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
Ω|ψ|pdx exists and finite. Moreover, the associated norm on Lp(Ω) with 1 ≤ p < ∞, is
Forp=∞,L∞(Ω) is a Banach space of all Lebesgue measurable and essentially bounded functions, equipped with the norm
Again, for p= 2, L2(Ω) is a Hilbert space equipped with the inner product (ψ1, ψ2) =
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.,
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.,
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
, 1≤p <∞.
For p=∞, we define the norm on Wk,∞(Ω) by kψkk,∞:= max
Further, we denote the semi-norm on Wk,p(Ω) by | · |Wk,p(Ω) which is defined as
|ψ|Wk,p(Ω) := X
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
(ψ1, ψ2)k := X
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
(ψ1, 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
<∞ for 1≤p <∞, kψkL∞(0,T;B) := ess sup
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
Z T 0
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
Young’s inequality with -form. For all non-negative real numbers a,b and every >0, the following inequality
2 + b2 2 holds.
Discrete version of H¨older’s inequality. Let ai, bi, i= 1, . . . , d be the positive real numbers. Then
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
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 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
→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,
CHAPTER 1. Introduction 13 and
−Φt+A∗Φ =f in ΩT, Φ(·, T) = 0 in Ω,
Φ = 0 on ΓT,
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
kΨtkL2(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.