Basic functional analysis and Sobolev spaces

Let Ω ⊂ R^d with d ∈ {1,2,3} denote an open, non-empty, bounded and Lebesgue-measurable set. We refer to Ω as the spatial domain and define its boundary by Γ =∂Ω = Ω\Ω, where Ω denotes the closure of the spatial domain.

Definition 1.1.1 (Lebesgue space)For 1≤p <∞ the spaceL^p(Ω) is defined as the space of all (equivalence classes of) functions f : Ω→R for which thep-th power of the absolute value is Lebesgue integrable:

L^p(Ω) =n

f : Ω→R

f is measurable and R

Ω

|f(x)|^pdx <∞o

. (1.1)

TheL^p-norm of f ∈L^p(Ω)is defined by

kfk_Lp(Ω) =



 Z

Ω

|f(x)|^pdx





1/p

. (1.2)

The spaceL^p(Ω)endowed with the normk · k_Lp(Ω) for1< p <∞ becomes a Banach space that is reflexive.

In this context, we identify those functions with each other, that only differ on a set of zero measure and consider them to belong to the same equivalence class. We will not distinguish between the function and its equivalence class, except the theory necessitates a precise distinction.

For p = ∞ the definition of the Lebesgue space L^∞(Ω) needs the notion of the essential maximum or supremum of the measurable function f : Ω → R on the measure space (Ω,A, µ) with σ-algebraAover Ωand measure µ,

Verbally, those maxima are excluded, that change upon the removal of single, in a certain sense, isolated and therefore not essential points. We say that f is essentially bounded onΩ, if

ess sup

Ω

|f|<∞. (1.5)

Definition 1.1.2 The spaceL^∞(Ω)is defined as the Lebesgue space of almost everywhere essen-tially bounded and measurable functions f on Ω,

L^∞(Ω) =

f is measurable andess sup

Ω space. In a Banach space an inner product of two elements as in Rⁿ does not necessarily exist.

However, for p= 2 we define for the space L²(Ω)ofsquare-integrable functions an inner product of two functions f, g∈L²(Ω)by

We define by C^m(Ω), m ∈ N, the linear space of all real-valued continuous functions on Ω, whose partial derivatives D^αf up to order |α| ≤ m exist and are continuous. By C^m(Ω) we denote for any m∈N0 the linear space of all elements of C^m(Ω)that together with their partial derivatives up to order m can be continuously extended to Ω. The spaces C^m(Ω) are Banach spaces with respect to the following norms:

kfk_C(Ω)= max

Ω |f|, kfk_Cm(Ω) = X

|α|≤m

kD^αfk_C(Ω), for m∈N. (1.9)

1.1 Basic functional analysis and Sobolev spaces The set

supp f =

x∈Ω|f(x)6= 0 (1.10)

is called thesupportoff, that is the smallest closed set outside of whichf vanishes identically. Our special interest is inC₀^∞(Ω)of so-calledtest functions, i.e., functions that vanish on the boundary Γ by yielding zero boundary integrals upon integration by parts, and that can be differentiated up to an arbitrary order.

The theory of partial differential equations requires spatial domainsΩwith a sufficiently smooth boundary Γ. Therefore, the affiliation of the domain and its boundary to a certain class C^k,1, k∈N∪{0}is utilized, which presupposes thatΩ, in a geometrical interpretation, is locally situated on exactly one side of the boundary Γ. For more details we refer, e.g., to Tröltzsch [Tro10] and are settled for the following compact definition.

Definition 1.1.4 (Lipschitz domain and boundary)Domains and boundaries of class C^0,1 are called Lipschitz domains (or regular domains) and Lipschitz boundaries, respectively. Boundaries of classC^k,1 are referred to as C^k,1-boundaries.

We introduce thesurface measureby the(d−1)-dimensional Lebesgue measure on the boundary Γ and denote it byds(x)or ds.

For a satisfactory theory for our governing partial differential equations, it is necessary to replace the classical function spaces C^k(Ω) by the so-called Sobolev spaces. They are based on the concept ofweak derivatives which we will present next. We denote byL¹_loc(Ω)the set of alllocally integrable functions inΩ, i.e., all functions that are Lebesgue integrable on every compact subset of Ω, and denote with

D^αf(x) = ∂^|α|f

∂x^α₁¹∂x^α₂²... ∂x^α_d^d =D^α₁¹... D_d^αdf(x) (1.11) its partial derivative of total order |α| = Pd

i=1αi with multi-index α = (α1, α2, ..., αd)^> ∈ N^d0. We put D⁽⁰⁾f :=f.

Definition 1.1.5 (Weak derivative)Let f ∈ L¹_loc(Ω) and some multi-index α be given. If a function g∈L¹_loc(Ω)satisfies

Z

Ω

f(x)D^αϕ(x) = (−1)^|α|

Z

Ω

g(x)ϕ(x)dx ∀ϕ∈C₀^∞(Ω), (1.12) theng is called the weak derivative of f (associated withα) and we setg=D^αf.

The property of weak differentiability can be understood as an extension of the concept of classical (or strong) differentiability. Herewith it is possible to assign (weak) derivatives to functions that are not differentiable in the strong sense. If the weak derivative exists, it may belong to spaces L^p(Ω) with more regularity than only L¹_loc(Ω). It is easy to see that L^p(Ω) ⊂ L¹_loc(Ω) for all p∈[1,∞].

Definition 1.1.6 (Sobolev space) Let1≤p <∞andk∈N. We denote byW^k,p(Ω)the linear space of all functions f ∈ L^p(Ω) having weak derivatives D^αf in L^p(Ω) for multi-indices α of length|α| ≤k,

W^k,p(Ω) =

f ∈L^p(Ω)

D^αf ∈L^p(Ω)with|α| ≤k , (1.13)

endowed with the norm

For p=∞we define W^k,∞(Ω)analogously, equipped with the norm kfk_Wk,∞(Ω)= max

|α|≤kkD^αfk_L^∞_(Ω). (1.15)

For the particular case p= 2 we write

H^k(Ω) :=W^k,2(Ω). (1.16)

The spaces W^k,p(Ω) are Banach spaces. Of special importance for our purposes is the space H¹(Ω) =W^1,2(Ω). Therefore we repeat the definition explicitly for the case k= 1,p= 2: Definition 1.1.7 We denote by H¹(Ω) the linear space of all functions f ∈ L²(Ω) with weak derivativeDif in L²(Ω)fori= 1, ..., d,

Remark 1.1.8 The spaceH¹(Ω)supplied with the inner product hf, gi_H¹_(Ω) =

Z

Ω

f g+∇f· ∇g dx, f, g∈H¹(Ω), (1.19) and the induced normk·k_H1(Ω) =q

h·,·i_H1(Ω)becomes a Hilbert space. With the existence of the structure of an inner product, length and angle can be measured and the concept of orthogonality is available.

We state a useful embedding result that is a direct consequence of Tröltzsch [Tro10, Theorem 7.1] about the existence and continuity of embeddings in Sobolev spaces.

Proposition 1.1.9 Let Ω ⊂ R² be a bounded Lipschitz domain. Then H¹(Ω) is continuously embedded in L^q(Ω)for all1≤q <∞, especially

H¹(Ω),→L²(Ω).

For a more comprehensive insight into the theory of Lebesgue and Sobolev spaces we refer the interested reader, e.g., to the monograph of Adams and Fournier [AF03] and to the textbook of Evans [Eva02].

Further we want to introduce another important concept in functional analysis, the dual space, and the thereon defined dual or adjoint operators.

1.1 Basic functional analysis and Sobolev spaces Definition 1.1.10 (Linear operator)Let X and Y be normed real vector spaces with norms k · k_X andk · k_Y.

1. A mappingA:X →Y is said to belinear or a linear operator if it satisfies

A(λu+µv) =λAu+µAv ∀u, v∈X, λ, µ∈R. (1.20) A linear mapping f :X →Ris calledlinear functional.

2. A linear operator A:X→Y is bounded if there is a constant c(A) such that

kAukY ≤c(A)kukX ∀u∈X. (1.21)

Then we call kAk_X,Y ∈Rthe (operator) norm ofA and it holds kAk_X,Y := sup

kukX=1

kAuk_Y <∞. (1.22)

3. We denote by L(X, Y) the normed space of all bounded linear operators from X into Y, endowed with the operator normk · k_X,Y.

A linear operator is bounded if and only if it is continuous (see Tröltzsch [Tro10, Theorem 2.8]).

If Y is complete, the space L(X, Y) is complete and hence a Banach space.

Definition 1.1.11 (Dual space)The spaceX^∗ :=L(X,R) of all bounded linear functionals on X is called dual space with associated operator norm

kfk_X^∗ = sup

kuk_X=1

|f(u)|= sup

u∈X\{0}

|f(u)|

kukX

, with f ∈X^∗. (1.23)

By (·,·)X^∗,X we denote the dual pairing of X^∗ andX, defined by

(f, u)_X^∗_,X :=f(u). (1.24)

Since the space of real numbers Ris complete,X^∗ is always a Banach space.

Definition 1.1.12 (Adjoint or dual operator) Let Banach spaces X,Y, a continuous linear operatorA :X →Y and linear functionals f ∈Y^∗ andg =f ◦A:X →R, g(x) =f(Ax), be given. The mapping A^∗ :Y^∗ → X^∗ defined byf 7→ g =f ◦A is called the adjoint operator or dual operator ofA, fulfilling

(A^∗f)(x) =f(Ax) ∀x∈X, (1.25)

kA^∗fk_X^∗ ≤ kAk_X,Ykfk_Y^∗ ∀f ∈Y^∗. (1.26) Further A^∗ is continuous, so that A^∗ ∈ L(Y^∗, X^∗) and kA^∗k_Y^∗_,X^∗ ≤ kAk_X,Y holds. The readability is improved using the duality pairing notation:

f(Ax) = (f, Ax)Y^∗,Y = (A^∗f, x)X^∗,X =: (x, A^∗f)X,X^∗ ∀f ∈Y^∗,∀x∈X. (1.27)

Although this form suggestsA^∗ to be already explicitly determined, this cannot be expected due to the fact that a functional f^∗ ∈ Y^∗ can take several completely different representations. An important result, which will help us in deriving explicit expressions for adjoint operators in Hilbert space, is the following theorem.

Theorem 1.1.13 (Riesz representation theorem)Let H be a real Hilbert space. The dual spaceH^∗ is isometric to H itself, it means that for any functionalF ∈H^∗ there exists a uniquely determinedf ∈H such that

F(v) = (F, v)H^∗,H =hf, viH ∀v ∈H (1.28) andkFk_H^∗=kfk_H. Due to this we can identifyH^∗ with H and write H =H^∗.

With this at hand we can formulate the following definition.

Definition 1.1.14 (Hilbert space adjoint)Let the real Hilbert spaces H and V as well as an operatorA∈L(H, V) be given. An operator A^∗ is calledHilbert space adjoint or adjoint of A, if

hv, Ahi_V =hA^∗v, hi_H ∀h∈H,∀v∈V. (1.29) Here we followed the suggestion in Tröltzsch [Tro10, p. 62, Remark] and skipped the distinction between the adjoint operator in Banach or Hilbert space and use the uniformly marking by ^∗.

Let us further introduce the notion of another useful construct that helps us to ease the argu-mentation in Hilbert space.

Definition 1.1.15 (Gelfand triple)LetH be a Hilbert space withV ⊂Ha dense subspace which is a Hilbert space in turn, endowed with an inner product h ·, · i_V, and continuous embedding E :V →H. By identifying H with its dual spaceH^∗, the adjoint embedding is given by

E^∗:H =H^∗ →V^∗,

and for any f ∈H = H^∗ ⊂V^∗ and all v ∈ V ⊂ H the duality pairing (f, v)_V^∗_,V is compatible with the inner product of H in the following sense

(f, v)V^∗,V =hf, viH. The chain of dense and continuous embeddings

V −→^E H=H^∗ −→^E∗ V^∗ is calledGelfand triple (V, H, V^∗).

The later introduced necessary and sufficient optimality conditions involve (higher order) deriva-tives. Therefore we extend the notion and concept of differentiability to operators between Banach spaces.

Definition 1.1.16 (Directional, Gâteaux and Fréchet derivative)LetX andY be real Banach spaces and X an open subset of X. A mappingF :X ⊂X→Y is called

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