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## ABSTRACT

Let H be a real Hilbert space and A : D(A) H ! H be an unbounded, linear,

self-adjoint, and maximal monotone operator. The aim of this thesis is to solve

u0(t) + Au(t) = 0, when A is linear but not bounded. The classical theory of

differential linear systems cannot be applied here because the exponential formula

exp(tA) does not make sense, since A is not continuous. Here we assume A is

maximal monotone on a real Hilbert space, then we use the Yosida approximation

to solve. Also, we provide many results on regularity of solutions. To illustrate the

basic theory of the thesis, we propose to solve the heat equation in L2(

). In order

to do that, we use many important properties from Sobolev spaces, Green’s formula

and Lax-Milgram’s theorem.

** **

## TABLE OF CONTENTS

Abstract i

Acknowledgment ii

Dedication iii

Table of Contents v

Introduction vi

1 Hilbert Spaces and Sobolev Spaces 1

1.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Maximal Monotone Operators on Hilbert spaces 8

2.1 Examples of maximal monotone operators . . . . . . . . . . . . . . . 11

2.2 Yosida Approximation of a maximal monotone operator . . . . . . . . 14

2.3 Self adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

iv

Bibliography 35

## CHAPTER ONE

Hilbert Spaces and Sobolev Spaces

The aim of this chapter is to recall some results on Lp spaces, distributions and

Sobolev spaces that we use in the next chapter.

1.1 Hilbert spaces

A normed vector space is closed under vector addition and scalar multiplication.

The norm defined on such a space generalises the elementary concept of the length

of a vector. However, it is not always possible to obtain an analogue of the dot

product, namely

a:b = a1b1 + a2b2 + a3b3

which yields

jaj =

p

a:a

which is an important tool in many applications. Hence, the question arises whether

the dot product can be generalised to arbitrary vectors spaces. In fact, this can be

done and leads to inner product spaces and complete inner product spaces, called

Hilbert spaces.

Definition 1.1. Let H be a linear space. An inner product on H is a function

h:; :i : H H ! R

1

defined on H H with values in R such that the following conditions are satisfied.

For x; y; z 2 H; ; 2 R

a) hx; xi 0 and hx; xi = 0 if and only if x = 0

b) hx; yi = hy; xi

c) hx + y; zi = hx; zi + hy; zi

The pair (H; h:; :i) is called an inner product space. A Hilbert space, H is a complete

inner product space ( complete in the metric defined by the inner product ).

1.1.1 Examples

1. Euclidean space Rn.

The space Rn is a Hilbert space with inner product defined by

hx; yi =

Xn

i=0

xiyi

where,

x = (x1; x2; :::; xn) and y = (y1; y2; :::; yn)

We obtain

jjxjj =

p

hx; xi = (

Xn

i=0

x2i

)

1

2

2. Space L2(

):

L2(

) := ff :

! R : f is measurable and

R

f2dx < 1g, where

is an open

set in Rn; is a Hilbert space with the inner product defined

hf; gi =

Z

f(x)g(x)dx

and

jjfjj = (

Z

jf(x)jdx)

1

2

3. Hilbert sequence space l2.

l2 := f(xn)n0 R :

1P

i=0

jxij2 < 1g is a Hilbert space with inner product

defined by

hx; yi =

X1

i=0

xiyi

2

Convergence of this series follows from Cauchy-Schwar’z inequality and the fact that

x; y 2 l2, by assumption.

The norm is defined by

jjxjj = (

X1

i=0

jxij2)

1

2

An inner product on H defines a norm on H given by

jjxjj =

p

hx; xi

and a metric on H given by

d(x; y) = jjx yjj =

p

hx y; x yi

Hence, inner products are normed spaces and Hilbert spaces are Banach space.

A norm on an inner product space satisfies the important parallelogram equality

jjx + yjj2 + jjx yjj2 = 2(jjxjj2 + jjyjj2) for all x; y 2 H

Not all normed spaces are inner product spaces.

4. Space lp.

Let 1 p < 1 be a fixed real number, we define lp space as

lp = f(xn)n0 R :

X1

i=0

jxijp < 1g:

When p 6= 2, lp is not a Hilbert space.

5. Space C([a; b];R).

The space C([a; b];R) provided with supremum norm is not a Hilbert space.

Proposition 1.2. Let (H; h:; :i) be an inner product space. Then, for all x; y 2 H

a. jhx; yij jjxjjjjyjj (Schwar0z inequality) where the equality holds if and

only if x,y are linearly dependent.

b. jjx + yjj jjxjj + jjyjj (triangle inequality) where the equality holds if

and only if x=cy (c 0)

Proposition 1.3. (Continuity of inner product). Let (xn)n0; (yn)n0 be sequences

in H, such that xn ! x and yn ! y, then

hxn; yni ! hx; yi:

3

1.2 Function Spaces

Here, we recall the definitions of functions spaces used in this thesis.

1.2.1 Lp Spaces

Definition 1.4. Let

be a nonempty open set in Rn, for 1 p < 1, we define

Lp(

) := ff :

! R : f is measurable and

Z

jf(x)jpdx < 1g

Remark 1.5. We say two functions f and g are equivalent if f = g almost everywhere.

Then we define Lp(

) spaces as the equivalent classes for this relationship.

The space Lp(

) can be seen as a space of functions. We do however, need to be

careful sometimes. For example, saying that f 2 Lp(

) is continuous means that f

is equivalent to a continuous function. Now, for f 2 Lp(

), we define

jjfjjp = (

Z

jf(x)jpdx)

1

p ; 1 p < 1

The Lp(

) is a Banach space.

1.2.2 Test functions

Definition 1.6. Let f :

! R be a continuous function. The support is

supp(f) := fx 2

: f(x) 6= 0g

The function is said to be of compact support on

if the support is a compact set

contained inside

.

Definition 1.7. The space of test functions in

, denoted by D(

) is the space of

all C1 functions defined on

which have compact supports in

.

C1(

) denotes the space of all real-valued functions on

of class C1.

= (1; 2; :::; n) 2 Nn is called multi-index with length jj =

Pn

i=1

i.

Let x = (x1; x2; :::; xn) 2 Rn. We write D = @jj

@

1

x1 :::@n

xn

and it acts on the space

C1(

). Thus, for f 2 C1(

), Df = @jjf

@

1

x1 :::@n

xn

is it partial derivatives of order jj.

Definition 1.8. Let f ngn0 be a sequence in D(

) and 2 D(

).

n ! in D(

) if

1. 9 a compact set K

: supp( ); supp( n) K; for all n 1

2. D n ! D uniformly on K; 8 2 Nn:

4

1.2.3 Distributions

Definition 1.9. A distribution on

is any continuous linear mapping T : D(

) !

R. The set of all distributions is denoted by D0(

).

Remark 1.10. By linearity, to show that T is continuous, it is enough to show that,

if n ! 0 in D(

), then it is enough to show that (T; n) ! 0 in R:

Definition 1.11. A function f :

! R is locally integrable if for any compact set,

K

, we have that Z

K

jf(x))jdx < 1

The collection of all locally integrable functionals on

is denoted by L1l

oc(

)

If f 2 C(

), then f 2 L1l

oc(

). For any f 2 L1l

oc(

), f gives a distribution Tf defined

by

(Tf ; ) =

Z

f(x) (x)dx; for all 2 D(

)

Definition 1.12. If T 2 D0(

) is a distribution on an open set

Rn, and if

is any multi-index, we define the distribution DT by

(DT; ) = (1)jj(T;D ) (1.1)

and it is the th partial derivative of T.

So, the map D : D0(

) ! D0(

) defined in (1.1) is linear and continuous.

1.3 Sobolev spaces

Sobolev spaces are based on the concept of weak (distributional) derivatives. It gives

us a modern approach to the study of differential equations.

Definition 1.13. Let 1 p < 1 and k be a non-negative integer. Then, Sobolev

space Wk;p(

) is defined by

Wk;p(

) := fu 2 LP (

) : Du 2 Lp(

); 8 0 jj kg

The space is equipped with the norm

jjujjWk;p(

) := (

X

0jjk

jjDujjp

LP (

))

1

p

5

WK;p

0 (

) = D(

)

Wk;p(

) i.e., WK;p

0 (

) is the closure of D(

) with respect to the

norm jj:jjWk;p(

).

When p=2, we write Hk(

) = Wk;2(

) and Hk

0 (

) = Wk;2

0 (

) and these are real

Hilbert spaces with the following inner product

hu; viHk(

) =

X

0jjk

Z

DuDvdx

and the norm

jjujjHk(

) = (

X

0jjk

jjDujj2

L2(

))

1

2

For, k=0,

W0;p(

) = LP (

):

Wk;p(

) are Banach spaces.

Given that

is smooth, then:

Wk;p

0 (

) := fu 2 Wk;p(

) : u = Du = ::: = Dk1u = 0 on @

g:

For p=2, we have

Wk;2

0 (

) := fu 2 Wk;2(

) : u = Du = ::: = Dk1u = 0 on @

g

For p=2,and k=1 , we have

W1;2

0 (

) := fu 2 W1;2(

) = H1(

) : u = 0 on @

g

and we denote it by H1

0(

)

For p=2, k=2, we write

W2;2(

) = H2(

):

Theorem 1.14. Let

be smooth and u 2 L2(

) such that u 2 L2(

). Then

u 2 H2(

):

6

Green’s Formula

Theorem 1.15. Let

be bounded and smooth. Let u 2 H2(

) and v 2 H1(

),

then Z

ru:rvdx =

Z

@

v

@u

@n

ds

Z

vudx

where @u

@n denotes the normal derivative defined by @u

@n = ru:!n

:

where !n

denotes the normal vector.

if u = v, then

Z

jjrujj2dx =

Z

@

u

@u

@n

ds

Z

uuds

=

Z

@

u

@u

@n

ds +

Z

u(u)ds

Then,

Z

(u)udx =

Z

jjrujj2dx

Z

@

u

@u

@n

ds

Theorem 1.16. (Lax-Milgram). Let a : V V ! R be a bilinear, continuous,

and coercive functional. Then, for each f 2 V 9! u 2 V :

a(u; v) = (f; v); for all v 2 V

Proposition 1.17. (Poincaré’s inequality). Suppose

is a bounded set. Then

there exists a constant C(

) > 0 such that

jjujjL2(

) C(

)jjrujjL2(

); for all u 2 W1;2

0 (

):