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

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66 II Basic Function Spaces and Related Inequalities

for 0 ≤ η

< η ≤ 1 we have

|φ(η) −φ(η

η −η

≤

η − η



∂U

∂λ

(λ, η

)



dλ +

η − η



∂U

∂µ

(η, µ)



dµ

and thus, by (II.3.3),

f(η, η

) ≡



φ(η) − φ(η

)

η − η



≤ 2

q−1



|η − η



∂U

∂λ

(λ, η

)



dλ





|η −η



∂U

∂µ

(η, µ)



dµ





(II.4.13)

We now recall the following inequalities due to G.H. Hardy (Hardy, Littlewood

and Polya 1934, p. 240):



x − a

f(t)dt



≤



q − 1



|f(t)|

dt, x > a, q > 1



b − x

f(t)dt



≤



q − 1



|f(t)|

dt, x < b, q > 1.

(II.4.14)

Integrating (II.4.13) ﬁrst in η ∈ (η

, 1] and then in η

∈ [0, 1] and using (I I.4.14)

we obtain



f(η, η

)dη



dη

≤ 2

q−1



q − 1





dη



∂U

∂λ

(λ, η

)



dλ

dη



∂U

∂µ

(η, µ)



dµ



≤ ck ∇uk

q,S

(II.4.15)

with c a suitable constant. Interchanging the roles of η and η

in (II.4.15) and

noticing that f(η, η

) = f(η

, η) one also has



f(η, η

)dη



dη ≤ ck∇uk

q,S

. (II. 4.16)

Adding (II.4.15) and (II.4.16) we ﬁnd

I ≤ 2ck∇uk

q,S

Since the other integrals on the left-hand side of (II.4.12) can be analogously

increased, the proof of (II.4.1 2) is accomplished. 

Exercise II.4.2 According to the method just described, the case q = 1 of Theorem

II.4.3 is excluded because Hardy’s inequalities (II .4.14) hold if q > 1. Show, by means

II.4 Boundary Inequalities and the Trace of Functions of W

m,q

of a counterexample, that ( II.4.14) does not hold when q = 1. Hint (Gagliardo 1957):

Take f (t) = (t − a)

−1

(log(t − a))

−2

. (For the characterization of the trace when

m = q = 1, see Gagliardo (1957, Teorema 1.II)).

The extension of Theorem II.4. 3 to the space W

m,q

(Ω), m ≥ 2, is for-

mally analogous, provided we introduce a suitable generalization of the space

1−1/q,q

(∂Ω). To this end, assume Ω of class C

m−1,1

and let {B

} and

{ζ

}, k = 1, 2, . . ., s, be a family of open balls centered at x

∈ ∂Ω with

∂Ω ⊂ B

, and of functions of class C

m−1,1

), respectively, deﬁning the

m−1,1

−regularity of ∂Ω in the sense of Deﬁnition II.1.1. Assuming that

(k)

= ζ

(k)

, . . . , x

(k)

n−1

), (x

(k)

, . . . , x

(k)

n−1

) ∈ D

is the representation of ∂Ω ∩ B

, for a function u on ∂Ω we set

= u(x

(k )

, . . . , x

(k)

n−1

, ζ

(k)

, . . . , x

(k)

n−1

))

and deﬁne

kuk

m−1/q, q(∂Ω)

≡

k=1

m−1/q, q,D

(II.4.17)

where

m−1/q, q,D

≡

0≤|α|≤m−1

q,D

+ hhu

m−1/q, q

hhu

m−1/q, q

≡

|α|=m−1



u(y) − D

u(y

|y − y

n−2+q

dydy



1/q

(II.4.18)

We next denote by W

m−1/q, q

(∂Ω) the linear space of functions u for which

the functional deﬁned by (I I.4.17)–(II.4 .18) is ﬁnite. It can be shown that

the deﬁnition of W

m−1/q, q

(∂Ω) does not depend on the particular choice

of the local representation {B

}, {ζ

} of the boundary. In fact, if {B

{ζ

} is another such a representation and kuk

m−1/q, q(∂Ω)

is the corresponding

functional associated to u, there exist constants c

, c

> 0 such that

kuk

m−1/q, q(∂Ω)

≤ c

kuk

m−1/q,q(∂Ω)

≤ c

kuk

m−1/q, q(∂Ω)

(Neˇcas 1967, Chapitre 3, Lemme 1.1). As in the case of W

1−1/q,q

(∂Ω), one

shows that the space W

m−1/q,q

(∂Ω) is a dense subset o f L

(∂Ω), which is

complete in the norm (II.4.17)–(II.4 .17), separable for q ∈ [1, ∞) and reﬂexive

for q ∈ (1, ∞) (Neˇcas 19 67, Chapitre 2, Proposition 3.1).

Set

m,q

(∂Ω) ≡ W

m−1/q, q

(∂Ω) × W

m−1−1/q,q

(∂Ω) × . . . ×W

1−1/q,q

(∂Ω).

We then have the foll owing characterization of the trace operator Γ

(m)

deﬁned

in (II.4.6)–(II.4.7) (Neˇcas 1967, Chapitre 2, Th´eor`eme 5.5, 5.8).

68 II Basic Function Spaces and Related Inequalities

Theorem II.4.4 Let Ω be of class C

m−1,1

, m ≥ 2. If

u ∈ W

m,q

(Ω), 1 < q < ∞,

then

(m)

(u) ∈ W

m,q

(∂Ω)

and for all ` = 0, 1, . . ., m −1 it is

kγ

(u)k

m−`−1/q,q(∂Ω)

≤ c

kuk

m,q,Ω

. (II.4.19)

Conversely, i f Ω is of class C

m,1

, given

w ∈ W

m,q

(∂Ω)

there exists u ∈ W

m,q

(Ω) with

(m)

(u) = w

and the following inequality holds

kuk

m,q,Ω

≤ c

m−1

`=0

kγ

(u)k

m−`−1/q,q(∂Ω)

. (II. 4.20)

The constants c

, i = 1, 2, depend only on n, m, q, and Ω.

As in the case of the operator γ, the operator Γ

(m)

can also be charac-

terized, in view of Theorem II.4.2 and Theorem II.4.4, as a bounded linear

bijection of W

m,q

(Ω) /W

m,q

(Ω) onto W

m,q

(∂Ω) (topologized in the obvious

way).

Remark II.4.2 If Ω is not globally smooth but has a smooth boundary

portion σ, we can still deﬁne the trace on σ of functions from W

m,q

(Ω) and

the space W

m,q

(σ). In particula r, inequali ty (II.4.19) continues to hold with

σ in place of ∂Ω (see Neˇcas, loc. cit.). 

Remark II.4.3 Problems o f trace on the plane {x

= 0} for functions de-

ﬁned in R

n−1

will be considered in Section II.10. 

Exercise II.4.3 (Neˇcas 1967, Chapitre 3, Th´eor`eme 1.1). Let Ω be bounded and

locally Lipschitz. Show the following Gauss identity:

Ω

Φ∇ · u =

∂Ω

Φ u · n −

Ω

u · ∇Φ (II.4.21)

for all vectors u with components in W

1,q

(Ω) and scalars Φ from W

1,r

(Ω) where q

and r satisfy

(i) q

−1

+ r

−1

≤ (n + 1)/n if 1 ≤ q < n, 1 ≤ r < n;

(ii) r > 1 if q ≥ n;

(iii ) q > 1 if r ≥ n;

Hint: Use Lemma II.4.1 and Theorem II.3.3 and Theorem II.4.1.

Remark II.4.4 An extension of (II.4.21) to functions u with less regular-

ity tha n that required in Exercise II.4.3 will be given in Section III.2, see

(III.2.14). 

II.5 Further Inequalities and Compactness Criteria in W

m,q

II.5 Further Inequalities and Compactness Criteria in

m,q

We begin to prove some inequali ties relating the L

-norm of a function with

that of its ﬁrst derivatives (Poincar´e 1894, §III, and Friedrichs 1933). Through-

out this section we shall denote by L

a layer of width d > 0, namely

= {x ∈ R

: −d/2 < x

< d/2}.

Theorem II.5.1 Assume Ω ⊂ L

, for some d > 0. Then, for all u ∈ W

1,q

(Ω),

1 ≤ q ≤ ∞,

kuk

≤ (d/2)k∇uk

. (II.5.1)

Proof. It is enough to show the theorem for u ∈ C

∞

(Ω). For such functions

one has

|u(x

, . . . , x

)| =



−d/2

∂u(x

, . . . , ξ)

∂ξ

dξ



d/2

∂u(x

, . . . , ξ)

∂ξ

dξ



which implies

|u(x)| ≤ (1/2)

d/2

−d/2

|∇u|dx

. (II.5.2)

From this relation we at once recover (II.5.1) for q = ∞. If q ∈ [1, ∞), em-

ploying the H¨older i nequality in the right-hand side of (II.5.2) yields

|u(x)|

≤ (d

q−1

)

d/2

−d/2

|∇u|

which, after integrating over L

, proves (II.5.1). ut

Exercise II.5.1 Inequality (II.5.1) fails, in general, if Ω is not contained in some

layer L

. Suppose, for instance, Ω ≡ R

and consider the sequence

= exp[− |x| /(m + 1)], m ∈ N.

Show that

k∇u

m + 1

Modify this example to prove the invalidity of (II.5. 1) for Ω an arbitrary exterior

domain or a half-space.

The special case q = 2 in (II.5.1) plays an important role in several applica-

tions. In particular, it is of great interest in uniqueness and stability questions

to determine the smallest constant µ such that

70 II Basic Function Spaces and Related Inequalities

kuk

≤ µk∇uk

. (II.5.3)

The constant µ (sometimes called the Poincar´e constant) depends on the

domain Ω, and when Ω is bounded one easily shows that µ = 1/λ

, where λ

is the smallest eigenvalue of the probl em

−∆u = λu in Ω , u = 0 at ∂Ω ; (II.5.4)

see Sobolev 1963a, Chapter II, §16. An estimate of λ

comes from (II.5 .1) and

one has

≥ 4/[δ(Ω)]

However, a better estimate can be obtained as a consequence of the following

simple argument due to E. Picard (Pi cone 1946, §160).

In fact, assume as

befo re Ω ⊂ L

for some d > 0 and consider the function

U(x) =

u(x)

sin[π(x

+ d/2)/d]

, u ∈ C

∞

(Ω).

Since U (x) is bounded in L

and vanishes at −d/2, d/2, integrating by parts

we ﬁnd

0 ≤

d/2

−d/2



∂u

∂x

−

u(x) cot



π(x

+ d/2)



d/2

−d/2



∂u

∂x



−

d/2

−d/2



sin

−2



π(x

+ d/2)



−cot



π(x

+ d/2)



Hence

d/2

−d/2

≤ (d/π)

d/2

−d/2



∂u

∂x



which implies

kuk

≤ (d/π)k∇uk

Therefore, one deduces

µ ≤ d

/π

and, if Ω is bounded,

µ ≤ [δ(Ω)/π]

Notice that these estimates a re sharp in the sense that when n = 1 and

Ω = L

we have from (II.5.4 ) µ

−1

= λ

= [π/δ(Ω)]

= (π/d)

Generalizations of (II.5.1) and (II.5.3) are considered in the following ex-

ercises.

This pro of was brought to my attention by Professor Luigi Pepe.

II.5 Further Inequalities and Compactness Criteria in W

m,q

Exercise II.5.2 Let Ω ⊂ {x ∈ R

: −d/2 < x

< d/2, i = 1, . . . , n}. Use Picard’s

argument to show the following estimate for the Poincar´e constant µ:

µ ≤ d

/nπ

Exercise II.5.3 Let Ω ⊂ L

, for some d > 0. Show that

k∇uk

≤ (d/π)k∆uk

for all u ∈ W

1,2

(Ω) ∩W

2,2

(Ω). Thus, i n particular,

kuk

≤ (d/π)

k∆uk

Hint: Consider the identity: (u, ∆u) = −k∇uk

Exercise II.5.4 Let Ω be of ﬁnite measure and let u ∈ W

1,q

(Ω), 1 ≤ q < ∞. Show

the inequality

kuk

≤ β|Ω|

1/n

k∇uk

(II.5.5)

where

β =

q(n −1)

2(n − q)

√

if q < n

√

if q ≥ n .

Hint: Use (II.3.5) and the inequality

kuk

≤ |Ω|

(1/q)−(1/r)

kuk

, r > q.

Exercise II.5.5 Let Ω be bounded and let u ∈ W

1,q

(Ω), q > n. Show that, for all

∈ (n, q), the following inequality holds

kuk

≤ c kuk

1−q/q

k∇uk

q/q

with c = c(n, q, q

, Ω). Hint: From (II.3.18) and (I I.5.1) we ﬁnd kuk

≤ c k∇uk

Exercise II.5.6 Let Ω be bounded and C

-smooth, and let u be a vector function

with components in W

1,q

(Ω), 1 ≤ q < ∞, and u ·n = 0 at ∂Ω (n being the outer

normal). Show the inequality

kuk

≤ C k∇uk

, C ≤ δ(Ω)(|q − 2| + n + 1).

Hint (due to L.H. Payne): Integrate the identity:

i,j=1

|u|

q−2

] −(D

|u|

q−2

− |u|

−u

|u|

q−2

]

= 0.

An inequality of the type (II. 5.1) continues to hold even though u is not

zero at the boundary, provided one replaces u with u −u

Ω

. We shall begin to

prove the following result which traces back to Poincar´e (1894).

72 II Basic Function Spaces and Related Inequalities

Lemma II.5.1 For a > 0 let

C = {x ∈ R

: 0 < x

< a}. (II.5.6)

Then, for all u ∈ W

1,q

(C), 1 ≤ q < ∞,

ku −u

≤ nak∇uk

. (II.5.7)

Proof. For simplicity, we shall gi ve the proof in the case n = 3 . Clearly, i n

view of Theorem II.3.1, it is enough to show (II.5. 6) for u ∈ C

(Ω). Consider

the identity

u(x

, x

) − u(y

, y

) =

∂u

∂ξ

(ξ, x

, x

)dξ +

∂u

∂η

, η, x

)dη

∂u

∂ζ

, y

, ζ)dζ.

Integrating over the y-variables and raising to the qth power, we deduce

|u(x

, x

) −u

≤ |C|

−q



|∇u(ξ, x

, x

)|dξ

|∇u(y

, η, x

)|dy

dη +a

|∇u|dC



Employing in this relation the inequality (II. 3.3) along with the H¨older in-

equality and integrating over the x-variables we obtain

|u − u

≤ 3

|∇u|

which completes the proof. ut

Remark II.5.1 An extension of (II.5.7) to arbitrary locally Li pschitz do-

mains will be given in Theorem II. 5.4. Here, however, we wish to observe

that, unlike Theorem II. 5.1, some regularity a ssumptions on Ω are strictly

necessary fo r inequalities o f type (II.5.7) to hold, as shown by means of coun-

terexample in Courant & Hilbert (1937, Kapitel VII, §8.2); see also Fraenkel

(1979, a nd §2 in particula r). 

Let us now analyze some consequences o f Lemma II.5.1. Suppose Ω is a

cube of side a and sub divide it into N equal cubes C

, each having sides of

length a/N

1/n

. Applying (II.5.7) to each cube C

and using the Minkowski

inequality and (II.3. 3) one recovers

II.5 Further Inequalities and Compactness Criteria in W

m,q

kuk

q,Ω

≤

i=1

q−1



1/n



n(1−q)



udC



(2na)

q/n

k∇uk

q,Ω

Therefore, i ntroducing the N independent functions

(x) = 2

(q−1)/q



1/n



n(1−q)/q

(x),

with χ

characteristic function of the cube C

, from the previous inequality

one has the fol lowing result due to Friedrichs (1933).

Lemma II.5.2 Let C be the cube (II.5.6) and let

u ∈ W

1,q

(C), 1 ≤ q < ∞.

Then, given an arbitrary positive integer N, there exist N independent func-

tions ψ

∈ L

∞

kuk

q,C

≤

i=1



(2na)

q/n

k∇uk

q,C

. (II.5.8)

Inequality (II.5.8) is very useful in proving compactness results, as we are

about to show. In fact, let Ω be bounded and let {u

} ⊂ W

1,q

(Ω), 1 ≤ q < ∞,

be uniformly bounded in the norm k·k

1,q

. Extending u

by zero outside Ω and

denoting again by u

such an extension, we thus have tha t {u

} is uniformly

bounded in W

1,q

(C), for some cube C (see Exercise II.3.11), and therefore,

by Lemma II.5.2, Theorem II.2.4(ii) and Theorem II.3.2, it is not diﬃcult to

show the existence of a subsequence {u

} that is Cauchy in L

a consequence, converges strongly in L

(Ω). On the other hand, by Lemma

II.3.2 and by Exercise II.5.5, it follows that {u

} converges also in L

(Ω), for

all r ∈ [1, nq/(n −q)), if q < n, for all r ∈ [1, ∞) if q = n, while it converges in

C(Ω) if q > n. We have proved the following compact embedding result (see

Rellich 1930).

Theorem II.5.2 Assume Ω bounded, a nd let q ∈ [1, ∞). Then

1,q

(Ω) ,→,→ L

(Ω) ,

with arbitrary r ∈ [1, nq/(n −q)), if q < n, and arbitrary r ∈ [1, ∞), if q = n.

Finally, if q > n, then W

1,q

(Ω) ,→,→ C(Ω)

In Theorem II.5.2, when q < n, the exponent q

∗

= nq/(n −q) is excluded.

Actually one proves by means of counterexamples tha t the strong convergence

is, in general, ruled out in this case. For, in the ball B

consider the sequence

of functions

(x) =







(n−q)/q

(1 − m|x|) if |x| < 1/m

0 if |x| ≥ 1/m

m = 1, 2, . . .

74 II Basic Function Spaces and Related Inequalities

with q < n. One has

k∇u

= C

, ku

q∗

= C

with C

and C

independent of m. Since

lim

m→∞

(x) = 0 a.e. i n B

it follows that no subsequence can converge strongly in L

q∗

Theorem II.5.2 admits the following counterpart in negative Sobolev

spaces.

Theorem II.5.3 Let Ω be bounded. Then L

(Ω) ,→,→ W

−1,q

(Ω), for any

1 < q < ∞. Precisely, if {u

} ⊂ L

(Ω) is uniformly bounded, there exists a

subsequence {u

} and u ∈ L

(Ω) such that

lim

→∞

ku − u

−1,q

= 0 .

Proof. In view of inequality (II.5.1), we may endow W

1,q

(Ω) with the equiv-

alent norm k∇(·)k

. We observe next that, by assumption and by Theorem

II.2.4(ii i), there are u ∈ L

(Ω) and a subsequence {u

} such that u

→ u.

Set U

= u − u

. By Theorem II.3.5 and Theorem II.1.4, for each m

∈ N,

we can ﬁnd w

∈ W

1,q

(Ω) such that

−1,q

= |(U

, w

)|, k∇w

= 1 . (II.5.9)

Then, by Theorem II.5.2 and Theorem II.1.3(i i), there exist a subsequence

} and w ∈ W

1,q

(Ω) such that w

→ w in L

(Ω), and so (II.5.9)

delivers

−1,q

≤ |(U

, w)|+kU

−wk

≤ |(U

, w)|+C kw

−wk

which, i n turn, gives the desired result since U

→ 0 in L

(Ω) and w

→ w

in L

(Ω). ut

Some generalizations of Theorem II.5.2 are proposed to the reader in the

following exercises.

Exercise II.5.7 Assume Ω bounded and let q ∈ [1, ∞), m ≥ 1. Show that

m,q

(Ω) ,→,→ L

(Ω)

with arbitrary r ∈ [1, nq/(n − mq)) if mq < n and all r ∈ [1, ∞) if mq = n.

Finally, show that if mq > n, then W

m,q

(Ω) ,→,→ C

(Ω), for all k ∈ N such that

0 ≤ k < 1 − mq/n.

II.5 Further Inequalities and Compactness Criteria in W

m,q

Exercise II.5.8 Prove that, when Ω is bounded and locally Lipschitz, Theorem

II.5.2 and Exercise II.5.7 continue to hold if W

m,q

(Ω) is replaced by W

m,q

(Ω).

Hint: Use Theorem II.3.3 and (II.3.19).

We want now to obtain further inequalities as a consequence of the com-

pactness results just derived. The following theorem extends the Poincar´e

inequality (II.5.7) to more general domains.

Theorem II.5.4 Let Ω be bounded and locally Lipschitz. Then, for all u ∈

1,q

(Ω), 1 ≤ q < ∞, we have

ku − u

Ω

≤ ck ∇uk

, (II.5.10)

where c = c(n, q, Ω).

Proof. To simplify notation, we omit the subscript Ω. If (II.5.10) were not

true, a sequence {u

} ⊂ W

1,q

(Ω) would exist such that for all m ∈ N

= 0, ku

= 1, k∇u

≤ 1/m. (II.5.11)

Therefore, from (II.5.11)

2,3

and Exercise II.5.8 there is a subsequence con-

verging in the norm of W

1,q

(Ω) to some u ∈ W

1,q

(Ω) which, by (I I.5.11),

should have ∇u = 0, u = 0, namely, u ≡ 0 a.e. in Ω and kuk

= 1. This gives

a contradiction that proves the theorem. ut

Theorem II.5 .4 admits several interesting consequences, some of which are

left to the reader in the following exercises.

Exercise II.5.9 Let Ω be an arbitrary domain and let u ∈ W

1,1

loc

(Ω). Show that, if

Du = 0, then there is u

∈ R such that u = u

a.e. in Ω. Using this result, show

that, more generally, if u ∈ W

m,1

loc

(Ω) with D

u = 0, |α| = m, then u = P a.e in Ω,

where P is a polynomial of degree ≤ m − 1. Hint: Use Lemma II.1.1.

Exercise II.5.10 Assume Ω bounded and locall y Lipschitz and let u ∈ W

1,q

(Ω).

If q ∈ [1, n), prove the foll owing Poincar´e-Sobolev inequality :

ku − u

Ω

≤ ck∇uk

, (II.5.12)

where r = nq/(n−q) and c = c(n, q, Ω). Moreover, show that, if q > n, the fol lowing

inequality holds

ku − u

Ω

≤ c

k∇uk

. (II.5.13)

Hint: Use Theorem II.5.4 and (II.3.16)

1,3

Exercise II.5.11 Let u ∈ W

1,q

)), q > n. Show that the following inequality

holds

max

x∈B

)

|u(x) − u(x

)| ≤ c r

1−n/q

k∇uk

q,B

)

with c = c(n, q). Hint: Use (II.5.13) on the unit ball and then rescale the result for

a ball of radius r.