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

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266 IV Steady Stokes Flow in Bounded Domains

q,Ω

+ k∇p

q,Ω

≤ c (kf

+ kv

1,q,Ω

−Ω

+ kp

q,Ω

−Ω

) .

Letting ε → 0 into this inequality and recalling the properties o f the molli-

ﬁer (II.2.9) along with the deﬁnition of weak derivative, one thus proves the

theorem for m = 0. The general case is now treated by induction. Assuming

that the theorem holds for m = ` − 1, ` ≥ 1, we shall show it for m = `. By

hyp othesis we then have

v ∈ W

`+1,q

loc

(Ω), p ∈ W

`,q

loc

(Ω)

and, moreover,

|v|

`+1,qΩ

] + |p|

`,q,Ω

] ≤ c



kfk

`,q,Ω

+ kvk

1,q ,Ω

−Ω

] + kpk

q,Ω

−Ω

]



(IV.4.8)

where Ω

⊂ Ω

]

, Ω

]

⊂ Ω

. We now choose ϕ in (IV.4.5) as a C

∞

function

that is one in Ω

and zero outside Ω

]

. Appl ying Theorem IV.2.1 to solutions

to (IV.4 .5) and recalling (IV.4.8) we thus deduce

`+2,q,Ω

+ |p

`+1,q,Ω

≤ c



kfk

`,q,Ω

] + kv

`+1,q,Ω

]

−Ω

0 + kp

q,Ω

]

−Ω



≤ c

(kfk

`,q,Ω

+ kvk

1,q,Ω

−Ω

+ kpk

q,Ω

−Ω

) .

This inequality, in the limit ε → 0, then proves the validity of the theorem for

arbitrary m ≥ 0. ut

Remark IV.4.1 It is of some interest to observe that if Ω

− Ω

satisﬁes

the cone condition, the term containing the pressure on the right-hand side of

(IV.4.4) can be removed, provided we modify p by the addition of a suitable

constant. Furthermore, if Ω

000

⊃ Ω

with Ω

000

−Ω

satisfying the cone condi-

tion, then the term kvk

1,q ,Ω

−Ω

can be replaced by kvk

q,Ω

000

−Ω

; see Remark

IV.4. 2. 

The next result provides a sharpened version of that just proved. In thi s

respect, we observe that if v ∈ W

1,r

loc

(Ω) satisﬁes (IV. 1.2) fo r al l ϕ ∈ D(Ω),

with f ∈ W

−1,q

(ω), for all bounded subdomains ω with ω ⊂ Ω, and where

a priori r 6= q, by Lemma IV.1.1 we can asso ci ate to v a pressure ﬁeld p

satisfying (IV.1.2) with p ∈ L

loc

(Ω), µ = min(r, q). We have

Theorem IV.4.2 Let Ω satisfy the assumption of Theorem IV.4.1. Assume

v i s weakly di vergence-free with ∇v ∈ L

loc

(Ω), 1 < r < ∞, and satisﬁes

identity (IV.1.3). Then, if

f ∈ W

m,q

loc

(Ω), m ≥ 0, 1 < q < ∞,

it follows that

v ∈ W

m+2,q

loc

(Ω), p ∈ W

m+1,q

loc

(Ω),

where p is the pressure ﬁeld associated to v by Lemma IV.1.1.

IV.4 Interior L

-Estimates 267

Proof. By Theorem IV.4.1, it is enough to show

∇v ∈ L

loc

(Ω).

(IV.4.9)

If r ≥ q the assertion is obvious. Therefore, take q > r. Then f ∈ L

loc

(Ω)

and, by Theorem IV.4.1 and the embedding Theorem II.3. 4, we deduce

v ∈ W

1,r

loc

(Ω)

with r

= nr/(n − r) (> r) if r < n and for arbitrary r

> 1 if r ≥ n. In the

latter case (IV.4.9) follows. If q ≤ r

< n we again draw the same conclusion.

So, assume 1 < r

< q. Then f ∈ L

loc

(Ω), and T heorem IV.4.1 and Theorem

II.3.4 imply

v ∈ W

1,r

loc

(Ω),

with r

= nr

/(n − r

) = nr/(n − 2r) (> r

) if 1 < r

< n and arbi trary

> 1, whenever r

≥ n. If either r

≥ q o r r

≥ n, (IV.4.9) follows; other-

wise we iterate the above procedure a ﬁnite number of times until (IV.4 .9) is

established. ut

Combining the result just proved with Theorem II.3.4 (speciﬁcally, in-

equality (II.3.18)) we at once obtain the following theorem concerning interior

regularity of q-weak solutions.

Theorem IV.4.3 Let v be a q-weak solution to the Stokes problem (IV.0.1),

(IV.0.2) corresponding to f ∈ C

∞

(Ω). Then, v, p ∈ C

∞

(Ω) where p is the

pressure ﬁeld associated to v by Lemma IV.1.1.

Intermediate regularity results are directly obtainable from T heorem IV.4.1

and the embedding Theorem II.3.4 and are left to the reader as an exercise.

Other regularity results i n H¨older norms can be obtained from the results of

Section IV.7.

Exercise IV.4.1 (Ladyzhenskaya 1969). In the case where q = 2, Theorem IV.4.1 is

obtained in an elementary way. Denote by ϕ the “cut-oﬀ” function of that theorem,

multiply (IV.4.3)

) by ϕ

∆v

and integrate by parts to show (IV.4.4) with m = 0,

q = 2. (Observe that if ζ ∈ C

(Ω), kD

ζk

= k∆ζk

.) Use then the induction

procedure to prove the general case m ≥ 0.

Exercise IV.4.2 Show that Theorem IV.4.1 also holds when ∇ · v = g 6≡ 0, pro-

vided g ∈ W

m+1,q

loc

(Ω). In such a case, the term

kgk

m+1,q,Ω

must be added to the right-hand side of (IV.4.4).

We shall next consider interior estimates for q-generalized solutions. Specif-

ically, we have the following theorem.

See footnote 1 in this section.

268 IV Steady Stokes Flow in Bounded Domains

Theorem IV.4.4 Let Ω, Ω

, Ω

, and v be as in Theorem IV.4.1. Suppose

f ∈ W

−1,q

(ω), for all bounded domains ω with ω ⊂ Ω. Then the following

inequality holds:

kvk

1,q,Ω

+ kpk

q,Ω

≤ c (kfk

−1,q,Ω

+ kvk

q,Ω

−Ω

+ kpk

−1,q,Ω

−Ω

) ,

(IV.4.10)

where p is the pressure ﬁeld associated to v by Lemma IV.1.1.

Proof. Let ϕ be as i n Theorem IV.4.1 and set

u = ϕv, π = ϕp .

From the assumption and Lemma IV.1.1, we know that (v, p) satisﬁes (IV.1.3)

for all ψ ∈ D

1,q

(Ω

). Therefore, choosing into this relation ψ = ϕφ, φ ∈

1,q

), we then readily deduce that u, π satisfy the identities

(∇u, ∇φ) − (π, ∇ · φ) = −[f

, φ], for all φ ∈ D

1,q

(u, ∇χ) = −(g, χ), for all χ ∈ D

1,q

)

(IV.4.11)

with

= f

− p∇ϕ + 2∇ϕ · ∇v + v∆ϕ

, φ] : = hf , ϕ φi

g = ∇ϕ ·v.

(IV.4.12)

Applying Theorem IV.2.2 to the above problem we then obtain that u and π

obey the inequality

|u|

1,q

+ kπk

≤ c(|f

−1,q

+ kgk

). (IV.4.13)

On account of (IV.4.12), it foll ows that

−1,q

≤ c



−1,q

+ |p∇ϕ + 2∇ϕ · ∇v + v∆ϕ|

−1,q



(IV.4.14)

and

kgk

≤ k∇ϕ ·vk

≤ c

kvk

q,Ω

−Ω

. (IV.4. 15)

Let φ be arbitrary element from D

1,q

). We distinguish the two cases:

(i) n/(n − 1) < q < ∞,

(ii) 1 < q ≤ n/(n − 1).

In case (i), q

< n and so, from the Sobolev inequality (II.3.7), we derive

kφk

,Ω

−Ω

≤ c

|φ|

1,q

which, after a simple calculation, allows us to show that

IV.4 Interior L

-Estimates 269

−1,q

+ |p∇ϕ+ 2∇ϕ · ∇v + v∆ϕ|

−1,q

≤ c

(kfk

−1,q,Ω

+ kvk

q,Ω

−Ω

+ kpk

−1,q,Ω

−Ω

) .

(IV.4.16)

In case (ii) choosing ψ = ϕ e

into (IV.1.3) delivers

(∇v

, ∇ϕ) − (p, D

ϕ) = −[f

, ϕ] , for all i = 1, . . . , n.

As a consequence, observing that

(2∇ϕ · ∇v + v∆ϕ) =

∇ϕ · ∇v

we ﬁnd

−p∇ϕ + 2∇ϕ · ∇v + v∆ϕ, φ] = [f

−p∇ϕ + 2∇ϕ ·∇v + v∆ϕ, φ + C],

(IV.4.17)

for a ny constant vector C. Choosing

C = −

|M|

φ, M ≡ Ω

−Ω

from Poincar´e’s inequality (II.5.10) we deduce

kφ + Ck

≤ c

|φ|

1,q

. (IV.4.18)

Since





−p∇ϕ + 2 ∇ ϕ · ∇v + v∆ϕ, φ + C]|

≤ c

(kfk

−1,q,Ω

+ kvk

q,M

+ kpk

−1,q,M

) kφ + Ck

(IV.4.19)

from (IV.4.17)–(IV.4.19) we conclude, also in case (ii), the validity of (IV.4.16).

The theorem becomes then a consequence of (IV.4.13)–(IV.4.1 6) and of the

properties of the function ϕ. ut

Remark IV.4.2 If, in the previous theorem, we modif y p by the addition of

a constant, then we can remove the term involving the pressure on the right-

hand side of (IV.4.10), provided Ω

−Ω

satisﬁes the cone condition. Therefore,

under these conditions, we obtain, in particular, the following estimate

kvk

1,q,Ω

+ inf

k∈R

kp + kk

q,Ω

≤ c (kfk

−1,q,Ω

+ kvk

q,Ω

−Ω

) . (IV.4.20)

To see this, set, for simplicity, D = Ω

−Ω

, and let g ∈ W

1,q

(D) be arbitrary.

Moreover, let φ ∈ C

∞

(D) be a ﬁxed function with

φ = |D|. We then

consider the problem

∇· ψ = g − φ g

, ψ ∈ W

2,q

(D) , kψk

2,q

≤ c kgk

1,q

. (IV.4.21)

270 IV Steady Stokes Flow in Bounded Domains

In view of Theorem III.3.3, (IV.4.21) has at least one solution. We thus replace

this solution in relation (IV.1. 3) and integrate by parts to obtain

(p + k, g) = hf , ψi −(v, ∆ψ) , (IV.4.22)

where k = −(pφ)

. Thus, from (IV.4.22), with the help of the estimate in

(IV.4.21), and by the arbitrariness of g, we a t once deduce

kp + kk

−1,q,D

≤ c (kfk

−1,q,D

+ kvk

q,D

) ,

which is what we wanted to prove. 

The result to foll ow is a sharpened version of the preceding one. I n this

respect, we recall that if v ∈ W

1,r

loc

(Ω) satisﬁes (IV.1.2) for all ϕ ∈ D(Ω)

with f ∈ W

−1,q

(ω), ω as in Theorem IV.4.4 and a priori r 6= q, we can

always associate with v a pressure ﬁeld p satisfying (IV.1.3). In particular,

p ∈ L

loc

(Ω) where µ = min(r, p). We have

Theorem IV.4.5 Let Ω be an arbitrary domain in R

, n ≥ 2. Suppose

v ∈ W

1,r

loc

(Ω), 1 < r < ∞, is weakly divergence-free and satisﬁes (IV.1.2 ) for

all ϕ ∈ D(Ω). Then, if f ∈ W

−1,q

(ω) , 1 < q < ∞, for all bounded domains ω

with ω ⊂ Ω it follows that

v ∈ W

1,q

loc

(Ω), p ∈ L

loc

(Ω),

where p is the pressure ﬁeld associated to v by Lemma IV.1.1.

Proof. If r ≥ q there is nothing to prove. We then take q > r. Consider

problem (IV.4.11)–(IV.4.12). If r ≥ n, by the embedding Theorem II.3.4 we

easily deduce

v ∈ L

loc

(Ω), p ∈ W

−1,q

(ω)

for all bounded ω with ω ⊂ Ω. and consequently, repeating the reasonings

used in the proof of Theorem IV.4.4 we obtain, in particula r,

−1,t

≤ c (kf k

−1,t,Ω

+ kvk

t,Ω

+ kpk

−1,t,Ω

)

kgk

≤ ckvk

t,Ω

(IV.4.23)

with t = q. Therefore, by Theorem IV.2.2, we deduce the assertion of the

theorem. If r < n, again by Theorem II.3.4, we obtain

v ∈ L

loc

(Ω), p ∈ W

−1,r

(ω) , r

= rn/(r − n)

and so the assertion again follows from (IV. 4.23) with t = r

and Theorem

IV.2. 2 if r

≥ q. If r

< q, from (IV.4.23) it still follows that

∈ D

−1,r

)

and so, using once more Theorem IV.2.2,

IV.5 L

-Estimates Near the Boundary 271

v ∈ W

1,r

loc

(Ω), p ∈ L

loc

(Ω) (r

> r).

This “bootstrap” argument becomes of the same type as that of Theorem

IV.4. 2 and then, proceeding as i n the proof o f that theorem, we arrive at the

desired conclusion. ut

Exercise IV.4.3 Show that Theorem IV.4.4 continues t o hold if ∇ · v = g 6≡ 0,

where g ∈ L

loc

(Ω). The inequality in the theorem is then modiﬁed by adding the

term

kgk

q,Ω

to its right-hand side.

Exercise IV.4.4 It is just worth noting that interior estimates of the type proved in

Theorem IV.4.1 and Theorem IV.4.4 are also valid for the “scalar” case, namely, the

Poisson equation ∆u = f . In fact, l et u ∈ W

1,q

loc

(Ω), q ∈ (1, ∞), satisfy (∇u, ∇ψ) =

hf, ψi for all ψ ∈ C

∞

(Ω). Show that, if f ∈ W

m,q

loc

(Ω), m ≥ −1, then necessarily

u ∈ W

m+2,q

loc

(Ω), and the following estimate holds

kuk

m+2,q,Ω

≤ c (kfk

m,q,Ω

+ kuk

q,Ω

) ,

for all Ω

and Ω

as in the above theorems, and with c independent of v and f .

IV.5 L

-Esti mates Near the Boundary

We wish now to determine L

-estimates analogous to those of Theorem IV.4.1 ,

but in a subdomain of Ω abutting a suitably smooth porti on σ o f the bound-

ary. This will then allow us, in particular, to obtain regularity results for

generalized solutions up to the boundary. Following the method outlined by

Cattabriga (1961) and based on the work of Ag mon, Douglis, & Nirenberg

(1959), the strategy we shall adopt is to introduce a suitable change of vari-

ables so that, locally “near” the boundary, the Stokes problem goes over into

a simila r problem in the half-space. The desired estimate will then follow

directly from Theorem IV.3.2 and Theorem IV.3.3.

Assume Ω has a boundary portion σ of class C

. Without loss, we may

rotate the coordinate system with the origin at a point x

∈ σ in such a way

that, if we denote by ζ = ζ(x

, . . . , x

n−1

) the function representing σ,

∇ζ(0) = 0. (IV.5.1)

(This means that the axes x

, . . . , x

n−1

are in the tangent plane at σ, at

the point x

.) Next, we denote by Ω

any bounded subdomain of Ω with

σ = ∂Ω

∩ ∂Ω and consider a pair of functions

v ∈ W

2,q

(Ω

), p ∈ W

1,q

(Ω

), 1 < q < ∞,

solving a.e. the Stokes problem in Ω

corresponding to f ∈ L

(Ω

) and

∗

∈ W

2−1/q,q

(σ). If we di rect the positive x

-axis into the i nterior of Ω,

for suﬃciently small d > 0 the cylinder

272 IV Steady Stokes Flow in Bounded Domains

ω = {x ∈ Ω : |x

| < d, ζ < x

< ζ + 2d}

is contained in Ω

. Let ϕ ∈ C

∞

) with ϕ = 0 in Ω − ω and ϕ = 1 in ω

with

= {x ∈ Ω : |x

| < δ, ζ < x

< ζ + 2δ, δ < d}.

If we make the change of va riables

= x

, y

= x

− ζ, (IV.5.2)

the functions v, p, f, v

∗

, and ϕ go over into

v, bp,

∗

, and bϕ, respectively,

while ω and ω

are transformed into the cubes

bω = {y ∈ R

: |y

| < d, 0 < y

< 2d}

and

bω

= {y ∈ R

: |y

| < δ, 0 < y

< 2δ},

respectively. Moreover, setting

u = bϕ

v, π = bϕbp

and extending all the ﬁelds to zero outside bω, after a simple calculation we

ﬁnd

∆u = ∇π + F

∇ · u = g

)

in R

u = Φ at Σ ≡ R

n−1

×{0},

(IV.5.3)

where

≡ bϕ

+ D

π) + D

) + α

bp + βbv

+ γ

g ≡ D

) + η

≡ bϕbv

∗i

(IV.5.4)

and a

, b

, c

, α

, β, γ

, and η

are continuously diﬀerentiable functions

in the closure of bω. Moreover, the f unctions a

, b

, and c

are bounded by

A|∇ζ| with a constant A independent of d, while α

, β, γ

, and η

are zero

outside bω. Notice that by (IV.5.1)

Notice that if f and

f are related by transformation (IV.5.2),

∂f

∂x

∂

∂y

−

∂

∂y

∂ζ

∂x

, i = 1, . . . n − 1;

∂f

∂x

∂

∂y

;

and

∂

∂y

∂f

∂x

−

∂f

∂x

∂ζ

∂y

, i = 1, . . . n − 1.

IV.5 L

-Estimates Near the Boundary 273

(0) = b

(0) = c

(0) = 0 . (IV.5.5)

From (IV.5.4) we readily obtain

kF k

q,R

≤ c



q,bω

+ kbpk

q,bω

1,q ,bω

+ akD

q,R

+ bk∇πk

q,R



kgk

1,q,R

≤ c



1,q,bω

+ ckD

q,R



(IV.5.6)

where c

and c

can b e taken independent of d, while

a = max

j,k

max

bω

)

b = max

j,k

max

bω

)

c = ma x

j,k

max

bω

Furthermore,

Φ ∈ W

1,q

(Σ), hh∇Φii

1−1/q,q

< ∞,

see Exercise IV.5.1, and

u ∈ L

We may thus applyTheorem IV.3.2 with m = 0 to system (IV. 5.3) to deduce,

in particular, that u and π satisfy inequality (IV.3.30). We then have

q,bω

+ k

1,q,bω

+ kbpk

q,bω

≤ c

(kfk

q,ω

+ kvk

1,q ,ω

+ kpk

q,ω

) (IV.5.7)

and, by Exercise IV.5.1, (II.4.19), and the properties of the function ϕ, we

have also

hh∇Φii

1−1/q,q

≤ c

kbϕ

∗

2−1/q,q(Σ)

≤ c

∗

2−1/q,q(σ)

, (IV.5. 8)

where c

, c

, and c

do not depend on d, f, v, p, and Φ. Collecting (IV.5.6)–

(IV.5.8) and using (IV.3.30) we deduce

[1 −c

(a + c)]kD

q,R

+(1 −c

b)k∇πk

q,R

≤ c



kf k

q,ω

+ kv

∗

2−1/q,q(σ)

+ kvk

1,q,ω

+ kpk

q,ω



(IV.5.9)

where, agai n, c

, c

, and c

do not depend on d, f, v, p, and Φ. Recalling

(IV.5.5), we can take d as smal l to satisfy the conditions

(a + c) < 1/c

, b < 1/c

Thus, from (IV.5.9) and the obvious inequalities:

274 IV Steady Stokes Flow in Bounded Domains

kwk

`,q,ω

≤ kϕwk

`,q,ω

≤ c

kbϕ bwk

`,q,R

≤ c

kϕwk

`,q,ω

(IV.5.10)

holdi ng for all ` ≥ 0 and for a suitable choice of c

, c

, we ﬁnally obtain

kvk

2,q,ω

+ kpk

1,q,ω

≤ c



kfk

q,ω

+ kv

∗

2−1/q,q(σ)

+ kvk

1,q,ω

+ kpk

q,ω



with c

independent of v, p, f and v

∗

. It is now easy to generalize this

estimate to the case when

v ∈ W

m+2,q

(Ω

), p ∈ W

m+2,q

(Ω

), m > 0.

The corresponding assumptions on the data are then

f ∈ W

m,q

(Ω

), v

∗

∈ W

m+2−1/q,q

(σ) (IV.5.11)

and, further, σ of class C

m+2

. To this end, it is suﬃcient to use Theorem

IV.3. 2 in its generality along with an inductive argument entirely analo gous

to that employed in Theorem IV.4.1. If we do this we obtain

kvk

m+2,q,ω

+ kpk

m+1,q,ω

≤ c



kfk

m,q,ω

+ kv

∗

m+2−1/q,q(σ)

+ kvk

1,q ,ω

+ kpk

q,ω



(IV.5.12)

which is the general estimate we wanted to show.

Following Cattabrig a (1961), we shall now prove that for (IV.5.12) to hold

it is enough to assume only

v ∈ W

1,q

(Ω

), p ∈ L

(Ω

). (IV.5.13)

More precisely, suppose v, p is such a pair o f ﬁelds satisfying identity (IV.1 .3)

for all ψ ∈ C

∞

(Ω

) and some f , along with the condition v = v

∗

at σ

(in the trace sense). Then if f and v

∗

verify (IV.5.12) for m ≥ 0 and σ is

of cla ss C

m+2

it follows tha t the norms of v and p on the left-hand side

of (IV.5.13) are ﬁnite and (IV.5. 13) holds. We shall prove this assertion for

m = 0, the general case being treated by a simple iteration and therefore left

to the reader. By performing on v and p the same kind of transformation

made previously to arrive at (IV.5 .3), we can deduce that this time u and π

are q-generalized soluti ons to the inhomo geneous Stokes problem in the hal f-

space (see the deﬁnition befo re Theorem IV.3.3) correspondi ng to the data

F , g and Φ deﬁned in (IV.5.4)

. In particular we have

(∇u, ∇ψ) − (π, ∇·ψ) = −(F , ψ), for all ψ ∈ D

1,q

). (IV.5.14)

Replace ψ by the di ﬀerence quotient ∆

ψ, where the variatio n is taken with

respect to any of the ﬁrst n − 1 coordinates, see Exercise II.3.13. Since

IV.5 L

-Estimates Near the Boundary 275

∆

w = −

∆

−h

we immediately deduce that ∆

u and ∆

π are agai n a q-generali zed solution

corresponding to the free terms ∆

F , ∆

g and to the boundary data ∆

Φ.

Recalling that

k∆

≤ k∇wk

see Exercise II.3.13(iii), from (IV.5.4) and (IV.5.7) it follows that

|∆

F |

−1,q,R

≤ c

(kfk

q,ω

+ kpk

q,ω

+ kvk

1,q,ω

+ak∆

∇uk

q,R

+ bk∆

πk

q,R



k∆

q,R

≤ c



kvk

1,q,ω

+ ck∆

∇uk

q,R



with c

, c

independent of d and h. Since

∆

u ∈ D

1,q

), ∆

π ∈ L

we may use this latter inequality, the results of Exercise IV.5.1 and estimate

(IV.3.32) together with the property

∇∆

u = ∆

∇u

to deduce

[1 −c

(a + c)]k∆

∇uk

q,R

+(1 − c

b)k∆

πk

q,R

≤ c



kf k

q,ω

+ kv

∗

2−1/q,q(σ)

+ kvk

1,q,ω

+ kpk

q,ω



(IV.5.15)

with c

independent of h and d. Taking d so small that

(a + c) < 1/c

, b < 1/c

from (IV.5.15) it follows that

k∆

∇uk

q,R

+k∆

πk

q,R

≤ c



kf k

q,ω

+ kv

∗

2−1/q,q(σ)

+ kvk

1,q,ω

+ kpk

q,ω



with c

independent of h. From the properties of the diﬀerence quotient (see

Exercise II.3.1 3), we then conclude that ∇

∇u and ∇

π exist

and belong to

). Moreover,

k∇

∇uk

q,R

+ k∇

πk

q,R

≤ c



kfk

q,ω

+ kv

∗

2−1/q,q(σ)

+ kvk

1,q ,ω

+ kpk

q,ω



(IV.5.16)

From (IV.5.3)

and (IV.5 .16) it is not hard to show

We recall that ∇

is deﬁned below (IV.3.10).