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

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316 V Steady Stokes Flow in Exterior Domains

On the other hand, from (IV.8.14) we deduce

, q

)(x −y)| = O(|x|

1−n−|α|

), |α| ≥ 0, (V.3.23)

uniform ly with respect to y in a bounded set. Likewise, since

(x − y) − U

(x)) | =



∂

∂x

(x − βy))



where β ∈ [0, 1], by (IV.2.6) it follows

(x − y) − U

(x)) | = O(|x|

1−n−|α|

), |α| ≥ 0, (V.3.24)

again uniforml y in y in a bounded set. Thus, by observing that v

(1)

(x) is

inﬁnitely diﬀerentiable fo r all x ∈ Ω − S, relations (V.3.19)

, (V.3.20), and

(V.3.21) foll ow from (V.3.22 )–(V.3.24). The analo gous estimate for p can be

shown in an entirely similar way. To prove (V.3.17), we take R so large that

S ⊂ B

(x). Therefore, for such an R,

Ω

(R)

(x − y)f

(y)dy =

Ω

(x − y)f

(y)dy. (V.3.25)

From Exercise V.3.1 we know that, under the stated assumptions, v(x) obeys

(V.3.6) with α = 0 for almost all x ∈ Ω. Therefore, from (V.3.22) and (V.3.25)

we ﬁnd, for almost all x ∈ Ω,

v(x) = v

(1)

(x) + v

(2)

(x), (V.3.26)

where

(2)

(x) ≡ −

Ω

(R)

(x − y)v

(y)dy. (V.3.27)

Since v − v

(1)

is independent of R, so is v

(2)

. Let us show that

(2)

≡ 0. (V.3.28)

Actually, from (V.3.21) and (V.3.27) we deduce for a suitable constant c in-

dependent of R

(2)

(x)| ≤ c

log

n+2

Ω

R/2,R

(x)

|v|, (V.3.29 )

where α = 1 if n = 2 , α = 0 if n > 2 and Ω

R,2R

(x) = {y ∈ Ω : R/2 < |x−y| <

R}. It is easy to prove that under the assumptions (i) and (ii ) the right-hand

side of (V.3.2 9) tends to zero as R → ∞. In fact,

log

n+2

Ω

R/2,R

(x)

|v| ≤











log

(assumption (i))

(log R)

α+1/t

(assumption (ii ))

V.3 Representation of Solutions and Behavior at Large Distances 317

with c

, c

independent of R. So, there exists an n × n matrix A with

trace(A) = 0 and a vector v

∞

such that

(2)

= A · x + v

∞

On the other hand, by using either (i) or (ii) and observing that

(1)

(x) = o(|x|) as |x| → ∞,

we readily show

A = 0,

thus establishing (V.3.17). Finally, since

Ω

∆H

(R)

(x − y)v

(y)dy = ∆v

(2)

(x),

identity (V.3.18) follows from (V.3.7), with α = 0, and (V.3.28). ut

From the proof just given it comes out that one may weaken assumptions

(i) or (ii) on condition that pol ynomials in v of suitable degree are added to

the right-hand sides of (V.3.17) and (V. 3.18). In particular, we wish to single

out the following result, which will be of interest for later purposes.

Theorem V.3.3 Repla ce assumptions (i) and (ii) of Theorem V.3.2 with

v ∈ L

(Ω), for some q ∈ [1, ∞), (V.3.30)

the other assumptions remaining unaltered. Then, there exist a scalar p

∞

, a

vector v

∞

, and an n × n matrix V

∞

with trace(V

∞

) = 0, such that

v(x) = v

∞

+ V

∞

· x + v

(1)

(x)

p(x) = p

∞

+ p

(1)

(x),

(V.3.31)

where v

(1)

and p

(1)

are deﬁned in (V.3.19).

Proof. To show (V.3 .31) it is enough to show that (V.3.30) implies (V.3.28).

In this respect we have

(2)

(x)| ≤

Ω

R/2,R

(x)

(R)

(x −y)||D

v(y)|dy

≤ c

log

n/q

q,Ω

R/2,R

(x)

where c does not depend on R and α = 1 if n = 2 and α = 0 if n > 2. Letting

R → ∞ into this relation proves (V.3.28). ut

Remark V.3.4 In Theorem V. 3.2 and Theorem V.3.3 no hypothesis is made

about the behavior of the pressure at inﬁnity; rather, it is derived as a conse-

quence of the behavior assumed on the velocity ﬁeld. 

318 V Steady Stokes Flow in Exterior Domains

Remark V.3.5 As pointed out in Remark V.2.2, for Ω a two-dimensional

exterior domain, by the method of Theorem V.2.1 we can construct a ﬁeld v

satisfying (i)-(iii) and (v) of Deﬁnition V.1.1, with q = 2. However, we are not

able to check condition (iv), for a prescrib ed v

∞

∈ R

. Nevertheless, if f is of

compact support, Theorem V.3.2 implies that v does tend to a certain vector

∞

∈ R

. In fact, since v ∈ D

1,2

(Ω), it is then simple to show that

∂Ω

T ( v, p) · n = 0, (V.3.32)

where R is taken so large that Ω

includes the support of f and p is the

pressure ﬁeld associated to p by Lemma V.1.1. To prove (V.3.32) we notice

that from (V.0.1) and from the deﬁnition of T we have, for all S > R,

∂Ω

T ( v, p) · n =

∂Ω

T (v, p) · n =

∂Ω

(2D(v) · n + pn).

However, since v ∈ D

1,2

(Ω

) and, by Lemma V.1.1, the corresponding pres-

sure ﬁeld p ∈ L

(Ω

), we can ﬁnd a sequence {S

}, S

→ ∞ as k → ∞, a long

which the last integral on the right-hand side of the preceding identities tends

to zero, thus proving (V.3.32). Theorem V.3. 2 then ensures the existence of

a well-deﬁned vector v

∞

to which v tends at large distances. In general, v

∞

cannot be prescribed a priori (in particular, cannot be zero) unless the data

verify a suitable restriction, see Section V.7. Notice, also, that, in the partic-

ular case when v

∗

≡ f ≡ 0, describing the slow, plane mo tion of the liquid

past a cyli nder, we can ta ke Ω

≡ Ω in (V.3.32), thus obtaining that the total

force exerted by the liquid on the cylinder is zero. This is ano ther fo rm of the

Stokes paradox; see al so Section V.7. 

Let us now derive some signiﬁcant implications of Theorem V.3.2. We

begin with a uniqueness result for q-generalized solutions.

Theorem V.3.4 Let v be a q-generalized solution to the Stokes problem

(V.0.1), (V.0.2) in an exterior, three-dimensional

domain of class C

, corre-

sponding to f ≡ v

∗

≡ 0. Then v ≡ 0.

Proof. From Lemma V.1.1 and the regularity results of Theorem IV.4.1 and

Theorem IV.5.1, we derive

v ∈ W

2,q

loc

(Ω) ∩ C

∞

(Ω), p ∈ W

1,q

loc

(Ω) ∩ C

∞

(Ω), for all q ∈ (1, ∞).

We may then apply Theorem V.3.2 to deduce

v = O(|x|

−1

), p, ∇v = O(|x|

−2

). (V.3.33)

Observe that v, p ∈ C

∞

(Ω

The result continues to hold in any space dimension n ≥ 3.

V.3 Representation of Solutions and Behavior at Large Distances 319

For ﬁxed R > δ(Ω

), we dot-multiply (V.0.1)

by v, integrate by pa rts over

Ω

(this is allowed by Exercise II.4.3), and take into account (V .0.1)

. We

thus deduce

Ω

∇v : ∇v =

∂B

n · (∇v · v − pv).

Estimating the surface integral through (V.3.3 3), and letting R → ∞ then

proves the result. ut

Similar uni queness results can be obtained for regular solutions possessing

a suitable behavior at large distances. For example, we have the following

result which for space dimension n = 2 furnishes another fo rm of the Stokes

paradox, already considered for generalized solutions in Theorem V.2.1. The

proof is much like that of Theorem V.3.4 and, therefore, it will be omitted.

Theorem V.3.5 Let v, p be a regular solution to the Sto kes system (V.0 .1),

in a C

-smooth exterior domai n of R

, corresponding to f ≡ v

∗

≡ 0. Then,

if as |x| → ∞

v(x) =

(

o(log |x|) if n = 2

o(1) if n > 2,

it follows that v ≡ 0.

Other consequences of Theorem V.3.2 are left to the reader in the following

exercises.

Exercise V.3.3 Let v, f , and Ω satisfy the assumptions of Theorem V.3.2. Show

that kv −v

= |v|

1,r

= ∞ for all q ∈ (1, n] and all r ∈ (1, n/(n−1)], unless T = 0.

Exercise V.3.4 Prove the f ollowing result of Liouville type. Let v, p be a regular

Stokes ﬂow in R

, corresponding to zero or, more generally, potential-like body force.

Then if v is bounded, it follows that v = const.

Exercise V.3.5 Let Ω ≡ R

. Prove that if v and f satisfy the assumptions of

Theorem V.3.2 the following asymptotic formulas hold:

v(x) = v

∞

+ U (x) ·

f + σ(x),

p(x) = p

∞

− q(x) ·

f + η(x),

where v

∞

, p

∞

are vector and scalar constants, while σ and η satisfy (V.3.21).

Exercise V.3.6 Show the following “scalar” version of Theorem V.3.2. Let Ω be a

domain of class C

and let u ∈ W

2,q

loc

(Ω), some q ∈ (1, ∞), be a solution to ∆u = f

in Ω, where f ∈ L

(Ω) is of bounded support. Then, if at least one of the following

conditions is satisﬁed as |x| → ∞:

320 V Steady Stokes Flow in Exterior Domains

(i) |u(x)| = o(|x|)

(ii)

|x|≤r

|u(x)|

(1 + |x|)

n+t

dx = o(log r) , some t ∈ (1, ∞),

there exist u

∞

∈ R such that for almost all x ∈ Ω

u(x) = u

∞

Ω

E(x − y) f (y)dy −

∂Ω

[E(x − y)

∂u

∂y

(y) − u(y)

∂E

∂y

(x −y)]n

(y)dσ

≡ u

∞

+ u

(1)

(x)

Moreover, as |x| → ∞, u

(1)

(x) is inﬁnitely diﬀerentiable and there the following

asymptotic representations hold:

(1)

(x) = a E(x) + σ(x) ,

where

a = −

∂Ω

∂u

∂x

Ω

and, for all |α| ≥ 0,

σ(x) = O(|x|

1−n−|α|

) .

Hint: Reproduce the same type of argument adopted in the proof of Theorem V.3.2,

by replacing the (tensor) Stokes-Fujita t runcated fundamental solution with the

(scalar) Laplace truncated fundamental solution deﬁned in (V.3.9)–(V.3.13).

V.4 Ex istence, Uniqueness, and L

-Esti mates: Strong

Solutions

Our next objective is to investigate to what extent the results proved in Sec-

tion IV. 6 can be generalized to the case when the region of motion is an

exterior one. Speciﬁcally, in the present section we shall be concerned with ex-

istence, uniqueness, and L

-estimates of strong solutions to the Stokes problem

(V.0.1), (V. 0.2), i.e., solutions with velocity ﬁelds po ssessing at least second

derivatives, while in Section 5 we analyze the same question for q-generalized

solutions.

To begin, we shall study some properties of solutions {v, p} to the Stokes

system

∆v = ∇p + f

∇ · v = 0

)

in Ω

v = v

∗

at ∂Ω,

(V.4.1)

with Ω an exterior domain in R

(n ≥ 2). Notice that the velocity ﬁeld v

need not satisfy a priori any prescribed condition at inﬁnity. Fo r this reason

we prefer to call (V.4.1 ) a Stokes “system” instead of a Stokes “problem” .

We have

V.4 Existence, Uniqueness, and L

-Estimates: Strong Solutions 321

Lemma V.4.2 Let v, p be a solution to (V.4.1)

1,2

. Assume v ∈ W

2,q

loc

(Ω),

p ∈ W

1,q

loc

(Ω) for some q ∈ (1, ∞), and for some r ∈ (1, ∞) and some R >

2δ(Ω

)

v ∈ D

2,r

(Ω

). (V.4.2)

Then, if f ∈ L

(Ω) it follows that

v ∈ D

2,q

(Ω), p ∈ D

1,q

(Ω).

Proof. Denote by ϕ = ϕ(|x|) a C

∞

-function in Ω that is zero for |x| ≤ ρ and

equals one for |x| ≥ R/2, δ(Ω

) < ρ < R/2. Setting u = ϕv and π = ϕp we

then have that u and π solve in R

the system

∆u = ∇π + f

∇ · u = g,

(V.4.3)

where

= ϕf

+ T

(v, p) D

ϕ + D

ϕ + v

ϕ), g = v · ∇ϕ, (V.4.4)

with T deﬁned in (IV.8.6). Clearly, f

∈ L

) and g ∈ W

1,q

) and so we

may a pply Theorem V.2.1 to prove the existence of a sol ution u

∗

∈ D

2,q

∗

∈ D

1,q

). Letting w = u − u

∗

, τ = π − π

∗

, we show

w(x) = ∇τ(x) = 0, for all x ∈ R

. (V.4.5)

Actually, i n R

∆w = ∇τ

∇ · w = 0

(V.4.6)

and, therefore, ∆(∇τ) = 0 in R

, which implies ∆(∆w) = 0 in R

Denot-

ing by ψ

the ith component of ∆w, we apply the mean value theorem for

harmonic functions (e.g., Gilbarg & Trudi nger 1983, Theorem 2.1) to deduce

for a ll x ∈ R

and with s = |x − y|

(x) =

(x)

{∆u

− ∆u

∗

}dy .

So, by the H¨older inequality, (V.4.2), and the fa ct that D

∗

∈ L

), we

get

|ψ

(x)| ≤ c[s

−n(1−1/r

)

+ s

−n(1−1/q

)

]

for i = 1, . . . , n. Letting s → ∞ in this relati on gives ∆w ≡ 0 in R

, which, in

turn, by (V.4.6)

, delivers (V.4.5 )

. As a consequence, from (V.4.6)

it follows

that

Notice that, by Theorem V.1.1, w, τ ∈ C

∞

322 V Steady Stokes Flow in Exterior Domains

∆(D

w) = 0 in R

Thus, arguing as before, one shows (V.4.5)

, completing the proof of the

lemma . ut

Remark V.4.6 If Ω is of class C

, the conclusion of the preceding lemma

can be reached under the assumptions that v satisﬁes (V.4 .1) and, moreover,

f ∈ L

(Ω), v ∈ D

2,r

(Ω), v

∗

∈ W

2−1/q,q

(∂Ω). (V.4.7)

To this end, it is enough to show that (V. 4.7) implies

v ∈ W

2,q

loc

(Ω), p ∈ W

1,q

loc

(Ω). (V.4.8)

If r ≥ q the assertion is obvious. Therefore, take q > r. From the embedding

Theorem II.3.4 and hyp othesis (V.4.7) on v we readily conclude v ∈ W

1,r

loc

(Ω)

with 1 < r

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

> 1 if r ≥ n. In

the latter case it follows that v ∈ W

1,q

loc

(Ω) and by Theorem IV.4. 1

v ∈ W

2,q

loc

(Ω), p ∈ W

1,q

loc

(Ω). (V.4.9)

If q ≤ r

< n we again draw the same conclusion. So, a ssume 1 < r

< q.

Then f ∈ L

(Ω) and Theorem IV.4. 1 along with Theorem II.3.4 impli es

v ∈ W

1,r

loc

(Ω) with 1 < r

≤ nr

/(n − r

) (> r

) if 1 < r

< n and for

arbitrary r

> 1, whenever r

≥ n. If either r

≥ q or r

≥ n we recover

(V.4.9); otherwise we iterate the above procedure as many times as needed,

until (V.4.9) is established. Properties (V.4.9) and the trace Theorem II.4.4

furnish v ∈ W

2−1/q,q

(∂Ω

) for all R > δ(Ω

). By Theorem IV.6.1 there

exists a solution v

, p

to the Stokes problem in Ω

corresponding to the

body force f , which equals v at the boundary ∂Ω

such that v

∈ W

2,q

(Ω

∈ W

1,q

(Ω

). Thus, u ≡ v − v

is a solution to the homogeneous Stokes

problem in Ω

with u ∈ W

1,r

(Ω

), since q > r. By Lemma IV.6. 2 we then

have u ≡ 0 and (V.4.8) is accomplished. 

We shall next establi sh for solutions to (V.4.1) a n estimate that is the coun-

terpart for exterior domains of estimate (IV.6.3) already proved for b ounded

domains. Precisely we have

Lemma V.4.3 Let v, p be a solution to (V. 4.1) in an exterior domain Ω ⊆

of cla ss C

m+2

, n ≥ 2, m ≥ 0, corresponding to f ∈ W

m,q

(Ω), v

∗

∈

m+2−1/q,q

(∂Ω), 1 < q < ∞. Assume

v ∈ D

2,q

(Ω).

Then v ∈ D

k+2,q

(Ω), p ∈ D

k+1

(Ω) for all k = 0, 1, . . ., m, and for any

R > δ(Ω

) it holds that

kvk

1,q,Ω

k=0

{|v|

k+2,q

+ |p|

k+1,q

}

≤ c



kf k

m,q

+ kv

∗

m+2−1/q,q(∂Ω)

+ kvk

q,Ω

+ kpk

q,Ω



(V.4.10)

V.4 Existence, Uniqueness, and L

-Estimates: Strong Solutions 323

where c = c(n, m, q, R).

Proof. As in the proof of the previous lemma, we transform (V.4.1) into

(V.4.3). Since u ∈ D

2,q

), from Theorem IV.2.1 it follows that u ∈

k+2,q

), π ∈ D

k+1,q

) for every k = 0, . . . , m. Furthermore,

k=0

{|u|

k+2,q

+ |π|

k+1,q

}

≤ c

(kfk

m,q

+ k∇ϕ · vk

m+1,q

+ kv∆ϕk

m,q

+ kp∇ϕk

m,q

) ,

(V.4.11)

where c

= c

(n, m, q). Inequality (V.4.11) then implies

k=0

{|v|

k+2,q,Ω

R/2

+ |p|

k+1,q,Ω

R/2

}

≤ c



kfk

m,q

+ kvk

m+1,q,Ω

R/2

+ kpk

m,q,Ω

R/2



(V.4.12)

Consider now probl em (V.4.1) in Ω

and use estimate (IV.6.3) to deduce

kvk

m+2,q,Ω

+ kpk

m+1,q

≤ c

(kfk

m,q,Ω

+ kvk

m+2−1/q,q(∂Ω

)

+ kvk

q,Ω

+ kpk

q,Ω

(V.4.13)

Setting σ ≡ ∂Ω

∩ Ω, by the trace Theorem II.4.4 we have

kvk

m+2−1/q,q(σ)

≤ c



|v|

m+2,q,Ω

R/2

+ kvk

m+1,q,Ω



. (V.4.14)

Combining (V.4. 12)–(V.4.14) we derive

kvk

1,q ,Ω

k=0

{|v|

k+2,q

+ |p|

k+1,q

}

≤ c



kf k

m,q

+ kv

∗

m+2−1/q,q(∂Ω)

+ kvk

m+1,q,Ω

+ kpk

m,q,Ω



and therefore applying Ehrling’s inequality (see Exercise II.5 .16) to the last

two terms on the right-hand side of this last inequality, we ﬁnally deduce

(V.4.10) and the lemma is proved. ut

In a complete analogy to the case where Ω is bounded, we wish now to

investigate whether the local norms involving v and p on the right-hand side

of (V.4.10) can be dropped out. Proceeding as in Section IV.6 (see the proof

of Lemma IV.6.1) we may try to use a contradiction a rgument to show the

inequality

kvk

q,Ω

+ kpk

q,Ω

≤ c



kfk

m,q

+ kv

∗

m+2−1/q,q(∂Ω)



which in turn would imply

324 V Steady Stokes Flow in Exterior Domains

kvk

1,q,Ω

k=0

{|v|

k+2,q

+ |p|

k+1,q

} ≤ c



kfk

m,q

+ kv

∗

m+2−1/q,q(∂Ω)



(V.4.15)

However, this a rgument needs the uni queness of solutions to the homogeneous

Stokes problem (V.4.1)

( i.e., (V.4.1) with f ≡ v

∗

≡ 0) in the class of those

functions for which the norms appearing on the left-hand side of (V.4.15) are

ﬁnite. On the other hand, it is hopeless to determine uniqueness in such a

class, unless we can control in some sense the behavior of v at inﬁnity. Now,

if 1 < q < n/2, this can be done as a consequence of the double application o f

Theorem II.6.1. However, if q ≥ n/2 we do not have this control any more, and

there could be nonzero solutions to (V.4.1)

in D

2,q

(Ω). We shall call these

solutions exceptional. A typical exampl e of an exceptional solution is given

by h(x), π(x), with h ≡ v

− v

, π ≡ p

, and v

, p

Stokes solution past a

sphere; see (V. 0.4). We emphasize that the existence of exceptional solutions

is related to the fact tha t a function in D

2,q

(Ω), even though approximable

by functions of bounded support (see Theorem II.7.4), need not “recall” the

zero value at inﬁnity of the approximating functions, since the approximating

procedure has been performed in a norm which, in general, does not control

the behavior at inﬁnity.

Notwithstanding this diﬃculty, we are able to characterize the space of

exceptional solutions and to determine its di mension d = d(n, q). Speciﬁcally,

it comes out that d is always ﬁnite and that d = 0 if 1 < q < n/2, d = n if

n/2 ≤ q < n, and d = n

+ n −1 if q ≥ n, see Lemma V.4.4. On the strength

of this result we then show the existence of solutions v, p ∈ D

2,q

(Ω)×D

1,q

(Ω)

that satisfy estimate (V.4.15) modulo exceptional sol utions; see Lemma V.4.5.

However, because d = 0 if 1 < q < n/2, for these values of q the validity of

(V.4.15) is established.

We begin to characterize the space of exceptional solutions. To this end,

we set

2,q

(Ω) = D

2,q

(Ω), if q ≥ n

2,q

(Ω) =



u ∈ D

2,q

(Ω) : |u|

1,r

< ∞, r =

n −q



≤ q < n

2,q

(Ω) =



u ∈ D

2,q

(Ω) : kuk

+ |u|

1,r

< ∞, s =

n − 2q

, r =

n − q



if 1 < q <

(V.4.16)

From Theorem II.7.4 we know that, f or Ω exterior and locally Lipschitz, every

function from

2,q

(Ω) can be approximated by functions from C

∞

(Ω) in the

seminorm | ·|

2,q

V.4 Existence, Uniqueness, and L

-Estimates: Strong Solutions 325

If a solution v to (V.4.1)

is in D

2,q

(Ω) for some q ≥ 1, the correspond-

ing pressure ﬁeld p is evidently in D

1,q

(Ω). Denote by Σ

the subspace of

2,q

(Ω) × D

1,q

(Ω) formed by solutions v, p to (V.4.1)

. We have

Lemma V.4.4 Let Ω be an exterior domain of class C

and set d = dim(Σ

Then,

d =











n + n

− 1 if q ≥ n

n if q ∈ [n/2 , n)

0 if q ∈ (1, n/2).

Proof. Let us ﬁrst consider the case where n > 2. We begin to show the

following two assertions:

(i) Fo r any v

∞

∈ R

− {0} there is a unique (nonzero) solution v, p ∈

∞

(Ω) to (V.4.1)

such that

lim

|x|→∞

|v(x) − v

∞

| = 0.

This solution veriﬁes the condition

v ∈

2,q

(Ω), for q ≥ n/2.

(ii) For any second order tensor A ≡ {A

}, A

6≡ 0, with trace(A) = 0,

there is a unique (nonzero) solution v, p ∈ C

∞

(Ω) to (V.4.1)

such that

lim

|x|→∞

|v(x) −A · x| = 0.

This solution veriﬁes the condition

v ∈

2,q

(Ω), for q ≥ n.

To prove (i ), we observe that, if we denote by v

, p

the solution con-

structed in Theorem V.2.1 corresponding to f ≡ 0, and v

∗

= −v

∞

, the pair

v ≡ v

+ v

∞

, p ≡ p

, satisﬁes all requirements. In fact, it belongs to C

∞

, by

Theorem IV.4.3, and by Theorem IV.5.1,

v ∈ W

2,t

loc

(Ω), p ∈ W

1,t

loc

(Ω), for any t ≥ 1. (V.4.17)

Also, v and p satisfy the asympto tic expansion (V.3.1 7)–(V.3.21) which, in

particular, furnishes that v → v

∞

uniform ly. Finally, again by (V.3.17)–

(V.3.21) and (V.4. 17), we deduce

v ∈ D

1,r

(Ω) ∩ D

2,q

(Ω), for all r > n/(n − 1) (V.4.18)

and since

v 6∈ L

(Ω), for any s ∈ (1, ∞),