Project Materials

GENERAL

Maximal Monotone Operators on Hilbert Spaces and Applications

click here to get this project topic material with complete chapters 1-5 for just ₦3000 flat rate.

Do You Have New or Fresh ic? Send Us Your Topic

Maximal on

ABSTR

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 S

Abstract i
Acknowledgment ii
Dedication iii
Table of Contents v
Introduction vi
1 Hilbert s and Sobolev s 1
1.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Function s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Lp s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Maximal 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Bibliography 35

CHAPTER ONE

Hilbert s and Sobolev s

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

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. L2(
):
L2(
) := ff :
! R : f is measurable and
R

f2dx

Not What You Were Looking For? Send Us Your Topic

click here to get this project topic material with complete chapters 1-5 for just ₦3000 flat rate.

INSTRUCTIONS AFTER

After making payment, kindly send the following:
  • 1.Your Full name
  • 2. Your Active Email Address
  • 3. Your Phone Number
  • 4. Amount Paid
  • 5. ect ic
  • 6. Location you made payment from

» Send the above details to our email; contact@premiumresearchers.com or to our support phone number; (+234) 0813 2546 417 . As soon as details are sent and payment is confirmed, your project will be delivered to you within minutes.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Advertisements