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

Sujets

• Mathematics

Informations

• Publié par: universitat_ulm
• Publié le: 01 janvier 2010
• Nombre de lectures: 28
• Langue: English

Extrait

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