Galdi G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems

36 II Basic Function Spaces and Related Inequalities

Finally, for λ ∈ (0, 1] and k ∈ N, by C

k,λ

(Ω) we denote the closed subspace

of C

(Ω) consisting of all functions u whose derivatives up to the kth order

inclusive are H¨older continuous (Lipschitz continuous if λ = 1) in Ω, that is,

[u]

k,λ

≡ max

0≤|α|≤k

sup

x,y∈Ω,x6=y

u(x) − D

u(y)|

|x −y|

< ∞.

k,λ

(Ω) is a Banach space with respect to the norm

kuk

k,λ ≡ kuk

k + [u]

k,λ

, (II.1.12)

(Miranda 1978, §54).

Exercise II.1.1 Assuming Ω bounded, use the Ascoli-Arzel`a theorem (see, e.g.,

Rudin 1987, p. 245) to show that from every sequence of functions uniformly

bounded in C

k+1,λ

(Ω) it is always possible to select a subsequence converging in

the space C

k,λ

(Ω).

II.1.4 Classes of Domains and the ir Properties

We begin wi th a simple but useful result ho lding for arbitrary domains of R

Lemma II.1.1 Let Ω be an arbitrary domain of R

. Then there exists an

open covering, O, of Ω satisfying the following properties

(i) O is constituted by an at most countable number of open ball s {B

k ∈ I ⊆ N, such that

⊂ Ω , for al l k ∈ I , ∪

k∈I

= Ω ;

(ii) For any fami ly F = {B

}, l ∈ I

with I

( I, there is B ∈ (O − F) such

that [∪

l∈I

] ∩ B 6= ∅;

(iii) For any B, B

∈ O, there exists a ﬁnite number of open balls B

∈ O,

i = 1, . . ., N, such that

B ∩ B

6= ∅, B

∩ B

6= ∅, B

∩B

j+1

6= ∅, j = 1, . . . N − 1 .

Proof. Since Ω is open, for each x ∈ Ω we may ﬁnd an open ball B

(x) ⊂

Ω. Clearly, the collection C ≡ {B

(x)}, x ∈ Ω, satisﬁes ∪

x∈Ω

(x) =

Ω. However, since Ω is separable, we may determine an at most countable

subcovering, O, of C satisfying condition (i) in the lemma. Next, assume (ii)

is not true. Then, there would be at least one fa mily F

= {B

}, k

∈ I

, with

( I such that

II.1.4 Classes of Domains and their Properties 37

[

∈I

B = ∅, for al l B ∈ (O − F

) .

Consequently, the sets

≡

[

∈I

, A

≡

[

k∈(I−I

)

are open, disjoint and satisfy A

∪A

= Ω, contradicting the assumption that

Ω is connected. Finally, let B, B

∈ O and denote their centers by x and x

respectively. Since Ω is open and connected, it is, i n particular, arc-connected.

Therefore, we may ﬁnd a curve, γ, jo ining x and x

, that is homeomorphic

to the interval [0, 1]. Let O

⊂ O be a covering of γ. Since γ is compact, we

can extract from O

a ﬁnite covering that satisﬁes the property stated in the

lemma . ut

We next present certain classes of domains of R

, along with their relevant

properties. We begin with the following.

Deﬁnition II.1.1. Let Ω be a domain with a bounded boundary, namely, Ω

is either a bounded domain or it is a domain complement in R

of a compact

(not necessarily connected) set, namely, Ω is an exterior domain.

Assume

that for each x

∈ ∂Ω there is a ball B = B

) and a real function ζ deﬁned

on a domain D ⊂ R

n−1

such that in a system of coordinates {x

, . . . , x

} with

the origin at x

(i) The set ∂Ω ∩ B can be represented by an equation of the type x

ζ(x

, . . . , x

n−1

);

(ii) Each x ∈ Ω ∩ B satisﬁes x

< ζ(x

, . . . , x

n−1

Then Ω is said to be of class C

(or C

-smooth) [respectively, of cl ass C

k,λ

(or C

k,λ

-smooth), 0 < λ ≤ 1] if ζ ∈ C

(D) [respectively, ζ ∈ C

k,λ

(D)]. If, in

particular, ζ ∈ C

0,1

(D), we say that Ω is locally Lipschitz. L ikewise, we shall

say that σ ⊂ ∂Ω is a boundary portion of class C

[respectively, of class C

k,λ

]

if σ = ∂Ω ∩ B

), for some r > 0, x

∈ ∂Ω a nd σ admits a representation

of the form described in (i), (ii) with ζ of class C

[respectively of class C

k,λ

If, in particular, ζ ∈ C

0,1

(D), we say tha t σ is a l ocally Lipschitz boundary

portion.

If Ω is suﬃciently smoo th, of class C

, for example, then the unit outer

normal , n, to ∂Ω is well deﬁned and continuous. However, in several inter-

esting cases, we need less regularity on Ω, but still would like to have n

well-deﬁned. In this regard, we have the following result, for whose proof we

refer to Neˇcas (1967, Chapitre II, Lemme 4.2).

Lemma II.1.2 Let Ω be locally Lipschitz. Then the unit outer normal n

exists almost everywhere on ∂Ω .

Hereafter, the whole space R

will be considered a particular exterior domain.

38 II Basic Function Spaces and Related Inequalities

We shall now consider a special class of bounded domains Ω called star-

shaped (or star-like) with respect to a point. For such domains, there exist

x ∈ Ω (which we may, occasionally, assume to be the origin of coo rdinates)

and a continuous, positi ve function h on the unit sphere such that

Ω =



x ∈ R

: |x −x| < h



x − x

|x −x|



. (II.1.13)

Some elementary properties of star-shaped domains are collected in the fol-

lowing exercises.

Exercise II.1.2 Show that Ω is star-shaped with respect to x if and only if every

ray starting from x intersects ∂Ω at one and only one p oint.

Exercise II.1.3 Assume Ω star-shaped with respect to the ori gin and set

Ω

(ρ)

= {x ∈ R

: x = ρy, for some y ∈ Ω}. (II.1.14)

Show that Ω

(ρ)

⊂ Ω if ρ ∈ (0, 1) and Ω

(ρ)

⊃ Ω if ρ > 1.

The following useful result holds.

Lemma II.1.3 Let Ω be locally Lipschitz. Then, there exist m locally Lip-

schitz bounded domai ns G

, . . . , G

such that

(i) ∂Ω ⊂ ∪

i=1

;

(ii) The domains Ω

= Ω ∩ G

, i = 1, . . . , m, are (locally Lipschitz and)

star-shaped with respect to every po int of a ball B

with B

⊂ Ω

Proof. Let x

∈ ∂Ω. By assumption, there is B

) and a function ζ = ζ(x

= (x

, . . . , x

n−1

) ∈ D ⊂ R

n−1

such that

|ζ(ξ

) −ζ(η

)| < κ|ξ

−η

|, ξ

, η

∈ D,

for som e κ > 0 and, moreover, points x = (x

, x

) ∈ ∂Ω ∩ B

) satisfy

= ζ(x

), x

∈ D,

while points x ∈ Ω ∩B

) satisfy

< ζ(x

), x

∈ D.

We may (and will) take x

to be the origin of coordi nates. Denote next, by

≡ (0, . . ., 0, y

) the point of Ω intersection of the x

-axis with B

)

and consider the cone Γ (y

, α) with vertex at y

, axis x

, and semiaperture

α < π/2. It is easy to see that, taking α suﬃciently small, every ray ρ starting

from y

and lying in Γ (y

, α) intersects ∂Ω ∩ B

) at (one and) only one

point. In fact, assume ρ cuts ∂Ω ∩ B

) at two points z

(1)

and z

(2)

and

II.1.4 Classes of Domains and their Properties 39

denote by α

< α the angle formed by ρ with the x

-axis. Possibly rotating

the coordinate system around the x

-axis we may assume without loss

(1)

= (z

(1)

, 0 , . . ., 0, ζ(z

(1)

, 0 , . . ., 0)), z

(1)

> 0

(2)

= (z

(2)

, 0 , . . ., 0, ζ(z

(2)

, 0 , . . ., 0)), z

(2)

> 0

and so, at the same time,

tan α

(1)

ζ(z

(1)

, 0 , . . ., 0) − y

tan α

(2)

ζ(z

(2)

, 0 , . . ., 0) − y

implying

|ζ(z

(1)

, 0 , . . ., 0) − ζ(z

(2)

, 0 , . . ., 0)|

(1)

− z

(2)

tan α

≥

tan α

Thus, i f (say)

tan α ≤

2κ

ρ will cut ∂Ω ∩ B

) at only one point. Next, denote by σ = σ(z) the

intersection of Γ (y

, α/2) with a plane orthogonal to x

-axis at a point z =

(0, . . ., z

) with z

> y

, and set

R = R(z) ≡ dist (∂σ, z).

Clearly, taking z suﬃciently close to y

(z = z, say), σ(z) will be entirely

contained in Ω and, further, every ray starting from a point of σ(z) and lying

within Γ (y

, α/2) will f orm with the x

-axis an angle less than α and so, by

what we have shown, it will cut ∂Ω ∩ B

) at only one point. Let C be a

cylinder with axis coincident with the x

-axis and such tha t

C ∩ ∂Ω = Γ (y

, α/2) ∩∂Ω.

Then, setting

G = C ∩ B

we have that G is locally Lipschitz and that G ∩Ω is star-shaped with respect

to all points of the ball B

R(z)

(z). Since x

∈ ∂Ω is arbi trary, we may form an

open covering G of ∂Ω constituted by domains of the type G. However, ∂Ω

is compact and, therefore, we may select from G a ﬁnite subset {G

, . . . , G

}

satisfying all conditions in the lemma, which is thus completely proved. ut

Clearly, the Lipschitz constant κ is invariant by this transformation.

40 II Basic Function Spaces and Related Inequalities

Other relevant properties related to star-shaped domains are described in

the following exercises.

Exercise II.1.4 Assume that the function h in (II.1.13) i s Lipschitz continuous,

so that, by Lemma II.1.2, the outer unit normal n = n(x) on ∂Ω exists for a.a. x.

Then, setting F (x) ≡ n(x) · (x − x), show that ess inf

x∈∂Ω

F (x) > 0.

Exercise II.1.5 Assume Ω bounded and locally Lipschitz. Prove that

Ω =

[

i=1

Ω

where each Ω

is a locally Lipschitz and star-shaped domain with respect to every

point of a ball B

with B

⊂ Ω

. Hint: Use Lemma II.1.3.

We end this section by recalling the following classical result, whose proof

can b e found, e.g., in Neˇcas (1967, Chapitre 1, Proposition 2.3).

Lemma II.1.4 Let K be a compa ct subset of R

, and let O = {O

, ··· , O

}

be an o pen covering of K. Then, there exist functions ψ

, i = 1, . . . , N satis-

fying the following properties

(i) 0 ≤ ψ

≤ 1 , i = 1, . . ., N ;

(ii) ψ

∈ C

∞

) , i = 1, . . ., N ;

(iii)

i=1

(x) = 1 , for all x ∈ K .

The fa mily {ψ

} is referred to as partition of unity in K subordinate to the

covering O.

II.2 The Lebesgue Spaces L

For q ∈ [1, ∞), let L

= L

(Ω) denote the linear space of all (equivalence

classes of) real Leb esgue-measurable functions u deﬁned in Ω such that

kuk

≡



Ω

|u|



1/q

< ∞. (II.2.1)

The functional (II.2.1) deﬁnes a no rm in L

, wi th respect to which L

becomes

a Banach space. Likewise, denoting by L

∞

= L

∞

(Ω) the linear space of all

(equivalence classes of) Lebesgue-measurable real-valued functions u deﬁned

in Ω with

kuk

∞

≡ ess sup

Ω

|u| < ∞ (II.2.2)

one shows that (II.2 .2) is a norm and that L

∞

endowed with this norm is a

Banach space. For a proo f of the ab ove properties see, e.g., Miranda (1978,

§47). For q = 2, L

is a Hilbert space under the scalar product

II.2 The Lebesgue Spaces L

(u, v) ≡

Ω

uv, u, v ∈ L

Whenever confusion of domains might occur, we shall use the notation

k · k

q,Ω

, k · k

∞,Ω

, and (· , ·)

Ω

Given a sequence {u

} ⊂ L

(Ω) and u ∈ L

(Ω), 1 ≤ q ≤ ∞, we thus have

that u

→ u, namely, {u

} converges (strongly) to u, if and only if

lim

k→∞

− uk

= 0 .

The following two basic properties, collected in as many lemmas, will be

frequently used throughout. The ﬁrst one is the classical Lebesgue dominated

convergence theorem (Jones 2001, Chapter 6 §C), while the other one relates

convergence in L

with pointwise convergence; see Jones (2001, Corollary at

p. 234)

Lemma II.2.1 Let {u

} be a sequence of measurable functions on Ω, and

assume that

u(x) ≡ lim

m→∞

(x) exists for a.a. x ∈ Ω ,

and that there is U ∈ L

(Ω) such that

(x)| ≤ |U(x)| for a.a x ∈ Ω .

Then u ∈ L

(Ω) and

lim

m→∞

Ω

u .

Lemma II.2.2 Let {u

} ⊂ L

(Ω) and u ∈ L

(Ω), 1 ≤ q ≤ ∞, with u

→ u.

Then, there exists {u

} ⊆ {u

} such tha t

lim

→∞

(x) = u(x) , for a.a. x ∈ Ω .

We want now to collect som e inequalities in L

spaces that will be fre-

quently used throughout. Fo r 1 ≤ q ≤ ∞, we set

= q/(q − 1);

one then shows the H¨older inequality

Ω

|uv| ≤ kuk

kvk

(II.2.3)

for all u ∈ L

(Ω), v ∈ L

(Ω) (Miranda 1978, Teorema 47.I). The number

is called the H¨older conjugate of q. In particular, (II.2.3) shows that the

42 II Basic Function Spaces and Related Inequalities

bilinear fo rm (u, v) is meaningful whenever u ∈ L

(Ω) and v ∈ L

(Ω). In

case q = 2, inequality (II.2.3) is referred to as the Schwarz inequality. More

generally, one has the generalized H¨older inequalit y

Ω

. . .u

| ≤ ku

· . . . · ku

, (II.2.4)

where

∈ L

(Ω) , 1 ≤ q

≤ ∞, i = 1, . . ., m ,

i=1

−1

= 1 .

Both inequalities (II.2.3) and (II.2.4) are an easy consequence of the Young

inequali ty:

ab ≤

εa

+ ε

−q

(a, b, ε > 0) (II.2.5)

holdi ng for all q ∈ (1, ∞). When q = 2, relation (II.2.5) is known as the

Cauchy inequality.

Two noteworthy consequences of inequality (II.2.3) are the Minkowski in-

equality:

ku + vk

≤ kuk

+ kvk

, u, v ∈ L

(Ω), (II.2.6)

and the interpolation (o r convexity) inequality:

kuk

≤ kuk

kuk

1−θ

(II.2.7)

valid for all u ∈ L

(Ω) ∩ L

(Ω) with 1 ≤ s ≤ q ≤ r ≤ ∞, and

−1

= θs

−1

+ (1 − θ)r

−1

, θ ∈ [0, 1].

Another important inequality is the generalized Minkowski inequality re-

ported in the following lemma, and for whose proof we refer to Jones (2001,

Chapter 11, §E).

Lemma II.2.3 Let Ω

, and Ω

be domains of R

and R

, respectively, with

m, n ≥ 1. Suppose that u : Ω

× Ω

→ R is a Lebesgue measurable function

such that, for some q ∈ [1, ∞],

Ω



Ω

|u(x, y)|



1/q

dy < ∞.

Then,



Ω



Ω

u(x, y) dy





1/q

< ∞,

and the following inequality holds

Actually, it can be proved that (II.2.6) is just a particular case of (II.2.8), hence

the adjective “generalized”; see Jones (2001, p. 272).

II.2 The Lebesgue Spaces L



Ω



Ω

u(x, y) dy





1/q

≤

Ω



Ω

|u(x, y)|



1/q

dy . (II.2.8)

Exercise II.2.1 Assume Ω bounded. Show that if u ∈ L

∞

(Ω), then

lim

q→∞

kuk

= kuk

∞

Exercise II.2.2 Prove inequality (II.2.5). Hint: Minimize the function

/q − t + 1/q

Exercise II.2.3 Prove inequalities (II.2.6) and (II.2.7).

We shall now list some of the basic properties of the spaces L

. We begi n

with the following (see, e.g. Miranda 1978, §51).

Theorem II.2.1 For 1 ≤ q < ∞, L

is separable, C

(Ω) being, in pa rticular,

a dense subset

Note tha t the above property is not true if q = ∞, since C(Ω) is a closed

subspace of L

∞

(Ω)); see Miranda, loc. cit..

Concerning the density of smooth functions in L

, one can prove something

more than what stated in Theorem II.2.1, namely, that every function in L

1 ≤ q < ∞, can be approximated by functions from C

∞

(Ω). This fact follows

as a particular case of a general smoothing procedure that we are going to

describe. To this end, given a real (measurable) function u in Ω, we shall

write

u ∈ L

loc

(Ω)

to mean

u ∈ L

(Ω

), for any bounded domain Ω

with Ω

⊂ Ω.

Likewise, we write

u ∈ L

loc

(Ω)

to mean

u ∈ L

(Ω

), for any bounded domain Ω

⊂ Ω.

Clearly, for Ω bounded we have L

loc

(Ω) = L

(Ω). Now, let j ∈ C

∞

(Ω) be a

non-negative function such tha t

(i) j(x) = 0, for |x| ≥ 1,

(ii)

j = 1.

44 II Basic Function Spaces and Related Inequalities

A typical exampl e is

j(x) =

(

c exp[−1/(1 −|x|

)] if |x| < 1

0 if |x| ≥ 1,

with c chosen i n such a way that property (ii) is satisﬁed. The regular izer (or

molliﬁer) in the sense o f Friedrichs u

of u ∈ L

loc

(Ω) is then deﬁned by the

integral

(x) = ε

−n



x − y



u(y)dy, ε < dist (x, ∂Ω) .

This function has several interesting properties, some of which will be recalled

now here. First o f all, we observe that u

is inﬁnitely diﬀerentiabl e at each

x ∈ Ω with dist (x, ∂Ω) > ε. Moreover, if u ∈ L

loc

(Ω) we may extend it by

zero outside Ω, so that u

becomes deﬁned f or all ε > 0 and all x ∈ R

. Thus,

in particular, if u ∈ L

(Ω), 1 ≤ q < ∞, one can show (Miranda 1978, §51; see

also Exercise II.2.10 for a generalization)

≤ kuk

for all ε > 0 ,

lim

ε→0

−uk

= 0.

(II.2.9)

Exercise II.2.4 Show that for u ∈ C

(Ω),

lim

ε→0

(x) = u(x) holds uniformly in x ∈ Ω.

Exercise II.2.5 For u ∈ L

(Ω), 1 ≤ q < ∞, show the inequality

sup

(x)| ≤ ε

−n/q−|α|

kuk

q,Ω

, |α| ≥ 0.

We next observe that, by writing u

(x) as follows:

(x) = ε

−n

|ξ|<ε





u(x + ξ)dξ,

it becomes apparent that, if u is of compact support i n Ω and ε is chosen less

than the distance of the support of u from ∂Ω, then u

∈ C

∞

(Ω). The latter,

together with (II.2 .9)

and the density of C

in L

, yields that C

∞

(Ω) is a

dense subspace of L

(Ω), 1 ≤ q < ∞. The proof of this property, along with

some of its consequences, is left to the reader in the following exercises.

Exercise II.2.6 Prove that C

∞

(Ω) is dense in L

(Ω), 1 ≤ q < ∞. Hint. Use the

density of C

(Ω) in L

(Ω) (Miranda 1978, §51) along with the properties of the

molliﬁer.

II.2 The Lebesgue Spaces L

Exercise II.2.7 Prove the existence of a basis in L

(Ω) constituted by functions

from C

∞

(Ω). Hint: Use the separability of L

along with the density of C

∞

into L

Exercise II.2.8 Let u ∈ L

(Ω), 1 ≤ q < ∞. Extend u to zero in R

− Ω and

continue to denote by u the extension. Show the following continuity in the mean

property: Given ε > 0 there is δ > 0 such that for every h ∈ R

with |h| < δ the

following inequality holds

Ω

|u(x + h) −u(x)|

dx < ε

Hint: Show the property f or u ∈ C

∞

(Ω), then use the density of C

∞

in L

Exercise II.2.9 Assume u ∈ L

loc

(Ω). Prove that

Ω

uψ = 0, for all ψ ∈ C

∞

(Ω), implies u ≡ 0, a.e. in Ω.

Hint: Consider the function

sign u =

1 if u > 0

−1 if u ≤ 0.

For a ﬁxed bounded Ω

with Ω

⊂ Ω,

sign u ∈ L

(Ω

)

and so sign u can be approximated by functions from C

∞

(Ω

Exercise II.2.10 Let u ∈ L

), 1 ≤ q < ∞, and for z ∈ R

and k ≤ n set

(k)

= (z

, . . . , z

) , z

(k)

= (z

k+1

, . . . , z

) .

Moreover, deﬁne

(k),ε

(x) = ε

−k

„

(k)

− y

(k)

u(y

(k)

, y

(k)

) dy

(k)

Show the following properties, for each y

(k)

∈ R

n−k

(k),ε

q,R

≤ ku(·, y

(k)

q,R

for all ε > 0 ,

lim

ε→0

(k),ε

− u(·, y

(k)

q,R

= 0.

Hint: Use the generalized Minkowski inequality, the result in Exercise II.2.8 and

Lebesgue dominated convergence theorem (Lemma II.2.1).

Let v ∈ L

(Ω), with q

the H¨older conjugate of q. Then, by (II.2 .3), the

integral

`(u) =

Ω

vu, u ∈ L

(Ω) (II.2.10)

deﬁnes a linear functional on L

. However, for q ∈ [1, ∞), every li near func-

tional must be of the form (II.2.10). Actually, we have the following Riesz

representation theorem for whose proof we refer to Miranda (1978, §48).

Galdi G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems

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