to the inner product on V.
if y0 V , then
 ( x)  x, y0 
?
define a bounded linear functional in V
Is the other way true???
Riesz Representation Theorem:
If  is any bounded linear functional on a Hilbert space H, then there
is a unique y0  H such that
 ( x)  x, y0 
Riesz Representation Theorem:
Every bounded linear functional on a Hilbert space can be written as
an inner product with some fixed vector in the space.
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Bilinear Form
a function or functional of two variables that is linear with
respect to each variable when the other variable is held
fixed.
1) a(u,v1  v2 )  a(u, v1 )  a(u, v2 )
2) a(u1  u2 , v)  a(u1, v)  a(u2 , v)
Example
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Bounded Form or continuous
A bilinear form a(.,.) on a Hilbert space H is said to be
continouos, if there exists a positive constant C such that
a(u, v)  C u
H
v
H
u, v  H
Coercive Form
A bilinear form a(.,.) on a Hilbert space H is said to be
coercive, if there exists a positive constant  such that
a(u, u )   u
2
H
u  H
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Consider the following problem
Find u  H such that
a(u, v)  F (v) v  H
Variational
Problem
Where
H is a Hilbert space with (. , .) inner product
F () is a bounded linear functional on H ( F  H' )
a(,) bounded, coercive, bilinear form
We want to study this problem
(existence and uniquence)
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
The Lax-Milgram Theorem
Given:
a Hilbert space H,(.,.),
a continuous, coercive bilinear form a(.,.) and
a continuous linear functional F in H’,
There exists a unique u in H such that
a(u, v)  F (v) v  H
Chapter 7 : Existence Theorems
7.2 : A Hilbert Space Approach
Example
1) Bilinear form
2) Bounded (cont.)
3) Coercive