Albert-LudwigsUniversität Freiburg im
Breisgau
Fakultät für Mathematik
und Physik
Mathematisches Institut • Abteilung für Angewandte Mathematik
Monotone Operators in Spaces
with Variable Exponents
Diplomarbeit
im Studiengang Mathematik
vorgelegt von
Philipp Nägele
bei
Prof. Dr. M. Růžička
Dezember 2009
ii
Erklärung
Hiermit versichere ich, dass ich die vorliegende Diplomarbeit selbständig
und nur mit den angegebenen Hilfsmitteln angefertigt habe und dass alle
Stellen, die dem Wortlaut oder dem Sinne nach anderen Werken entnommen
sind, durch Angabe der Quellen als Entlehnung kenntlich gemacht worden sind.
Ort, Datum
Unterschrift
iii
iv
Contents
Introduction
1
1 Lebesgue spaces with variable exponent
7
2 The Hardy-Littlewood maximal operator
13
2.1 Boundedness of the maximal operator . . . . . . . . . . . . . . . 13
2.2 Mollification on Lp(·) (Rd ) . . . . . . . . . . . . . . . . . . . . . . 17
3 Function spaces and integration by parts
3.1 The space W (QT ) . . . . . . . . . . . . . .
3.2 The dual space W (QT )0 . . . . . . . . . .
3.3 The space Z(QT ) . . . . . . . . . . . . . .
3.4 Integration by parts . . . . . . . . . . . . .
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21
24
34
38
43
4 Existence of a weak solution
49
4.1 Weak formulation of the problem . . . . . . . . . . . . . . . . . 50
4.2 Passage to the limit . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 Outlook
61
5.1 Description of the dual space YD (QT )0 . . . . . . . . . . . . . . . 80
6 Appendix
83
6.1 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
v
vi
Contents
Introduction
The aim of the present work is to show the existence of a weak solution
u : (0, T ) × Ω ⊂ Rd+1 → Rd of the system
∂t u − divS(∇u) = f in (0, T ) × Ω
u = 0 on (0, T ) × ∂Ω
u(0) = u0 in Ω,
(1)
where the stress tensor S is given by
S(∇u)(τ, x) := (1 + |∇u|)p(τ,x)−2 ∇u(τ, x),
and p, f, u0 are known functions defined on QT := (0, T ) × Ω and on Ω,
respectively. We instantly remark that S exhibits two special features. Apart
from its nonlinear character, the most important peculiarity is that the
exponent p(·, ·) itself depends on the space and time variables and the
system is therefore said to have a variable nonlinearity. Equations of this type
have their application in image restoration and also appear in the
mathematical description of non-newtonian and electrorheological fluids.
We also want to mention that the above system represents a generalization of
the so-called parabolic p-Laplacian equation which reads
∂t u − divP (∇u) = f in (0, T ) × Ω
u = 0 on (0, T ) × ∂Ω
u(0) = u0 in Ω.
(2)
Here, the stress tensor P is defined by
P (∇u)(τ, x) := |∇u|p−2 ∇u(τ, x),
the exponent p ∈ (1, ∞) is a fixed real number and the functions f and u0 are
given data.
To deduce the existence of weak solutions of this system, one normally works
within the setting of spaces like X := Lp (I, W01,p (Ω)d ). The first advantage of
this space is that the norm,
 T
 p1
Z
||u||X :=  ||∇u(τ )||pLp (Ω)d×d dτ  ,
0
1
2
Introduction
is related to the natural energy of the system which is given by
ZT
||∇u(τ )||pLp (Ω)d×d dτ .
0
Now, the energy of our generalized system is formally given by
ZT Z
0
(1 + |∇u|)p(τ,x)−2 |∇u|2 dx dτ .
Ω
Thus, in a quite natural way we are led to use generalized Lebesgue spaces
Lp(·,·) (QT )d .
These spaces, also called Lebesgue spaces with variable
exponent, have first been studied systematically in a survey article by Kováčik
and Rákosnı́k in [24], where they proved basic properties such as completeness,
reflexivity and separability, and later on by Fan and Zhao in [17]. The spaces
Lp(·) (Ω) were then used by Růžička to propose a model for electrorheological
fluids. These fluids have the ability to change their mechanical properties
when an electric field is applied. In this model, the variable exponent p(·)
explicitly depends on the electric field and thus usually varies on its domain.
For a more detailed discussion we refer to [31]. Major progress was then made
by Diening in [9], [11] and Cruz-Uribe, Fiorenza, Martell and Pérez in [7],
where boundedness of the Hardy-Littlewood maximal operator on generalized
Lebesgue spaces is proved. It is the continuity of this operator that allows the
treatment of more sophisticated issues such as density of smooth functions and
singular integrals like the Riesz potential or Calderon-Zygmund operators on
Lp(·) (Rn ), see also [10] and [8]. For an in-depth treatment we also want to refer
to [12], where these topics are discussed in the yet even broader context of
generalized Orlicz spaces. An overview concerning state-of-the-art techniques
and problems can be found in [21] and [13].
So up to now, the theory of generalized Lebesgue and Sobolev spaces is well
developed but there are still major problems regarding their applicability to
partial differential equations. For instance, in the treatment of time-dependent
problems it is often helpful to use the following perception:
a function u(·, ·) : I × Ω → Rd can also be interpreted as t 7→ u(t, ·) : Ω → Rd
and together with the identity
Lq (QT )d = Lq (I, Lq (Ω)d )
(3)
one can also capture this distinction in terms of a function space. In fact, (3)
is an easy consequence of the definition of the norm on Lq (QT ) and Fubini’s
theorem and, as a further generalization, leads to classical Bochner spaces like
Lr (I, Lq (Ω)d ),
3
Introduction
where the functions u(t, ·) : Ω → Rd and u(·, x) : I → Rd even differ with
respect to their integrability. In the setting of generalized Lebesgue spaces like
Lp(·,·) (QT ), a distinction regarding integrability isn’t achieved as easily as in
the classical one, but yet would be very useful. As a matter of fact, an identity
like
Lp(·,·) (QT ) = Lpt (·) (I, Lpx (·) (Ω))
does not hold in general because of the definition of the (Luxemburg) norm on
Lebesgue spaces with variable exponent.
Another nice feature of Bochner spaces like X = Lp (I, W01,p (Ω)d ), especially
in connection with the time-dependent p-Laplacian, is that these spaces can
help to ”decouple” the elliptic part of the equation induced by P from the
time derivative. After defining the right notion of a (weak) time derivative for
functions in X and using the structure of the weak formulation of the system
(2), the problem is accessible to the theory of monotone operators and certain
compactness methods. For a detailed description of this approach we refer to
[36], [37] and [32], but we will also pick up some of these ideas later on, in
order to motivate our methods. Though, as already mentioned, the situation
changes if generalized Lebesgue spaces come into play since it is not even
clear how to distinguish between the functions t 7→ u(t, ·) and x 7→ u(·, x) as
to their integrability, and thus, Bochner spaces and methods connected with
these linebreak cannot directly be adapted.
For this reason, a main part of this work deals with the problem of defining
the appropriate function spaces and elaborating their properties, in order to
generalize the theory of monotone operators that has succesfully been used for
the existence theory of weak solutions of the parabolic p-Laplacian equation.
The paper is organized as follows.
In the first chapter we introduce Lebesgue and Sobolev spaces with variable
exponent as a starting point. The next chapter comprises the most important
facts about the Hardy-Littlewood maximal operator on generalized Lebsegue
spaces and its connection with approximate identities and the Riesz potential.
At the beginning of the third chapter, we take the classical heat equation as an
initial point to give a brief summary of so-called Gelfand triples (V, H, V 0 ), the
” and some special function spaces
idea of a ” generalized time derivative du
dt
connected with these issues, in order to explain how these constructions can
be applied to the parabolic p-Laplacian equation. In particular we define the
space
du
p
p0
0
W := u ∈ L (I, V ) ∈ L (I, V ) ,
dt
which plays a central role in the treatment of this equation. Being of great
importance in the proof of the existence of weak solutions of (2), we also cite
4
Introduction
the formula
ZT
ZT
du
dv
h (τ ), v(τ )iV dτ = hu(T ), v(T )iH − hu(0), v(0)iH − h (τ ), u(τ )iV dτ , (4)
dt
dt
0
0
which is valid for every u, v ∈ W . After pointing out the difficulties in adopting
the classical approach, we define and completely describe our basic function
space W (QT ). To our best knowledge, this space was first used by Antontsev
and Shmarev in [4] and [3], in order to prove existence and uniqueness of weak
solutions for general parabolic Dirichlet problems with variable nonlinearities.
In particular, we are going to show density of smooth functions in W (QT )
by means of a mollification method that makes use of the boundedness of
the Hardy-Littlewood maximal operator. This in turn opens the door to a
complete characterization of the dual space W (QT )0 . Then we will define
n
o
Z(QT ) := u ∈ W (QT ) ut ∈ W (QT )0 ,
as a substitute for W . We already want to mention that we use distributional
. This is due
time derivatives ut instead of generalized time derivatives du
dt
to the fact that the notion of generalized time derivatives is tailor-made for
problems involving Bochner spaces built on Gelfand triples. However, as we
neither want to use Gelfand triples nor Bochner spaces, it appears to be more
natural to work with distributional time derivatives.
Chapter 3 also contains a rule for integration by parts in time which reads
hut , χ[s,t] viW (QT ) = hu(t), v(t)iL2 (Ω)d − hu(s), v(s)iL2 (Ω)d − hvt , χ[s,t] uiW (QT )
and generalizes (4) to our framework. In the fourth chapter we will define the
notion of a weak solution of (1) and then, based on a Galerkin approximation,
our representation of the dual space W (QT )0 and our rule for integration
by parts, give the actual proof of the existence of weak solutions of (1) by
imitating the classical approach.
Finally, we try to give an outlook on the possibility of carrying over certain
ideas to the treatment of the p(t,x)-Navier-Stokes equations in Chapter 5.
The last chapter contains basic facts concerning functional analysis and
distributions.
Chapter 1
Lebesgue spaces with variable
exponent
In this chapter we shall introduce generalized Lebesgue and Sobolev spaces
and state some of their basic properties. As a starting point we therefore give
Definition 1.1. Let Ω ⊂ Rd be a measurable subset and let p : Ω → [1, ∞]
be a measurable function. Then p(·) will be called a variable exponent and we
define P(Ω) to be the set of all variable exponents. Furthermore, we define
+
+
+
p− := p−
Ω := essinf y∈Ω p(y) and p := pΩ := esssupy∈Ω p(y). If p is finite, p(·)
is called a bounded exponent.
Definition 1.2. Let p : Ω → [1, ∞] be a variable exponent defined on a measurable subset Ω ⊂ Rd . For a measurable function f : Ω → R,
Z
ρp(·) (f ) := |f (x)|p(x) dx
Ω
is called the modular of f with respect to p(·).
With the help of a modular, Lebesgue spaces with variable exponent are
defined as follows.
Definition 1.3. Let Ω ⊂ Rd be measurable and let p(·) ∈ P(Ω) be a variable
exponent. The set
n
o
Lp(·) (Ω) := f : Ω → R f measurable , ρp(·) (f ) < ∞
is called Lebesgue space with variable exponent p(·) or generalized Lebesgue
space.
Now we introduce the so-called Luxemburg norm on Lp(·) (Ω) which is
defined similar to the Minkowski functional on convex and absorbing sets.
5
6
Chapter 1. Lebesgue spaces with variable exponent
Definition 1.4. Let Ω ⊂ Rd and let p(·) ∈ P(Ω) be a variable exponent. For
f ∈ Lp(·) (Ω) we define its Luxemburg norm with respect to p(·) by
n
o
||f ||Lp(·) (Ω) := inf λ > 0 ρp(·) (λ−1 f ) ≤ 1 ,
where we use the convention inf ∅ = ∞.
Remark 1.5. The Luxemburg norm is monotone in the sense that f ≤ g
almost everywhere in Ω and g ∈ Lp(·) (Ω) imply ||f ||Lp(·) (Ω) ≤ ||g||Lp(·) (Ω) .
The last definition enables us to list some basic properties of the spaces
Lp(·) (Ω).
Theorem 1.6. Let Ω ⊂ Rd be a measurable subset and let p(·) ∈ P(Ω) be a
variable exponent. The spaces Lp(·) (Ω) are real vector spaces. Moreover, the
functional || · ||Lp(·) (Ω) truly defines a norm on Lp(·) (Ω) and, equipped with this
norm, Lp(·) (Ω) is a Banach space. If additionally p(·) is a bounded exponent,
then Lp(·) (Ω) is separable.
Proof. These facts are proved in [12], chapter 2.
Hence, we note that many nice properties of the spaces Lr (Ω), where r is
constant, carry over to Lebesgue spaces with variable exponent. We can say
even more: if p(y) = r with r ∈ [1, ∞) for every y ∈ Ω, then still p(·) is a
variable exponent in the sense of Definition 1.1 and in this case the spaces
Lp(·) (Ω) and Lr (Ω) coincide, as do the norms || · ||Lp(·) (Ω) = || · ||Lr (Ω) .
Thus, the spaces Lp(·) (Ω) really constitute a generalization of the well-known
classic Lebesgue spaces.
On Lp(·) (Ω) there are two natural kinds of convergence: the first one is convergence with respect to the norm and the second one is convergence with respect
to the modular induced by the variable exponent.
Definition 1.7. Suppose p(·) ∈ P(Ω) and fn , f ∈ Lp(·) (Ω). Then we say fn is
modular convergent to f for n → ∞ if
ρp(·) (fn − f ) → 0 as n → ∞.
The relationship between these two kinds of convergence is reflected in
Lemma 1.8. Norm convergence always implies convergence with respect to
the modular. If p(·) ∈ P(Ω) is bounded, then the converse also holds true and
in this case we have
n
o
n
o
−
+
−
+
min ||f ||pLp(·) (Ω) , ||f ||pLp(·) (Ω) ≤ ρp(·) (f ) ≤ max ||f ||pLp(·) (Ω) , ||f ||pLp(·) (Ω)
and
−
+
||f ||pLp(·) (Ω) − 1 ≤ ρp(·) (f ) ≤ ||f ||pLp(·) (Ω) + 1.
Proof. This is Theorem 2.4 in [24].
7
For later applications we provide another useful result concerning
convergence almost everywhere, namely, Lebesgue’s theorem on dominated
convergence. This is a very important tool in the framework of classic Lebesgue
spaces and fortunately, the same result is valid in Lp(·) (Ω) under some
additional assumptions on the exponent.
Proposition 1.9. Let Ω ⊂ Rd be measurable and let p(·) ∈ P(Ω) be a bounded
exponent i.e. 1 ≤ p− ≤ p(·) ≤ p+ < ∞.
Suppose we are given measurable functions fn , f : Ω → R satisfying fn → f
almost everywhere for n → ∞. If there exists g ∈ Lp(·) (Ω) such that |fn | ≤ g
holds almost everywhere for every n, then we have fn , f ∈ Lp(·) (Ω) and fn
converges to f with respect to the norm in Lp(·) (Ω).
Proof. Corollary 2.25 in [23].
Another main tool in studying Banach spaces and their structure is the
description of their dual spaces and, closely connected with that, the question
of reflexivity. For this purpose we give
Definition 1.10. For a bounded variable exponent p(·) ∈ P(Ω) with 1 < p−
we define the dual exponent with respect to p(·) by
p0 (x) =
p(x)
,
p(x) − 1
for every x ∈ Ω.
Hölder’s inequality is another important tool that can also be retrieved.
Lemma 1.11. Let p(·) ∈ P(Ω) be a bounded variable exponent such that 1 <
0
p− . Then, for every f ∈ Lp(·) (Ω) and g ∈ Lp (·) (Ω) we have
Z
|f (x)g(x)| dx ≤ rp ||f ||p(·) ||g||p0 (·) ,
Ω
with rp = 1 +
1
p−
−
1
.
p+
Proof. Theorem 1 in [24].
We are now able to characterize the dual spaces (Lp(·) (Ω))0 and get the same
result as in the classical case. Moreover, we can answer the issue of reflexivity.
8
Chapter 1. Lebesgue spaces with variable exponent
Theorem 1.12. For a bounded variable exponent p(·) ∈ P(Ω) with 1 < p− ,
the mapping
0
I : Lp (·) (Ω) → (Lp(·) (Ω))0
defined by
Z
hIg, f iLp(·) (Ω) :=
g(x)f (x) dx for f ∈ Lp(·) (Ω),
Ω
provides an isomorphism that additionally shows the reflexivity of Lp(·) (Ω). For
every functional G ∈ (Lp(·) (Ω))0 , the duality between (Lp(·) (Ω))0 and Lp(·) (Ω) is
thus given by
Z
hG, f iLp(·) (Ω) = g(x)f (x) dx ,
Ω
for some g ∈ L
p0 (·)
(Ω) and f ∈ Lp(·) (Ω).
Proof. Theorem 2.6 and Corollary 2.7 in [24].
The use of different variable exponents induces different modulars and thus,
the resulting generalized Lebesgue spaces are likely to differ from each other
as well. But if the underlying domain has bounded measure we can recapture
classic embedding properties.
Lemma 1.13. Assume that Ω ⊂ Rd satisfies |Ω| < ∞ and that p(·) and q(·)
both belong to P(Ω). Then p(·) ≤ q(·) almost everywhere in Ω implies the
continuous embedding Lq(·) (Ω) ,→ Lp(·) (Ω), since
||f ||Lp(·) (Ω) ≤ c (|Ω|) ||f ||Lq(·) (Ω)
holds true for every f ∈ Lq(·) (Ω).
For a single p(·) ∈ P(Ω) with 1 ≤ p− ≤ p(·) ≤ p+ < ∞ we have a chain of
continuous embeddings:
+
−
L∞ (Ω) ,→ Lp (Ω) ,→ Lp(·) (Ω) ,→ Lp (Ω) ,→ L1 (Ω) ,→ L1loc (Ω)
Proof. A proof can be found in [24], Theorem 2.8.
Of course, even for unbounded domains Ω ⊂ Rd the inclusion
Lp(·) (Ω) ⊂ L1loc (Ω) remains valid. Thus, the notion of distributional
derivatives still makes sense for f ∈ Lp(·) (Ω) and we can define generalized
Sobolev spaces built from Lebesgue spaces with variable exponent.
9
Definition 1.14. Let Ω ⊂ Rd be open and let p(·) ∈ P(Ω) be a variable
exponent. We define the Sobolev space with variable exponent p(·) by
n
o
W 1,p(·) (Ω) := f ∈ Lp(·) (Ω) | ∂j f ∈ Lp(·) (Ω) j = 1, ..., d ,
where ∂j f stands for the j-th partial derivative of f in the distributional sense.
If f belongs to W 1,p(·) (Ω) we define its norm by
||f ||W 1,p(·) (Ω) := ||f ||Lp(·) (Ω) + ||∇f ||Lp(·) (Ω) .
1,p(·)
The subspace W0
W 1,p(·) (Ω) norm.
(Ω) is defined as the closure of C0∞ (Ω) with respect to the
We summarize the most important facts about generalized Sobolev spaces
and refer to [12], Theorem 3.54, for the proof.
Proposition 1.15. For p(·) ∈ P(Ω), the space W 1,p(·) (Ω) is a Banach space
and it is separable if p+ < ∞. If 1 < p− ≤ p(·) ≤ p+ < ∞, then W 1,p(·) (Ω) is
reflexive.
Remark 1.16. We will see that, similar to the case of classical Sobolev
1,p(·)
spaces, the space W0 (Ω) can be equipped with an equivalent norm, if
the exponent p(·) satisfies a special continuity property that is going to be
introduced in the next chapter.
Remark 1.17. In the following chapters of this work we mainly have to use
function spaces consisting of vector valued and tensor valued functions.
For u = (u1 , ..., ud )> : Ω → Rd we say that u belongs to Lp(·) (Ω)d if and only
if ui ∈ Lp(·) (Ω) for i = 1, ..., d. If ∇u exists for such a function it belongs to
Lp(·) (Ω)d×d if and only if ∂j ui is an element of Lp(·) (Ω) for i, j = 1, ..., d.
All the results mentioned above can directly be extended to the spaces Lp(·) (Ω)d
and Lp(·) (Ω)d×d , respectively. The only thing we have to do is to replace the
2
modulus | · | by an euclidean norm on Rd or Rd×d ∼
= Rd . Since there will be
no confusion we keep using | · | to denote this norm.
In the context of vector and tensor valued functions we also want to
introduce some common notations.
For two P
vectors x, y ∈ Rd the euclidean inner product is denoted by
d
x · y :=
A = (aij ), B = (bij ) ∈ Rd×d ,
i=1 xi yi . Similarly, for two tensors
Pd
their inner product is defined by A : B := i,j=1 aij bij .
For vectors x, y ∈ Rd we also have to define their tensor product given by
x ⊗ y := (x ⊗ y)ij = (xi yj )ij ∈ Rd×d .
Moreover, if we do not need to know the explicit value of constants that
appear in our calculations, these constants are denoted by c or C and may
change their values from line to line.
10
Chapter 1. Lebesgue spaces with variable exponent
Chapter 2
The Hardy-Littlewood maximal
operator
In this chapter we introduce the Hardy-Littlewood maximal operator which is
a powerful tool in harmonic analysis. The importance of the Hardy-Littlewood
maximal operator is based on the fact that it can be used to control several
other operators pointwise. Especially the technique of approximate identities
is intimately connected with the maximal operator as we will see below.
2.1
Boundedness of the maximal operator
Definition 2.1. For f ∈ L1loc (Rd ) we define
Z
1
M f (x) := sup d
r>0 |Br (x)|
|f (y)| dy,
Brd (x)
where |Brd (x)| denotes the Lebesgue measure of a d-dimensional ball centered
at x with radius r. The operator M : f 7→ M f is called Hardy-Littlewood
maximal operator or simply maximal operator.
We summarize the most important facts about the maximal operator in
Theorem 2.2. For f ∈ L1loc (Rd ) the function M f : Rd → [0, ∞] is lower semicontinuous and dominates f , i.e. we have |f (x)| ≤ M f (x) almost everywhere.
For f ∈ Lr (Rd ), 1 ≤ r ≤ ∞, the function M f is finite almost everywhere and
M is of weak type (1, 1), that is, for every λ > 0 we have
||λχ{M f >λ} ||L1 (Rd ) ≤ c||f ||L1 (Rd )
Here, χ{M f >λ} is an abbreviation for the indicator function of the set
{ x ∈ Rd | M f (x) > λ }. In addition, if f belongs to Lr (Rd ), 1 < r ≤ ∞, then
11
12
Chapter 2. The Hardy-Littlewood maximal operator
the operator M : f 7→ M f is bounded on Lr (Rd ), which means that we have
||M f ||Lr (Rd ) ≤ Ar ||f ||Lr (Rd ) .
The constant Ar > 0 only depends on r and the dimension d.
As there is extensive literature on the maximal operator we only want to
refer to [34], [35] or [20] for a more detailed review.
All these results raise the question whether they are still valid if Lr (Rd ) is
replaced by Lp(·) (Rd ). In other words, are there any requirements on our
variable exponent ensuring the boundedness of the Hardy-Littlewood maximal
operator on Lp(·) (Rd ) ?
Fortunately, we can give an affirmative answer in terms of a special continuity
property we have to impose on our exponent.
Definition 2.3. Let p(·) ∈ P(Rd ) be a variable exponent. Assume the
following:
1
c1
1 −
is valid for all x, y ∈ Rd with some
a) ≤
p(x)
p(y)
ln(e + 1/|x − y|)
constant c1 > 0.
1
1 c2
b) There exist constants c2 , p∞ > 0 such that −
≤
p(x)
p∞
ln(e + |x|)
holds for each x ∈ Rd .
The first condition is called local log-Hölder continuity, the second one is called
log-Hölder decay condition and we define clog (p) := max{c1 , c2 } as the log1
is said to be
Hölder constant of p(·). If both conditions are satisfied, then p(·)
globally log-Hölder continuous and we set
log
d
d 1
P (R ) := p(·) ∈ P(R ) is globally log-Hölder continuous
p(·)
Remark 2.4. Every constant r ∈ [1, ∞) belongs to P log (Rd ). Moreover,
p(·) ∈ P log (Rd ) implies p0 (·) ∈ P log (Rd ).
We can now restate Theorem 2.2 in a more general setting and refer to [7]
and [12] for the proof.
Theorem 2.5. Let p(·) ∈ P log (Rd ) be a variable exponent with p− > 1. Then
the Hardy-Littlewood maximal operator M : f 7→ M f is bounded on Lp(·) (Rd ),
that is
||M f ||Lp(·) (Rd ) ≤ Ap(·) ||f ||Lp(·) (Rd )
holds true with a constant Ap(·) only depending on the log-Hölder constant of
the variable exponent p(·).
2.1 Boundedness of the maximal operator
13
This result is crucial for the theory of generalized Lebesgue and Sobolev
spaces. For example, Sobolev embeddings and many questions concerning
singular integrals on Lp(·) (Rd ) can be reduced to estimates involving the HardyLittlewood maximal operator. As a consequence, some results just follow from
the continuity of M .
Now, most of the above statements treat cases where p(·) is defined on the
whole space. But dealing with partial differential equations, we will encounter
situations where our exponents are only defined on a subset Ω ⊂ Rd . Therefore,
we cannot directly use the above statements. In order to solve this problem,
we give the following extension result.
Proposition 2.6. For every p(·) ∈ P log (Ω) there exists an extension
p̃(·) ∈ P log (Rd ) of p(·) with the same log-Hölder constant. This extension
satisfies p̃− = p− and p̃+ = p+ .
Proof. See Lemma 3.5 in [12].
The last proposition enables us to formulate a local version of Theorem 2.5
by extending the problem to Rd .
Corollary 2.7. For p(·) ∈ P log (Ω), p− > 1, and f ∈ Lp(·) (Ω) we have
||M f ||Lp(·) (Ω) ≤ Ap(·) ||f ||Lp(·) (Ω)
with a constant Ap(·) only depending on the log-Hölder constant of p(·).
Proof. By Proposition 2.6 we can extend p(·) ∈ P log (Ω) to p̃(·) ∈ P log (Rd )
without changing p(·)‘s significant properties. Setting f = 0 on Ωc and
denoting the extension of f by f˜ yields f˜ ∈ Lp̃(·) (Rd ) and we can make use of
Theorem 2.5 to estimate
||M f ||Lp(·) (Ω) ≤ ||M f˜||Lp̃(·) (Rd ) ≤ Ap(·) ||f ||Lp̃(·) (Rd ) = Ap(·) ||f ||Lp(·) (Ω) .
As the treatment of parabolic equations often demands to distinguish between spatial and time variables, we introduce two additional operators acting
on functions defined on Rd+1 and examine their connection with the usual
Hardy-Littlewood maximal operator .
Given g ∈ L1loc (Rd+1 ), we know that up to null-sets K ⊂ R and L ⊂ Rd ,
we have g(t, ·) ∈ L1loc (Rd ) for every t ∈ K c and g(·, x) ∈ L1loc (R) for every
x ∈ Lc . Therefore we may define two more operators, one acting on the spatial
variables and the other one acting on the time variable, i.e.
Z
1
S
|g(t, y)| dy,
M g(t, x) := sup
r>0 |Br (x)|
Br (x)
Z
1
M T g(t, x) := sup
|g(s, x)| ds .
ρ>0 |Iρ (t)|
Iρ (t)
14
Chapter 2. The Hardy-Littlewood maximal operator
Here, |Iρ (t)| means the Lebesgue measure of the interval Iρ (t) := [t − ρ, t + ρ].
In order to guarantee boundedness of these operators we have to impose higher
integrability on the functions involved.
Namely, for g ∈ Lr (Rd+1 ), r > 1, we have g(t, ·) ∈ Lr (Rd ) for almost every
t ∈ R and g(·, x) ∈ Lr (R) for almost every x ∈ Rd . Therefore, Theorem 2.2
implies M S g(t, ·) ∈ Lr (Rd ), because
||M S g(t, ·)||Lr (Rd ) ≤ ASr ||g(t, ·)||Lr (Rd )
is valid for almost every t ∈ R. Integration with respect to t then leads to
||M S g||Lr (Rd+1 ) ≤ ASr ||g||Lr (Rd+1 ) .
The same reasoning applies to the spatial variables:
||M T g(·, x)||Lr (R) ≤ ATr ||g(·, x)||Lr (R)
holds true for almost every x ∈ Rd . But this in turn implies the boundedness
of M T on Lr (Rd+1 ). That is
||M T g||Lr (Rd+1 ) ≤ ATr ||g||Lr (Rd+1 ) .
This allows us to use M T and M S successively and to define for a function
g ∈ Lr (Rd+1 )
M F g := M T (M S g).
According to
||M F g||Lr (Rd+1 ) = ||M T (M S g)||Lr (Rd+1 ) ≤ ATr ASr ||g||Lr (Rd+1 ) ,
we have proved
Theorem 2.8. The operator M F maps Lr (Rd+1 ) to Lr (Rd+1 ) whenever r > 1.
But the situation changes if we use variable exponents, although Theorem
2.5 ensures the boundedness of the Hardy-Littlewood maximal operator on
Lp(·,·) (Rd+1 ) for p(·, ·) ∈ P log (Rd+1 ) with p− > 1. To derive a contradiction we
give the following
Example 2.9. We set Ω := [0, 1] × [0, 1] ⊂ R2 , Ω1 := [0, 31 ] × [0, 1] and
Ω2 := [ 32 , 1] × [0, 1].
Assume p(·, ·) ∈ P log (Ω) satisfies p|Ω1 = 2 and increases to p|Ω2 = 3. Now take
an arbitrary h ∈ L2 ([0, 1]) and define f (x, y) := h(y)χΩ1 (x, y). It follows that
Z
Z
p(x,y)
|f (x, y)|
dx dy = |h(y)χΩ1 (x, y)|2 dx dy ≤ c||h||2L2 ([0,1]) .
Ω
Ω
2.2 Mollification on Lp(·) (Rd )
15
Thus, by definition of the Luxemburg norm we get ||f ||Lp(·,·) (Ω) ≤ c||h||L2 ([0,1]) .
For x ∈ ( 13 , 1] we have
1
1
Z1
Z3
Z3
1
X
M f (x, y) ≥ − |f (z, y)| dz ≥ |f (z, y)| dz = |h(y)| dz = |h(y)|,
3
0
0
0
where M X denotes the maximal operator acting on the first variable.
Now suppose M X is bounded on Lp(·,·) (Ω), then using the last inequality we
could estimate
||h||L3 ([0,1]) ≤ c||M X f ||Lp(·,·) (Ω) ≤ c||f ||Lp(·,·) (Ω) ≤ c||h||L2 ([0,1]) ,
which in turn implies L2 ([0, 1]) ,→ L3 ([0, 1]), since h ∈ L2 ([0, 1]) was arbitrary.
But this is the desired contradiction.
2.2
Mollification on Lp(·)(Rd)
In the next section we want to examine mollification on Lp(·) (Rd ) since this is
a standard tool to derive density of smooth functions. In the context of the
classical Lr -theory, one frequently uses the following inequality.
Proposition 2.10. Suppose f belongs to Lr (Rd ) for 1 ≤ r ≤ ∞ and that g is
an element of L1 (Rd ). Then Young’s inequality for convolutions holds true:
||f ∗ g||Lr (Rd ) ≤ ||f ||Lr (Rd ) ||g||L1 (Rd )
(2.1)
Proof. The inequality is proved in [2] , Theorem 2.12.
The proof of this fact is based on the continuity of the translation operator
on Lr (Rd ) and Hölder’s inequality. Unfortunately, it was shown in Theorem
2.37 in [12] that the translation operator is not bounded on Lp(·) (Rd ). As a
consequence (2.1) only holds for constant r and convolution with an arbitrary
L1 - function does not yield a continuous mapping on Lp(·) (Rd ) in general. But
if we restrict ourselves to convolution with so-called ”bell-shaped” functions,
it is still possible to get a result that is similar to Proposition 2.10 and thus,
to derive boundedness of the convolution operator on this class of functions.
To this end, we adopt
Definition 2.11. A real valued function ψ ∈ L1 (Rd ), ψ ≥ 0, is called bellshaped if it is radially decreasing and radially symmetric i.e. ψ(x) = ψ(|x|).
The function
Ψ(x) := sup ψ(y)
y:|y|≥|x|
16
Chapter 2. The Hardy-Littlewood maximal operator
is called the least bell-shaped majorant of ψ and we define
n
o
B(Rd ) := ψ bell-shaped | Ψ ∈ L1 (Rd ) .
B(Rd ) is the suitable set of functions for the convolution operator to be
bounded on and it will turn out to be the appropriate set for the mollification
method we will use in the sequel. Again, this is due to the fact that convolution
with bell-shaped functions can be controlled by the Hardy-Littlewood maximal
operator.
For a proof of the next theorem we refer to [34] or [20], but we want to mention
that they only treat the case of classic Lebesgue spaces Lr (Rd ). Fortunately,
the proof can easily be transfered to the case of generalized Lebesgue spaces
Lp(·) (Rd ), see also [12], Theorem 3.50.
Theorem 2.12. Suppose p(·) ∈ P log (Rd ), p− > 1, ψ ∈ B(Rd ) and that f
belongs to Lp(·) (Rd ). Then ψε (x) := ε−d ψ(x/ε) also defines a function in B(Rd )
for every ε > 0, and we have the following facts:
a) There is a K > 0 that only depends on ||Ψ||L1 (Rd ) and the log-Hölder
constant of p(·) such that ||ψε ∗ f ||Lp(·) (Rd ) ≤ K||f ||Lp(·) (Rd ) .
b) Setting A := ||Ψ||L1 (Rd ) , we have sup |(ψε ∗ f )(x)| ≤ 2AM f (x).
ε>0
R
c) If Rd ψ(x)dx = 1 then ψε ∗ f → f almost everywhere for ε → 0. And if
p(·) is bounded, we even have ψε ∗ f → f in Lp(·) (Rd ) as h → 0.
Remark 2.13. Since P log (Rd ) contains every constant exponent 1 ≤ r < ∞,
the results stated above clearly hold true for p(·) replaced by some constant
r ∈ (1, ∞).
The last theorem is the key step in carrying over the standard mollification
method to Lp(·) (Rd ), and for our purpose, it serves as one of the most important applications of the Hardy-Littlewood maximal operator on generalized
Lebesgue spaces, because it enables us to prove density of smooth functions in
the function spaces we use in this work. As a first result we show
Corollary 2.14. Let p(·) ∈ P log (Rd ) be a bounded exponent. Then the set
C0∞ (Rd ) is dense in Lp(·) (Rd ).
R
Proof. Take ψ ∈ B(Rd )∩C0∞ (B1d (0)) with Rd ψ(x)dx = 1 and a cut-off function
η ∈ C0∞ (B1d (0)) with 0 ≤ η ≤ 1. For ψε , 0 < ε ≤ 1, as above and
ηj (x) := η(x/j), j ∈ N, we define fε,j := ψε ∗ (ηj f ).
d
Thus, supp (fε,j ) ⊂ supp(ψε ) + supp(ηj f ) ⊂ Bj+1
(0) and the standard
theory on convolutions shows that the smoothness of ψε transfers to fε,j and
2.2 Mollification on Lp(·) (Rd )
17
fε,j therefore belongs to C0∞ (Rd ). Now,
||f − fε,j ||Lp(·) (Rd ) = ||f − ψε ∗ (ηj f )||Lp(·) (Rd )
≤ ||f − ηj f ||Lp(·) (Rd ) + ||ψε ∗ (ηj f ) − ηj f ||Lp(·) (Rd ) .
(2.2)
Suppose we are given δ > 0.
As limj→∞ (1 − ηj )f = 0 almost everywhere and |(1 − ηj )f | ≤ |f | ∈ Lp(·) (Rd ),
we can use dominated convergence to deduce the existence of k ∈ N such that
||f − ηk f ||Lp(·) (Rd ) = ||(1 − ηk )f ||Lp(·) (Rd ) ≤ 2δ .
Furthermore, ||ηk f ||Lp(·) (Rd ) ≤ K||f ||Lp(·) (Rd ) yields ηk f ∈ Lp(·) (Rd ). As a result,
Theorem 2.12 c) provides ε0 > 0 such that ||ψε0 ∗(ηk f )−ηk f ||Lp(·) (Rd ) is bounded
by 2δ . Plugging in (2.2) shows ||f − fε0 ,k ||Lp(·) (Rd ) ≤ δ and we are done since
δ > 0 was arbitrary.
Another nice application we will frequently encounter, is the connection
between the Hardy-Littlewood maximal operator and the so-called Riesz potential operator.
Definition 2.15. For a function f ∈ C0∞ (Rd ) and 0 < α < d we define its
Riesz potential Rα f : Rd → [0, ∞] by
Z
|f (y)|
Rα f (x) :=
dy .
|x − y|d−α
Rd
The condition f ∈ C0∞ (Rd ) can be weakened and the integral yet remains
well-defined. This is the content of the next Theorem. For a more detailed
description of Riesz potentials and their properties we refer to [34], but here
we only cite
Theorem 2.16. Suppose we have 0 < α < d and 1 ≤ r < s < ∞ such that
1
= 1r − αd . If f belongs to Lr (Rd ), then Rα f is finite almost everywhere.
s
Proof. The proof can be found in [34], p.119 Theorem 1.
The Riesz potential not only is finite almost everywhere for certain classes
of functions, but it can also be controlled in terms of the Hardy-Littlewood
maximal operator. This is captured in the next theorem. For a proof we refer
to [29], Theorem 1.32, or [12], Lemma 3.57.
Theorem 2.17. For 0 < α < d, r > 0, x ∈ Rd and f ∈ L1loc (Rd ) we have
Z
|f (y)|
dy ≤ c(α)rα M f (x),
|x − y|d−α
Brd (x)
where M f is the Hardy-Littlewood maximal operator.
18
Chapter 2. The Hardy-Littlewood maximal operator
With the help of the Riesz potential operator and the boundedness of the
1,p(·)
maximal operator, it is possible to obtain the Poincaré inequality on W0 (Ω)
which in turn yields the existence of an equivalent norm, as already mentioned
in Remark 1.16.
Theorem 2.18. Let Ω ⊂ Rd be a bounded domain and let p(·) ∈ P log (Ω) be a
1,p(·)
variable exponent. For every function u ∈ W0 (Ω) we have
||u||Lp(·) (Ω) ≤ c diam(Ω)||∇u||Lp(·) (Ω)d ,
1,p(·)
thus, ||∇ · ||Lp(·) (Ω)d constitutes an equivalent norm on W0 (Ω). The constant
c only depends on the dimension d and the log-Hölder constant of the variable
exponent.
Proof. The proof can be found in [14].
The last result is very important in applications, because the functional
ρp(·) (∇u) frequently appears in connection with the natural energy in the
weak formulation of partial differential equations involving generalized Sobolev
spaces. We keep using the equivalent norm for the rest of this work without
mentioning it particularly.
Chapter 3
Function spaces and integration
by parts
Based on the nonhomogeneous heat equation we first of all want to highlight
some ideas we are going to seize on in what follows. Remember that this
equation is given by
∂t u − ∆u = f in I × Ω
u = 0 on I × ∂Ω
u(0) = u0 on Ω,
(3.1)
for a bounded domain Ω ⊂ R3 with smooth boundary, a bounded time interval
I := (0, T ) ⊂ R+ and an unknown function u : I × Ω → R. Assume for the
moment, we are given a smooth solution u of (3.1) for f ∈ L2 (I, L2 (Ω)) and
u0 ∈ L2 (Ω). Then u is also a weak solution, which means that
ZT Z
ZT Z
∇u · ∇ϕ dx dτ =
ut ϕ dx dτ +
0
Ω
ZT Z
0
f ϕ dx dτ
0
Ω
(3.2)
Ω
is valid for every ϕ ∈ L2 (I, W01,2 (Ω)).
With this equation in mind we can on the one hand interpret ∂t u as an element
of L2 (I, (W01,2 (Ω))0 ) which is the dual space of L2 (I, W01,2 (Ω)). On the other
hand, the choice ϕ = u in (3.2), together with an integration by parts in time
and Young’s inequality, yields the a-priori estimate
2
2
||u||C(I,L
¯ 2 (Ω)) ≤ c ||u0 ||L2 (Ω) + ||f ||L2 (I×Ω) .
So in a quite natural way, we have to use the three different spaces W01,2 (Ω),
L2 (Ω) and (W01,2 (Ω))0 , that are linked by a chain of continuous embeddings
W01,2 (Ω) ,→,→ L2 (Ω) ,→ (W01,2 (Ω))0 .
19
20
Chapter 3. Function spaces and integration by parts
This construction can be generalized in the following way.
Suppose a Banach space V with norm || · ||V satisfies a continuous and dense
embedding V ,→ H into a Hilbert space H with norm || · ||H . Then, by
the Riesz representation theorem, see Theorem 6.2 in the Appendix, we can
identify H and its dual space H 0 . But now we can restrict f ∈ H 0 to V to
obtain H 0 ,→ V 0 , and overall we then have
V ,→ H ,→ V 0 .
In this situation (V, H, V 0 ) is called Gelfand triple or evolution triple.
For all v, w ∈ V there holds
hv, wiV = (v, w)H = (w, v)H = hw, viV .
Therefore, the notion of a generalized time derivative is defined as follows:
given a Gelfand triple (V, H, V 0 ) and u ∈ Lp (I, V ), 1 < p < ∞, a function
0
g ∈ Lp (I, V 0 ) is called generalized time derivative of u if and only if
ZT
ZT
hg(τ ), viV ϕ(τ ) dτ = −
0
(u(τ ), v)H ϕ0 (τ ) dτ
0
0
holds true for every v ∈ V and for every ϕ ∈ C0∞ (I). If such g ∈ Lp (I, V 0 )
:= g.
exists it is unique and we can define du
dt
Similar to classic Sobolev spaces, one may now introduce the Banach space
du
p0
0
p
∈ L (I, V ) ,
W := u ∈ L (I, V ) dt
with the associated norm
du ||u||W := ||u||Lp (I,V ) + 0
.
dt Lp (I,V 0 )
It is easy to show that W contains smooth functions with values in V , but
even more is true.
Theorem 3.1. The space C ∞ (I, V ) is dense in W and we have the continuous
¯ H). Moreover,
embedding W ,→ C(I,
Zb
Zb
du
dv
h (τ ), v(τ )iV dτ = (u(b), v(b))H − (u(a), v(a))H − h (τ ), u(τ )iV dτ (3.3)
dt
dt
a
is valid for all u, v ∈ W and all a, b ∈ I.
a
21
The proof of Theorem 3.1 is based on mollification together with a
subsequent limiting process and can be found in [16] and [18]. Now consider
again the parabolic p-Laplacian which reads
∂t u − div(|∇u|p−2 ∇u) = f in I × Ω
u = 0 on I × ∂Ω
u(0) = u0 in Ω.
Assuming I × Ω ⊂ R1+3 ,
would choose
6
5
0
0
≤ p < 3, f ∈ Lp (I, Lp (Ω)3 ) and u0 ∈ L2 (Ω)3 , we
V := W01,p (Ω)3 with ||u||V := ||∇u||Lp (Ω)3×3 and H := L2 (Ω)3 ,
to put us in a situation where we can use the appropriate evolution triple
W01,p (Ω)3 , L2 (Ω)3 , (W01,p (Ω)3 )0 .
A function u that is contained in the resulting space
1,p
1,p
p
3 du
p0
3 0
W = u ∈ L (I, W0 (Ω) ) ∈ L (I, (W0 (Ω) ) ) ,
dt
is then called a weak solution of the parabolic p-Laplacian if and only if
Z Z
Z Z
Z
du
p−2
|∇u| ∇u(τ ) : ∇ϕ(τ ) dτ =
f (τ ) · ϕ(τ ) dτ
h (τ ), ϕ(τ )iV dτ +
dt
I
I
Ω
I
Ω
is satisfied for every ϕ ∈ C0∞ (I × Ω)3 , and u(0) = u0 . Notice that the initial
¯ L2 (Ω)3 ). With the
value is well-defined due to the embedding W ,→ C(I,
theory of monotone operators it is now possible to prove existence of a weak
solution u ∈ W and in doing so, formula (3.3) plays a major role.
We also want to mention that the space Lp (I, W01,p (Ω)3 ) is the natural choice in
view of the elliptic part induced by −div(|∇u|p−2 ∇u), while the information
coming from the time derivative is captured in the corresponding Gelfand
triple.
But as already explained, we neither want to use Bochner spaces nor spaces
built on Bochner spaces for the treatment of our generalized parabolic p(t,x)Laplacian equation. However, we would like to have similar results as above,
namely, a rule for integration by parts and some sort of continuous embedding
in order to justify certain initial values.
We want to briefly discuss another problem we have to face.
On the one hand the notion of a “generalized time derivative” perfectly fits
situations where one can use Gelfand triples. But on the other hand the
existence of a suitable Gelfand triple not always occurs as natural as in the
22
Chapter 3. Function spaces and integration by parts
case of the prototype heat equation above. In case of the parabolic p-Laplacian,
we also have to keep in mind that the whole construction strongly depends on
the value of p and the necessary embedding theorems. When dealing with
variable exponents and equations where these exponents play a significant
role, the situation can even get worse. For example, think of a problem where
an exponent is given on a domain Ω ⊂ R3 such that p− < 6/5. Now, due
to the Sobolev embedding theorem on generalized Sobolev spaces, see [14],
1,p(·)
W0 (Ω)3 ,→ L2 (Ω)3 is true if and only if p− ≥ 6/5. Thus, the only chance to
get back into the above setting is to enforce an evolution triple by presuming
p− ≥ 6/5 nevertheless, which isn’t very natural.
We therefore construct a space similar to W without using a Gelfand triple and
by working with the more natural concept of distributional time derivatives.
The next step will then be to show density of smooth functions in this new
space without major constraints on the exponent. As we will heavily rely on
the boundedness of the Hardy-Littlewood maximal operator on generalized
Lebesgue spaces, we only require that p(·, ·) belongs to P log (QT ) and p− > 1.
In addition, the same construction can be used to treat the case of constant
p ∈ (1, ∞) even for 1 < p < 6/5. The desired embedding result and the rule
for integration by parts will follow by an approximation argument similar to
that in [18] or [16].
3.1
The space W (QT )
On a bounded domain Ω ⊂ Rd let us now consider the initial-boundary value
problem
∂t u − divS(∇u) = f in I × Ω
u = 0 on I × ∂Ω
u(0) = u0 in Ω,
for a nonlinear operator S having the special form
S : I×Ω × Rd×d → Rd×d
S(τ, x, ∇u) := (1 + |∇u|)p(τ,x)−2 ∇u(τ, x).
As already mentioned above, this operator represents a generalization of the
p-Laplacian operator since the variable exponent p(·, ·), p− > 1, may depend
on τ ∈ (0, T ) and x ∈ Ω. The unknown function u : I × Ω → Rd is to be
determined subject to given data f and u0 .
Since classical Bochner spaces cannot properly reflect the effects induced by
the variable exponent, we have to approach more carefully in defining the right
function spaces.
23
3.1 The space W (QT )
In view of a weak formulation of our problem and the structure of S, it certainly
makes sense to require ∇u ∈ Lp(·,·) (QT )d×d , since S can be interpreted as a
0
monotone operator mapping Lp(·,·) (QT )d×d to its dual space Lp (·,·) (QT )d×d .
Furthermore, the method presented in [18] leads to the conjecture, that an
integration by parts applied to the term with the time derivative could yield
the a-priori estimate ||u||C(I,L
¯ 2 (Ω)d ) ≤ K. Thus, the idea is, to already exploit
these information and to justify them in the aftermath. Since we want to keep
the perception of the unknown u as a function depending on the time variable
with values in a certain function space, we make the following ansatz.
Definition 3.2. Given an open, bounded subset Ω ⊂ Rd with Lipschitz
boundary and a bounded time interval I = (0, T ) ⊂ R, we define for fixed
τ ∈ I and a bounded exponent p(·, ·) ∈ P log (I × Ω) with p− > 1
o
n
Vτ (Ω) := u ∈ L2 (Ω)d ∩ W01,1 (Ω)d ∇u ∈ Lp(τ,·) (Ω)d×d ,
equipped with the norm
||u||Vτ (Ω) := ||u||L2 (Ω)d + ||∇u||Lp(τ,·) (Ω)d×d .
Lemma 3.3. For every τ ∈ I the set Vτ (Ω) is a reflexive and separable Banach
space since
1,p(τ,·)
Vτ (Ω) = L2 (Ω)d ∩ W0
(Ω)d .
Proof. Being the intersection of two separable Banach spaces, Vτ (Ω) again is
a separable Banach space, see Lemma 6.7 in the Appendix. Hence, we only
1,p(τ,·)
(Ω)d and this is the
have to show that every u ∈ Vτ (Ω) is contained in W0
only time we use the norm
1,p(τ,·)
||v||W 1,p(τ,·) (Ω)d = ||v||Lp(τ,·) (Ω)d + ||∇v||Lp(τ,·) (Ω)d×d , for v ∈ W0
(Ω)d .
0
As u ∈ Vτ (Ω) implies ∇u ∈ Lp(τ,·) (Ω)d×d , the problem is reduced to showing
u ∈ Lp(τ,·) (Ω)d and that every u ∈ Vτ (Ω) is a trace-zero function, see [14]. To
this end, we note that u ∈ Vτ (Ω) ⊂ W01,1 (Ω)d satisfies the following Poincarétype inequality
Z
|∇u(y)|
diam(Ω)d
dy,
|u(x) − uΩ | ≤
d
|x − y|d−1
Ω
R
where uΩ := − Ω u(x) dx.
The expression uΩ is well defined since
2
d
1
d
u ∈ L (Ω) ,→ L (Ω) and for a proof of the result we refer to [29], Lemma 1.50.
Since Ω is bounded, we have Ω ⊂ BR (x) for every x ∈ Ω with R := diam(Ω)
and after extending u by zero outside of Ω we get
Z
diam(Ω)d
|∇u(y)|
|u(x) − uΩ | ≤
dy.
d
|x − y|d−1
d (x)
BR
24
Chapter 3. Function spaces and integration by parts
Appealing to Theorem 2.17 we obtain
|u(x) − uΩ | ≤ c(R, d)M (|∇u|)(x).
The monotonicity of the Luxemburg norm and the boundedness of the
maximal operator show
||u − uΩ ||Lp(τ,·) (Ω)d = || |u − uΩ | ||Lp(τ,·) (Ω)
≤ c(R, d)||M (|∇u|)||Lp(τ,·) (Ω)
≤ C(R, d, p(τ, ·))||∇u||Lp(τ,·) (Ω)d×d .
So the first assertion follows from
||u||Lp(τ,·) (Ω)d ≤ ||u − uΩ ||Lp(τ,·) (Ω)d + ||uΩ ||Lp(τ,·) (Ω)d
≤ C(R, d, p(τ, ·))||∇u||Lp(τ,·) (Ω)d×d + ||uΩ ||Lp(τ,·) (Ω)d < ∞,
which is valid for every u ∈ Vτ (Ω). Since every u ∈ Vτ (Ω) is contained in
W01,1 (Ω)d as well, u is a trace-zero function and in combination with the first
1,p(τ,·)
assertion we obtain u ∈ W0
(Ω)d .
The spaces Vτ (Ω) will now serve as building blocks for the basic space we
are going to use.
Definition 3.4. For Ω and I as in Definition 3.2, ΓT := (0, T ) × ∂Ω and a
bounded exponent p(·, ·) ∈ P log (I × Ω), p− > 1, we set
n
o
2
d
p(·,·)
d×d
W (QT ) := u : I → Vτ (Ω) u ∈ L (QT ) , ∇u ∈ L
(QT )
, u|ΓT = 0
and equip it with the norm
||u||W (QT ) := ||u||L2 (QT )d + ||∇u||Lp(·,·) (QT )d×d .
Here, ∇u stands for the distributional gradient of u which is well-defined since
u ∈ L2 (QT )d ,→ L1loc (QT )d , see Definition 6.14 in the Appendix.
Remark 3.5. The prescribed values on the lateral boundary u|ΓT are now
icorporated in the function space. It will follow from a density result we are
going to prove that they are well defined in the sense of traces.
Lemma 3.6. The space W (QT ) is a Banach space.
25
3.1 The space W (QT )
Proof. Let (uk )k∈N be a Cauchy sequence in W (QT ). By definition of the norm
in W (QT ), (uk )k∈N then is a Cauchy sequence in L2 (QT )d and (∇uk )k∈N is a
Cauchy sequence in Lp(·,·) (QT )d×d . According to the completeness of these two
spaces, there exist u ∈ L2 (QT )d and U ∈ Lp(·,·) (QT )d×d such that uk converges
to u in L2 (QT )d and ∇uk to U in Lp(·,·) (QT )d×d , respectively, and it remains
to show that ∇u = U holds true. So let uki denote the i-th component of the
vector valued function uk . For j ∈ {1, ..., d} and ϕ ∈ C0∞ (QT ) we have by
definition of the distributional derivative
Z
Z
k
ui ∂j ϕ dx dτ = − ∂j uki ϕ dx dτ ,
QT
QT
because uk ∈ W (QT ) for every k ∈ N. Since strong convergence implies weak
convergence we can pass to the limit k → ∞ in the last equation to get
Z
Z
ui ∂j ϕ dx dτ = − Uji ϕ dx dτ .
QT
QT
But this is equivalent to Uji = ∂j ui in the distributional sense and since
i, j ∈ {1, ..., d} were arbitrary we get ∇u = U .
The space W (QT ) seems awkward to handle but we will see below that it
perfectly fits our problem. As a first step towards an easier handling, we verify
density of smooth functions in W (QT ). But before we come to the proof we
need a pointwise estimate near the boundary of Ω.
Proposition 3.7. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary.
Then for every u ∈ W01,1 (Ω) there holds
Z
|∇u(y)|
d(x, ∂Ω)d
dy
|u(x)| ≤ c
|B2d(x,∂Ω) (x)|
|x − y|d−1
B2d(x,∂Ω) (x)
for almost every x ∈ Ω, after extending u by zero to Ωc .
Proof. First assume u ∈ C0∞ (Ω) ∩ W01,1 (Ω), the general case u ∈ W01,1 (Ω) then
follows by approximation.
As ∂Ω is Lipschitz, the complement of Ω has the following property: there
exists a constant cf such that
|B2d(x,∂Ω) (x) ∩ Ωc | ≥ cf |B2d(x,∂Ω) (x)|.
A domain with this property is said to have a fat complement, see [26]. So if
we extend ∇u by 0 outside of Ω, fix x ∈ Ω and pick y ∈ B2d(x,∂Ω) (x) ∩ Ωc we
get
|x−y|
Z
u(x) = u(x) − u(y) = −
∇u(x + sω) · ω ds
ω ∈ S d−1 .
0
26
Chapter 3. Function spaces and integration by parts
Integrating over B2d(x,∂Ω) (x) ∩ Ωc with respect to y and dividing by
|B2d(x,∂Ω) (x) ∩ Ωc | yields
Z
Z
−
u(x) dy −
−
u(y) dy
B2d(x,∂Ω) (x)∩Ωc
B2d(x,∂Ω) (x)∩Ωc
|x−y|
Z
Z
−
=−
∇u(x + sω)ω ds dy .
B2d(x,∂Ω) (x)∩Ωc
0
Since u vanishes identically outside of Ω we can resume by
|x−y|
Z
Z
1
|u(x)| ≤
|B2d(x,∂Ω) (x) ∩ Ωc |
|∇u(x + sω)| ds dy
B2d(x,∂Ω) (x)∩Ωc
≤
|x−y|
Z
Z
1
|∇u(x + sω)| ds dy
cf |B2d(x,∂Ω) (x)|
B2d(x,∂Ω) (x)∩Ωc
≤
0
2d(x,∂Ω)
Z
Z
1
0
|∇u(x + sω)| ds dy,
cf |B2d(x,∂Ω) (x)|
0
B2d(x,∂Ω) (x)
and changing to polar coordinates leads to
2d(x,∂Ω)
Z
Z
1
|∇u(x + sω)| ds dy
cf |B2d(x,∂Ω) (x)|
0
B2d(x,∂Ω) (x)
=
Z
1
2d(x,∂Ω)
2d(x,∂Ω)
Z
Z
|∇u(x + sω)|rd−1 ds drdo(ω)
cf |B2d(x,∂Ω) (x)|
S d−1
d(x, ∂Ω)d
≤ c
|B2d(x,∂Ω) (x)|
d(x, ∂Ω)d
|B2d(x,∂Ω)(x) |
0
2d(x,∂Ω)
Z
Z
S d−1
= c
0
|∇u(x + sω)| d−1
s
ds do(ω)
sd−1
0
Z
B2d(x,∂Ω) (x)
|∇u(y)|
dy .
|x − y|d−1
27
3.1 The space W (QT )
We now have all the tools to state one of the main theorems of this section.
Theorem 3.8. The space C0∞ (QT )d is dense in W (QT ).
Proof. For convenience, we explain our strategy before we go into details.
First of all we construct a suitable cut-off function in order to “gain some space
to mollify”. Then, given u ∈ W (QT ), we will construct a sequence of smooth
functions by mollifying a truncated version of u with a bell-shaped function.
In a last step we show that a subsequence actually converges to u in W (QT )
by means of Lebesgue’s theorem on dominated convergence.
For a small h > 0 let
n
o
Ωh := x ∈ Ω d(x, ∂Ω) > h ,
Ih := (h, T − h),
QhT := Ih × Ωh .
In order to define the right
R cut-off function, we take a positive mollifier
ϕ ∈ C0∞ (B1d+1 (0)) with Rd+1 ϕ(τ, x) dx dτ = 1 and set
ηh := ϕ h ∗ χQhT ,
4
where χQhT denotes the characteristic function of QhT . Then the cut-off function
ηh belongs to C0∞ (QT ) for every h > 0, and we also note that this function
cuts off in space and time direction. Besides, ηh has three useful properties
that can easily be verified:
3h
supp ηh ⊂ QT4 ⊂⊂ QT , 0 ≤ ηh ≤ 1 and ||∇ηh ||L∞ (QT )d ≤ c/h.
For a positive mollifier ω ∈ B(Rd+1 ) ∩ C0∞ (B1d+1 (0)),
we now define
uh := ω h ∗ (ηh u).
R
Rd+1
(3.4)
ω(τ, x) dx dτ = 1,
4
By the method of construction, uh is an element of C0∞ (QT )d which obviously
is a subspace of W (QT ).
We are now going to prove limh→0 uh = u in W (QT ), for a suitable subsequence.
Since the norm in W (QT ) was defined by
|| · ||W (QT ) := || · ||L2 (QT )d + ||∇ · ||Lp(·,·) (QT )d×d ,
we first of all show convergence in L2 (QT )d . By Young’s inequality for convolutions, see Theorem 2.10, we have
||uh ||L2 (QT )d = ||ω h ∗ (ηh u)||L2 (QT )d ≤ ||ηh u||L2 (QT )d ≤ ||u||L2 (QT )d ,
4
28
Chapter 3. Function spaces and integration by parts
and that is why uh belongs to L2 (QT )d for h > 0. Again we can use Young’s
inequality to estimate
||uh − u||L2 (QT )d = ||ω h ∗ (ηh u) − u||L2 (QT )d
4
≤ ||ω h ∗ (ηh u − u)||L2 (QT )d + ||ω h ∗ u − u||L2 (QT )d
4
4
≤ ||(ηh − 1)u||L2 (QT )d + ||ω h ∗ u − u||L2 (QT )d .
4
But the right-hand side converges to zero for h → 0 thanks to Lebesgue’s
theorem on dominated convergence and the classical theory of convolutions.
Our next goal will be to show that limh→0 ∇uh = ∇u is valid in Lp(·,·) (QT )d×d ,
but for that we have to approach more subtle. First of all we can calculate
∇uh = ∇(ω h ∗ (ηh u))
4
= ω h ∗ ∇(ηh u)
4
= ω h ∗ (u ⊗ ∇ηh ) + ω h ∗ (ηh ∇u).
4
4
The second term on the right-hand side can be treated similarly as above. The
only difference is that we have to rely on the fact that ω is bell-shaped, in order
to use Young’s inequality in the form of Theorem 2.12. But then
||ω h ∗ (ηh ∇u) − ∇u||Lp(·,·) (QT )d×d
4
≤ ||ω h ∗ (ηh ∇u − ∇u)||Lp(·,·) (QT )d×d + ||ω h ∗ ∇u − ∇u||Lp(·,·) (QT )d×d
4
4
≤ K||(1 − ηh )∇u||Lp(·,·) (QT )d×d + ||ω h ∗ ∇u − ∇u||Lp(·,·) (QT )d×d ,
4
and we can now use Proposition 1.9 and Theorem 2.12 to show that the righthand side of the last inequality converges to zero as h tends to zero.
Hence, it remains to prove
lim ω h ∗ (u ⊗ ∇ηh ) = 0 in Lp(·,·) (QT )d×d .
h→0
4
By the method of construction we know ∇ηh ≡ 0 on QhT , and as we want to use
dominated convergence again, we will therefore only need an integrable upper
bound for ωh/4 ∗ (u ⊗ ∇ηh ) on the set where ∇ηh ”behaves badly”. But this
bad set is contained in (0, T ) × Ωch , or near the boundary of Ω, respectively.
For every x ∈ Ωch and τ ∈ (0, T ) we get
Z
|(ω h ∗ (u ⊗ ∇ηh ))(τ, x)| ≤
ω h (τ − s, x − y)|∇ηh (s, y)| |u(s, y)|dyds.
4
4
Bhd+1 ((τ,x))
By definition of W (QT ) we have u(s, ·) ∈ W01,1 (Ω)d for almost every s ∈ (0, T ).
So we can apply Proposition 3.7 and Theorem 2.17 together with (3.4) and
29
3.1 The space W (QT )
c
hd+1
||ωh ||L∞ (Rd+1 ) ≤
Z
to estimate
ω h (τ − s, x − y)|∇ηh (s, y)| |u(s, y)| dyds
4
Bhd+1 ((τ,x))
Z
c
≤
h
Bhd+1 ((τ,x))
=
Z
c
h
Bhd+1 ((τ,x))
Z
−
≤c
hd
d
hd+1 |B2h
(y)|
Z
1
1
d
(y)|
|B2h
|∇u(s, z)|
dzdyds
|y − z|d−1
d (y)
B2h
|∇u(s, z)| dsdz
|∇u(s, z)| dsdz
Bhd+1 ((τ,x))
≤ cM (|∇u|)(τ, x),
where, in the third row we used Fubini’s theorem together with the fact that
Z
1
dy =
|y − z|d−1
Z2π Z2h
0
d (y)
B2h
r1−d rd−1 drdϕ = 4πh.
0
As the maximal function is non-negative, we have thus particularly proved
|ω h ∗ (u ⊗ ∇ηh )| ≤ cM (|∇u|) almost everywhere in QT for every u ∈ W (QT ),
4
and cM (|∇u|) represents the desired integrable upper bound. Due to the
monotonicity of the Luxemburg norm and the boundedness of the maximal
operator, the last inequality yields
ω h ∗ (u ⊗ ∇ηh ) ∈ Lp(·,·) (QT )d×d for every h > 0.
4
Now that we have the right expression to control ω h ∗ (u ⊗ ∇ηh ) it remains
4
to establish
lim ω h ∗ (u ⊗ ∇ηh ) = 0 almost everywhere in Ω.
h→0
4
Notice that for every (τ, x) ∈ QT there is an h > 0 such that (τ, x) lies in
QhT . This in turn implies that u ⊗ ∇ηh converges to 0 almost everywhere in
QT for h → 0. Moreover, for every τ ∈ (0, T ) and x ∈ Ωch , an application of
Proposition 3.7 and ||∇ηh ||L∞ (QT )d ≤ hc show
Z
c
|∇u(τ, y)|
|(u ⊗ ∇ηh )(τ, x)| ≤
dy .
h
|x − y|d−1
d (x)
B2h
30
Chapter 3. Function spaces and integration by parts
Again, we want to control the last integral in terms of a suitable maximal
function. But Example 2.9 has shown that we cannot proceed by simply
applying the standard Hardy-Littlewood maximal operator to spatial and
time coordinates one after another. So the idea will be to make use of the
−
boundedness of the operator M F on Lp (Rd )d×d instead.
−
The function ∇u(τ, ·) obviously belongs to Lp(τ,·) (Ω)d×d ,→ Lp (Ω)d×d , p− > 1,
and after extending ∇u by 0 to QcT we can continue as follows.
c
|(u ⊗ ∇ηh )(τ, x)| ≤
h
Z
d (x)
B2h
|∇u(τ, y)|
c
dy ≤ C2hM S (|∇u|)(τ, x)
d−1
|x − y|
h
≤ CM S (|∇u|)(τ, x)
≤ CM T (M S (|∇u|))(τ, x)
= CM F (|∇u|)(τ, x),
where we again used Theorem 2.17 in the first row and Theorem 2.8 for the
rest.
−
So |(u ⊗ ∇ηh )| is controlled by CM F (|∇u|) ∈ Lp (QT ) and thus converges
−
to zero in Lp (QT ) as h → 0, again by dominated convergence. Young’s
inequality immediately yields
lim ||ω h ∗ (u ⊗ ∇ηh )||Lp− (QT )d×d ≤ lim ||u ⊗ ∇ηh ||Lp− (QT )d×d = 0
h→0
4
h→0
and we can finally deduce limh→0 ωh/4 ∗ (u ⊗ ∇ηh ) = 0 almost everywhere in
QT , at least for a not relabled subsequence, and especially limh→0 ∇uh = ∇u
in Lp(·,·) (QT )d×d .
All in all, for an arbitrary u ∈ W (QT ) we have found a sequence (uh )h>0 that
is contained in C0∞ (QT )d and converges to u with respect to the W (QT ) norm.
But this is equivalent to the density of C0∞ (QT )d in W (QT ).
The proof of the last theorem contained an estimate that is going to be
used very often in the rest of this work. For this reason, we capture it in an
extra lemma.
Lemma 3.9. Let ωh and ηh satisfy the same assumptions as in Theorem 3.8.
For every u ∈ W (QT ) we then have
|ω h ∗ (u ⊗ ∇ηh )| ≤ cM (|∇u|) almost everywhere in QT
4
Theorem 3.8 can also be used to give an alternative proof of Lemma 3.3.
Lemma 3.10. For every u ∈ Vτ (Ω) there exists a sequence (uh )h>0 in C0∞ (Ω)d
1,p(τ,·)
such that uh → u in L2 (Ω)d ∩ W0
(Ω)d as h tends to zero.
31
3.1 The space W (QT )
Proof. Define Ωh := { x ∈ Ω | d(x, ∂Ω) R> h }, choose a positive bell-shaped
mollifier µ ∈ B(Rd ) ∩ C0∞ (B1d (0)) with Rd µ(x) dx = 1 and define a cut-off
function by
νh := µ h ∗ χΩh .
4
Then
take a positive bell-shaped mollifier ω ∈ B(Rd ) ∩ C0∞ (B1d (0)) with
R
ω(x) dx = 1 and define
Rd+1
uh := ω h ∗ (νh u).
4
Clearly, uh ∈ C0∞ (Ω)d for h > 0 and some obvious modifications of the proof
of Theorem 3.8 yield the assertion.
As an immediate consequence of Theorem 3.8 we can furthermore state
¯ C ∞ (Ω)d ) is dense in W (QT ).
Corollary 3.11. The set C ∞ (I,
0
Proof. In Theorem 3.8 we constructed a special sequence (uh )h>0 ∈ C0∞ (QT )d
by setting
uh := ω h ∗ (ηh u)
4
with a smooth, bell-shaped and positive mollifier ω and a suitable cut-off
function ηh .
¯ C0∞ (Ω)d ) in W (QT ), we will also explicitly
In order to prove density of C ∞ (I,
¯ C ∞ (Ω)d ) converging to u ∈ W (QT ). In
construct a sequence (ūh )h>0 ∈ C ∞ (I,
0
a first step we set
n
o
Ωh : = x ∈ Ω d(x, ∂Ω) > h .
Now
we choose a positive bell-shaped mollifier µ ∈ B(Rd ) ∩ C0∞ (B1d (0)) with
R
µ(x) dx = 1 and define a cut-off function by
Rd
νh := µ h ∗ χΩh .
4
Notice that the function νh only cuts off in space directions whereas ηh from
Theorem 3.8 acted in space and time direction. This special construction was
necessary to get zero boundary values in space and time. Now that we only
require zero boundary values in space, it is natural to work with a cut-off
function like νh .
In the next step we extend u ∈ W (QT ) to the larger interval 3I := (−T, 2T )
by means of reflection:

u(−t, ·)
for t ∈ (−T, 0]



u(t, ·)
for t ∈ (0, T )
ũ(t, ·) :=

u(2T − t, ·) for t ∈ [T, 2T )



0
for t ∈
/ (−T, 2T ).
32
Chapter 3. Function spaces and integration by parts
This extension is necessary for the regularization we want to apply to νh ũ.
d+1
d+1
∞
We take
R the above bell-shaped mollifier ω ∈ B(R ) ∩ C0 (B1 (0)), ω > 0,
with Rd+1 ω(τ, x) dx dτ = 1 and define
u˜h := ω h ∗ (νh ũ).
4
By the method of construction, u˜h belongs to C ∞ (3I, C0∞ (Ω)d ), and
consequently ūh := u˜h |QT defines an element of C ∞ (I, C0∞ (Ω)d ).
Now that we have found the right candidate ūh , it remains to show that ūh
actually converges to u in W (QT ).
However, in view of the proof of Theorem 3.8 we only want to mention that we
could exactly repeat the arguments we have used there to prove the claim.
Remark 3.12. The last corollary can also be proved directly by using the
inclusions
C0∞ (QT )d ⊂ C ∞ (I, C0∞ (Ω)d ) ⊂ W (QT ).
We will though make use of the explicit form of the sequence constructed in
Corollary 3.11 later on.
3.2
The dual space W (QT )0
The density of C0∞ (QT )d in W (QT ) enables us to provide a thorough
description of W (QT )0 , the dual space of W (QT ). The following results are
similar to those in [1] concerning the dual spaces of the classical Sobolev spaces
W0k,r (Ω).
Proposition 3.13. A functional g belongs to W (QT )0 if and only if there exist
0
g 0 ∈ L2 (QT )d and G ∈ Lp (·,·) (QT )d×d such that the following identity holds true
for every φ ∈ W (QT ):
Z
hg, φiW (QT ) := g(φ) =
Z
0
g · φ dx dτ +
QT
G : ∇φ dx dτ .
(3.5)
QT
Proof. Suppose the functional g is given by
Z
0
Z
g · φ dx dτ +
g(φ) :=
QT
G : ∇φ dx dτ
QT
for every φ ∈ W (QT ). Then clearly φ 7→ g(φ) is linear, moreover, we can
3.2 The dual space W (QT )0
33
estimate with Hölder’s inequality
Z
Z
0
|g(φ)| ≤ |g ||φ| dx dτ + |G||∇φ| dx dτ
QT
QT
0
≤ ||g ||L2 (QT )d ||φ||L2 (QT )d + c||G||Lp0 (·,·) (QT )d×d ||∇φ||Lp(·,·) (QT )d×d
≤ c ||g 0 ||L2 (QT )d + ||G||Lp0 (·,·) (QT )d×d ||φ||W (QT ) .
Thus, g belongs to W (QT )0 since
||g||W (QT )0 =
sup
||φ||W (QT ) ≤1
0
|g(φ)| ≤ c ||g ||L2 (QT )d + ||G||Lp0 (·,·) (QT )d×d < ∞.
For the opposite direction we define
E := L2 (QT )d × Lp(·,·) (QT )d×d ,
endowed with the norm ||w||E := ||w0 ||L2 (QT )d +||W ||Lp(·,·) (QT )d×d for an element
w := (w0 , W ) ∈ E. We also define a mapping Π by
Π : W (QT ) → E
u 7→ (u, ∇u).
The completeness of W (QT ) and the definition of the norms in W (QT ) and E
assert that Π is an isometric isomorphism from W (QT ) onto a closed subspace
H := Π(W (QT )) ⊂ E. Therefore, Π−1 : H → W (QT ) exists and is linear and
continuous due to Corollary 6.4 in the Appendix.
For g ∈ W (QT )0 fixed, consider the linear mapping H 3 h 7→ g(Π−1 h). Since
Π−1 is continuous, H 3 h 7→ g(Π−1 h) belongs to H 0 and due to the HahnBanach theorem, see Theorem 6.1, it can be extended to E. That means,
there exists Fg ∈ E 0 that satisfies Fg (h) = g(Π−1 h) for each h ∈ H. But
with the help of Theorem 1.12 one easily checks that for Fg ∈ E 0 there exist
0
g 0 ∈ L2 (QT )d and G ∈ Lp (·,·) (QT )d×d , such that
Fg (w) = hFg , wiE = hg 0 , w0 iL2 (QT )d + hG, W iLp0 (·,·) (QT )d×d
Z
Z
0
0
= g · w dx dτ + G : W dx dτ .
QT
QT
As h ∈ H implies h = Πφ for some φ ∈ W (QT ), the proof is finished since
g(φ) = hg, Π−1 ΠφiW (QT )
= hFg , ΠφiE
= hg 0 , φiL2 (QT )d + hG, ∇φiLp(·,·) (QT )d×d
Z
Z
0
= g · φ dx dτ + G : ∇φ dx dτ .
QT
QT
34
Chapter 3. Function spaces and integration by parts
Remark 3.14. For a, b ∈ I, a < b, let χ(a,b) denote the characteristic function
of (a, b) ⊂ I. Then we set, for g ∈ W (QT )0 and φ ∈ W (QT ),
Zb Z
0
Zb Z
G : ∇φ dx dτ .
g · φ dx dτ +
hχ(a,b) g, φiW (QT ) := hg, χ(a,b) φiW (QT ) =
a
a
Ω
Ω
Corollary 3.15. The space W (QT ) is reflexive.
Proof. The proof of Proposition 3.13 showed that W (QT ) can be identified
with a closed subspace of the reflexive space L2 (QT )d × Lp(·,·) (QT )d×d . Hence,
W (QT ) is reflexive due to Theorem 6.5 in the Appendix.
The next three results help us to clarify the connection between the space
W (QT )0 and the space of distributions on QT .
Lemma 3.16. Every functional g ∈ W (QT )0 of the form (3.5) is an extension
to W (QT ) of a distribution T ∈ D0 (QT )d .
0
Proof. Take (g 0 , G) ∈ L2 (QT )d × Lp (·,·) (QT )d×d from (3.5) and define a
distribution T ∈ D0 (QT )d by setting
T := Tg0 − divTG ,
where −divTG , Tg0 ∈ D0 (QT )d are given by
Z
0
hTg0 , φi := hg , φi := g 0 · φ dx dτ
QT
Z
h−divTG , φi := hG, ∇φi :=
G : ∇φ dx dτ ,
QT
for φ ∈ C0∞ (QT )d . For more details about notation we refer to the Appendix.
T then is well-defined as the difference of two regular distributions and for
every φ ∈ C0∞ (QT )d we have
hT, φi = hTg0 , φi − hdivTG , φi
= hg 0 , φi + hG, ∇φi
Z
Z
0
= g · φ dx dτ + G : ∇φ dx dτ
QT
QT
= hg, φiW (QT ) .
Therefore, g ∈ W (QT )0 really is an extension of T ∈ D0 (QT )d .
3.2 The dual space W (QT )0
35
The opposite direction is reflected in the next
Lemma 3.17. Suppose T ∈ D0 (QT )d has the form T = Tg0 − divTG for some
0
g 0 ∈ L2 (QT )d and G ∈ Lp (·,·) (QT )d×d . Then T can be uniquely extended to
W (QT ).
Proof. We have proved in Theorem 3.8 that for every u ∈ W (QT ) there exists
a sequence (uh )h>0 ∈ C0∞ (QT )d converging to u in W (QT ). Applying T to the
difference uh − ul and using Hölder’s inequality we get
|hT, uh − ul i| ≤ |hg 0 , uh − ul i| + |hG, ∇uh − ∇ul i|
0
≤ c||uh − ul ||W (QT ) ||g ||L2 (QT )d + ||G||Lp0 (·,·) (QT )d×d .
Since (uh )h>0 is a Cauchy sequence in W (QT ), (hT, uh i)h>0 is a
Cauchy sequence in R and therefore converges to an unique limit g(u).
This notation is not ambiguous, for, if (vh )h>0 ⊂ C0∞ (QT )d is another sequence
converging to u in W (QT ) it is readily checked that (hT, uh − vh i)h>0 tends to
zero as h → 0.
But then the functional g is linear and continuous since
|g(u)| = lim |hT, uh i| ≤ lim ||uh ||W (QT ) ||g 0 ||L2 (QT )d + ||G||Lp0 (·,·) (QT )d×d
h→0
h→0
= ||u||W (QT ) ||g 0 ||L2 (QT )d + ||G||Lp0 (·,·) (QT )d×d .
The combination of the last two lemmas proves
Theorem 3.18. The dual space W (QT )0 is isomorphic to the subspace of
D0 (QT )d consisting of regular distributions T of the form T = Tg0 − divTG ,
0
for some g 0 ∈ L2 (QT )d and G ∈ Lp (·,·) (QT )d×d .
Exactly the same reasoning can be used to characterize Vτ (Ω)0 which is the
dual space of Vτ (Ω).
Corollary 3.19. A functional g̃ belongs to Vτ (Ω)0 if and only if there exist
0
g̃ 0 ∈ L2 (Ω)d and G̃ ∈ Lp (τ,·) (Ω)d×d such that
hg̃, ϕ̃iVτ (Ω) := g̃(ϕ) = hg̃ 0 , ϕ̃iL2 (Ω)d + hG̃, ∇ϕ̃iLp(τ,·) (Ω)d×d
Z
Z
0
= g̃ · ϕ̃ dx + G̃ : ∇ϕ̃ dx
Ω
Ω
for every ϕ̃ ∈ Vτ (Ω).
The dual space Vτ (Ω)0 is isomorphic to the subspace consisting of those
distributions T̃ ∈ D0 (Ω)d that satisfy T̃ = Tg̃0 − divTG̃ for some g̃ 0 ∈ L2 (Ω)d
0
and G̃ ∈ Lp (τ,·) (Ω)d×d .
36
Chapter 3. Function spaces and integration by parts
With the help of the last statements we get another nice representation for
the duality h·, ·iW (QT ) .
Proposition 3.18 ensures the existence of functions g 0 ∈ L2 (QT )d and
0
G ∈ Lp (·,·) (QT )d×d representing a given g ∈ W (QT )0 . Now take φ ∈ W (QT )
and fix τ ∈ I. By definition of W (QT ) we have φ(τ, ·) ∈ Vτ (Ω), as well as
0
g 0 (τ, ·) ∈ L2 (Ω)d and G(τ, ·) ∈ Lp (τ,·) (Ω)d×d . So in view of Corollary 3.19, each
g ∈ W (QT )0 yields functionals g(τ ) ∈ Vτ (Ω)0 , and the representation is given
by
Z
Z
0
hg, φiW (QT ) = g · φ dx dτ + G : ∇φ dx dτ
QT
QT
ZT
ZT
0
hg (τ ), φ(τ )iL2 (Ω)d dτ +
=
0
hG(τ ), ∇φ(τ )iLp(τ,·) (Ω)d×d dτ
0
ZT
hg(τ ), φ(τ )iVτ (Ω) dτ .
=
0
3.3
The space Z(QT )
If we combine the spaces W (QT ) and W (QT )0 , it is possible to build up an
analogue of the space W appearing in the treatment of the parabolic pLaplacian equation.
Definition 3.20. For W (QT ) and W (QT )0 defined above, we set
o
n
0
Z(QT ) := u ∈ W (QT ) ut ∈ W (QT ) ,
and endow Z(QT ) with the norm
||u||Z(QT ) := ||u||W (QT ) + ||ut ||W (QT )0 .
Remark 3.21. Remember that the distributional time derivative ut is defined
by
Z
hut , φi := − u · φt dx dτ , for φ ∈ C0∞ (QT )d .
QT
By definition of the space W (QT ) we have u ∈ L2 (QT )d ,→ L1loc (QT )d . Hence,
ut is well-defined in the distributional sense and due to our characterization of
W (QT )0 in Theorem 3.18, the space Z(QT ) is well-defined, too.
37
3.3 The space Z(QT )
Lemma 3.22. The space Z(QT ) is a Banach space.
Proof. Let (un )n∈N be a Cauchy sequence in Z(QT ). By completeness of
W (QT ) and W (QT )0 we deduce the existence of u ∈ W (QT ) and v ∈ W (QT )0
such that limn→∞ un = u and limn→∞ unt = v, respectively. Hence, we must
show ut = v in W (QT )0 . For φ ∈ C0∞ (QT )d we have
Z
hut , φi = −
Z
u · φt dx dτ = − lim
n→∞
QT
QT
un · φt dx dτ = lim hunt , φi = hv, φi
n→∞
That is, ut = v is valid in D0 (QT )d . Since v belongs to W (QT )0 , Proposition
3.13 states
Z
Z
0
hut , φi = hv, φi = v · φ dx dτ + V : ∇φ dx dτ , φ ∈ C0∞ (QT )d ⊂ W (QT ),
QT
QT
0
for some v 0 ∈ L2 (QT )d and V ∈ Lp (·,·) (QT )d×d . But now Lemma 3.17 allows
us to extend ut uniquely to W (QT ) and as v ∈ W (QT )0 already is an extension
we have ut = v ∈ W (QT )0 .
We now prove another density result which will turn out to be essential in
view of a rule for integration by parts.
¯ C0∞ (Ω)d ) is dense in Z(QT ).
Theorem 3.23. The set C ∞ (I,
¯ C ∞ (Ω)d )
Proof. In Corollary 3.11 we constructed a sequence (u¯h )h>0 ⊂ C ∞ (I,
0
converging to u ∈ W (QT ). Remember that the sequence was given by
u¯h := ω h ∗ (νh u),
4
(3.6)
where ω is a suitable bell-shaped mollifier and ν is a cut-off function only
depending on space variables, see Corollary 3.11 for details. Now, if u belongs
to Z(QT ) then ut belongs to W (QT )0 and we want to prove that the sequence
¯ C0∞ (Ω)d ) converges to ut in W (QT )0 . For this purpose we
(u¯ht )h>0 ⊂ C ∞ (I,
will have to use some facts from the theory of distributions that can all be
found in the Appendix or in any classical book that treats this topic, such as
for instance [22].
For every fixed h > 0, the function u¯h defines a distribution by setting
Z
hu¯h , φi :=
QT
u¯h · φ dx dτ for φ ∈ C0∞ (QT )d .
38
Chapter 3. Function spaces and integration by parts
Hence, the distributional time derivative of u¯h is well-defined and acts on φ in
the following way:
hu¯ht , φi = −hu¯h , φt i
= −hω h ∗ (νh u), φt i
4
= −hνh u, ω̌ h ∗ φt i
4
= −hu, νh (ω̌ h ∗ φt )i
4
= −hu, νh (ω̌ h ∗ φ)t i,
4
where ω̌h/4 (·) := ωh/4 (−·). This result can also be established by writing
down the appropriate integrals, but the above presentation appears to be
clearer in what follows. Also note that the term −hu, νh (ω̌h/4 ∗ φ)t i is
meaningful since u ∈ L2 (QT )d can be interpreted as a distribution and the
function νh (ω̌h/4 ∗ φ)t belongs to C0∞ (QT )d . We now want to bring ut into play
and invoke the representation we had proved in Theorem 3.18. Since ν does
not depend on the time variable we have
−hu, νh (ω̌ h ∗ φ)t i = −hu, (νh (ω̌ h ∗ φ))t i = hut , νh (ω̌ h ∗ φ)i,
4
4
(3.7)
4
by definition of the distributional time derivative. But now Theorem 3.18 tells
us that the distribution ut ∈ W (QT )0 has the special form ut = Tu0 −divTU , for
0
some u0 ∈ L2 (QT )d and U ∈ Lp (·,·) (QT )d×d . By using this fact and identifying
the functions u0 and U with their corresponding distributions we get
hut , νh (ω̌ h ∗ φ)i = hTu0 − divTU , νh (ω̌ h ∗ φ)i
4
4
0
= hu , νh (ω̌ h ∗ φ)i + hU, ∇(νh (ω̌ h ∗ φ))i
4
4
0
= hu , νh (ω̌ h ∗ φ)i + hU, νh (ω̌ h ∗ ∇φ)i + hU, ∇νh ⊗ (ω̌ h ∗ φ)i
4
=
hu0h , φi
4
4
+ hUh , ∇φi + hU, ∇νh ⊗ (ω̌ h ∗ φ)i.
4
Here u0h and Uh are defined as in (3.6) and the last step is again an easy
calculation if one uses the integral representation for hu0 , νh (ω̌h/4 ∗ φ)i and
hU, νh (ω̌h/4 ∗ ∇φ)i, respectively. In conclusion, we arrive at
hu¯ht , φi = hu0h , φi + hUh , ∇φi + hU, ∇νh ⊗ (ω̌ h ∗ φ)i,
4
(3.8)
which is valid for every φ ∈ C0∞ (QT )d . As we want to pass to the limit h → 0
in (3.8), we first check that every single limit on the right hand side exists.
The first two terms do not really cause problems, for, notice that
u0h = ωh/4 ∗ (νh u0 ) for u0 ∈ L2 (QT )d . So using the same arguments as in
the first part of the proof of Theorem 3.8 we deduce limh→0 u0h = u0 in L2 (QT )d
and therefore limh→0 u0h = u0 in D0 (QT )d . The same reasoning applied to
39
3.3 The space Z(QT )
0
Uh and U ∈ Lp (·,·) (QT )d×d yields limh→0 divUh = divU in D0 (QT )d since
div : D0 (QT )d×d → D0 (QT )d is continuous and the convergence limh→0 Uh = U
0
in Lp (·,·) (QT )d×d also implies limh→0 Uh = U in D0 (QT )d×d .
For the third term we notice that
hU, ∇νh ⊗ (ω̌ h ∗ φ)i = hω h ∗ (U ∇νh ), φi.
4
(3.9)
4
0
Hölder’s inequality then shows ωh/4 ∗ (U ∇νh ) ∈ Lp (·,·) (QT )d ,→ L1loc (QT )d for
every h > 0, witnessing that ωh/4 ∗(U ∇νh ) is a well-defined regular distribution.
For every φ ∈ C0∞ (QT )d there exists h1 > 0 such that dist(supp(φ), ∂QT ) = h1 .
Hence, for h < h1 we end up with supp(ωh/4 ∗ (U ∇νh )) ∩ supp(φ) = ∅, and
therefore hωh/4 ∗ (U ∇νh ), φi converges to zero for every function φ ∈ C0∞ (QT )d ,
which is the same as limh→0 ωh/4 ∗ (U ∇νh ) = 0 in D0 (QT )d . Thus, we may now
pass to the limit in (3.8) and with the help of Theorem 6.21 in the Appendix
we obtain
lim hu¯ht , φi = lim hu0h , φi + lim hUh , ∇φi + lim hU, ∇νh ⊗ (ω̌ h ∗ φ)i
h→0
h→0
0
h→0
h→0
4
= hu , φi + hU, ∇φi = hut , φi.
As an intermediate result this shows that (u¯ht )h>0 ⊂ C ∞ (I, C0∞ (Ω)d ) converges to ut in D0 (QT )d . It also turns out that this will help us to verify that
limh→0 u¯ht = ut is actually valid in W (QT )0 . However, first of all we have to
show that u¯ht is an element of W (QT )0 , since the representation of u¯ht in (3.8)
slightly obscures this fact.
Theorem 3.18 states that w belongs to W (QT )0 , if w = Tw0 − divTW for
0
w0 ∈ L2 (QT )d and W ∈ Lp (·,·) (QT )d×d , and, appealing to (3.9) and (3.8) we
have
u¯ht = Tu0h − divTUh + Tω h ∗(U ∇νh )
4
u0h
2
p0 (·,·)
d
0
d×d
with
∈ L (QT ) , Uh ∈ L
(QT )
and ω h ∗ (U ∇νh ) ∈ Lp (·,·) (QT )d .
4
Notice that for φ ∈ W (QT ) the linear mapping
φ 7→ hFh , φiW (QT ) :=hTω h ∗(U ∇νh ) , φiW (QT )
4
=hU, ∇νh ⊗ (ω̌ h ∗ φ)iLp(·,·) (QT )d×d
4
is well-defined. To witness this assertion, remember that the proof of Theorem
3.8 showed that for every φ ∈ W (QT ) we have
|∇νh ⊗ (ω̌ h ∗ φ)| ≤ cM (|∇φ|) almost everywhere in QT .
4
So the monotonicity of the Luxemburg norm together with the boundedness
of the Hardy-Littlewood maximal operator M yields
||∇νh ⊗ (ω̌ h ∗ φ)||Lp(·,·) (QT )d×d ≤ c||M (|∇φ|)||Lp(·,·) (QT )
4
≤ c||∇φ||Lp(·,·) (QT )d×d ≤ c||φ||W (QT ) .
40
Chapter 3. Function spaces and integration by parts
Moreover, by definition of the cut-off function νh , see Corollary 3.11, the
integral representation of hFh , φiW (QT ) reduces to
Z
hFh , φiW (QT ) = U : ∇νh ⊗ (ω̌ h ∗ φ) dx dτ
4
QT
Z
U : ∇νh ⊗ (ω̌ h ∗ φ) dx dτ ,
=
4
Gh
with Gh := I × { x ∈ Ω | d(x, ∂Ω) ≤ h }, since ∇νh is equal to 0 on the set
{ x ∈ Ω | d(x, ∂Ω) > h }. Now we can apply Hölder’s inequality to the last
integral to obtain
|hFh , φi| ≤ c||U ||Lp0 (·,·) (Gh )d×d ||∇νh ⊗ (ω̌ h ∗ φ)||Lp(·,·) (Gh )d×d
4
≤ c||U ||Lp0 (·,·) (Gh )d×d ||φ||W (QT ) .
This shows that Fh belongs to W (QT )0 for every h > 0, too. Therefore,
u¯ht = Tu0h − divTUh + Tωh/4 ∗(U ∇νh ) truly defines an element of W (QT )0 and it
remains to prove limh→0 u¯ht = ut in W (QT )0 . But for φ ∈ W (QT ) we have
Z
Z
0
0
|hut − u¯ht , φiW (QT ) | ≤ |u − uh ||φ| dx dτ + |U − Uh ||∇φ| dx dτ
QT
QT
Z
|U ||∇νh ⊗ (ω̌ h ∗ φ)| dx dτ
+
4
Gh
≤ c ||u0 − u0h ||L2 (QT )d + ||U − Uh ||Lp0 (·,·) (QT )d×d ||φ||W (QT )
+ c||U ||Lp0 (·,·) (Gh )d×d ||φ||W (QT ) ,
where we again used Hölder’s inequality for the second inequality. This in turn
implies
||ut − u¯ht ||W (QT )0 ≤ c||u0 − u0h ||L2 (QT )d + c||U − Uh ||Lp0 (·,·) (QT )d×d
+ c||U ||Lp0 (·,·) (Gh )d×d .
Now we can pass to the limit in the last inequality. Since the first two terms
can be treated similarly as in the proof of Theorem 3.8 we only show how to
handle the second one.
||U − Uh ||Lp0 (·,·) (QT )d×d = ||U − ω h ∗ (νh U )||Lp0 (·,·) (QT )d×d
4
≤||ω h ∗ (νh U − U )||Lp0 (·,·) (QT )d×d + ||ω h ∗ U − U ||Lp0 (·,·) (QT )d×d
4
4
≤K||(1 − νh )U ||Lp0 (·,·) (QT )d×d + ||ω h ∗ U − U ||Lp0 (·,·) (QT )d×d ,
4
41
3.4 Integration by parts
where the last two terms tend to zero for h → 0, due to dominated convergence
and Theorem 2.12.
Now we have to take care of ||U ||Lp0 (·,·) (Gh )d×d . Since p0 (·, ·) ∈ P log (QT ) is a
bounded exponent we know that modular convergence implies convergence
with respect to the norm, see Lemma 1.8. It is therefore sufficient to establish
Z
0
lim |U |p (·,·) dx dτ = 0.
h→0
Gh
The set Gh was defined by Gh := I × { x ∈ Ω | d(x, ∂Ω) ≤ h }. Hence, we
0
0
have |U |p (·,·) χGh → 0 almost everywhere in QT for h → 0. Since |U |p (·,·) χGh is
0
bounded by |U |p (·,·) ∈ L1 (QT )d×d , Lebesgue’s theorem on dominated
convergence proves the claim, and we finally arrive at
lim u¯ht = ut in W (QT )0 ,
h→0
¯ C0∞ (Ω)d ) in Z(QT ).
which was the missing part for the density of C ∞ (I,
Remark 3.24. It is not possible to derive density of C0∞ (QT )d in Z(QT ) with
the technique we used above. The reason for this lies in the construction of
the approximate sequence. In Theorem 3.8 we set
uh := ω h ∗ (ηh u),
4
with a cut-off function ηh also depending on the time variable. If we tried to
perform the same calculation as in Theorem 3.23 with uh , we would have had
to produce the term hu, (ηh )t (ω̌h/4 ∗ φ)i instead of being able to directly use the
definition of the distributional time derivative like we did in equation (3.7).
Although the structure of hu, (ηh )t (ω̌h/4 ∗ φ)i is similar to hU, ∇νh ⊗ (ω̌h/4 ∗ φ)i
in equation (3.10), it is not possible in general to control (ηh )t (ω̌h/4 ∗ φ) by
M (|φt |), for φ ∈ W (QT ). But such an estimate would be necessary to show
covergence of hu, (ηh )t (ω̌h/4 ∗ φ)i. This circumstance is due to the fact that,
for φ ∈ W (QT ), φt is only defined in the distributional sense, and therefore,
necessary Poincaré-type inequalities cannot be established.
But this is not a strong setback, since also the classical result in Theorem 3.1
¯ V ) in the space W introduced there, and for the
only yields density of C ∞ (I,
¯ C0∞ (Ω)d ) in Z(QT ).
following results we require density of C ∞ (I,
3.4
Integration by parts
As already explained in the beginning of this chapter, the formula for an
integration by parts cited in Theorem 3.1 is a crucial tool for proving the
existence of weak solutions of the parabolic p-Laplacian equation. In order
42
Chapter 3. Function spaces and integration by parts
to generalize these results, particularly with regard to an application to our
p(t,x)-Laplacian, we use the density result from Theorem 3.23 to prove a rule
for integration by parts in the space Z(QT ) and an embedding result to justify
initial values in L2 (Ω)d .
n
o
0
Theorem 3.25. For Z(QT ) = u ∈ W (QT ) ut ∈ W (QT ) , see Definition
3.20 , we have the continuous embedding
¯ L2 (Ω)d ).
Z(QT ) ,→ C(I,
Furthermore, for all u, v ∈ Z(QT ) and s, t ∈ I¯ the following rule for integration
by parts is valid
Zt
hut (τ ), v(τ )iVτ (Ω) dτ =hu(t), v(t)iL2 (Ω)d − hu(s), v(s)iL2 (Ω)d
s
Zt
−
hvt (τ ), u(τ )iVτ (Ω) dτ .
s
By means of our representation for the duality in W (QT ), the last formula is
equivalent to
hut , χ[s,t] viW (QT ) = hu(t), v(t)iL2 (Ω)d − hu(s), v(s)iL2 (Ω)d − hvt , χ[s,t] uiW (QT ) .
Choosing v = u ∈ Z(QT ), equation (3.10) becomes
Zt
1
1
hut (τ ), u(τ )iVτ (Ω) dτ = ||u(t)||2L2 (Ω)d − ||u(s)||2L2 (Ω)d .
2
2
s
Remark 3.26. This theorem justifies the fact that ”in advance”, the space
W (QT ) incorporates the L2 -information that is likely to come from the time
derivative in applications. This trick enables us to avoid (maybe non-existing)
evolution triples and nevertheless to derive the same results as in the classical
framework, even without the typical ”p ≥ 6/5”-restriction that is necessary in
case of the parabolic p-Laplacian in three space dimensions.
¯ C ∞ (Ω)d ) ⊂ Z(QT ).
Proof of Theorem 3.25. First assume u ∈ C ∞ (I,
0
Then classical and distributional time derivatives coincide and we have
¯ C0∞ (Ω)) ⊂ L2 (QT )d ⊂ W 0 (QT ),
ut ∈ C ∞ (I,
which implies
Zt
hut , χ[s,t] uiW (QT ) =
Zt
hut (τ ), u(τ )iVτ (Ω) dτ =
s
hut (τ ), u(τ )iL2 (Ω)d dτ .
s
43
3.4 Integration by parts
Thus,
Zt
1
1
hut (τ ), u(τ )iL2 (Ω)d dτ = ||u(t)||2L2 (Ω)d − ||u(s)||2L2 (Ω)d
2
2
s
Zt
−
hut (τ ), u(τ )iL2 (Ω)d dτ
s
1
1
= ||u(t)||2L2 (Ω)d − ||u(s)||2L2 (Ω)d
2
2
− hut , χ[s,t] uiW (QT ) .
It follows that
||u(t)||2L2 (Ω)d = ||u(s)||2L2 (Ω)d + 2hut , χ[s,t] uiW (QT )
≤ ||u(s)||2L2 (Ω)d + 2||ut ||W 0 (QT ) ||u||W (QT )
≤ ||u(s)||2L2 (Ω)d + 2||u||2Z(QT )
implies
||u(t)||L2 (Ω)d ≤ ||u(s)||L2 (Ω)d +
√
2||u||Z(QT ) .
Integrating the last inequality from 0 to T with respect to s and applying
Hölder’s inequality immediately yields
√
1
||u(t)||L2 (Ω)d ≤ √ ||u||L2 (QT )d + 2||u||Z(QT )
T
≤ c||u||Z(QT ) ,
where c only depends on T, and in conclusion
||u||C(I,L
¯ 2 (Ω)d ) := sup ||u(t)||L2 (Ω)d ≤ c||u||Z(QT ) .
(3.10)
t∈I¯
For an arbitrary u ∈ Z(QT ), Theorem 3.23 asserts the existence of a sequence
¯ C ∞ (Ω)d ) converging to u in Z(QT ). If we now apply the last
(uh )h>0 ∈ C ∞ (I,
0
¯ C0∞ (Ω)d ) we arrive at
estimate to uh − ul ∈ C ∞ (I,
||uh − ul ||C(I,L
¯ 2 (Ω)d ) ≤ c||uh − ul ||Z(QT ) .
This shows that, (uh )h>0 being a Cauchy sequence in Z(QT ), (uh )h>0 is a
¯ L2 (Ω)d ), too. By completeness of C(I,
¯ L2 (Ω)d ) there
Cauchy sequence in C(I,
¯ L2 (Ω)d ).
thus exists ũ such that limh→0 uh = ũ in C(I,
But limh→0 uh = u in Z(QT ) implies limh→0 uh (τ ) = u(τ ) in L2 (Ω)d for almost
44
Chapter 3. Function spaces and integration by parts
every τ ∈ I, at least for a suitable subsequence. Therefore we can assume this
sort of convergence to already hold for the whole sequence (uh )h>0 . This in
turn implies u(τ ) = ũ(τ ) for almost every I and we may identify u and its
continuous representative ũ. With this identification in mind, plugging uh in
(3.10) and taking limits on both sides yields
||u||C(I,L
¯ 2 (Ω)d ) ≤ c||u||Z(QT )
and our first assertion follows.
The second claim now follows by approximation:
for u, v ∈ Z(QT ) there exist representatives, again denoted by u and v, such
¯ L2 (Ω)d ), and (uh )h>0 , (vh )h>0 ∈ C ∞ (I,
¯ C ∞ (Ω)d ) that satisfy
that u, v ∈ C(I,
0
limh→0 uh = u in Z(QT ) and limh→0 vh = v in Z(QT ), respectively.
As (uh )h>0 and (vh )h>0 are smooth there holds
Zt
huht (τ ), vh (τ )iVτ (Ω) dτ = huh (t), vh (t)iL2 (Ω)d − huh (s), vh (s)iL2 (Ω)d
s
(3.11)
Zt
hvht (τ ), uh (τ )iVτ (Ω) dτ
−
s
¯ It follows that
for every s, t ∈ I.
Zt
huht (τ ), vh (τ )iVτ (Ω) dτ = lim huht , χ[s,t] vh iW (QT )
lim
h→0
h→0
s
= hut , χ[s,t] viW (QT )
Zt
= hut (τ ), v(τ )iVτ (Ω) dτ ,
s
since we have limh→0 uh = u in Z(QT ) and limh→0 vh = v in Z(QT ),
respectively, and the same reasoning shows
Zt
Zt
h(vh )t (τ ), (uh )(τ )iVτ (Ω) dτ =
lim
h→0
s
hvt (τ ), u(τ )iVτ (Ω) dτ .
s
After identifying u and v with their representatives, we have limh→0 uh = u in
¯ L2 (Ω)d ), limh→0 vh = v in C(I,
¯ L2 (Ω)d ), and of course this implies
C(I,
lim huh (t), vh (t)iL2 (Ω)d − lim huh (s), vh (s)iL2 (Ω)d
h→0
h→0
= hu(t), v(t)iL2 (Ω)d − hu(s), v(s)iL2 (Ω)d .
3.4 Integration by parts
45
In conclusion, taking limits in (3.11) proves the second claim. The equivalent
representation for the formula then follows from the calculation subsequent to
Corollary 3.19 and the third claim is now obvious.
46
Chapter 3. Function spaces and integration by parts
Chapter 4
Existence of a weak solution
Remember that we initially considered the parabolic Cauchy problem
∂t u − divS(∇u) = f in QT
u = 0 on ΓT
u(0, ·) = u0 in Ω,
(4.1)
where the operator S is given by
S : I×Ω × Rd×d → Rd×d
S(τ, x, ∇u) : = (1 + |∇u|)p(τ,x)−2 ∇u(τ, x)
for a known variable exponent p(·, ·) ∈ P log (I × Ω). The initial data u0 and
the right-hand side f are given functions, too.
Now that we have all necessary instruments, we are going to show existence
of a weak solution for our equation under some additional assumption on the
data. We want to mention that we follow the method proposed in [32] or [4].
To be more precise, the actual proof is based on a Galerkin approximation that
provides a sequence of approximate solutions which is uniformly bounded in
the appropriate function space W (QT ). Due to the reflexivity of W (QT ), we
can find a weak limit u ∈ W (QT ) for the sequence of approximate solutions
and we can thus pass to the limit in the Galerkin system. Our representation
for the dual space W (QT )0 , the monotonicity of the operator induced by S and
the formula for integration by parts in time then allow us to identify the weak
limit as a weak solution of (4.1).
For this reason, this approach can be seen as a generalization of the theory of
monotone operators to spaces with variable exponents.
47
48
Chapter 4. Existence of a weak solution
4.1
Weak formulation of the problem
Definition 4.1. For f ∈ W (QT )0 and initial values u0 ∈ L2 (Ω)d , a function
u ∈ Z(QT ) is called a weak solution of (4.1) if
hut , ϕiW (QT ) + hS̃u, ϕiW (QT ) = hf, ϕiW (QT )
is satisfied for every ϕ ∈ W (QT ) and if the initial condition u(0, ·) = u0 (·) is
fulfilled. Here, the operator S̃ : W (QT ) → W (QT )0 is defined by
ZT Z
S(τ, x, ∇u) : ∇ϕ dx dτ
hS̃u, ϕiW (QT )0 : =
0
Ω
ZT
Z
0
Ω
=
(1 + |∇u|)p(·,·)−2 ∇u : ∇ϕ dx dτ .
Remark 4.2. The operator S̃ is a Nemyckii operator induced by the
Carathéodory function S, we refer to [23] p.49-50 for further details. It
0
follows that S̃ maps W (QT ) to Lp (·,·) (QT )d×d which shows that S̃ is well-defined
due to our representation theorem for W (QT )0 . Furthermore, according to the
¯ L2 (Ω)d ), the initial values make sense.
embedding Z(QT ) ,→ C(I,
For our Galerkin approximation we introduce another function space.
Lemma 4.3. The set
+
V + (Ω) := L2 (Ω)d ∩ W01,p (Ω)d ,
which we endow with the norm
||ψ||V + (Ω) := ||ψ||L2 (Ω)d + ||∇ψ||Lp+ (Ω)d×d
is a separable Banach space and we have a continuous embedding
V + (Ω) ,→ Vτ (Ω) for every τ ∈ (0, T ).
Proof. Being defined as the intersection of two separable spaces, V + (Ω) is
separable itself. The norm properties and the completeness follow by similar
arguments as in the proof of the completeness of the space W (QT ).
+
Now, p+ ≥ p(τ, ·) for every τ ∈ (0, T ) yields Lp (Ω)d×d ,→ Lp(τ,·) (Ω)d×d for
every τ ∈ (0, T ). For this reason, the last claim holds true due to the definition
of || · ||V + (Ω) and || · ||Vτ (Ω) .
As V + (Ω) is separable, it contains a countable and dense subset
Ψ := { ψk | k ∈ N } ⊂ V + (Ω). By virtue of the Gram-Schmidt theorem
we can, without loss of generality, assume it to be orthonormal in L2 (Ω)d .
The advantage of Ψ lies in the fact that we can use it to perform a Galerkin
approximation uniformly for every τ ∈ (0, T ). That’s the content of the next
theorem.
49
4.1 Weak formulation of the problem
Theorem 4.4. Given f ∈ W (QT )0 and u0 ∈ L2 (Ω)d , for every l ∈ N there
exists a function ul ∈ W (QT ) of the form
ul =
l
X
¯ R) and ψk ∈ Ψ
clk ψk , with clk ∈ C 1 (I,
k=1
such that ul satisfies the following Galerkin system for ψk ∈ {ψ1 , ..., ψl } and
every τ ∈ I:
hult (τ ), ψk iL2 (Ω)d + hS̃ul (τ ), ψk iVτ (Ω) = hf (τ ), ψk iVτ (Ω)
(4.2)
with initial values
l
u (0) =
ul0
l
:= P u0 :=
l
X
hu0 , ψk iL2 (Ω)d ψk .
(4.3)
k=1
Moreover, the following energy inequality holds true for every l ∈ N
l
+
||ul ||C(I,L
.
¯ 2 (Ω)d ) + ||∇u ||Lp(·,·) (QT )d×d ≤ K ||u0 ||L2 (Ω)d , ||f ||W 0 (QT ) , T, QT , p
As a consequence, the sequence (ul )l∈N is uniformly bounded in W (QT ), that
is
||ul ||L2 (QT )d + ||∇ul ||Lp(·,·) (QT )d×d ≤ K.
Proof. First of all, each expression in in the above system is well-defined. This
may be seen by definition of ul and S̃ and the assumption on f , together
with the observation that every functional g ∈ W (QT )0 yields a functional
g(τ ) ∈ Vτ (Ω)0 due to Theorem 3.18 and Corollary 3.19. Finally, ul (0) is the
projection of u0 ∈ L2 (Ω)d on the subspace spanned by ψ1 , ..., ψl , which clearly
is meaningful since ||ul (0)||L2 (Ω)d ≤ ||u0 ||L2 (Ω)d by Bessel’s inequality. Thanks
to the L2 -orthonormality of the functions in Ψ, (4.2) and (4.3) represent a
system of equations of the form
!
l
X
(clj )0 (τ ) = Fj τ,
clk (τ )ψk
j = 1, ..., l
(4.4)
k=1
clj (0) = hu0 , ψj iL2 (Ω)d
j = 1, ..., l
with the abbreviation
!
*
l
X
Fj τ,
clk (τ )ψk = hf (τ ), ψj iVτ (Ω) − S̃
k=1
l
X
k=1
!
clk (τ )ψk
+
, ψj
.
Vτ (Ω)
In order to prove existence of a solution of (4.4), we only note that the
function F := (F1 , ..., Fl ) meets the requirements of Carathéodory’s (existence)
50
Chapter 4. Existence of a weak solution
theorem for ordinary differential equations. Hence, for every l ∈ N there is
a solution of (4.4) on a somewhat smaller interval [0, T 0 ), T 0 ≤ T , and the
a-priori estimate announced above would allow us to iterate the procedure to
get a solution on [0, T ].
So suppose ul is a solution of (4.4) on [0, T 0 ), then ul solves (4.2) as well and
after multiplying (4.2) with clj (τ ), summing from 1 to l with respect to j and
integrating from 0 to s, s ≤ T 0 , with respect to τ we arrive at
hult , χ(0,s) ul iW (QT ) + hS̃ul , χ(0,s) ul iW (QT ) = hf, χ(0,s) ul iW (QT ) .
(4.5)
Since the functions clk belong to C 1 ([0, T 0 ], R) for k = 1, ..., l,
we have ul ∈ C 1 ([0, T 0 ], V + (Ω)) and therefore classical and distributional time
derivative coincide. Together with the definition of S̃ this implies that (4.5) is
equivalent to
Zs
0
hult (τ ), ul (τ )iVτ (Ω)
Z
dτ +
(1 + |∇ul |)p(·,·)−2 |∇ul |2 dx dτ = hf, χ(0,s) ul iW (QT )
Qs
with Qs := [0, s) × Ω.
By the method of construction we have ult (τ ), ul (τ ) ∈ L2 (Ω)d for every
τ ∈ [0, T 0 ), so the duality hult (τ ), ul (τ )iVτ (Ω) equals hult (τ ), ul (τ )iL2 (Ω)d and
integration by parts in the first term of the left-hand side yields
1 l
||u (s)||2L2 (Ω)d +
2
Z
(1 + |∇ul |)p(·,·)−2 |∇ul |2 dx dτ
Qs
1
= hf, χ(0,s) ul iW (QT ) + ||ul (0)||2L2 (Ω)d .
2
R
In the next step we want to estimate Qs (1 + |∇ul |)p(·,·)−2 |∇ul |2 dx dτ in terms
of ρp(·,·) (|∇ul |χQs ). First consider the case 1 < p+ < 2. By definition of ρp(·,·)
we have
l
Z
ρp(·,·) (|∇u |χQs ) =
|∇ul |p(·,·) dx dτ .
Qs
Since we want to produce the term (1 + |∇ul |)p(·,·)−2 |∇ul |2 , we rewrite the last
51
4.1 Weak formulation of the problem
integral as
Z
p(·,·)−2
2−p(·,·)
|∇ul |p(·,·)−1 (1 + |∇ul |) 2 (1 + |∇ul |) 2 |∇ul | dx dτ
Qs
Z
≤
(1 + |∇ul |)p(·,·)−1 (1 + |∇ul |)
2−p(·,·)
2
(1 + |∇ul |)
p(·,·)−2
2
|∇ul | dx dτ
Qs
Z
=
(1 + |∇ul |)
p(·,·)
2
(1 + |∇ul |)
p(·,·)−2
2
|∇ul | dx dτ
Qs
Z
≤
1
(1 + |∇ul |)p(·,·) dx dτ +
2
Z
1
(1 + |∇ul |)p(·,·)−2 |∇ul |2 dx dτ ,
2
Qs
Qs
where we used Young’s inequality in the last step. Since p(·, ·) > 1, the
convexity of a 7→ ap(·,·) applied to the first integral in the last inequality shows
Z
Z
1
+
l p(·,·)
+
(1 + |∇u |)
dx dτ ≤ c(|Qs |, p ) + 2p −2 |∇ul |p(·,·) dx dτ . (4.6)
2
Qs
Qs
Plugging
(4.6) in the above inequality and absorbing
R p+ −2
2
|∇ul |p(·,·) dx dτ in the left-hand side, we conclude that
the
term
(1 + |∇ul |)p(·,·)−2 |∇ul |2 dx dτ .
(4.7)
Qs
Z
c
l p(·,·)
|∇u |
+
Z
dx dτ −c(|Qs |, p ) ≤
Qs
Qs
The second case where 2 ≤ p− follows from the pointwise estimate
|∇ul |p(·,·) = |∇ul |p(·,·)−2 |∇ul |2 ≤ (1 + |∇ul |)p(·,·)−2 |∇ul |2 .
Hence, (4.7) remains true for both cases and we can use this result to proceed
in the proof of our a-priori estimate. Remembering that
Z
1 l
2
||u (s)||L2 (Ω)d + (1 + |∇ul |)p(·,·)−2 |∇ul |2 dx dτ
2
Qs
1
= hf, χ(0,s) ul iW (QT ) + ||ul (0)||2L2 (Ω)d ,
2
we then have
1 l
||u (s)||2L2 (Ω)d + c
2
Z
|∇ul |p(·,·) dx dτ − c(|Qs |, p+ )
Qs
1
≤ hf, χ(0,s) ul iW (QT ) + ||ul (0)||2L2 (Ω)d .
2
52
Chapter 4. Existence of a weak solution
We now take care of hf, χ(0,s) ul iW (QT ) . As f belongs to W (QT )0 , Theorem 3.18
again tells us that in W (QT )0 , f can be expressed as f = Tf 0 − divTF with
0
f 0 ∈ L2 (QT )d and F ∈ Lp (·,·) (QT )d×d . It follows that
Z
Z
0
l
l
hf,χ(0,s) u iW (QT ) = f · u dx dτ + F : ∇ul dx dτ
Qs
Qs
Z
0
Z
l
|F |
f · u dx dτ +c(ε)
≤
Qs
Qs
Z
0
l
p0 (·,·)
Z
Qs
p+
f · u dx dτ +c(ε) ||F ||Lp0 (·,·) (Q
≤
|∇ul |p(·,·) dx dτ
dx dτ +ε
d×d
T)
+1
Qs
Z
+ε
|∇ul |p(·,·) dx dτ .
Qs
Here we used Lemma 1.8 and Young’s inequality with ε. After choosing ε
sufficiently small the last inequality implies
Z
1 l
2
||u (s)||L2 (Ω)d + c̃ |∇ul |p(·,·) dx dτ
2
Qs
Zs
≤
1
||f 0 (τ )||L2 (Ω)d ||ul (τ )||L2 (Ω)d dτ + ||ul (0)||2L2 (Ω)d + K̃
2
(4.8)
0
Zs
≤
1
||f 0 (τ )||L2 (Ω)d ||ul (τ )||L2 (Ω)d dτ + ||u0 ||2L2 (Ω)d + K̃,
2
0
and Gronwall’s lemma then shows
Zτ
1
1
l
||u (τ )||L2 (Ω)d ≤ c0 +
||f 0 (τ 0 )||L2 (Ω)d dτ 0 ≤ c0 + T 1/2 ||f 0 ||L2 (QT )d .
2
2
0
Combining with (4.8) and simplifying, we finally obtain
Z
1 l
2
||u (s)||L2 (Ω)d + c̃ |∇ul |p(·,·) dx dτ ≤ K,
2
Qs
where the constant K only depends on the data, the measure of QT , p+ and
T . Since the right-hand side does not depend on s ∈ [0, T 0 ], we can iterate
the whole procedure to construct a sequence ul that satisfies all the above
estimates on QT , especially
l
||ul ||C(I,L
¯ 2 (Ω)d ) + ||∇u ||Lp(·,·) (QT )d×d ≤ K
53
4.2 Passage to the limit
¯ L2 (Ω)d ) ,→ L2 (QT )d then shows that
The continuous embedding C(I,
||ul ||W (QT ) ≤ K is valid for every l ∈ N, in particular, the sequence (ul )l∈N is
uniformly bounded in W (QT ) and the proof is finished.
4.2
Passage to the limit
We will now use the boundedness of (ul )l∈N to obtain candidates for a
subsequent limiting process and to show existence of a weak solution in the
sense of Definition 4.1.
Lemma 4.5. There exists a subsequence of (ul )l∈N , for the sake of clarity again
denoted by (ul )l∈N , and u ∈ W (QT ) such that
a) ul * u in W (QT ) for l → ∞, which in turn implies
ul * u in L2 (QT ) and ∇ul * ∇u in Lp(·,·) (QT )d×d ,
0
b) (1 + |∇ul |)p(·,·)−2 ∇ul * U for l → ∞ and U ∈ Lp (·,·) (QT )d×d ,
c) ul (0, ·) → u0 in L2 (Ω)d and ul (T, ·) * u∗ in L2 (Ω)d .
Proof. Since ||ul ||W (QT ) ≤ K for every l ∈ N, (ul )l∈N is a bounded sequence
in the reflexive Banach space W (QT ). The Eberlein-Šmuljan theorem, see
Theorem 6.9 in the Appendix, thus guarantees the existence of u ∈ W (QT )
such that ul * u in W (QT ) for l → ∞, at least for a subsequence. By
definition of the norm in W (QT ) this implies the second claim in a).
From the boundedness of ||∇ul ||Lp(·,·) (QT )d×d and the boundedness of the
mapping
0
N : Lp(·,·) (QT )d×d → Lp (·,·) (QT )d×d
V 7→ N (V ) := (1 + |V |)p(·,·)−2 V,
0
we deduce that (1 + |∇ul |)p(·,·)−2 ∇ul is bounded in Lp (·,·) (QT )d×d as well. So
0
0
the reflexivity of Lp (·,·) (QT )d×d ensures the existence of U ∈ Lp (·,·) (QT )d×d
satisfying b).
The first part in c) immediately follows from the method of construction
and basic Hilbert space theory. For the second claim we notice that the
l
energy estimate ||ul ||C(I,L
¯ 2 (Ω)d ) + ||∇u ||Lp(·,·) (QT )d×d ≤ K in particular reveals
||ul (T )||L2 (Ω)d ≤ K. Hence the assertion again follows from the same argument
as in a) and b).
54
Chapter 4. Existence of a weak solution
It remains to show that the weak limit we found in Lemma 4.5 really is a
weak solution. This will now be proved in the next theorem.
Theorem 4.6. For the weak limit u ∈ W (QT ) we have ut ∈ W (QT )0 and
therefore u belongs to Z(QT ). The function U is equal to (1 + |∇u|)p(·,·)−2 ∇u
0
in Lp (·,·) (QT )d×d , u(0) = u0 , u∗ = u(T ) and u is a weak solution in the sense
of Definition 4.1.
Proof. First of all we notice that ut is well-defined in the sense of
distributions because ul * u in L2 (QT )d implies ult → ut in D0 (QT )d .
Now fix w ∈ Vk := span{ψ1 , ..., ψk }. By the method of construction we then
have
hult (τ ), wiVτ (Ω) + hS̃ul (τ ), wiVτ (Ω) = hf (τ ), wiVτ (Ω)
¯ R) and integrate
for every l ≥ k . If we multiply this equation with η ∈ C 1 (I,
from 0 to T with respect to τ we get
Z
Z
l
− u · (ηt w) dx dτ + (1 + |∇ul |)p(·,·)−2 ∇ul : ∇(ηw) dx dτ
QT
QT
Z
Z
0
f · (ηw) dx dτ +
=
QT
F : ∇(ηw) dx dτ
Q
T
Z h
i
l
l
u (T ) · (η(T )w) − u (0) · (η(0)w) dx .
−
Ω
Due to Lemma 4.5 we can now pass to the limit l → ∞ to obtain
Z
Z
− u · (ηt w) dx dτ + U : ∇(ηw) dx dτ
QT
QT
Z
Z
0
f · (ηw) dx dτ +
=
QT
F : ∇(ηw) dx dτ
(4.9)
QT
Z h
i
−
u∗ · (η(T )w) − u0 · (η(0)w) dx .
Ω
¯ R) and every w ∈ V + (Ω)
Now theSlast equality holds true for every η ∈ C 1 (I,
+
because k≥1 Vk is dense in V (Ω). So if we choose w ∈ C0∞ (Ω)d ⊂ V + (Ω) and
¯ R) we actually get
η ∈ C0∞ (I, R) ⊂ C 1 (I,
Z
Z
Z
Z
0
− u · ηt w dx dτ = f · ηw dx dτ + F : ∇(ηw) − U : ∇(ηw) dx dτ .
QT
QT
QT
QT
55
4.2 Passage to the limit
As products of the form ηw, η ∈ C0∞ (I, R) and w ∈ C0∞ (Ω)d , are dense in
C0∞ (QT )d , see Theorem 6.11, the last equality is even valid for every
ϕ ∈ C0∞ (QT )d . That is,
Z
Z
Z
Z
0
− u · ϕt dx dτ = f · ϕ dx dτ + F : ∇ϕ dx dτ − U : ∇ϕ dx dτ .
QT
QT
QT
QT
By definition of the distributional time derivative, this is equivalent to
ut = Tf 0 − divT(F −U ) in D0 (QT )d .
This in turn implies ut ∈ W (QT )0 thanks to Theorem 3.18, since
0
f 0 ∈ L2 (QT )d and F − U belongs to Lp (·,·) (QT )d×d . So up to now, we have
shown u ∈ Z(QT ).
In the next step we want to prove u(0) = u0 and u∗ = u(T ) in L2 (Ω)d .
¯ R) and w ∈ V + (Ω), we can interpret ηt w ∈ L2 (QT )d as an
For η ∈ C 1 (I,
element of W (QT )0 , and by using our rule for integration by parts in Z(QT )
we find
Z
Z
Z
0
− (ηt w) · u dx dτ − f · (ηw) dx dτ − (F − U ) : ∇(ηw) dx dτ
QT
QT
QT
= −hηt w, uiW (QT ) − hut , ηwiW (QT )
= −η(T )hu(T ), wiL2 (Ω)d + η(0)hu(0), wiL2 (Ω)d .
On the other hand, recall that (4.9) states
Z
Z
Z
0
− u · (ηt w) dx dτ − f · (ηw) dx dτ − (F − U ) : ∇(ηw) dx dτ
QT
Q
T
Z h
i
∗
u · (η(T )w) − u0 · (η(0)w) dx
=−
QT
Ω
= −η(T )hu∗ , wiL2 (Ω)d + η(0)hu0 , wiL2 (Ω)d .
¯ R) with η(0) = ±1 and η(T ) = ±1 to establish
We can now choose η ∈ C 1 (I,
∗
hu , wiL2 (Ω)d = hu(T ), wiL2 (Ω)d and hu0 , wiL2 (Ω)d = hu(0), wiL2 (Ω)d for every
w ∈ V + (Ω) and thus for every w ∈ Ψ. But now the claim follows by density
of Ψ in L2 (Ω)d .
0
It remains to show U = (1 + |∇u|)p(·,·)−2 ∇u in Lp (·,·) (QT )d×d . For this purpose
we use our rule for integration by parts and rely on the monotonicity of the
operator S̃ and Minty’s trick. As a first step we show
Z
Z
l p(·,·)−2
l 2
lim sup (1 + |∇u |)
(4.10)
|∇u | dx dτ ≤ U : ∇u dx dτ .
l→∞
QT
QT
56
Chapter 4. Existence of a weak solution
To this end, we take ul as an admissible test function and compute
hS̃(ul ), ul iW (QT ) = hf, ul iW (QT ) − hult , ul iW (QT ) ,
which, by definition of the opeator S̃, is equivalent to
Z
(1 + |∇ul |)p(·,·)−2 |∇ul |2 dx dτ = hf, ul iW (QT ) − hult , ul iW (QT ) .
QT
By means of integration by parts, the last equality is the same as
Z
(1 + |∇ul |)p(·,·)−2 |∇ul |2 dx dτ = hf, ul iW (QT )
QT
1
1
− ||ul (T )||L2 (Ω)d + ||ul (0)||L2 (Ω)d ,
2
2
and taking lim sup on both sides yields
l→∞
Z
(1 + |∇ul |)p(·,·)−2 |∇ul |2 dx dτ
lim sup
l→∞
QT
1
1
≤ hf, uiW (QT ) − ||u(T )||L2 (Ω)d + ||u(0)||L2 (Ω)d
2
2
= hf, uiW (QT ) − hut , uiW (QT )
Z
Z
0
= f · u dx dτ + F : ∇u dx dτ
QT
QT
Z
−
0
Z
f · u dx dτ −
QT
(F − U ) : ∇u dx dτ
QT
Z
U : ∇u dx dτ .
=
QT
Here we used the weak convergence ul * u in W (QT ) together with the lower
semi-continuity of the norm for the second row, integration by parts in Z(QT )
for the third one and the representation for ut ∈ W (QT )0 we had established
above for the rest.
Now, S̃ : W (QT ) 7→ W (QT )0 is a monotone mapping since
(1 + |A|)p(·,·)−2 A − (1 + |B|)p(·,·)−2 B : (A − B) ≥ 0
is valid pointwise for all A, B ∈ Rd×d and 1 < p− ≤ p(·, ·) ≤ p+ < ∞. For a
detailed proof of this inequality we refer to [28], chapter 5.
57
4.2 Passage to the limit
Moreover, the mapping ε 7→ hS̃(u + εv), wiW (QT ) is continuous on [0, 1] for all
u, v, w ∈ W (QT ) which means that S̃ is hemi-continuous. So if we take an
arbitrary v ∈ W (QT ) we have
Z
0≤
(1 + |∇ul |)p(·,·)−2 ∇ul − (1 + |∇v|)p(·,·)−2 ∇v : ∇ul − ∇v dx dτ .
QT
Applying lim sup to both sides shows
l→∞
Z
0≤
U − (1 + |∇v|)p(·,·)−2 ∇v : (∇u − ∇v) dx dτ ,
QT
according to Lemma 4.5 and (4.10). This in turn is equivalent to
0 ≤ hU − S̃(v), u − viW (QT ) ,
and the special choice v = u ± εw together with the hemi-continuity of S̃
reveals
hU − S̃(u), wiW (QT ) = 0 for every w ∈ W (QT ),
0
or U = (1 + |∇u|)p(·,·)−2 ∇u in Lp (·,·) (QT )d×d , respectively. If we gather all the
limits we finally obtain
hut , ϕiW (QT ) + hS̃u, ϕiW (QT ) = hf, ϕiW (QT )
for every ϕ ∈ C0∞ (QT )d , and u(0) = u0 . By density of C0∞ (QT )d in W (QT ) we
deduce that u is a weak solution in the sense of Definition 4.1.
Remark 4.7. The techniques we used in the foregoing chapters and the proof
of the existence of weak solutions of the generalized p(t,x)-Laplacian equation
can be adopted to prove existence of weak solutions for the classic parabolic
p-Laplacian equation without the restriction p ≥ 6/5 in the three-dimensional
case.
This is due to the fact in case of constant p, the only restriction we have
to impose on our exponent is p > 1, in order to guarantee boundedness of
the Hardy-Littlewood maximal operator. The mollification methods from the
third chapter then immediately carry over. As a consequence, we can construct
function spaces that do not make use of an underlying evolution triple and
yet allow to derive a formula for integration by parts in time. A Galerkin
approximation as in the above proof and the combination of integration by
parts with the Minty trick then yield existence of a weak solution for the
parabolic p-Laplacian. To our best knowledge, this can be seen as a new
result.
58
Chapter 4. Existence of a weak solution
Chapter 5
Outlook
A variant of the parabolic p(t,x)-Laplacian we treated in the previous chapters
also appears as one component of the so-called p(t,x)-Navier-Stokes equations.
These equations model the unsteady flow of an incompressible
electrorheological fluid in a bounded domain Ω ⊂ Rd and read as
∂t u − divS(Du) + [∇u]u + ∇π = f in QT
divu = 0 in QT
u = 0 on ΓT
u(0) = u0 in Ω.
(5.1)
Here, the stress tensor S has a structure similar to the p(t,x)-Laplacian,
S : I × Ω × Rd×d → Rd×d
S(τ, x, Du) := (1 + |Du|)p(τ,x)−2 Du(τ, x),
but now depends on the symmetric part of the gradient
1
Du := (∇u + (∇u)T ).
2
Pd
The expression ([∇u]u)i :=
j=1 uj ∂j ui , i = 1, ..., d is referred to as the
convective term and π stands for the pressure, whereas the dependence on
the electric field is captured in the variable exponent with 1 < p− ≤ p(·, ·) ≤
p+ < ∞. For the physical motivation and a detailed derivation of the model
we refer to [31].
We now want to turn our attention to the possibility of carrying over some of
the ideas of the previous chapters to treat the p(t,x)-Navier-Stokes
equations, and we then highlight the problems that occur in this new situation.
Under the assumption that p ∈ (1, ∞) is constant and that Ω ⊂ R3 is a
bounded domain with a sufficiently smooth boundary, we first of all describe
59
60
Chapter 5. Outlook
the classical approach of finding weak solutions by means of evolution triples
and the monotone operator theory. In view of a weak formulation of the
problem, we want to introduce some typical function spaces. We define
V := { u ∈ C0∞ (Ω)3 | divu = 0 }, the space of divergence-free test functions,
Vp := V
H := V
Us := V
W01,p (Ω)3
L2 (Ω)3
, the closure of V with respect to the W01,p (Ω)3 norm,
, the closure of V with respect to the L2 (Ω)3 norm and
W0s,2 (Ω)3
, the closure of V with respect to the W0s,2 (Ω)3 norm.
As an immediate consequence of the definition of
n
1,p
(Ω)3 := u ∈ W01,p (Ω)3
Vp = W0,div
Vp we get
o
divu = 0 .
Here and for the rest of this work, the subscript ”div” means divergence-free
or solenoidal. Notice that the divergence constraint is now already captured
in the function spaces. To obtain an evolution triple and simultaneously give
sense to the convective term in the weak formulation, we now have to make
another special assumption on the exponent p, namely p ≥ 11/5. With this
restriction at hand, we gain the Gelfand triple
Vp ,→,→ H ,→ Vp0 .
Also, we introduce the space
du
p
p0
0
¯ H),
X := u ∈ L (I, Vp ) ∈ L (I, Vp ) ,→ C(I,
dt
where the embedding result follows from Theorem 3.1. In order to define
a weak formulation of (5.1), we use the following three operators, each one
mapping Y := Lp (I, Vp ) into its dual space:
ZT
h
hLu, ϕiY :=
du
, ϕiVp dτ
dt
0
ZT Z
hS̃u, ϕiY :=
0
Ω
ZT
Z
hBu, ϕiY :=
(1 + |Du|)p−2 Du : Dϕ dx dτ
[∇u]u · ϕ dx dτ .
0
Ω
Notice that due to the restriction on p, the operator B is well-defined. With
the help of these operators and the spaces mentioned above, we now state the
existence theorem.
61
Theorem 5.1. Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary, let
I = (0, T ) be a bounded time interval and suppose p ≥ 11/5.
0
For f ∈ Lp (I, (W01,p (Ω)3 )0 ) and u0 ∈ H, there exists at least one weak solution
u ∈ X of the p-Navier-Stokes system in the sense that
hLu, ϕiY + hS̃u, ϕiY + hBu, ϕiY = hf, ϕiY
holds for every ϕ ∈ Lp (I, Vp ), and u(0) = u0 .
Remark 5.2. As we make use of divergence-free functions, the pressure π does
not appear in the weak formulation. However, it can be reconstructed with
the help of a time-dependent version of deRham’s theorem, see for instance
[33] chapter IV Lemma 1.41.
The proof of the last theorem can be obtained by imitating a proof of Lions
in [27] or Ladyženskaja in [25] , although they considered the existence of weak
solutions for a different variant of the Navier-Stokes equations. For the sake
of clarity and since we want to generalize these ideas for the treatment of the
p(t,x)-Navier-Stokes equations, we briefly describe the most important steps
in the proof. The first step consists of choosing a basis of eigenfunctions of the
Stokes operator on the space Us for s = 3. With this basis at hand, a Galerkin
approximation is performed to transform our equation into an initial-value
problem for a system of ordinary differential equations which is then solved
by a sequence (ul )l∈N . Since these solutions only yield short-time existence,
we have to establish a first a-priori estimate by testing the Galerkin system
with a fixed ul . Although the operator S̃ depends on the symmetric gradient,
we chose a norm on Y that uses the full gradient, so we first have to fix this
difference. The necessary tool is Korn’s inequality, which plays an important
role in the mathematical theory of fluid dynamics and elasticity, because it
relates the norm of the full gradient to the norm of the symmetric gradient. It
states that for a bounded domain Ω ⊂ Rd and u ∈ W01,p (Ω)d , 1 < p < ∞, we
have
||∇u||Lp (Ω)d×d ≤ c||Du||Lp (Ω)d×d .
(5.2)
As a consequence, we can proceed with the a-priori estimate. We have by
definition of S̃
hS̃ul , ul iY =
ZT Z
0
(1 + |Dul |)p−2 |Dul |2 dx dτ .
Ω
An algebraic manipulation as in the proof of Theorem 4.4 then yields
ZT
0
||Dul (τ )||pLp (Ω)3×3 dτ −c(|QT |) ≤ c
ZT Z
0
Ω
(1 + |Dul |)p−2 |Dul |2 dx dτ .
62
Chapter 5. Outlook
Hence, Korn’s inequality and an integration by parts imply
||ul ||C(I,H)
+ ||∇ul ||Lp (QT )3×3 ≤ c(f, u0 , |QT |).
¯
(5.3)
The special properties of our basis then allow to bound the sequence of
l
generalized time derivatives du
in a certain space that depends on the Stokes
dt
operator. Since we do not want to blur the main ideas by calculating
every necessary embedding and exponent, we only explain the general idea
that is behind this approach. Inequality (5.3) already contains most of the information we need for the limiting process in the Galerkin approximation. The
0
boundedness of (ul )l∈N in Y and the reflexivity of Lp (QT )3×3 and Y guarantee
0
the existence of U ∈ Lp (QT )3×3 and u ∈ Y such that
0
(1 + |Dul |)p−2 Dul * U in Lp (QT )3×3 and
ul * u in Y as l → ∞
holds for a suitable subsequence. In order to prove
lim hBul , ϕiY = hBu, ϕiY ,
l→∞
we notice that the operator induced by B represents a compact perturbation of
the monotone operator S̃. Since it has no additional p-structure, monotonicity
methods fail and only a simple convergence in Lr (QT ), of course for a suitable r,
will help. Fortunately, a compactness result by Aubin and Lions is tailor-made
for this problem, but it requires control over the generalized time derivative.
l
)
Thus, the purpose of the special basis and the resulting boundedness of ( du
dt l∈N
was, to enable us to use a lemma the proof of which can be found in [27], section
1.5 or in [32], Lemma 3.74.
Lemma 5.3. Let 1 < α, β < ∞ and let V be a Banach space. Suppose we
are given two more separable and reflexive Banach spaces V0 and V1 , such that
V0 ,→,→ V ,→ V1 . Then we have
dv
β
α
∈ L (I, V1 ) ,→,→ Lα (I, V ).
v ∈ L (I, V0 ) dt
A subsequent application of a parabolic interpolation result, see [32] p.124125, ensures the convergence in the B-part. It is then possible to pass to
the limit in the Galerkin approximation and to identify du
as an element of
dt
p0
0
L (I, Vp ). Finally, the formula for integration by parts in X ensures u(0) = u0
and Minty’s trick then allows us to deduce that the function U is actually equal
to S(Du) = (1 + |Du|)p−2 Du. Now this finishes the proof since it additionally
implies
hLu, ϕiY + hS̃u, ϕiY + hBu, ϕiY = hf, ϕiY .
63
We now return to the p(t,x)-Navier-Stokes equations. If we wanted to work
out a proof similar to that in the previous chapters or as in Lions’ book, we
would certainly have to adjust our function spaces. In particular, we also want
to incorporate the divergence constraint and we somehow have to deal with
the symmetric gradient that appears in the stress tensor. Moreover, in order to
possibly gain a rule for integration by parts without using evolution triples, it
turned out to be useful to presume the potential L2 -information coming from
the time derivative. In the spirit of the foregoing chapters we thus make the
following ansatz.
Definition 5.4. Suppose we are given a bounded Lipschitz domain Ω ⊂ R3 ,
a bounded time interval I = (0, T ) ⊂ R+ and a bounded variable exponent
p(·, ·) ∈ P log (I × Ω) that satisfies p− ≥ 11/5. For a fixed τ ∈ I we set
o
n
Yτ (Ω) := u ∈ L2 (Ω)3 ∩ W01,1 (Ω)3 ∇u ∈ Lp(τ,·) (Ω)3×3 , divu = 0 ,
and define a norm on Yτ (Ω) by
||u||Yτ (Ω) := ||u||L2 (Ω)3 + ||∇u||Lp(τ,·) (Ω)3×3 .
The proof of Lemma 3.3 immediately shows that the space Yτ (Ω) forms a
1,p(τ,·)
closed subspace of the Banach space L2 (Ω)3 ∩W0
(Ω)3 . To be more precise,
we will see below that actually
Yτ (Ω) = H ∩ Vp(τ,·) ,
1,p
and we want to call attention to the similarity of Yτ (Ω) and the space W0,div
(Ω)3 .
Keep in mind that we want to use the L2 -part to simulate an evolution triple.
Fortunately, Korn’s inequality is also valid in the context of variable exponent
spaces and that’s the content of the next theorem.
Theorem 5.5. Let Ω ⊂ Rd be a bounded domain and let p(·) ∈ P log (Ω) be a
bounded exponent with 1 < p− . There exists a constant c > 0, such that for
1,p(·)
every u ∈ W0 (Ω)d there holds
||∇u||Lp(·) (Ω)d×d ≤ c||Du||Lp(·) (Ω)d×d .
Proof. This is Theorem 5.37 in [12].
From the last theorem and the definition of the symmetric gradient it
follows that
||∇u||Lp(·) (Ω)d×d ≤ c||Du||Lp(·) (Ω)d×d ≤ C||∇u||Lp(·) (Ω)d×d
and consequently,
|||u|||Yτ (Ω) := ||u||L2 (Ω)3 + ||Du||Lp(τ,·) (Ω)3×3
64
Chapter 5. Outlook
constitutes an equivalent norm on Yτ (Ω).
Similar to the space W (QT ), we now want to define a substitute for the space
Y = Lp (I, Vp ) we had introduced above.
Definition 5.6. For Ω, I and p(·, ·) as in Definition 5.4 we set
n
o
Y∇ (QT ) := u : I → Yτ (Ω) u ∈ L2 (QT )3 , ∇u ∈ Lp(·,·) (QT )3×3 , divu = 0
Notice that the subscript ∇ indicates that ∇u ∈ Lp(·,·) (QT )3×3 . Again, an
obvious choice for a norm on Y∇ (QT ) would be
||u||Y∇ (QT ) := ||u||L2 (QT )3 + ||∇u||Lp(·,·) (QT )3×3 .
But here is the first problem. Although the operator S̃ is well-defined on
Y∇ (QT ) and maps it into its dual space, the coercivity of S̃ cannot be
established as easily as in the case of constant p. The situation is even worse,
since the striking inequality
||∇u||Lp (QT )3×3 ≤ c||Du||Lp (QT )3×3
(5.4)
which, according to (5.2), holds for every u ∈ Lp (I, Vp ), generally fails in the
context of time-dependent variable exponents and therefore also for functions
in Y∇ (QT ). We give the following counterexample to illustrate this issue.
Example 5.7. For Ω := BR (0) ⊂ R3 and I = [0, T ] we define
Ω1
Ω2
Ω3
Ωε
:= { x ∈ Ω | 0 ≤ |x| ≤ R/3 }
:= { x ∈ Ω | R/3 < |x| ≤ 2R/3 }
:= { x ∈ Ω | 2R/3 < |x| ≤ R }
:= { x ∈ Ω | R − ε < |x| ≤ R },
where ε ∈ (0, R/3). Then there exists a smooth bump function η ∈ C0∞ (Ω)
such that
η ≡ 1 on Ω1 ∪ Ω2
η ≡ 0 on Ωε .
Moreover, we choose a variable exponent p̃(·) ∈ P log (Ω) satisfying p̃(x) = 4 on
Ω1 and that smoothly and radially decreases on Ω2 to p̃(x) = 2 on Ω3 . Then
we extend p̃(·) to QT := I × Ω by setting p(t, x) := p̃(x), which certainly yields
p(·, ·) ∈ P log (I × Ω). For a skew symmetric tensor A ∈ R3×3 we finally set
w(x) := Axη(x). Now suppose we had, as an analog of (5.4),
||∇u||Lp(·,·) (QT )3×3 ≤ c||Du||Lp(·,·) (QT )3×3
(5.5)
65
for every Du ∈ Lp(·,·) (QT )d×d . That is, we assume the operator
K : D(K) ⊂ Lp(·,·) (QT )3×3 → Lp(·,·) (QT )3×3
Du 7→ ∇u
to be continuous with constant c. As p(·, ·) is a bounded exponent, inequality
(5.5) remains true in its modular form and conversely, a modular form of the
inequality implies the norm inequality if the right-hand side is finite. For
h ∈ L2 (I) with ||h||L2 (I) ≤ 1 and u(τ, x) := h(τ )w(x) we get
Z Z
Z Z
p(·,·)
|h(τ )Dw(x)|p(·,·) dx dτ
|Du|
dx dτ =
I
Ω
ZI Z
Ω
=
|h(τ )Dw(x)|2 dx dτ = c̃(η, A)||h||2L2 (I)
I Ω3
due to the definition of w. Since the Luxemburg norm and the standard Lp norm coincide in case of constant p we deduce
||Du||Lp(·,·) (QT )3×3 = c(η, A)||h||L2 (I) .
But on the other hand we have
Z Z
Z Z
p(·,·)
|∇u|
dx dτ ≥
|∇u|p(·,·) dx dτ
I
Ω
I Ω
Z Z1
=
|h(τ )∇w(x)|4 dx dτ = C̃(A)||h||4L4 (I) ,
I Ω1
which in turn implies
C(A)||h||L4 (I) ≤ ||∇u||Lp(·,·) (QT )3×3 ,
where the right hand side is finite by assumption. If we now combine all the
inequalities, we arrive at
||h||L4 (I) ≤
c(η, A)c
C(A)
14
||h||L2 (I)
(5.6)
which would have to be valid for every h ∈ L2 (I). But since L4 (I) ( L2 (I),
1
) 4 ||h0 ||L2 (I) = 1 whereas
there exists a function h0 ∈ L2 (I) such that ( c(η,A)c
C(A)
||h0 ||L4 (I) > 1. Plugging h0 in (5.6) yields a contradiction.
Thus, the norm on Y∇ (QT ) is not suitable with regard to the properties of
S̃ and for this reason, we adjust our space once more.
66
Chapter 5. Outlook
Definition 5.8. For Ω, I and p(·, ·) as in Definition 5.4 we set
n
o
YD (QT ) := u : I → Yτ (Ω) u ∈ L2 (QT )3 , Du ∈ Lp(·,·) (QT )3×3 , divu = 0 ,
and use the norm
||u||YD (QT ) := ||u||L2 (QT )3 + ||Du||Lp(·,·) (QT )3×3 .
Notice that the space YD (QT ) is an equally obvious parabolic extension of
Yτ (Ω). The reason for this lies in the validity of Korn’s inequality on Yτ (Ω).
Hence, on Yτ (Ω) it makes no difference whether we use the norm of the full
gradient or the norm of its symmetric part.
Now that we apparently found the suitable function space in view of the
coercivity of S̃, we want to clarify our notion of a weak solution of the p(t,x)Navier-Stokes equations.
Definition 5.9. Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary,
let I = (0, T ) be a bounded time interval and suppose the variable exponent
p(·, ·) ∈ P log (Ω × I) satisfies p− ≥ 11/5. For given data u0 ∈ H and
0
f = f 0 − divF with f 0 ∈ L2 (QT )3 and F ∈ Lp (·,·) (QT )3×3 , a function
u ∈ YD (QT ) whose distributional time derivative ut belongs to YD (QT )0 is
called a weak solution of the p(t,x)-Navier-Stokes equations (5.1) if
hut , ϕiYD (QT ) + hS̃u, ϕiYD (QT ) + hBu, ϕiYD (QT ) = hf, ϕiYD (QT )
∞
(QT )d and if additionally u(0) = u0 .
holds for every ϕ ∈ C0,div
Similar to Lions’ proof, we would like to prove the existence of a weak
solution by means of a Galerkin approximation, using a special basis in the
space Yτ (Ω). If we would formally try such an approximation, without going
into too many details, we would first of all need the lower bound p− ≥ 11/5,
in order to give sense to the term
Z
[∇ul ]ul · ϕ dx dτ .
QT
Using ϕ = ul as a test function, it would certainly be possible to derive the
energy estimate
l
||ul ||C(I,L
¯ 2 (Ω)3 ) + ||Du ||Lp(·,·) (QT )3×3 ≤ C(f, u0 , |QT |),
which in turn implies
||ul ||YD (QT ) ≤ C for every l ∈ N.
(5.7)
67
Due to the divergence constraint we also have
Z
[∇ul ]ul · ul dx dτ = 0.
QT
That is why the convective term does not contribute to the energy inequality
and, in the first instance, we lose control over ||∇ul ||Lp(·,·) (QT )3×3 , since we do
not have Korn’s inequality in the form (5.5). But as the convective term does
not have a particular p(·, ·)-structure, we can fix this difficulty by using the
embedding
−
Lp(·,·) (QT )3×3 ,→ Lp (QT )3×3 ,
−
together with Korn’s inequality on Lp (QT )3×3 . The energy estimate then also
reveals
||∇ul ||Lp− (QT )3×3 ≤ C(f, u0 , |QT |),
−
that is, the sequence (ul )l∈N is bounded in Lp (I, Vp− ). Now, the usual plan
is to show boundedness of the sequence (ult )l∈N in the space YD (QT )0 which
is eventually included in the space Lr (I, Vs0 ), for some 1 < r, s < ∞. If we
additionally knew that
Vp− ,→,→ L2 (Ω)3 ,→ Vs0 ,
where the first embedding is assured since p− ≥ 11/5, we could use a variant of
the Aubin-Lions lemma to justify a passage to the limit in the convective term.
Moreover, YD (QT ) is reflexive since it is the nullspace of the continuous
mapping div in the reflexive space
n
o
V := u : I → Yτ (Ω) u ∈ L2 (QT )3 , Du ∈ Lp(·,·) (QT )3×3 .
For this reason, (5.7) guarantees that there is a function u ∈ YD (QT ) such
that ul * u in YD (QT ), of course for a suitable subsequence. The reflexivity of
0
0
Lp (·,·) (QT )3×3 and estimate (5.7) also imply the existence of U ∈ Lp (·,·) (QT )3×3
such that
0
(1 + |Dul |)p(·,·)−2 Dul * U in Lp (·,·) (QT )3×3 as l → ∞.
Hence, we could now pass to the limit in the Galerkin approximation. Though,
since up to now we don’t have a detailed representation for the space YD (QT )0
it is not obvious if ut ∈ YD (QT )0 holds true. Besides, the current lack of an
embedding result and an appropriate rule for integration by parts for functions
u ∈ YD (QT ) with ut ∈ YD (QT )0 has two unfavourable consequences. The first
one is twofold since at this stage, we neither know if the initial value u(0)
is well-defined nor can we use integration by parts to show u(0) = u0 in H.
68
Chapter 5. Outlook
The second problem is that we cannot rely on the interplay of integration by
parts and Minty’s trick. But this interplay is indispensable to prove that the
weak limit U actually equals S(Du) = (1 + |Du|)p(·,·)−2 Du. Gathering all these
problems, we have to say that it is not possible so far to identify u as a weak
solution of the p(t,x)-Navier-Stokes equations in the sense of Definition 5.9.
In order to overcome these problems, we would like to propose a rough plan
that could allow to tackle the p(t,x)-Navier-Stokes equations with some of
the tools we had succesfully used in the framework of the p(t,x)-Laplacian
equation.
1) Take YD (QT ) as the energy space.
2) Show density of smooth solenoidal functions in YD (QT ).
3) Give a full description of the dual space YD (QT )0 .
4) Use the description of the dual space to show density of smooth functions
in
n
o
0
Z = u ∈ YD (QT ) ut ∈ YD (QT ) ,
where ut again denotes the distributional time derivative.
¯ H) and a formula for integration by
5) Deduce the embedding Z ,→ C(I,
parts for functions u.v ∈ Z
hut , viYD (QT )0 = hu(T ), v(T )iH − hu(0), v(0)iH − hvt , uiYD (QT )0 .
6) Show the existence of a weak solution u ∈ Z of the p(t,x)-Navier-Stokes
equations in the sense of Definition 5.9 by means of a Galerkin approximation.
The first point does not need any further explanation, whereas already the
second point contains several major problems. First of all, we want to recall
that the space YD (QT ) was given by
n
o
YD (QT ) := u : I → Yτ (Ω) u ∈ L2 (QT )3 , Du ∈ Lp(·,·) (QT )3×3 , divu = 0
with a bounded exponent p(·, ·) ∈ P log (QT ), p− ≥ 11/5,
n
o
Yτ (Ω) := u ∈ L2 (Ω)3 ∩ W01,1 (Ω)3 ∇u ∈ Lp(τ,·) (Ω)3×3 , divu = 0 ,
and we chose the norm
||u||YD (QT ) := ||u||L2 (QT )3 + ||Du||Lp(·,·) (QT )3×3 .
69
To derive the required density result, we could try the same mollification
method we had used to prove density of C0∞ (QT )d in W (QT ). Remember,
we multiplied a given u ∈ W (QT ) with a suitable cut-off function and then we
mollified this product with a smooth bell-shaped kernel. The same approach
with v ∈ YD (QT ) instead, again yields a sequence of functions contained in
C0∞ (QT )d . But by construction, these functions aren’t divergence-free anymore and therefore simply do not belong to YD (QT ). Thus, the divergence
constraint deserves special attention.
To this end, we introduce the so-called Bogovskii operator, an instrument to
construct an explicit solution u of the equation
divu = f in Ω,
(5.8)
for an appropriate class of functions f on Lipschitz domains Ω. Fortunately, in
[23] Huber extended Bogovskii’s original solution on classic Lp -spaces, see [5],
to generalized Lebesgue spaces. Hence, we only cite the results concerning the
Lp(·) -theory and refer to [23] and [5] for the classical theory and an in-depth
treatment of the divergence equation (5.8).
As a first step we introduce two auxiliary spaces.
Definition 5.10. For a bounded subset Ω ⊂ Rd and a bounded exponent
p(·) ∈ P(Ω) we set
Z
n
o
p(·)
p(·)
L0 (Ω) := f ∈ L (Ω) f dx = 0 , and
Ω
Z
o
n
∞
f dx = 0
C0,0
(Ω) := f ∈ C0∞ (Ω) Ω
Remark 5.11. According to the boundedness of Ω, we have the embedding
p(·)
Lp(·) (Ω) ,→ L1 (Ω), witnessing that L0 (Ω) is well-defined. Also, the space
p(·)
∞
C0,0
(Ω) is dense in L0 (Ω).
In view of a solution of equation (5.8) we cite
Theorem 5.12. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and
let further p(·) belong to P log (Ω) with p− > 1. Then, there exists a continuous
and linear operator
p(·)
1,p(·)
B : L0 (Ω) → W0 (Ω)d ,
p(·)
called the Bogovskii operator, satisfying div(Bf ) = f for every f ∈ L0 (Ω).
∞
For every f ∈ C0,0
(Ω) there additionally holds Bf ∈ C0∞ (Ω)d .
Proof. The proof can be found in [23], Theorem 2.40.
We introduce yet another function space that already encountered us in a
less general form.
70
Chapter 5. Outlook
Definition 5.13. Given a bounded subset Ω ⊂ Rd and two variable exponents
p(·), q(·) ∈ P(Ω) we set
1,p(·)
Xp(·),q(·) (Ω) := W0
(Ω)d ∩ Lq(·) (Ω)d ,
which is a Banach space under the norm
|| · ||Xp(·),q(·) (Ω) := || · ||W 1,p(·) (Ω)d + || · ||Lq(·) (Ω)d .
0
Remark 5.14. For d = 3 and q(·) ≡ 2, the space Yτ (Ω) forms a closed
subspace of Xp(τ,·),2 (Ω)
The next theorem is the key result for the construction of smooth divergencefree test functions.
Theorem 5.15. Suppose we are given a bounded domain Ω ⊂ Rd with
Lipschitz boundary and two bounded exponents p(·), q(·) ∈ P log (Ω) with the
property p− , q − > 1. Then there exists a linear and continuous mapping
E : Xp(·),q(·) (Ω) → Xp(·),q(·) (Ω) with the following features:
1) For every f ∈ Xp(·),q(·) (Ω) we have div(Ef ) = divf .
2) There exists a positive constant Cp(·) such that
||Ef ||W 1,p(·) (Ω)d ≤ Cp(·) ||divf ||Lp(·) (Ω) .
0
3) For every f ∈ C0∞ (Ω)d we have Ef ∈ C0∞ (Ω)d .
Proof. See Theorem 2.43 in [23].
With the help of the last theorem, we are now able to prove
Lemma 5.16. With the same notation as in the first part of this chapter we
have
Yτ (Ω) = H ∩ Vp(τ,·) .
Proof. The inclusion H ∩ Vp(τ,·) ⊂ Yτ (Ω) follows from the definition of the
spaces. For the opposite inclusion, we have to show that for every u ∈ Yτ (Ω)
there exists a sequence (uh )h>0 ⊂ V that converges to u in H ∩ Vp(τ,·) . To
this end, we set Ωh :=
R { x ∈ Ω | d(x, ∂Ω) > h }, and for a positive mollifier
υ ∈ C0∞ (B1d (0)) with Rd υ(x) dx = 1 we define a cut-off function by
µh := υ h ∗ χΩh .
4
71
Now take a bell-shaped mollifier ζ ∈ B(Rd ) ∩ C0∞ (B1d (0)) with
and given u ∈ Yτ (Ω), define an approximate sequence by
R
Rd
ζ(x) dx = 1
uh := ζ h ∗ (µh u) − E(ζ h ∗ (µh u)).
4
4
Appealing to the properties of the operator E, it is obvious that the sequence
(uh )h>0 belongs to V. Now we show uh → u in H, as h → 0. We have
||u − uh ||L2 (Ω)3 ≤ ||u − ζ h ∗ (µh u)||L2 (Ω)3 + ||E(ζ h ∗ (µh u))||L2 (Ω)3 .
4
4
The first term on the right-hand side tends to zero according to Young’s inequality and the theory of convolutions. For the second term we can use the
1,p(τ,·)
embedding W0
(Ω)3 ,→ L2 (Ω)3 since p(τ, ·) ≥ p− > 2 together with the
second property of E and divu = 0 to get
||E(ζ h ∗ (µh u))||L2 (Ω)3 ≤ c||E(ζ h ∗ (µh u))||W 1,p(τ,·) (Ω)3
4
4
0
= c||∇E(ζ h ∗ (µh u))||Lp(τ,·) (Ω)3×3
4
≤ c||div(ζ h ∗ (µh u))||Lp(τ,·) (Ω)
4
= c||ζ h ∗ (∇µh · u)||Lp(τ,·) (Ω) .
4
The expression ζ h ∗ (∇µh · u) converges to 0 almost everywhere in Ω and we
4
have the upper bound
|ζ h ∗ (∇µh · u)| ≤ cM |∇u| ∈ Lp(τ,·) (Ω).
4
Hence, ζ h ∗ (∇µh · u) → 0 in Lp(τ,·) (Ω) as h → 0, witnessing that (uh )h>0
4
converges to u in L2 (Ω)3 . It remains to show uh → u in Vp(τ,·) .
||u − uh ||Vp(τ,·) ≤ c||∇u − ζ h ∗ (µh ∇u)||Lp(τ,·) (Ω)3×3
4
+ c||ζ h ∗ (∇µh ⊗ u)||Lp(τ,·) (Ω)3×3
4
+ c||∇E(ζ h ∗ (µh u))||Lp(τ,·) (Ω)3×3
4
≤ c||∇u − ζ h ∗ (µh ∇u)||Lp(τ,·) (Ω)3×3
4
+ c||ζ h ∗ (∇µh ⊗ u)||Lp(τ,·) (Ω)3×3
4
+ c||ζ h ∗ (∇µh · u)||Lp(τ,·) (Ω) .
4
The fourth expression on the right-hand side converges to 0, again by Young’s
inequality and the theory of convolutions on generalized Lebesgue spaces. The
remaining two terms vanish for h → 0 after applying the same trick as in the
first part of the proof.
72
Chapter 5. Outlook
Lemma 5.16 states that divergence-free test functions are dense in the space
Yτ (Ω), but it isn’t obvious at all, how to carry over the techniques we used, in
order to verify density of smooth solenoidal functions in YD (QT ). The proof
was essentially based on two tools we had at our disposal. The first one is the
inequality
||∇(Ef )||Lp(·) (Ω)3×3 ≤ Cp(·) ||divf ||Lp(·) (Ω) ,
(5.9)
1,p(·)
which is valid for every f ∈ W0 (Ω)3 . If we tried to adopt the above proof
in a straightforward manner, we would need an analogue version of (5.9) for
exponents also depending on the time variable. But we do not know without
fail if such an inequality is true in general.
The second tool was the estimate
|ζ h ∗ (∇µh · u)| ≤ cM |∇u| ∈ Lp(τ,·) (Ω),
4
which is a by-product of the fact that
|ω h ∗ (u ⊗ ∇ηh )| ≤ cM (|∇u|)
4
holds almost everywhere in QT for every u ∈ W (QT ), see Lemma 3.9.
This estimate provided the necessary upper bound for Lebesgue’s theorem on
dominated convergence in Theorem 3.8. The proof used a special Poincarétype inequality near the boundary of Ω in terms of ∇u, but keep in mind that
for functions in YD (QT ) we only have control over the symmetric part of the
gradient. Thus, in order to proceed in an analogous way, we need a tool that
allows us to estimate a function u ∈ W01,1 (Ω)d in terms of its symmetric part
of the gradient Du. But that’s the next missing piece.
However, in [30] Mosolov and Mjasnikov establish a formula that could help.
The starting point is Cesàro’s formula which reads
d
X
vi (x) = vi (x ) +
(Av)ij (x0 )(xj − x0j )
0
j=1
x0
−
"
d Z
X
k=1 x
#
d
X
∂(Dv)ik ∂(Dv)kj
dξk .
−
(Dv(ξ))ik +
(xj − ξj )
∂ξ
∂ξ
j
i
j=1
Here, Av := 21 (∇v − (∇v)T ) denotes the skew symmetric part of the gradient.
After some calculation we do not want to mention here, this leads them to
d Z
X
1
vi (x) = R
Kij (x, y)vj (y)dy
%(x0 )dx0 j=1
Ω0
1
+R
%(x0 )dx0
Ω0
Ω
d
X
Z
k,j=1 Ω
(Dv)kj (y)
B(r, |x − x0 |, x − x0 ) dy,
rd−1
73
where B is a bounded function. This formula looks encouraging, especially
with regard to the second part of the sum, because an appropriate choice of
the function % and the boundedness of B could in turn imply
Z
|Dv|
|vi (x)| ≤ c
dz.
|x − z|n−1
Ω
Nevertheless, a complete examination of this issue lies beyond the scope of
this chapter and even of this work.
∞
In conclusion, the required density of C0,div
(QT )3 already represents a major
obstacle if we wanted to follow our plan.
The next step, the description of the dual space, does not seem to be less
difficult. The reason for this is the divergence constraint captured in the space
YD (QT ) that prevents us from establishing a representation for functionals from
YD (QT )0 in a similar fashion as in the context of the space W (QT )0 . Certainly,
every regular distribution T of the form
T = Tf − divTF − ∇Tg ,
p0 (·,·)
(5.10)
p0 (·,·)
for f ∈ L2 (QT )3 , F ∈ L
(QT )3×3 and g ∈ L
(QT ), yields an element in
0
YD (QT ) , but we do not know with certainty if the converse is also true. That
is, we do not know if there is a representation of the form
o
n
0
0
YD (QT ) ' T ∈ D (QT ) T = Tf − divTF − ∇Tg , .
where f, F and g have the same properties as above. The restriction and
extension arguments we used in the third chapter do not directly apply in this
∞
(QT )d .
new setting since C0∞ (QT )d has to be replaced by the subspace C0,div
This has the effect that
Z
h−∇Tg , ϕi = gdivϕ dx dτ = 0
QT
∞
holds for every ϕ ∈ C0,div
(QT )d . Therefore, the third part of such a
distribution T cannot be uniquely determined from its values on the
∞
potentially dense subspace C0,div
(QT )d . Hence, it is not obvious, whether and
how it could be possible to identify the dual space YD (QT )0 with the subspace
consisting of those regular distributions of the form (5.10).
Without appropriate density results and a detailed description of the dual
space YD (QT )0 it seems to be impossible to derive an embedding result to
justify our initial condition u(0) = u0 ∈ H. Another consequence is that the
desired rule for integration by parts in time seems to be out of reach as well.
In summary, it can be said that the methods from the previous chapters cannot
be adapted in a straightforward and obvious way, since there still are several
open problems.
74
Chapter 5. Outlook
Theorem 5.8 also helps us to proceed in the proof of the density of smooth
divergence-free functions in the parabolic space YD (QT ). As in the proof of
Theorem
3.8 we take a positive mollifier ϕ ∈ B(Rd+1 ) ∩ C0∞ (B1d+1 (0)) with
R
ϕ(τ, x) dx dτ = 1 to set
Rd+1
QhT := I × { x ∈ Ω | dist(x, ∂Ω) > h } and
ηh := ϕ h ∗ χQhT ,
(5.11)
4
where χQhT denotes the characteristic function of QhT . For another positive
R
mollifier ω ∈ B(Rd+1 ) ∩ C0∞ (B1d+1 (0)),
ω(τ, x) dx dτ = 1, we then define
Rd+1
for u ∈ YD (QT )
uh := ω h ∗ ω h ∗ (ηh u) − ω h ∗ E(ω h ∗ (ηh u)).
4
4
4
(5.12)
4
The purpose of using two convolutions will later become clearer. Notice that
(ω h ∗ (ηh u))(τ, ·) belongs to C0∞ (Ω)d . Thus, uh is well-defined thanks to the
4
third property of E in Theorem 5.15. Moreover, the function uh is not only
smooth but also solenoidal:
divuh = ω h ∗ ω h ∗ div(ηh u) − ω h ∗ divE(ω h ∗ (ηh u))
4
4
4
4
= ω h ∗ ω h ∗ div(ηh u) − ω h ∗ div(ω h ∗ (ηh u))
4
4
4
(5.13)
4
= ω h ∗ ω h ∗ div(ηh u) − ω h ∗ ω h ∗ div(ηh u) = 0,
4
4
4
4
according to the first property of E.
∞
The sequence (uh )h>0 ⊂ C0,div
(QT )d ⊂ YD (QT ) seems to be the right candidate
with regard to the density result announced above. Thus, we first of all show
uh → u in L2 (QT )3 for h → 0. Now
||uh − u||L2 (QT )3 ≤ ||ω h ∗ ω h ∗ (ηh u) − u||L2 (QT )3 + ||ω h ∗ E(ω h ∗ (ηh u))||L2 (QT )3 ,
4
4
4
4
(5.14)
and the first term on the right hand side tends to zero due to a straightforward
application of Young’s inequality and the classical theory of convolutions.
For the second term we notice that ω h ∗E(ω h ∗(ηh u))(τ, ·) belongs to C0∞ (Ω)3 ⊂
4
−
W01,p (Ω)3 , and since p− ≥
11
5
ZT
||ω h ∗ E(ω h ∗ (ηh u))||L2 (QT )3 ≤ c
4
4
we have
4
||E(ω h ∗ (ηh u))(τ )||2L2 (Ω)3 dτ
4
0
ZT
≤c
||E(ω h ∗ (ηh u))(τ )||2
−
W01,p (Ω)3
4
dτ (5.15)
0
ZT
≤c
||div(ω h ∗ (ηh u))(τ )||2Lp− (Ω) dτ ,
4
0
75
where we used Youngs’s inequality for convolutions in the first line and the
second part of Theorem 5.15 in the last one. Due to the fact that u ∈ YD (QT )
is divergence-free, the last integral reduces to
ZT
||div(ω h ∗ (ηh u))(τ )||2Lp− (Ω) dτ =
ZT
4
0
||(ω h ∗ (∇ηh · u))(τ )||2Lp− (Ω) dτ , (5.16)
4
0
−
and we are done once we established (ω h ∗ (∇ηh · u))(τ, ·) → 0 in Lp (Ω) for
4
almost every τ ∈ I.
By the method of construction, we know that (ω h ∗ (∇ηh · u))(τ, x) tends
4
to zero for almost every x ∈ Ω as h → 0. Hence, we can conclude with
−
dominated convergence if we find a Lp (Ω)-majorant. But appealing to the
proof of Theorem 3.8, we know that
|(ω h ∗ (∇ηh · u))(τ, x)| ≤ cM (|∇u|)(τ, x).
(5.17)
4
By the boundedness of the Hardy-Littlewood maximal operator, that would
−
be the adequate majorant if |∇u|(τ, ·) belongs to Lp (Ω). Now, this in turn
follows from the next chain of inequalities, which holds true since u belongs to
−
YD (QT ) and Korn’s inequality in the form of (5.4) is valid on Lp (QT )3×3
||∇u||Lp− (QT )3×3 ≤ c||Du||Lp− (QT )3×3 ≤ C||Du||Lp(·,·) (QT )3×3 < ∞.
(5.18)
All in all, we get uh → u in L2 (QT )3 as h tends to zero.
The real problem, is to prove convergence of the symmetric gradient Duh in
Lp(·,·) (QT )3×3 . To make things clearer, we calculate
Duh = ω h ∗ ω h ∗ D(ηh u) − ω h ∗ D[E(ω h ∗ (ηh u))]
4
4
4
4
4
4
4
(5.19)
= ω h ∗ ω h ∗ (ηh Du) + ω h ∗ ω h ∗ (u ⊗ ∇ηh )sym − ω h ∗ D[E(ω h ∗ (ηh u))].
4
4
4
Here, (u ⊗ ∇ηh )sym denotes the symmetric part of the tensor u ⊗ ∇ηh , that is
explicitely given by (u ⊗ ∇ηh )sym = 21 (u ⊗ ∇ηh + ((u ⊗ ∇ηh ))T ).
As required, the term ω h ∗ (ηh Du) converges to Du in Lp(·,·) (QT )3×3 for h →
4
0. This is a direct consequence of Young’s inequality, Lebesgue’s theorem of
dominated convergence and the theory of convolutions on generalized Lebesgue
spaces.
We are thus forced, to prove that the remaining two expressions converge to
zero in Lp(·,·) (QT )3×3 . As a first step in this direction we consider the term
ω h ∗ ω h ∗ (u ⊗ ∇ηh )sym .
4
4
(5.20)
By Youngs’s inequality for convolutions on generalized Lebesgue spaces, which
is valid since ω h is bell-shaped, we get
4
||ω h ∗ω h ∗(u⊗∇ηh )sym ||Lp(·,·) (QT )3×3 ≤ c||ω h ∗(u⊗∇ηh )sym ||Lp(·,·) (QT )3×3 . (5.21)
4
4
4
76
Chapter 5. Outlook
The term ω h ∗ (u ⊗ ∇ηh )sym certainly converges to 0 almost everywhere in QT ,
4
according to the properties of the cut-off function ηh . In order to bound it by
a Lp(·,·) (QT )3×3 -function, the only promising idea is to use the fact that Du
belongs to this space. Remember, the same problem occured in the proof of
Theorem 3.8. But in that situation, we were able to use the crucial inequality
|(ω h ∗ (u ⊗ ∇ηh ))(τ, x)| ≤ cM (|∇u|)(τ, x) a.e. in QT .
4
(5.22)
Together with the boundedness of the maximal function on Lp(·,·) (QT )d×d and
the fact that ∇u belonged to Lp(·,·) (QT )d×d we could then use dominated convergence to conclude (ω h ∗ (u ⊗ ∇ηh )) → 0 in Lp(·,·) (QT )d×d . The whole idea
4
was based on the following Poincaré-type inequality, we had at our disposal:
Z
d(x, ∂Ω)d
|∇u(y)|
|u(x)| ≤ c
dy, for every u ∈ W01,1 (Ω)d .
d−1
|B2d(x,∂Ω) (x)|
|x − y|
B2d(x,∂Ω) (x)
(5.23)
Unfortunately, we have to confess that the treatment of the expression
ω h ∗ D[E(ω h ∗ (ηh u))]
4
(5.24)
4
seems to be even more delicate. The only tool that could help us along, is the
second estimate in Theorem 5.15. But as a similar estimate is not known for
time-dependent variable exponents, we cannot directly apply it. Again, this
is due to the fact that the norm on Lp(·,·) (QT )d×d cannot be decomposed like
the norm on the classic Bochner space Lp (QT )d×d = Lp (I, Lp (Ω)d×d ). Since in
case of a constant exponent there are less obstacles, we assume for the moment
, ∞), in order to get
that our exponent p(·, ·) is constant, i.e. p(t, x) ≡ p ∈ [ 11
5
a feeling for the problem. The task would then be to show
||ω h ∗ D[E(ω h ∗ (ηh u))]||Lp (QT )3×3 → 0 for h → 0.
4
4
(5.25)
Now, thanks to the definition of the norm on Lp (QT )d×d we have
||ω h ∗ D[E(ω h ∗
4
4
(ηh u))]||pLp (QT )3×3
ZT
≤c
||D[E(ω h ∗ (ηh u))](τ )||pLp (Ω)3×3 dτ
4
0
ZT
≤c
||∇[E(ω h ∗ (ηh u))](τ )||pLp (Ω)3×3 dτ
4
0
ZT
≤c
(5.26)
||div[E(ω h ∗ (ηh u))](τ )||pLp (Ω) dτ
4
0
ZT
=c
||[ω h ∗ (∇ηh · u))](τ )||pLp (Ω) dτ ,
4
0
77
and the last integral converges to zero.
There are two crucial ingredients, that make this technique work. The first
one is the application of Theorem 5.15 to get from the second to the third line.
But once again, that’s only possible because of the decomposition properties
of the Lp (QT )d×d -norm. The second one is inequality (5.17), that allowed us
to pass to the limit in the last integral. This in turn, was only possible since
we used a constant exponent in this sample calculation, which allows us to
switch from the symmetric gradient to the full gradient without problems.
The problem is, that we relied on structures which do not directly apply in
the context of time-dependent variable exponents.
The most ingenuous approach to show
ω h ∗ D[E(ω h ∗ (ηh u))] → 0 in Lp(·,·) (QT )3×3
4
4
(5.27)
probably is to try to argue with Lebesgue’s theorem of dominated convergence.
To this end we will finally make use of the second mollifier in the definition of
uh . Its only purpose is to spend another integral. We have
h
i
|ω h ∗ D[E(ω h ∗ (ηh u))](t, y)| ≤ ωh ∗ |∇[E(ω h ∗ (ηh u))]| (t, y)
4
4
4
Z
≤ c − |∇[E(ω h ∗ (ηh u))]| dx dτ , (5.28)
4
Bhd+1 (t,y)
since |ωh | ≤ c/hd+1 and the convolution can thus be transformed into an
average integral. We choose 1 < s ≤ p− , and with the help of Jensen’s
inequality we arrive at
Z Z
Z
s/s
− |∇[E(ω h ∗ (ηh u))]| dx
dτ
− |∇[E(ω h ∗ (ηh u))]| dx dτ = c −
4
4
Bhd+1 (t,y)
Ih (t)
Bhd (y)
Ih (t)
Bhd (y)
Z Z
1/s (5.29)
s
− |∇[E(ω h ∗ (ηh u))]| dx
≤c −
dτ .
4
But now we can use Theorem 5.15 on the space W01,s (Ω)3 , in order to switch
from the gradient to the divergence of E(ω h ∗ (ηh u)).
4
Z Z
1/s
c −
− |∇[E(ω h ∗ (ηh u))]|s dx
dτ
4
Ih (t)
Bhd (y)
Z Z
1/s
≤c −
dτ
− |div[E(ω h ∗ (ηh u))]|s dx
4
Ih (t)
Bhd (y)
Z Z
1/s
=c −
− |ω h ∗ (∇ηh · u))|s dx
dτ
4
Ih (t)
Bhd (y)
(5.30)
78
Chapter 5. Outlook
Now we apply Jensen’s inequality once more to the concave function t 7→ t1/s
to end up with
Z Z
1/s
dτ
c −
− |ω h ∗ (∇ηh · u))|s dx
4
Ih (t)
Bhd (y)
Z Z
1/s
≤ c − − |ω h ∗ (∇ηh · u))|s dx dτ
(5.31)
4
Ih (t) Bhd (y)
1/s
≤ c M (|ω h ∗ (∇ηh · u))|s )(t, y)
.
4
Inequality (5.17) and the boundedness of the Hardy-Littlewood maximal operator assure that the last expression is finite for almost all (t, y) ∈ QT . But
now we make use of the weak type (1, 1) estimate from Theorem 2.2 which
yields
Z
||λχ{M ((|ω h ∗(∇ηh ·u))|s )>λ} ||L1 (QT ) ≤ c |ω h ∗ (∇ηh · u))|s dx dτ → 0. (5.32)
4
4
QT
1/s
This in turn implies c M (|ω h ∗ (∇ηh · u))| )
→ 0 almost everywhere in
4
QT for h → 0, and thus, |ω h ∗ D[E(ω h ∗ (ηh u))]| → 0 almost everywhere, as h
4
4
tends to 0. In order to use dominated convergence, we need an Lp(·,·) (QT )3×3 majorant.
But again, this is the missing piece. However, according to the similarity of
the structure of ω h ∗ ω h ∗ (u ⊗ ∇ηh )sym and the expression
4
s
4
Z Z
1/s
− − |ω h ∗ (∇ηh · u))|s dx dτ
,
4
(5.33)
Ih (t) Bhd (y)
we conjecture, that a bound for |ω h ∗ (∇ηh · u))| in terms of Du would serve
4
as the required majorant for both expressions.
5.1
Description of the dual space YD (QT )0
In this subsection we want to try to work out a characterization of the space
YD (QT )0 . The purpose of this description is two fold. First of all, it is required
to possibly prove density of a certain class of smooth functions in YD (QT )0 .
This, in turn, would be necessary with regard to a generalized rule for integration by parts in the space Z, we mentioned in the fourth step of our plan.
Remember, in case of the space W (QT )0 wo got the desired representation by
5.1 Description of the dual space YD (QT )0
79
projecting W (QT ) onto the space L2 (QT )d × Lp(·,·) (QT )d×d in order to make
use of the Hahn-Banach extension theorem on this space. Then we used the
density of C0∞ (QT )d in W (QT ) to identify the dual space W (QT )0 as a subspace
of the space of regular distributions on QT , by means of some restriction and
extension arguements.
If we try the same approach in the context of YD (QT ), the divergence constraint
that is captured in the definition of the space again causes the problems. If
we take a closer look at the divergence of u ∈ YD (QT ), we obtain
1
1
divu = tr∇u = tr∇u + tr(∇u)T
2
2
1
1
= tr( ∇u + (∇u)T ) = trDu,
2
2
showing that |divu| ∈ Lp(·,·) (QT )
(5.34)
80
Chapter 5. Outlook
Chapter 6
Appendix
6.1
Functional analysis
This section contains some basic functional analytic facts we use throughout
this work. For a normed space X, we define its dual space by
o
n
X 0 := L(X, R) := f : X → R f is linear and continuous .
For f ∈ X 0 and x ∈ X we use the notation hf, xiX := f (x) to express the
duality between X 0 and X. The norm in X 0 is defined by
||f ||X 0 := sup |hf, xiX |,
||x||X ≤1
and a sequence (fn )n∈N ⊂ X 0 converges to f ∈ X 0 if and only if
lim ||fn − f ||X 0 = 0.
n→∞
Now we cite the Hahn-Banach theorem, which guarantees the existence of
non-trivial functionals.
Theorem 6.1. Let Y ⊂ X be a linear subspace and suppose g : Y → R is
linear and continuous. Then there exists an extension f ∈ X 0 of g such that
||f ||X 0 = ||g||Y 0 .
Proof. Corollary I.2 in [6].
For Hilbert spaces we have Riesz’ representation theorem.
Theorem 6.2. Let H be a real Hilbert space with scalar product (·, ·). For
every functional F ∈ H 0 there exists an unique element f ∈ H such that
hF, hiH = (f, h) for every h ∈ H.
In this sense, every Hilbert space H can be identified with its dual space H 0 .
81
82
Chapter 6. Appendix
Proof. Theorem V.5 in [6].
The next important tool we want to mention is the open-mapping principle.
Theorem 6.3. Suppose X and Y are Banach spaces. If a linear operator T
maps X continuously onto Y , then there is a constant r > 0 such that
BrY (0) ⊂ T (B1X (0)).
Proof. This is Theorem II.5 in [6].
Corollary 6.4. Again let X, Y be Banach spaces and let T ∈ L(X, Y ) be
one-to-one and onto. Then T −1 : Y → X is linear and continuous.
Proof. See [6], Corollary II.6.
For a normed space X, its bidual space X 00 is defined by
X 00 := (X 0 )0 .
We have a canonic isometry J : X → X 00 which is given by
hJx, f iX 0 := hf, xiX for all x ∈ X, f ∈ X 0 .
The fact that J is an isometry follows from the Hahn-Banach theorem, hence,
J(X) ⊂ X 00 is always a closed subspace. A normed space X is called reflexive
if and only if J is onto.
Theorem 6.5. Let X and Y be normed vector spaces.
If X is reflexive then any closed linear subspace U ⊂ X is reflexive.
If S : X → Y is an isomorphism, then X is reflexive if and only if Y is
reflexive. Also, X is reflexive if and only if X 0 is reflexive.
Proof. The proof can be found in [2], Theorem 6.8.
Definition 6.6. A normed space X is called separable if it contains a countable
and dense subset.
Lemma 6.7. Let X be separable. Then any subset U ⊂ X is separable, too.
Proof. Lemma 2.16 in [2].
On X we can introduce another type of convergence that is weaker than
convergence with respect to the norm || · ||X .
Definition 6.8. Suppose (xn )n∈N is a sequence in X. Then we say (xn )n∈N
converges weakly to an element x ∈ X and write xn * x, if and only if
lim hf, xn iX = hf, xiX for every f ∈ X 0 .
n→∞
83
6.2 Distributions
Since |hf, xn − xiX | ≤ ||f ||X 0 ||xn − x||X , norm convergence always implies
weak convergence.
The next result generalizes the theorem of Bolzano-Weierstass to infinite dimensional vector spaces and is due to Eberlein and Šmuljan.
Theorem 6.9. Let X be a reflexive Banach space. Then every bounded
sequence in X contains a subsequence that converges weakly in X.
Proof. This is Theorem III.28 in [6].
6.2
Distributions
In this section we want to briefly summarize the most important facts about
distributions. All the results and definitions we use here can also be found in
the monographs of Alt, Hörmander or Dobrowolski.
Definition 6.10. Let X ⊂ Rn be a bounded open domain. Then we define
\
C0m (X)
C0∞ (X) :=
m∈N
as the space consisting of all real valued functions that are differentiable
infinitely many times and have compact support in X. A sequence (ϕn )n∈N
is said to converge to ϕ in C0∞ (X), if there is a compact subset K ⊂⊂ X such
that supp(ϕn ), supp(ϕ) ⊂ K and limn→∞ supx∈K |Dα (ϕn (x) − ϕ(x))| = 0 for
every multiindex α ∈ Nn . If C0∞ (X) is equipped with this notion of convergence
it is also denoted by D(X).
Theorem 6.11. Suppose X = Rd = U × V for U = Rd−k and V = Rk . The
set
D̃(X) = span ϕ ∈ D(X) | ϕ(x) = ψ(u)τ (v), ψ ∈ D(U ) and τ ∈ D(V )
is dense in D(X).
Proof. [19], Theorem 8.19.
Definition 6.12. A linear functional T : D(X) → R is called a distribution
if and only if limn→∞ ϕn = ϕ in D(X) implies (T (ϕn ))n → T (ϕ) in R as
n → ∞. For a distribution T and ϕ ∈ D(X) we use the common notation
hT, ϕi := T (ϕ). Also, we define D0 (X) as the space of distributions on X.
An example of a special class of distributions arises from the following
construction.
84
Chapter 6. Appendix
For a function f ∈ L1loc (X) :=
we define
n
o
g : X → R g ∈ L1 (K) for every K ⊂⊂ X
Z
hTf , ϕi :=
f ϕ dx .
X
Then Tf clearly is linear and also continuous, for, if ϕn → ϕ in D(X) there
exists a subset K ⊂⊂ X such that supp(ϕn ), supp(ϕ) ⊂ K and since f is in
L1 (K) we have
Z
|hTf , ϕn − ϕi| ≤ ||ϕn − ϕ||C(K) |f | dx → 0 for n → ∞.
K
The mapping f 7→ Tf is one-to-one because of
R
Theorem 6.13. Suppose the function f ∈ L1loc (X) satisfies X f ϕ dx = 0 for
every ϕ ∈ D(X). Then we have f = 0 almost everywhere in X.
Proof. This is Theorem 5.1 in [15].
Hence, every function f ∈ L1loc (X) yields a distribution Tf and these are
called regular distributions. From now on we do not longer distinguish between
a function f ∈ L1loc (X) and the regular distribution Tf ∈ D0 (X).
Definition 6.14. Let T be a distribution and α ∈ Nn be a multiindex. Then
we define the distribution ∂ α T by setting
h∂ α T, ϕi := (−1)|α| hT, ∂ α ϕi for every ϕ ∈ D(X),
P
where |α| := ni=1 αi . A function f ∈ L1loc (X) is said to have an α-th partial
derivative in L1loc (X) in the distributional sense if there exists a function fα
belonging to L1loc (X) such that
h∂ α f, ϕi := h∂ α Tf , ϕi = (−1)|α| hTf , ∂ α ϕi = hTfα , ϕi = hfα , ϕi.
In this sense, every distribution has distributional derivatives of infinite
order.
Lemma 6.15. Let T ∈ D0 (X) be a distribution. For a function g ∈ C ∞ (X)
we get a well-defined distribution gT ∈ D0 (X) by setting hgT, ϕi := hT, gϕi for
ϕ ∈ D(X).
Lemma 6.16. Suppose we are given T ∈ D0 (X), f ∈ L1loc (X), g ∈ C ∞ (X)
and a multiindex α ∈ Nn with |α| = 1. Then we have
a) gTf = Tgf in D0 (X),
85
6.2 Distributions
b) ∂ α (gT ) = ∂ α gT + g∂ α T in D0 (X).
Proof. Part a) follows from the definitions. For part b) we calculate
h∂ α (gT ), ϕi = (−1)|α| hgT, ∂ α ϕi = (−1)|α| hT, g∂ α ϕi
= (−1)|α| hT, ∂ α (gϕ) − ∂ α gϕi
= (−1)|α| hT, ∂ α (gϕ)i − (−1)|α| hT, ∂ α gϕi
= (−1)2|α| h∂ α T, gϕi − (−1)|α| h∂ α gT, ϕi
= hg∂ α T + ∂ α gT, ϕi.
Definition 6.17. Suppose T ∈ D0 (X) is a distribution and Y ⊂ X is a subset.
Then the restriction of T to Y is defined by
hT |Y , ϕi := hT, ϕi for ϕ ∈ D(Y ).
The set
supp(T ) :=
n
o
x ∈ X T |X∩Bδ (x) 6= 0 ∀δ > 0
is called the support of T .
Proposition 6.18. We have the following facts about the support of a
distribution.
a) supp(T ) is closed and if supp(T ) ∩ supp(ϕ) = ∅, then hT, ϕi = 0.
b) If Tf is a regular distribution then f can be modified on a set of measure
zero such that supp(Tf ) ⊂ supp(f ).
c) If Tf is a regular distribution with f ∈ C(X) then we have supp(Tf ) =
supp(f ).
Proof. We only prove part a), for the rest we refer to [15], Theorem 9.10.
If x belongs to X \ supp(T ) then there exists a δ > 0 such that T|X∩Bδ (x) = 0.
This means Bδ (x) ⊂ X \ supp(T ) and X \ supp(T ) is thus open.
Now suppose supp(T ) ∩ supp(ϕ) = ∅. Then supp(ϕ) ⊂ X \ supp(T ) and
since supp(ϕ) is compact and X \ supp(T ) is open we can find an > 0 and
an open set K such that supp(ϕ) ⊂ K ⊂ X \ supp(T ). Hence, we have
hT, ϕi = hT |X∩K , ϕi = 0.
Definition 6.19. The space D0 (X) is equipped with the weak-∗ topology.
This means that a sequence (Tk )k∈N ⊂ D0 (X) converges to T in D0 (X) if and
only if lim hTk , ϕi = hT, ϕi is valid for any ϕ ∈ D(X).
k→∞
86
Chapter 6. Appendix
Lemma 6.20. The mapping ∂ α : D0 (X) → D0 (X) is continuous for any
multiindex α ∈ Nn .
Proof. Suppose the sequence (Tn )n∈N converges to T in D0 (X), then by
definition of the distributional derivative we have
lim h∂ α Tn − ∂ α T, ϕi = lim h∂ α (Tn − T ), ϕi = lim (−1)|α| hTn − T, ∂ α ϕi = 0,
n→∞
n→∞
n→∞
for every ϕ ∈ D(X).
Finally, we also need the following completeness result. A proof can be
found in [22], Theorem 2.1.8.
Theorem 6.21. Let (Tk )k∈N be a sequence in D0 (X). If the limit, given by
hT, ϕi := limk→∞ hTk , ϕi, exists for every ϕ ∈ D(X), then T defines an element
in D0 (X).
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