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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