Heat semigroups and diffusion of characteristic functions [Elektronische Ressource] / vorgelegt von Marc Preunkert

Sujets

• Mathematics

Informations

• Publié par: eberhard_karls_universitat_tubingen
• Publié le: 01 janvier 2006
• Nombre de lectures: 18
• Langue: English

Extrait

Heat Semigroups and Diffusion of
Characteristic Functions
Dissertation
der Fakult˜at fur˜ Mathematik und Physik
der Eberhard-Karls-Universit˜at Tubingen˜
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
vorgelegt von
Marc Preunkert
aus Gerabronn
2006Tag der mundlic˜ hen Qualiflkation: 8. Juni 2006
Dekan: Prof. Dr. Peter Schmid
1. Berichterstatter: Prof. Dr. Rainer Nagel
2. Berich Prof. Dr. Gerhard HuiskenContents
Introduction iii
1 Heat difiusion and the isoperimetric inequality 1
1.1 The heat semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Symmetric rearrangements, the Riesz-Sobolev inequality and
2an L -difiusion inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Functions of bounded variation, perimeter and measure theoretic notions 8
21.4 From the L -difiusion inequality to the isoperimetric inequality . . . . . 9
n1.5 Geometry of hypersurfaces inR . . . . . . . . . . . . . . . . . . . . . 11
2 Difiusion of characteristic functions: Short time behaviour 15
2.1 Asymptotic expansion for the evolution of level sets . . . . . . . . . . . 16
2.2 Heat difiusion into the complement . . . . . . . . . . . . . . . . . . . . 35
3 Difiusion of characteristic functions: Long time behaviour 45
3.1 Heat difiusion for ring domains . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Heat for arbitrary compact sets . . . . . . . . . . . . . . . . . 50
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 Final considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Bibliography 65
Zusammenfassung in deutscher Sprache 67
Lebenslauf 69
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Introduction
nWe start from a uniform distribution of heat in a compact subset D ofR represented
2 nby the characteristic function of D. Then the heat semigroup (T(t)) on L (R ),D t‚0
applied to , gives the unique solution u(x;t)=T(t) (x) of the heat equationD D
‰
d nu(x;t) = ¢u(x;t); x2R ;t‚0;
dt(HE)
u(x;0) = (x)D
nonR for all times t‚0 with initial data .D
2This heat ow in particular induces an evolution of the corresponding L -norms
t 7! kT(t) k 2; t‚0: (1)D L
If we adopt the notation
Z
hf;gi:= f(x)¢g(x)dx
nR
2 n 1 n 1 nboth for the inner product on L (R ) and for the duality pairing hL (R );L (R )i,
then by the semigroup property
T(t+s) =T(t)T(s) ; s;t‚0D D
and the self-adjointness of the operators T(t) we obtain the following alternative de-
scription for the evolution (1):
t t t 2hT(t) ; i=hT( ) ;T( ) i= kT( ) k ; t‚0: (2)D D D D D 2L2 2 2
nSince onR no heat is lost under difiusion, this also yields
t 2hT(t) ; ci =jDj¡hT(t) ; i= jDj¡kT( ) k ; t‚0: (3)2D D D D D2 L
Observe that, by
Z Z
chT(t) ; i= T(t) (x)dx and hT(t) ; i= T(t) (x)dx;D D D D D D
cD D
(2) describes the amount of heat that is, at time t, still inside the set D, while
c(3)es the heat that has owed into the complement D . In this sense the
2evolutionoftheL -norm(1)directlyre ectshowgoodtheset D keepstheheatinside.
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M. Ledoux [Led94] discovered that there are interesting connections between the heat
ow into the complement (3) and the perimeter P(D) of D, i.e., area measure of the
boundary @D in the sense of geometric measure theory. In particular he proved that
2the L -inequality
kT(t) k 2 •kT(t) k 2; t‚0; (4)D L B L
for a Euclidean ball B and a second compact set D of the same volume implies the
isoperimetric inequality
P(D)‚P(B);
i.e., the Euclidean ball has the smallest perimeter under all compact sets of the same
volume.
2This is in fact a very interesting conclusion since the L -inequality (4) can be derived
neasily from the Riesz-Sobolev inequality for symmetric rearrangements in R , a fact
that, remarkably, was not realized by Ledoux.
In this thesis we present a further systematic treatment of the connections between
properties of the heat ow (1)-(3) and the geometry of D. Our main results concern
the short time behaviour as well as large time phenomena.
In the flrst chapter we give a brief summary of the analytic and geometric concepts
and results that will be used in the following. We introduce the heat semigroup on
nR and its main properties. We recall the concept of symmetric rearrangements in
nR , the Riesz-Sobolev inequality and present the interesting, but rather unknown,
connections between the heat semigroup and rearrangement inequalities. These then lead to a proof of the isoperimetric inequality. Further, we brie y
present the necessary background on perimeters, relevant geometric measure theory
nand the basic notions of the geometry of smooth hypersurfaces inR .
In Chapter two we focus on the short time behaviour of the ow t 7! T(t) . WeD
start with a detailed treatment of the evolution of the level sets of T(t) and deter-D
mine the pointwise asymptotic behaviour for this evolution: We show that for short
1=2times the evolution of the level sets admits an asymptotic expansion in powers of t .
2We determine the coe–cients up to order t in terms of geometric invariants of the
boundary @D and give a general formula for the further coe–cients of higher order.
We then show that the short time behaviour of the ow (1)-(3) is controlled by the
perimeter of D. We prove this flrst for a compact set with smooth boundary using the
resultsobtainedfortheevolutionofthelevelsets, andthenforaCaccioppolisetusing
measure theoretic arguments. These results generalise what Ledoux [Led94] proved
for Euclidean balls.
Asaconsequenceweobtainacomparisonresultstatingthatfortwoarbitrarycompact
nsets A;D ‰ R of the same volume the one with smaller perimeter keeps for small
times the heat better than the other - a fact that corresponds to the isoperimetric
character of inequality (4) but now allows to compare two arbitrary compact sets of
the same volume and not only the Euclidean ball and a second set.
ivIn the third chapter we concentrate on large time phenomena of the ow (1)-(3). In
particular, we study the analogue of the question treated at the end of Chapter 2:
nGiven two compact sets A;D ‰R of the same volume. Which one keeps the heat
better for large times? We again prove a comparison theorem stating now that this
holdsfortheonewhichhasthesmallersecondcentralmoment. Thisagaincompatibly
correspondstotheinequality(4)sincetheEuclideanballminimizesthesecondcentral
moment under all sets of a given volume.
In addition we give further criteria on the fourth central moments and on the tensors
of inertia of A and D yielding an answer in case the second central moments are
equal.
We conclude with considerations on the question how much geometry of D is already
determined if we know the ow (1)-(3) on a (maybe small) time interval.
Acknowledgements. First of all I thank Prof. Rainer Nagel for his continuous and
strong encouragement and for his permanent interest in the progress of my work.
I am also deeply indepted to Prof. Gerhard Huisken who has spent much time with
mediscussingtopicsofthisthesisandwhoalwayshadanopenearforallmyquestions.
During the preparation of this work I had the privilege to meet many friends and
colleagues who showed great interest in my research. In particular I thank
Prof. Mark Ashbaugh, Prof. Giovanni Bellettini, Stefano Cardanobile, Prof. Thierry
Coulhon, Dr. Abdelhadi Es-Sarhir, Peter G˜ogelein, Prof. Jerry Goldstein, Prof.
Giorgio Metafune, Dr. Michele Miranda, Prof. Diego Pallara, Dr. Fabio Paronetto,
Prof. Abdelaziz Rhandi, Prof. Ulf Schlotterbeck, and Dr. Felix Schulze
for stimulating and helpful discussions, comments and hints.
Manyoftheideasbehindthisthesiswerebornduringhighlyproductiveandmotivating
stays at the University of Lecce (Prof. Giorgio Metafune and Prof. Diego Pallara), at
the Max Planck Institute for Gravitational Physics, Golm (Prof. Gerhard Huisken),
and at the University of Memphis (Profi. Jerry Goldstein and Gisµele Ruiz Goldstein).
I deeply thank for these invitations, the ofiered hospitality and all the help.
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Chapter 1
Heat difiusion and the
isoperimetric inequality
In this chapter we give a short summary of the notions and results we frequently need
inthefollowing. Weintroducetheheatsemigroup(Section1.1), symmetricrearrange-
mentsofsetsandfunctions,theRiesz-Sobolevinequality(Section1.2),perimeter,some
measure theoretic background (Section 1.3), connections between the heat semigroup
and the isoperimetric inequality (Section 1.4), and flnally the basic notions of the
ngeometry of smooth hypersurfaces inR (Section 1.5).
1.1 The heat semigroup
nWe consider the heat equation (HE) inR
‰
d nu(x;t) = ¢u(x;t); x2R ;t‚0;
dt(HE)
u(x;0) = (x);D
2 n 2 nwhere the Laplace operator ¢: D(¢)‰L (R )!L (R ) is given by
n 2X @
¢u(x;t) = u(x ;:::;x ;t) with domain1 n2@xii=1
2;2 nD(¢) = W (R );
2 n nand where 2L (R ) denotes the characteristic function of a compact set D‰R .D
2;2 nUsing, e.g., the Fourier transform we obtain that the Laplace operator (¢;W (R ))
generates a strongly continuous, analytic semigroup (T(t)) of linear operators, thet‚0
2 nheat semigroup, on L (R ). This semigroup is given