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 = ::: = Dk􀀀1u = 0 on @
g:
For p=2, we have
Wk;2
0 (
) := fu 2 Wk;2(
) : u = Du = ::: = Dk􀀀1u = 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 (
):

 

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