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268 M. Geissert and H. Heck

As noted above the case q = 2 is easier to handle. It is well known that

the Stokes operator is a semibounded self-adjoint operator in L

(Ω). Hence, it is

the generator of an analytic semigroup (e

)

t≥0

on L

(Ω). Note that L2(Ω)

(Ω) ⊕ G

(Ω) for all open sets Ω. For a proper deﬁnition of these spaces

see (1.2).

Assuming that the Helmholtz decomposition exists in L

(Ω)

there is associ-

ated a projection P

onto L

(Ω) called the Helmholtz projection. In this case and if

Ω has a suﬃciently smooth boundary we deﬁne the Stokes operator by A

= P

with domain D(A

)=W

2,q

(Ω) ∩ W

1,q

(Ω) ∩ L

(Ω) in L

(Ω).

We say that the problem (1.1) has maximal L

-regularity in L

(Ω), 1 <

p, q < ∞, if the operator

(∂

−A

):L

((0,T); D(A

)) ∩ W

1,p

((0,T); L

(Ω)) → L

((0,T); L

(Ω)

)

is an isomorphism.

An aﬃrmative answer to the above question for bounded or exterior domains

with smooth boundaries was ﬁrst given by Solonnikov ([Sol77]). His proof makes

use of potential theoretic arguments. Lateron, further proofs were obtained, e.g.,

by combining Giga’s result on bounded imaginary powers of the Stokes operator

([Gig85]) with the Dore-Venni theorem, by Giga and Sohr [GS91a], by Grubb and

Solonnikov [GS91b] using pseudo-diﬀerential techniques and by Fr¨ohlich [Fr¨o07]

making use of the concept of weighted estimates with respect to Muckenhoupt

weights. For related results see also [Gig81] and [FS94]. The half-space case was

studied, e.g., in [Uka87] and [DHP01]. For results concerning inﬁnite layers we

refer to the work of Abe and Shibata [AS03b] and [AS03a], Abels [Abe05] and

Abels and Wiegner [AW05]. In [Fra00], [His04] the case of an aperture domain

is discussed and in [FR08] it was shown that the Stokes Operator has maximal

-regularity in L

(Ω) on tube-like domains Ω. Moreover, by applying pseudo-

diﬀerential techniques, a rather large class of domains could be treated in [Abe10].

For applications of these results to the equations of Navier-Stokes, see, e.g., [Kat84],

[Ama00] and [Soh01].

One problem in treating domains with unbounded boundary is that the

Helmholtz decomposition does not exist in L

(Ω)

in general. For example Bogov-

ski˘ı gave in [Bog86] examples of unbounded domains Ω with smooth boundaries

for which the Helmholtz decomposition of L

(Ω)

exists only for certain values of

q. For details, see also [Gal94]. In [FKS07] the authors sail around this problem

by changing the basic Banach space to L

(Ω) + L

(Ω) or to L

(Ω) ∩ L

(Ω) for

p ≥ 2orp ≤ 2, respectively. The majority of the works cited above treat classes

of domains that yield maximal regularity in L

(Ω) for any 1 <q<∞.

In [GHHS09] the authors prove maximal L

-regularity in L

(Ω) for (1.1)

assuming that the Helmholtz decomposition exists in L

(Ω)

.Indeed,theirmain

result is the following

Proposition 1.1 ([GHHS09]). Let 1 <p,q<∞, J =(0,T) for some T>0

and Ω ⊂ R

be a domain with uniform C

-boundary. Assume that the Helmholtz

A Remark on Maximal Regularity of the Stokes Equations 269

projection P exists for L

(Ω)

. Then problem (1.1) has maximal L

-regularity in

(Ω).

In this paper we are concerned with the question to which spaces L

(Ω) we

can extend the maximal L

-regularity property in this case.

Let q ∈ (1, ∞)andΩ⊂ R

. We say that the Helmholtz decomposition on

(Ω)

exists if

(Ω)

= L

(Ω) ⊕ G

(Ω), (1.2)

where

(Ω) := {f ∈ L

(Ω)

: ∃g ∈ L

loc

(Ω) : f = ∇g} and L

(Ω) := G



(Ω)

⊥

Here M

⊥

⊂ X



denotes the annihilator of M ⊂ X for some Banach space X and



q−1

denotes the dual exponent of q. In this case, there exists the Helmholtz

projection P

from L

(Ω)

onto L

(Ω). Since G

(Ω) and L

(Ω) are both reﬂexive

we obtain

(Ω))

⊥



(Ω)

⊥



⊥

= G



(Ω).

Hence, P



is a projection from L



(Ω)

onto L



(Ω). However, it is not clear

whether P

and P



are consistent, i.e.,

f = P



f, f ∈ L

(Ω) ∩ L



(Ω).

The main assumption on the underlying domain Ω will be the (ε, ∞) property.

Let ε>0andδ ∈ (0, ∞]. Then we say that a domain Ω ⊂ R

is an (ε, δ) domain

if for any x, y ∈ Ω satisfying |x − y| <δthere exists a rectiﬁable curve γ ⊂ Ω

connecting x with y such that for any point z on γ

(γ) ≤

|x −y| (1.3)

dist(z,Ω

) ≥ ε

|x − z||y − z|

|x −y|

(1.4)

holds. Here (γ) denotes the length of the path γ and dist(z,M) is the distance of

the point z to the set M.

Now we can formulate the main result of this paper.

Theorem 1.2. Let Ω ⊂ R

be an (ε, ∞) domain for some ε>0 with uniform C

boundary. Moreover, assume that the Helmholtz decomposition exists in L

(Ω)

for some 1 <q<∞. Then the problem (1.1) has maximal L

-regularity in L

(Ω)

for s ∈ [min{q,q



}, max{q, q



}].

Remark 1.3. Consider the cone K ⊂ R

which is the set of all points between

the rays M

:= {x ∈ R

: x

> 0,x

= ±κx

} with (1, 0) ∈ K.Denotebyθ

the angle between the rays M

measured across the domain K. Using piecewise

circular paths connecting the points x, y ∈ K it is easy to see that K is an (ε, ∞)

domain. By a result of Bogovski˘ıforn = 2 the Helmholtz decomposition in L

(K),

where K is a cone with angle θ>π, exists if and only if 2/(1 + π/θ) <q<

270 M. Geissert and H. Heck

2/(1 −π/θ). The same statement is true for cones with smooth apex. This means

that the results in this paper yield maximal L

-regularity in L

(K)for(1.1)and

q ∈ (2/(1 + π/θ), 2/(1 − π/θ)).

2. Proof of the main result

In this section we give the proof of our main result. We start with some comments

about the Helmholtz projection. It is well known that existence of the Helmholtz

decomposition on L

(Ω)

is equivalent to the solvability of the following weak

Neumann problem, see [SS92] and the references therein:

There exists C>0 such that for any f ∈ L

(Ω)

there exists a unique

u ∈

1,q

(Ω) := {f ∈ L

loc

(Ω) : ∇f ∈ L

(Ω)

} satisfying u

1,q

(Ω)

≤ Cf

(Ω)

and



∇u, ∇ϕ =



f,∇ϕ,ϕ∈

1,q



(Ω). (2.1)

In this case, P

f = f −∇u.

If (2.1) is uniquely solvable the mapping J :

1,q

(Ω) →



1,q



(Ω)





deﬁned

Jϕ,Ψ :=



∇ϕ, ∇Ψ, Ψ ∈

1,q



(Ω), (2.2)

is an isomorphism (even the converse is true, see [SS92, Chapter 6] for a similar

result). Indeed, let ϕ ∈

1,q

(Ω). Then,

|Jϕ,Ψ| = |



∇ϕ, ∇Ψ| ≤ ϕ

1,q

(Ω)

Ψ

1,q



(Ω)

, Ψ ∈

1,q



(Ω).

Note that by assumption J is one-to-one. On the other hand, let ϕ ∈



1,q



(Ω)





Since ∇Ψ ∈ G



(Ω) ⊂ L



(Ω) there exists f

∈ L

(Ω)

such that

ϕ, Ψ =



f

, ∇Ψ =



∇u, ∇Ψ, Ψ ∈

1,q



(Ω),

where u ∈

1,q

(Ω) is the unique solution of (2.1). Therefore J is onto as well.

The next proposition allows us to apply [GHHS09, Theorem 1.2] to our sit-

uation.

Proposition 2.1. Let Ω ⊂ R

be an (ε, ∞) domain with a uniform C

boundary.

Assume that the Helmholtz projection P

exists for some p ∈ (1, ∞). Then the

Helmholtz projection P

exists for q ∈ [min{p, p



}, max{p, p



}] where 1/p



+1/p =1.

Moreover, P

= P

in L

(Ω)

∩ L

(Ω)

A Remark on Maximal Regularity of the Stokes Equations 271

In order to prove Proposition 2.1 we need some preparation. By adapting the

arguments given in [SS92, Lemma 3.8(b)] to our situation, we obtain the following

lemma.

Lemma 2.2. Let 1 <r<q<∞, Ω ⊂ R

an (ε, ∞) domain and ∇p ∈ G

(Ω) with

sup

0=∇v∈C

∞

(Ω)

∇p, ∇v

v

1,r



(Ω)

< ∞,

where r



is the dual exponent of r. Moreover, assume that the Helmholtz projection

exists in L

(Ω)

.Then∇p ∈ G

(Ω).

The next proposition states some basic properties of homogeneous Sobolev

spaces.

Proposition 2.3. Let p ∈ (1, ∞) and Ω ⊂ R

be an (ε, ∞) domain. Then C

∞

(Ω)

is dense in

1,p

(Ω). Moreover, for q ∈ (1, ∞) and u ∈

1,p

(Ω) ∩

1,q

(Ω) there

exists (u

)

n∈N

⊂ C

∞

(Ω) such that lim

n→∞

= u in

1,p

(Ω) ∩

1,q

(Ω).

Proof. Let f ∈

1,p

(Ω). By [Jon81, Theorem 2], there exists an extension operator

E ∈L(

1,p

(Ω),

1,p

)), independent of p ∈ (1, ∞). Since C

∞

)isdense

1,p

), see [Sob63] for instance, there exists (ϕ

) ⊂ C

∞

) such that

lim

n→∞

ϕ

− Ef

1,p

)

= 0. Hence,

lim

n→∞

ϕ

−f

1,p

(Ω)

≤ lim

n→∞

ϕ

− Ef

1,p

(Ω)

=0. 

Remark 2.4.

1. The proof of Proposition 2.3 is based on the existence of an extension opera-

tor E ∈L(

1,p

(Ω),

1,p

)) only. In particular, whenever there exists an

extension operator for some p ∈ (1, ∞)thenC

∞

(Ω) is dense in

1,p

(Ω).

2. O.V. Besov constructed in [Bes67] an extension operator

E ∈L(

1,p

(Ω),

1,p

)),p∈ (1, ∞),

for domains Ω ⊂ R

satisfying an interior horn condition.

3. Note that Proposition 2.3 does not hold for arbitrary domains as, for instance,

1,p

(Ω) = C

∞

(Ω)

1,p

(Ω)

for aperture domains, see [Fra00].

4. Let p ∈ (1, ∞)andθ ∈ (0, 1). Since the extension operator E given in Propo-

sition 2.3 is a coretraction it follows from real interpolation theory, see, e.g.,

[Tri78], that



1,p

(Ω),

1,p



(Ω)



θ,r

1,r

(Ω),

where 1/r = θ/p +(1− θ)/p



We are now in the position to prove Proposition 2.1.

Proof of Proposition 2.1. Let P

denote the Helmholtz projection on L

(Ω)

.Then

)



is a projection from L



(Ω)

onto L



(Ω). In order to show that P

and (P

)



272 M. Geissert and H. Heck

are consistent note that it is enough to show that the solution u of (2.1) is consis-

tent for q = p, p



. We may assume without loss of generality that p>2.

Let f ∈ L

(Ω)

∩L



(Ω)

and denote the corresponding solutions of (2.1) in

1,p

(Ω) and

1,p



(Ω) by u

and u



, respectively. Obviously, we have



∇u

, ∇ϕ =



f,∇ϕ≤f 



(Ω)

ϕ

1,p

(Ω)

,ϕ∈ C

∞

(Ω).

Hence, it follows from Lemma 2.2 that u

∈

1,p



(Ω). Since C

∞

(Ω) is dense in

1,p

(Ω) we obtain



∇u

, ∇ϕ =



f,∇ϕ,ϕ∈

1,p

(Ω).

Finally, by uniqueness of solutions to (2.1) in

1,p



(Ω), we obtain u

= u



Therefore P

and (P

)



are consistent and we may write P



=(P

)



.Now

the proposition follows from interpolation theory.



Remark 2.5. 1. In order to show that P



exists whenever P

exists, we might

use J :

1,p

(Ω) → (

1,p



(Ω))



and duality theory.

2. The result that P



exists whenever P

exists is due to [Bog86].

Finally, we are prepared to give the

Proof of Theorem 1.2. We assume w.l.o.g. that p>2. Once we establish the con-

sistency of the resolvent problem in L

(Ω) and in L



(Ω) we can use interpolation

theory and Proposition 1.1 (see [GHHS09]) in order to get the claim. The consis-

tency of the Helmholtz projection was proved in Proposition 2.1.

That means that the solution operator of the weak Neumann problem exists

in L

(Ω) for s ∈ [p



,p] and is consistent. The consistency of the Stokes resolvent

problem now follows from the solution formula for the Stokes resolvent problem

which was developed in [GHHS09]. In this representation the authors use an ex-

plicit solution formula for the half-space. Noting that in the half-space we have

consistent resolvents it is clear that the consistency is preserved in the general

case. 

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Matthias Geissert and Horst Heck

Fachbereich Mathematik

Technische Universit¨at Darmstadt

Germany

e-mail: geissert@mathematik.tu-darmstadt.de

heck@mathematik.tu-darmstadt.de

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 275–300

 2011 Springer Basel AG

On Linear Elliptic and Parabolic Problems

in Nikol’skij Spaces

Davide Guidetti

Dedicated to professor Herbert Amann, in occasion of his seventieth birthday

Abstract. We consider estimates depending on a parameter for general linear

elliptic boundary value problems, with nonhomogeneous boundary conditions,

in Nikol’skij spaces. These estimates are then employed to study general li-

near nonautonomous parabolic systems, again with nonhomogeneous boun-

dary conditions. Maximal regularity results are proved.

Mathematics Subject Classiﬁcation (2000). 35K30; 35J40; 46B70.

Keywords. Elliptic systems depending on a parameter; mixed parabolic sys-

tems; Nikol’skij spaces; maximal regularity.

0. Introduction and notations

The aim of this paper is to illustrate some improvements concerning estimates

depending on a parameter for elliptic boundary value problems with nonhomoge-

neous boundary conditions, and to apply them to mixed parabolic systems. The

main spaces we shall work with are Nikol’skij spaces, which are a category of the

class of Besov spaces: if β ∈ R, p ∈ [1, ∞] and Ω is a domain in R

, the Nikol’skij

space N

(Ω) coincides with the Besov space B

p,∞

(Ω). The main reason why we

are interested in N

(Ω) is that results of maximal regularity in closed subspaces

of B([0,T]; N

(Ω)), the space of bounded functions with values in N

(Ω), are

available. Maximal regularity estimates in L

(0,T; L

(Ω)) (1 <p<∞) (even for

a very general class of systems) were obtained by V.A. Solonnikov (see [11]). In

the case of homogeneous boundary conditions, the extension to L

(0,T; L

(Ω))

(1 <p,q<∞) is essentially due to G. Dore and A. Venni as an application

of their celebrated theorem (see [4]). An extension of this result to the case of

operator-valued coeﬃcients is proposed by R. Denk, M. Hieber, J. Pr¨uss in [2].

The case of nonhomogeneous boundary conditions (again in the L

-L

setting)

is treated by the same authors in [3]. They prove a remarkable intrinsic cha-

276 D. Guidetti

racterization of the boundary data in such a way that a solution in the class

1,p

(0,T; L

(Ω)) ∩L

(0,T; W

2m,q

(Ω)) exists. Such characterization is in terms of

vector-valued Triebel-Lizorkin spaces. However, for problems of convergence to a

stationary state and asymptotic expansion of the solutions, it seems preferable to

work in spaces of continuous, or, at least, bounded functions in the time variable.

If A is the inﬁnitesimal generator of an analytic semigroup in X, maximal regu-

larity results of this type for the corresponding Cauchy problem are available in

closed subspaces of the real interpolation space (X, D(A))

θ,∞

(see, for example,

[9]). If X = L

(Ω) and D(A) is a closed subspace of W

2m,p

(Ω), (X, D(A))

θ,∞

is typically a closed subspace of the Nikol’skiy space N

2mθ

(Ω) (see Section 1 for

precise deﬁnitions). This makes our choice quite natural.

We pass to describe the content of the paper.

In Section 1, we recall some deﬁnitions and results concerning Nikol’skij

spaces, which we shall use throughout the article. The Nikol’skij space N

(Ω)

(s ∈ R, p ∈ [1, ∞]) coincides with the Besov space B

p,∞

(Ω). We refer to [6] for

a summary of results concerning Besov spaces. Much more complete expositions

can be found (for example) in [14], [10], and (in the case s>0) in [1], Chapter

7. For the sake of simplicity, we have chosen to consider only the case of domains

with “smooth” (that is, C

∞

) boundaries, following the lines of [14], [10], [6].

In Section 2 we deal with estimates depending on a parameter for general

nonhomogeneous elliptic boundary value problems in Nikol’skij spaces. Here also

we shall limit ourselves to the case of smooth coeﬃcients. It is quite well known that

estimates of this type are basic to treat parabolic systems. The main result of this

Section is Theorem 2.3, in which we prove an estimate depending on a parameter

in the case that the solution annihilates the elliptic operator with the parameter

and the boundary conditions are nonhomogeneous, and extending Proposition 2.16

in [6]. Theorem 2.3 has Theorem 2.7, containing a general estimate depending on

a parameter, as a corollary.

In Section 3 we consider general parabolic systems. The estimates obtained

in Section 2 allow to prove maximal regularity results for problems with nonho-

mogeneous boundary conditions at ﬁrst in the case of autonomous systems. In

this direction, the main result is Theorem 3.6. The method of the proof follows

an old idea by B. Terreni (see [13]). Such idea was already employed in [5], where

I treated similar problems in little-Nikol’skij spaces (the closure of smooth func-

tions in Nikol’skij spaces). Apart the slightly diﬀerent setting, here I have been

able to extend considerably the range of indexes to which the results are appli-

cable. Roughly speaking, if N

(Ω) is the basic space, here β is allowed to vary

in the interval (μ + p

−1

− 2m, ν + p

−1

), with ν and μ, respectively, the minimum

and maximum order of the boundary conditions, while in [5] I treated only the

case β ∈ (0,p

−1

) (compare Theorem 3.6 with Theorem 2.1 in [5]). So we are able

to treat even cases with β<0 and the elements of the basic space are distribu-

tions. Finally we have put an extension, which does not seem completely trivial,

to nonautonomous parabolic systems.

On Linear Elliptic and Parabolic etc. 277

We conclude this introductory section with the indication of some notations

that we shall use in the paper.

If X and Y are Banach spaces, we shall indicate with L(X, Y ) the Banach

space of linear bounded operators from X to Y .IncaseX = Y , we shall write

L(X). If Ω is an open subset of R

with smooth boundary and x belongs to the

boundary ∂Ω, we shall indicate with ν(x) the unit vector, which is normal to ∂Ω

in x and points outside Ω, with

∂f

∂ν

(x) the normal derivative, and with T

(∂Ω) the

tangent space to ∂Ωinx.Givenθ ∈ (0, 1) and q ∈ [1, ∞], we shall indicate with

(., .)

θ,q

the real interpolation functor, W

m,p

(Ω) (m ∈ N

, p ∈ [1, ∞]) will be the

standard Sobolev spaces. If A is a set and X is a normed space, we shall indicate

with B(A; X) the space of bounded functions from A to X, with its natural norm.

If β ∈ R,weset

[β]:=max{j ∈ Z : j<β}, {β} := β − [β]. (0.1)

If ρ∈(0,1), T ∈R

, X is a Banach space with norm .,andf :[0,T]→X,weset

f

([0,T ];X)

:= max



f

B([0,T ];X)

, sup

0≤s<t≤T

f(t) −f(s)

(t −s)



Of course, C

([0,T]; X) is the set of functions f such that |f

([0,T ];X)

< +∞.

If m ∈ N, C

m+ρ

([0,T]; X) is the set of functions f in C

([0,T]; X) such that

(m)

∈ C

([0,T]; X). We shall equip it with the norm

f

m+ρ

([0,T ];X)

:= max{f

([0,T ];X)

, f

(m)



([0,T ];X)

Finally, C will be used to indicate a positive constant which we are not interested to

precise, and may be diﬀerent from time to time. We shall often use the notation C

,... if we have a sequence of estimates. Other notations, concerning Nikol’skij

spaces, will be introduced in Section 1.

1. Nikol’skij spaces

We start by introducing Nikol’skij spaces.

Deﬁnition 1.1. Let n ∈ N, p ∈ [1, +∞[, β ∈ (0, 1]. We deﬁne

):={f ∈ L

):[f ]

β,p,R

:= sup

h∈R

\{0}

|h|

−β

f(. + h) − 2f + f(. − h)

)

< +∞}.

(1.1)

If β ∈ (1, +∞), we set

):={f ∈ W

[β],p

):∀α ∈ N

, |α|≤[β],∂

f ∈ N

{β}

)} (1.2)

(we recall that {β}∈(0, 1]).

Remark 1.2. N

), with the norm

f

β,p,R

:= f

[β],p

)

|α|=[β]

[f]

{β},p,R