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Degenerate diffusion [Elektronische Ressource] : behaviour at the boundary and kernel estimates / vorgelegt von Michal Chovanec

122 pages
DEGENERATEDIFFUSIONBehaviour at the boundary andkernel estimatesMichal ChovanecApril 2010DEGENERATE DIFFUSIONBehaviour at the boundary and kernel estimatesDissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult at fur Math-ematik und Wirtschaftswissenschaften der Universit at Ulmvorgelegt von Michal Chovanec aus Bansk a BystricaGutachter: Prof. Dr. Wolfgang ArendtProf. Dr. Werner KratzProf. Tom ter Elst, PhD (Auckland)Dekan: Prof. Dr. Werner KratzTag der Promotion: 31. Mai 2010Abstract: We study evolution equations of the form:@u(t;x) =m(x)(4u)(t;x) t2R ; x2 ;+@tNwhere is a bounded domain in R and the function m : ! (0;1) is assumedto be measurable. Dirichlet boundary conditions are posed. We investigate underwhich conditions on m and @ the operator m4 generates a strongly continuoussemigroup on C ( ). In the second part of the thesis we obtain various estimates0pon the kernel of the semigroup generated by m4 on weighted L -spaces.ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Preliminaries 91.1 Sesquilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Associated operator, fundamentals of the semigroup theory . . . . . 111.3 Holomorphic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Beurling-Deny conditions . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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DEGENERATE
DIFFUSION
Behaviour at the boundary and
kernel estimates
Michal Chovanec
April 2010DEGENERATE DIFFUSION
Behaviour at the boundary and kernel estimates
Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult at fur Math-
ematik und Wirtschaftswissenschaften der Universit at Ulm
vorgelegt von Michal Chovanec aus Bansk a Bystrica
Gutachter: Prof. Dr. Wolfgang Arendt
Prof. Dr. Werner Kratz
Prof. Tom ter Elst, PhD (Auckland)
Dekan: Prof. Dr. Werner Kratz
Tag der Promotion: 31. Mai 2010
Abstract: We study evolution equations of the form:
@u
(t;x) =m(x)(4u)(t;x) t2R ; x2
;+
@t
Nwhere
is a bounded domain in R and the function m :
! (0;1) is assumed
to be measurable. Dirichlet boundary conditions are posed. We investigate under
which conditions on m and @
the operator m4 generates a strongly continuous
semigroup on C ( ). In the second part of the thesis we obtain various estimates0
pon the kernel of the semigroup generated by m4 on weighted L -spaces.Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1 Preliminaries 9
1.1 Sesquilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Associated operator, fundamentals of the semigroup theory . . . . . 11
1.3 Holomorphic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Beurling-Deny conditions . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Interpolation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7 Sobolev embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.8 Spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.9 Regularized distance function . . . . . . . . . . . . . . . . . . . . . . 29
1.10 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Elliptic operators on weighted spaces 35
2.1 Forms on a weighted space . . . . . . . . . . . . . . . . . . . . . . . . 35
12.2 Criteria for L -contractivity of elliptic operators . . . . . . . . . . . 41
2.3 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 The operator m4 - Introduction and generation on C ( ) 470
3.1 The operator m4 - intro and de nition . . . . . . . . . . . . 47
3.2 The operator m4 on C ( ) . . . . . . . . . . . . . . . . . . . . . . 510 0
3.3 Regular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Points of weak di usion . . . . . . . . . . . . . . . . . . . . . . . . . 56
mAt3.5 Generation theorem for e on C ( ) . . . . . . . . . . . . . . . . . 600
3.6 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Kernel estimates I - Ultracontractivity 63
4.1 Dunford-Pettis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Kernel representation . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Characterisations of Ultracontractivity . . . . . . . . . . . . . . . . . 67
4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
m4t5 Pseudo-Gaussian estimates for e 81
5.1 The twisted form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Ultracontractivity for the twisted form . . . . . . . . . . . . . . . . . 82
5.3 Gaussian estimates for m4 . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
34 Contents
6 Intrinsic ultracontractivity 85
6.1 Intrinsic ultracony - motivation . . . . . . . . . . . . . . . . 85
6.2 Positivity of the kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3 Intrinsic ultracontractivity . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 Rosen’s criterion for intrinsic ultracontractivity . . . . . . . . . . . . 92
m4t6.5 Intrinsic ultracontractivity for e . . . . . . . . . . . . . . . . . . 94
6.6 Applications of intrinsic ultracontractivity . . . . . . . . . . . . . . . 97
6.7 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
p7 L -maximal regularity for m4 101
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 R-boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.3 Maximal ergodic estimate for contractive semigroups . . . . . . . . . 104
p7.4 Maximal regularity for contractive semigroups on L -spaces . . . . . 110
7.5 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
A Appendix 115
A.1 Sesquilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.2 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
1A.3 Sobolev space H ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.4 Laplace operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Notation 5
Notation
C ( ) continuous functions with compact support in
c
N@
the boundary of an open set
R
N
the closure of
inR
N!
! is an open subset ofR such that !

^N maxfN; 2g
1 1D( ) = C ( ) test functions ( C functions with compact support)c
D( ) fv2D( ) : v 0g+
0D( ) the space of all distributions (continuous functionals onD( ))
@ thD = (weak) derivative with respect to the j coordinatej @xj
1 2 2H ( ) u2L ( ) : D u2L ( ) ; j = 1;:::;d -the rst Sobolev spacej
1 1H ( ) the closure of D( ) in H ( )0 R p pL ( ) u :
!R measurable: ju(x)j dx<1 whenever !
loc !
1 2 2H ( ) u2L ( ) : D u2L ( ) for j = 1;:::;d :jloc loc loc
C ( ) u2C( ) : u = 00 j@

characteristic function of a set AA
a^b minfa;bg
a_b maxfa;bg
fz2C : z = 0;j argzj<g

fz2C : z = 0;< argz<g
Ate the semigroup generated by an (unbounded) operator A
1R( ;A ) : = ( A) , the resolvent of A
666 Introduction
Introduction
In this work, the main object of investigation is the equation
@u
(t;x) =m(x)(4u)(t;x) t2R ; x2
; (1)+
@t
Nwhere
is a bounded domain (open connected set) inR and Dirichlet boundary
conditions are posed. The function m :
! (0;1) is assumed to be measurable
Nand the Laplace operator is understood to operate on functions of x2 R . It is
the desire to understand deeper the interplay of growth (or decay) properties of m
and properties of the kernels of semigroups generated by m4 on various function
spaces that motivates our work in this thesis.
There are two main themes that are elaborated. Firstly, it is the question
whether the operatorm4 (after a proper de nition) generates a strongly continuous
semigroup on C ( ), the space of continuous functions vanishing at the boundary0
of . More precisely, we study conditions on
and m that guarantee the existence
of such a semigroup. There are good reasons for studying the operator on the space
C ( ). One reason is that one obtains a Feller semigroup in this way with the0
corresponding relations to stochastic processes (see [27], [30], [32] and [66] for the
role ofC ( ) in the theory of Markov processes). Another reason concerns possible0
applications to non-linear problems and dynamical systems. For semilinear prob-
plems the space C ( ) is much better suited than L ( )-spaces since composition0
with a locally Lipschitz continuous function is locally Lipschitz continuous onC ( )0
pbut never onL ( ) unless the function is already globally Lipschitz continuous, see
the treatise of Cazenave-Haraux [18], for example. Studying arbitrary measurable
functions m seems to be useful for possible applications to quasilinear equations.
The second main theme of the work are kernel estimates for the semigroup on
1pweighted L -spaces (with the weight ). Here we rst give a condition on the
m
2functionm so that a bounded kernel for the semigroup (on the weightedL ) exists.
After demonstrating by an example that this condition is optimal we proceed to
re ne the estimates for bounded kernels. This re nement is of twofold nature -
2jx yj
c t rstly we prove estimates where a Gaussian factor e is incorporated and
secondly we obtain upper and lower kernel estimates depending on the behaviour
of the rst eigenfunction of the operator.
A few words of explanation concerning the title of this work should be said
before we describe the content of the chapters. We are motivated by the fact that
the evolution equation
u (t;x) = (4 u)(t;x) (+f(t;x)) (2)t x
is used in various models to describe di usion. If one changes the operator on the
right-hand side of (2) by a multiplicative factor m(x) one obtains the equation
u (t;x) =m(x)(4 u)(t;x): (3)t x
From the probabilistic point of view this perturbation results in the change of time
in the underlying Markov process (see [44], [45] and the references there). The
change is governed by the behaviour ofm andm may, in general, blow up or vanish
fast at the boundary of . So much to the word degenerate.
The operator on the right-hand side of (3) is a particular kind of a general second
order elliptic operator in non-divergence form having merely a principal part
@u
a (x) : (4)ij
@x @xi jIntroduction 7
Here, the coe cients a take a special forma =m(x) fori = 1;:::;N anda = 0ij ii ij
for i = j. The word isotropically refers to the fact that the coe cients do not
depend on a particular direction.
Note that the operator m4 is elliptic (in the terminology of [34]) since we assume
thatm(x)> 0, but we do not assume strict ellipticity (this would require m(x)>"
for some "> 0).
We comment on the content of the chapters.
We start in Chapter 1 by introducing all relevant notions and theorems which
are needed in the main body of the work. We try to keep the text as self-contained
as possible and thus prove some of the results even in this introductory chapter. In
particular, since the theory of submarkovian semigroups provides a most natural
setting for the operators analysed in this thesis we treat in detail the Beurling-Deny
criteria for the generation of such semigroups.
Similarly, we devote a whole section to the notion of a regularised distance func-
tion and prove in detail its basic properties. These are of decisive importance at
various places in the work; rstly when developing a local theory for points of weak
di usion in Section 3.4. It reappears later con rming optimality of the ultracontrac-
tivity result in Theorem 4.4.1 and once more when proving intrinsictrac-
tive estimates in Chapter 6. On the other hand, interpolation theorems, Sobolev
embeddings and the spectral theorem are given without proof here. There exists
abundant high-quality literature concerning all these topics. It is listed in Notes
and Comments to this chapter.
Our own work starts in Chapter 2. Here we introduce general forms acting on
pweighted L -spaces and prove basic properties of the sesquilinear forms in order to
be able to obtain an associated operator - the generator of a strongly continuous
semigroup. Furthermore, since we are interested in obtaining a submarkovian semi-
group, we study conditions which have to be posed on coe cients of the form in
1order to make the associated operator L -contractive.
The next four chapters constitute the core of the thesis.
In Chapter 3 we give a precise meaning to the operator m4 and develop both
global and local theory of regular points for this operator. We stress here that the
results of this chapter depend strongly on the fact that the operator in question is
isotropic. Our results show in particular that for m larger than a positive constant
maxfN;2g1 q(even 2L ( ) ; q > su ces) the regular points of m4 are the same as
m 2
for4. For general (non-isotropic) elliptic operators in non-divergence form this is
no longer true in both directions. In fact Miller [49] showed that there may be reg-
ular points for the Laplacian which are non-regular for a particular elliptic operator
and vice versa.
The theory of this chapter culminates in the Theorem 3.5.1 where we develop a
local for the generation on C ( ). In order to obtain a strongly continuous0
semigroup generated bym4 onC ( ) we require each point in @
to ful l (at least)0
one of the conditions: the point should be regular for the Laplace equation or the
di usion should be weak enough in the neighbourhood of the point (see Sections
3.3 and 3.4 for precise de nitions).
In Chapter 4 the existence of a bounded kernel for the semigroup on the weighted
spaces is investigated. We prove the abstract Dunford-Pettis theorem and reformu-
late the question in terms of the ultracontractivity of the semigroup. We con-
tinue by a positive result for the operator m4 with a perturbing function m s.t.
^1 Nq2L ( ;dx) for some q > . We show the optimality of the result by consider-
m 2
68 Introduction
2ing the operator 4, where is a regularised distance function. The semigroup
2 dxassociated to this operator is not ultracontractive on L ( ; ). We nish the2(x)
chapter by listing various consequences of ultracontractivity. In particular, we ob-
tain a representation of the kernel in terms of a series containing denumerably many
eigenfunctions of the operator.
In Chapter 5 we re ne the ultracontractive estimates of the previous chapter
and incorporate a Gaussian factor. This can be used to prove the existence of a
1holomorphic extension for the semigroup on the weighted L -space. However, the
strong continuity of the semigroup at zero is not guaranteed. Further possible con-
sequences include the investigation of degenerate operators on unbounded domains.
Although certainly interesting and useful, these topics lie beyond the scope of the
present thesis and must be left for the future work.
Chapter 6 re nes the ultracontractive estimates of Chapter 4 in a di erent way.
Namely, since we work with Dirichlet boundary conditions throughout this thesis
one expects the kernel of the semigroup to vanish at the boundary. One may also
ask how fast the convergence is once it takes place. For a particular class of opera-
tors we show that the behaviour of the kernel at the boundary is controlled by the
rst eigenfunction. It is also interesting to note that upper kernel estimates of this
form automatically imply corresponding lower ones.
Here we also show the following result interesting on its own. If the perturbing
function m is strictly positive on a bounded Dirichlet regular domain
and if
^1 Nq2L ( ;dx) for someq> then the rst eigenfunction of the operator m4 (on
m 2
dxpL ( ; )) is also strictly positive at all points of .
m(x)
pChapter 7 concludes the thesis. Here we prove L -maximal regularity for real-
pisations of the operator m4 on weighted L -spaces. We deduce our result from a
pmore general theorem guaranteeingL -maximal regularity for generators of positive
pcontractions onL -spaces. This result on its own is based on an estimate of ergodic
type for such generators.
At the end of the introduction we would like to express our acknowledgement.
I would like to thank Wolfgang Arendt, without whom the whole project might
have never started. His expertise was admirable as were his acute remarks and
suggestions to the preliminary versions of the work. For all this and much more,
I express my sincere gratitude. At this point I would also like to thank Jaroslav
Milota for leading me to the realm of higher mathematics.Chapter 1
Preliminaries
In this introductory chapter we collect the most important results and methods
which will be needed in the sequel.
1.1 Sesquilinear forms
We introduce basic terminology concerning form techniques. Note that throughout
this work we could do only with quadratic forms in most of the text, the only place
where we need sesquilinear forms in general is in Chapter 5 when working with
twisted forms.
LetH be a Hilbert space, denote byK eitherR orC and leta(;) be a sesquilinear
form i.e. a mapping de ned on a linear subspace D(a) of H satisfying
a(; ) :D(a)D(a)! K
a(u +v;w) =a(u;w) +a(v;w)
a(u;v +w) =a(u;v) +a(u;w)
for all 2 K and u;v;w 2 D(a). We de ne the adjoint form of a to be the
sesquilinear form a :
a (u;v) :=a(v;u) with the domain D(a ) :=D(a):
We call a symmetric if a =a.
We say that a is accretive if Re a(u;u) 0 for all u2D(a). If a is accretive we
may de ne a scalar product on D(a) in the following way:
1
hu;vi := [a(u;v) +a (u;v)] +hu;vi 8u;v2D(a):a
2
Equipped with this scalar product D(a) becomes a pre-Hilbert space. The expres-
q
2
sionk k := Re a(u;u) +kuk then de nes a norm on D(a). We say thata
a is densely de ned ifD(a) is dense in H.
a is continuous if it is accretive and there exists M 0 such that
ja(u;v)jMkuk kvk 8u;v2D(a): (1.1)a a
a is closed if a is accretive and (D(a);k k ) is a complete space.a
910 Preliminaries
We say that a linear subspaceDD(a) is a core ofa ifD is dense in (D(a);k k ).a
We note that instead of the norm k k we could also use an equivalent normaq
2
kuk := Re a(u;u) +wkuk for anyw> 0. Analogously for the scalar producta;w
h ; i .a
Any sesquilinear form a may be written as a sum of two symmetric forms. This is
easily accomplished as follows. De ne the forms a and a by1 2
1
a : = (a +a );1
2
1
a : = (a a )2
2i
withD(a ) :=D(a) =:D(a ). Then we have1 2
a =a +ia1 2
and also
a (u;u) = Re a(u;u) for any u2D(a):1
The form a is called the real part of a. Similarly, a is called the imaginary1 2
part of a. One may check continuity of a given sesquilinear form by verifying the
(sectoriality) assumption of the next proposition. The result is known as Schwarz’s
inequality.
Proposition 1.1.1 Leta be a sesqulinear form on a Hilbert spaceH. If there exists
a constant c such that
jIm a(u;u)jc Re a(u;u); 8u2D(a);
then the inequality
p p
ja(u;v)j (c + 1) Re a(u;u) Re a(v;v) (1.2)
is valid for all u;v2D(a).
Proof. Choose u;v2D(a) arbitrarily. Without loss of generality we assume that
i a(u;v) 2 R since we may replace u by e u, 2 (0; 2) without a ecting the
inequality (1.2). We have
1
a(u;v) = (a(u +v;u +v) a(u v;u v))
4
and thus by the assumption and the parallelogram identity (for the form a ) we1
obtain
c + 1
ja(u;v)j (Re a(u +v;u +v) + Re a(u v;u v))
4
c + 1
= (Re a(u;u) + Re a(v;v)):
2
1Using the last inequality for u and v with an arbitrary > 0 we have


c + 1 12
ja(u;v)j Re a(u;u) + Re a(v;v) :
22
1
2Rea(v;v)2In case Re a(u;u) = 0 we put := and get the result. If Re a(u;u) =
Rea(u;u)
0 we let tend to1 and obtain a(u;v) = 0. Thus (1.2) holds in both cases.
6