# Notations and preliminaries

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 Rd. We denote the boundary of Ω by Γ :=∂Ω. Let µ= (µ1, . . . , µd) be a non negative integer with d-components and the order of µ is denoted by |µ| =µ1+. . .+µd. Further, let x = (x1, x2, . . . , xd) ∈Ω and dx=dx1, . . . , dxd. Then, theµ-th derivative of a function ψ is denoted byDµψ with

Dµψ := ∂|µ|ψ

∂xµ11. . . ∂xµdd.

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

|ψ|pdx exists and finite. Moreover, the associated norm on Lp(Ω) with 1 ≤ p < ∞, is

kψkLp(Ω) :=Z

|ψ|pdx1p .

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

kψkL(Ω) :=esssup

x∈Ω

|ψ|<∞.

Again, for p= 2, L2(Ω) is a Hilbert space equipped with the inner product (ψ1, ψ2) =

Z

ψ1ψ2dx, ∀ψ1, ψ2 ∈L2(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 .

LetCk(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 C0k(Ω) be the space of all Ck(Ω) functions with compact support in Ω. Moreover, the space C0(Ω) consists of all infinitely differentiable functions with compact support in Ω. For a non-negative integerk, letWk,p(Ω) (1≤p <∞) be the collection of all equivalence class of functions inLp(Ω) such that the distributional derivatives upto orderkare also belongs toLp(Ω), i.e.,

Wk,p(Ω) :=

ψ ∈Lp(Ω) : Dµψ ∈Lp(Ω) for 0≤ |µ| ≤k .

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

kψkk,p :=Z

X

0≤|µ|≤k

|Dµψ|pdx1p

, 1≤p <∞.

For p=∞, we define the norm on Wk,∞(Ω) by kψkk,∞:= max

0≤|µ|≤kkDµψkL(Ω).

Further, we denote the semi-norm on Wk,p(Ω) by | · |Wk,p(Ω) which is defined as

|ψ|Wk,p(Ω) := X

|µ|=k

kDµψkLp(Ω).

When p= 2, we denote Wk,p(Ω) =Hk(Ω). Moreover, if k = 0 and p= 2 then we have W0,2(Ω) =H0(Ω) =L2(Ω). Let Hk(Ω) be the Hilbert space with inner product

1, ψ2)k := X

0≤|µ|≤k

Z

Dµψ1Dµψ2dx, ψ1, ψ2 ∈Hk(Ω).

Observe that the Sobolev space Hk(Ω) and H0k(Ω), respectively, are defined to be the closure ofC(Ω) andC0(Ω) with respect to the normk·kkand semi-norm|·|k. Further, we shall denote the space H−1(Ω) as the dual of H01(Ω) with the norm

1kH−1(Ω):= sup

ψ2∈H01(Ω), ψ26=0

1, ψ2) kψ2kH1(Ω).

For 1 ≤ p ≤ ∞, we now define the standard Bochner spaces Lp(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

kψkLp(0,T;B) := Z T 0

kψ(t)kpBdt1p

<∞ for 1≤p <∞, kψkL(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 H1(0, T;B) consisting of all the measurable functions ψ : (0, T) → B, such that

kψkH1(0,T;B) :=

Z T 0

n

kψ(t)k2B+

∂ψ

∂t(t)

2 B

o dt

12

<∞.

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 , and Grisvard .

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

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

ab ≤ ap p + bq

q holds.

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

ab≤ a2

2 + b2 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| ≤Xd

i=1

|ai|p1pXd

i=1

|bi|q1q ,

where p >1 and 1p + 1q = 1. In particular, for p= 2 and q = 2, the above inequality is known as the Cauchy-Schwarz inequality in Rd.

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

Z

1ψ2|dx≤ kψ1kLp(Ω)2kLq(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 (). Givenaˆ= (a0,a1, . . . ,ad)∈Rd+1,bˆ= (b0,b1, . . . ,bd)∈Rd+1 and ˆ

c ∈R such that

|ˆa|2 ≤ˆc2+ ˆa·ˆb, then we have the following

|ˆa| ≤ |ˆb|+|ˆc|,

where | · | and ˆa·ˆb denote the standard Euclidean vector norm and inner product on Rd+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 ⊂ Rd is a convex bounded domain. Then there exists a constant C >0 such that

kψkL2(Ω) ≤Ck∇ψkL2(Ω), f or every ψ ∈H01(Ω), where the constant C depends on the domain Ω.

The above inequality implies that, for any ψ ∈ H01(Ω), the L2-norm is bounded by the semi H1-norm.

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

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 : ˜X0 →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 =f0(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 = f0(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 Rd(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 x1,x2 ∈X˜0 such that

F(νx1+ (1−ν)x2)≤νF(x1) + (1−ν)F(x2).

A functional F is said to be strictly convex, if it satisfies the above condition with strict inequality whenever x1 6=x2, 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) := L2(0, T;H2(Ω)∩H01(Ω))∩H1(0, T;L2(Ω)), W2(0, T) := L2(0, T;H01(Ω))∩H1(0, T;H−1(Ω)).

We observe that W1(0, T),→ C(0, T;H01(Ω)) and W2(0, T),→ C(0, T;L2(Ω)) (cf., ).

We now introduce some auxiliary problems. For any f ∈L2(0, T;L2(Ω)), let Ψ, Φ∈ W1(0, T) be solutions of the following forward and backward in time standard parabolic problems













Ψt+AΨ =f in ΩT, Ψ(·,0) = 0 in Ω,

Ψ = 0 on ΓT,

(1.17)

CHAPTER 1. Introduction 13 and













−Φt+AΦ =f in ΩT, Φ(·, T) = 0 in Ω,

Φ = 0 on ΓT,

(1.18)

respectively.

The solutions Ψ and Φ satisfy the following regularity results .

Lemma 1.2.3. For a given function f ∈ L2(0, T;L2(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 CR such that

tkL2(0,T;L2(Ω))+kΨkL2(0,T;H2(Ω)) ≤ CRkfkL2(0,T;L2(Ω)), kΦtkL2(0,T;L2(Ω))+kΦkL2(0,T;H2(Ω)) ≤ CRkfkL2(0,T;L2(Ω)), and

kΨ(·, T)kH1(Ω) ≤CRkfkL2(0,T;L2(Ω)), and kΦ(·,0)kH1(Ω) ≤CRkfkL2(0,T;L2(Ω)). Throughout the thesis, we assume that Ω is convex polygonal bounded domain in Rd with Lipschitz boundary Γ := ∂Ω, where the dimension d will be specified in each chapter.

Outline

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