This section introduces some standard notations and preliminary materials, which
are important in the set-up of the thesis. All the functions considered in the thesis are
real-valued. Let Ω be a convex bounded domain in R^{d}. We denote the boundary of Ω
by Γ :=∂Ω. Let µ= (µ_{1}, . . . , µ_{d}) be a non negative integer with d-components and the
order of µ is denoted by |µ| =µ_{1}+. . .+µ_{d}. Further, let x = (x_{1}, x_{2}, . . . , x_{d}) ∈Ω and
dx=dx_{1}, . . . , dx_{d}. Then, theµ-th derivative of a function ψ is denoted byD^{µ}ψ with

D^{µ}ψ := ∂^{|µ|}ψ

∂x^{µ}_{1}^{1}. . . ∂x^{µ}_{d}^{d}.

We now recall some standard function spaces. Let L^{p}(Ω), 1 ≤ p < ∞, be the linear
space of equivalence classes of measurable functions ψ in Ω such that the measure of
R

Ω|ψ|^{p}dx exists and finite. Moreover, the associated norm on L^{p}(Ω) with 1 ≤ p < ∞,
is

kψk_{L}^{p}_{(Ω)} :=Z

Ω

|ψ|^{p}dx^{1}_{p}
.

Forp=∞,L^{∞}(Ω) is a Banach space of all Lebesgue measurable and essentially bounded
functions, equipped with the norm

kψk_{L}^{∞}_{(Ω)} :=esssup

x∈Ω

|ψ|<∞.

Again, for p= 2, L^{2}(Ω) is a Hilbert space equipped with the inner product
(ψ1, ψ2) =

Z

Ω

ψ1ψ2dx, ∀ψ1, ψ2 ∈L^{2}(Ω).

We denote the support of a functionψ bysupp(ψ) which is a closure of all pointsxwith ψ(x)6= 0, i.e.,

supp(ψ) :=

x: ψ(x)6= 0 .

LetC^{k}(Ω) be the collection of all functions with continuous derivatives upto and includ-
ing order k in Ω, where k is a non-negative integer. Further, let C_{0}^{k}(Ω) be the space
of all C^{k}(Ω) functions with compact support in Ω. Moreover, the space C_{0}^{∞}(Ω) consists
of all infinitely differentiable functions with compact support in Ω. For a non-negative
integerk, letW^{k,p}(Ω) (1≤p <∞) be the collection of all equivalence class of functions
inL^{p}(Ω) such that the distributional derivatives upto orderkare also belongs toL^{p}(Ω),
i.e.,

W^{k,p}(Ω) :=

ψ ∈L^{p}(Ω) : D^{µ}ψ ∈L^{p}(Ω) for 0≤ |µ| ≤k .

CHAPTER 1. Introduction 9
The space W^{k,p}(Ω) is known as the Sobolev space of order (k, p) on Ω and is endowed
with the norm

kψk_{k,p} :=Z

Ω

X

0≤|µ|≤k

|D^{µ}ψ|^{p}dx^{1}_{p}

, 1≤p <∞.

For p=∞, we define the norm on W^{k,∞}(Ω) by
kψkk,∞:= max

0≤|µ|≤kkD^{µ}ψk_{L}^{∞}_{(Ω)}.

Further, we denote the semi-norm on W^{k,p}(Ω) by | · |_{W}k,p(Ω) which is defined as

|ψ|_{W}^{k,p}_{(Ω)} := X

|µ|=k

kD^{µ}ψkL^{p}(Ω).

When p= 2, we denote W^{k,p}(Ω) =H^{k}(Ω). Moreover, if k = 0 and p= 2 then we have
W^{0,2}(Ω) =H^{0}(Ω) =L^{2}(Ω). Let H^{k}(Ω) be the Hilbert space with inner product

(ψ_{1}, ψ_{2})_{k} := X

0≤|µ|≤k

Z

Ω

D^{µ}ψ_{1}D^{µ}ψ_{2}dx, ψ_{1}, ψ_{2} ∈H^{k}(Ω).

Observe that the Sobolev space H^{k}(Ω) and H_{0}^{k}(Ω), respectively, are defined to be the
closure ofC^{∞}(Ω) andC_{0}^{∞}(Ω) with respect to the normk·k_{k}and semi-norm|·|_{k}. Further,
we shall denote the space H^{−1}(Ω) as the dual of H_{0}^{1}(Ω) with the norm

kψ_{1}k_{H}^{−1}_{(Ω)}:= sup

ψ2∈H_{0}^{1}(Ω), ψ26=0

(ψ_{1}, ψ_{2})
kψ_{2}k_{H}^{1}_{(Ω)}.

For 1 ≤ p ≤ ∞, we now define the standard Bochner spaces L^{p}(0, T;B), where B is a
real Banach space equipped with the norm k · kB, consisting of all measurable functions
ψ : [0, T]→ B for which

kψk_{L}^{p}(0,T;B) := Z T
0

kψ(t)k^{p}_{B}dt^{1}_{p}

<∞ for 1≤p <∞,
kψk_{L}^{∞}(0,T;B) := ess sup

t∈(0,T)

kψ(t)kB <∞ for p=∞.

In the succeeding chapters, we use the following spaces. For a given Banach space B,
the space H^{1}(0, T;B) consisting of all the measurable functions ψ : (0, T) → B, such
that

kψk_{H}^{1}_{(0,T};B) :=

Z T 0

n

kψ(t)k^{2}_{B}+

∂ψ

∂t(t)

2 B

o dt

^{1}_{2}

<∞.

CHAPTER 1. Introduction 10 Furthermore, let C(0, T;B) be the space of continuous functions ψ : [0, T] → B and the norm on C(0, T;B) is defined as kψkC(0,T;B) := maxt∈[0,T]kψ(t)kB <∞. For a more detailed discussion on Sobolev spaces, one may refer to Adams and Fournier [1], and Grisvard [36].

Time to time we need the following inequalities for our analysis. For a proof, see Hardy et al. [37].

Young’s inequality. For all non-negative real numbers a, b, and 1 < p < ∞, q > 1
with ^{1}_{p} + ^{1}_{q} = 1, the following inequality

ab ≤ a^{p}
p + b^{q}

q holds.

Young’s inequality with -form. For all non-negative real numbers a,b and every >0, the following inequality

ab≤ a^{2}

2 + b^{2}
2
holds.

Discrete version of H¨older’s inequality. Let ai, bi, i= 1, . . . , d be the positive real numbers. Then

d

X

i=1

|aibi| ≤X^{d}

i=1

|ai|^{p}^{1}_{p}X^{d}

i=1

|bi|^{q}^{1}_{q}
,

where p >1 and ^{1}_{p} + ^{1}_{q} = 1. In particular, for p= 2 and q = 2, the above inequality is
known as the Cauchy-Schwarz inequality in R^{d}.

Integral form of the H¨older’s inequality. Let 1 < p, q < ∞ with ^{1}_{p} + ^{1}_{q} = 1. For
real valued functions ψ1 ∈L^{p}(Ω) and ψ2 ∈L^{q}(Ω), we have

Z

Ω

|ψ_{1}ψ_{2}|dx≤ kψ_{1}k_{L}^{p}_{(Ω)}kψ_{2}k_{L}^{q}_{(Ω)}.

Forp=q= 2, the above inequality is known as the Cauchy-Schwarz inequality.

The following lemma is proved to be convenient for later use.

CHAPTER 1. Introduction 11
Lemma 1.2.1 ([48]). Givenaˆ= (a0,a1, . . . ,ad)∈R^{d+1},bˆ= (b0,b1, . . . ,bd)∈R^{d+1} and
ˆ

c ∈R such that

|ˆa|^{2} ≤ˆc^{2}+ ˆa·ˆb,
then we have the following

|ˆa| ≤ |ˆb|+|ˆc|,

where | · | and ˆa·ˆb denote the standard Euclidean vector norm and inner product on
R^{d+1}, respectively.

In this thesis, we shall use the Poincar´e inequality which is stated below.

Lemma 1.2.2 (Poincar´e inequality). Assume that the Ω ⊂ R^{d} is a convex bounded
domain. Then there exists a constant C >0 such that

kψk_{L}^{2}_{(Ω)} ≤Ck∇ψk_{L}^{2}_{(Ω)}, f or every ψ ∈H_{0}^{1}(Ω),
where the constant C depends on the domain Ω.

The above inequality implies that, for any ψ ∈ H_{0}^{1}(Ω), the L^{2}-norm is bounded by
the semi H^{1}-norm.

We now borrow some useful definitions from F. Tr¨oltzsch [96].

Definition 1.2.1 (Directional derivative). Assume that X˜ and Y˜ are two Banach
spaces. Let X˜_{0} be a non-empty open subset of X˜ and f be a given map from X˜_{0} to
Y˜. If the limit

δf(x,v) := lim

h→0

f(x+hv)−f(x)

h for x∈X˜_{0} and v∈X,˜

exists in Y˜, then it is called the directional derivative of f at x in the direction of v.

If this limit exists for all v ∈ X˜_{0}, then the mapping v 7→ δf(x,v) is termed the first
variation of f at x.

Definition 1.2.2 (Gˆateaux derivative). Suppose that the first variation at x ∈ X˜_{0}
exists, and suppose there exists a continuous linear operator T : ˜X_{0} →Y˜ such that

δf(x,v) = Tv ∀v∈X˜_{0}.

Then f is said to be Gˆateaux differentiable at x∈X˜_{0}, and T is referred as the Gˆateaux
derivative of f at x. We write T =f^{0}(x).

CHAPTER 1. Introduction 12
Definition 1.2.3 (Fr´echet derivative). Assume that X˜ and Y˜ are two Banach spaces,
and X˜0 is a non-empty open subset of X. A mapping˜ f : ˜X0 →Y˜ is said to be Fr´echet
differentiable atx∈X˜_{0} if there exists an operator T ∈ L( ˜X,Y˜) and a mappingR(x,·) :
X˜_{0} →Y˜ with the following properties: for all v∈X˜ such that x+v∈X˜_{0}, we have

f(x+v) = f(x) +Tv+R(x,v), where the remainder R satisfies the condition

kR(x,v)kY˜

kvkX˜

→0 as kvkX˜ →0.

Then operator T is called the Fr´echet derivative of f at x, and we write T = f^{0}(x).

If T is Fr´echet differentiable at every point of x ∈ X˜_{0}, then T is said to be Fr´echet
differentiable in X˜_{0}.

Definition 1.2.4 (Convex functional). LetX˜_{0} be a non-empty convex subset of R^{d}(d=
2, 3). A functional F from X˜_{0} to R is said to be convex if it satisfies the following
property: For all ν∈[0,1] and x_{1},x_{2} ∈X˜_{0} such that

F(νx_{1}+ (1−ν)x_{2})≤νF(x_{1}) + (1−ν)F(x_{2}).

A functional F is said to be strictly convex, if it satisfies the above condition with strict
inequality whenever x_{1} 6=x_{2}, ν∈(0,1).

Before introducing the auxiliary problems, we first collect some function spaces which will be useful in our subsequent analysis. For simplicity, we set

W1(0, T) := L^{2}(0, T;H^{2}(Ω)∩H_{0}^{1}(Ω))∩H^{1}(0, T;L^{2}(Ω)),
W_{2}(0, T) := L^{2}(0, T;H_{0}^{1}(Ω))∩H^{1}(0, T;H^{−1}(Ω)).

We observe that W_{1}(0, T),→ C(0, T;H_{0}^{1}(Ω)) and W_{2}(0, T),→ C(0, T;L^{2}(Ω)) (cf., [55]).

We now introduce some auxiliary problems. For any f ∈L^{2}(0, T;L^{2}(Ω)), let Ψ, Φ∈
W_{1}(0, T) be solutions of the following forward and backward in time standard parabolic
problems

Ψ_{t}+AΨ =f in Ω_{T},
Ψ(·,0) = 0 in Ω,

Ψ = 0 on Γ_{T},

(1.17)

CHAPTER 1. Introduction 13 and

−Φ_{t}+A^{∗}Φ =f in Ω_{T},
Φ(·, T) = 0 in Ω,

Φ = 0 on Γ_{T},

(1.18)

respectively.

The solutions Ψ and Φ satisfy the following regularity results [55].

Lemma 1.2.3. For a given function f ∈ L^{2}(0, T;L^{2}(Ω)), let Ψ, Φ ∈ W1(0, T) be the
solutions of the problems (1.17) and (1.18), respectively. Then there exists a positive
generic constant C_{R} such that

kΨ_{t}k_{L}^{2}_{(0,T}_{;L}^{2}_{(Ω))}+kΨk_{L}^{2}_{(0,T}_{;H}^{2}_{(Ω))} ≤ C_{R}kfk_{L}^{2}_{(0,T}_{;L}^{2}_{(Ω))},
kΦtk_{L}^{2}_{(0,T}_{;L}^{2}_{(Ω))}+kΦk_{L}^{2}_{(0,T}_{;H}^{2}_{(Ω))} ≤ CRkfk_{L}^{2}_{(0,T}_{;L}^{2}_{(Ω))},
and

kΨ(·, T)k_{H}^{1}_{(Ω)} ≤C_{R}kfk_{L}^{2}_{(0,T;L}^{2}_{(Ω))}, and kΦ(·,0)k_{H}^{1}_{(Ω)} ≤C_{R}kfk_{L}^{2}_{(0,T;L}^{2}_{(Ω))}.
Throughout the thesis, we assume that Ω is convex polygonal bounded domain in
R^{d} with Lipschitz boundary Γ := ∂Ω, where the dimension d will be specified in each
chapter.