# Maximal Monotone Operators on Hilbert Spaces and Applications

## Maximal Monotone Operators on Hilbert Spaces and Applications

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
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

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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. 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

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