An iterative approach to operators on manifolds with singularities [Elektronische Ressource] / von Jamil Abed

Jamil Abed

Sujets

• Mathematics

Informations

• Publié par: universitat_potsdam
• Publié le: 01 janvier 2010
• Nombre de lectures: 16
• Langue: English

Extrait

Institut für Mathematik
aus der Arbeitsgruppe
„Partielle Differentialgleichungen und Komplexe Analysis“
an der Universität Potsdam




An Iterative Approach to Operators
on Manifolds with Singularities









Dissertation
zur Erlangung des akademischen Grades
"doctor rerum naturalium"
(Dr. rer. nat.)
in der Wissenschaftsdisziplin "Mathematische Analysis"








eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultät
der Universität Potsdam




von
Jamil Abed





Potsdam, den 1. März 2010 This work is licensed under a Creative Commons License:
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Published online at the
Institutional Repository of the University of Potsdam:
URL http://opus.kobv.de/ubp/volltexte/2010/4475/
URN urn:nbn:de:kobv:517-opus-44757
http://nbn-resolving.org/urn:nbn:de:kobv:517-opus-44757 Contents
Introduction iii
Acknowledgment xi
1 The pseudo-differential cone calculus 1
1.1 Basics in pseudo-differential operators . . . . . . . . . . . . . . . . . . 1
1.1.1 Spaces of symbols . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Pseudo-differential operators and distributional kernels . . . . . 4
1.1.3 Kernel cut-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Elements of the calculus . . . . . . . . . . . . . . . . . . . . . . 7
1.1.5 Continuity in Sobolev spaces . . . . . . . . . . . . . . . . . . . 9
1.1.6 Ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.7 Mellin pseudo-differential operators . . . . . . . . . . . . . . . . 13
1.2 Operators on a manifold with conical exits to infinity . . . . . . . . . . 17
1.2.1 Manifolds with conical exits to infinity . . . . . . . . . . . . . . 17
1.2.2 Calculus in the Euclidean space . . . . . . . . . . . . . . . . . . 19
1.2.3 Invariance under push forwards . . . . . . . . . . . . . . . . . . 24
1.2.4 Classical symbols and operators with exit property . . . . . . . 30
1.2.5 Exit calculus on manifolds . . . . . . . . . . . . . . . . . . . . . 33
2 Operators on infinite cylinders 35
2.1 The behaviour of push forwards from cylinders to cones . . . . . . . . 35
2.1.1 Characterisation of push forwards . . . . . . . . . . . . . . . . 35
2.1.2 Estimates near the diagonal . . . . . . . . . . . . . . . . . . . . 43
iii CONTENTS
2.1.3 Global operators . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2 A new parameter-dependent calculus on infinite cylinders . . . . . . . 49
2.2.1 Operator-valued symbols with parameter . . . . . . . . . . . . 49
2.2.2 Continuity in Schwartz spaces . . . . . . . . . . . . . . . . . . . 53
2.2.3 Leibniz products and remainder estimates . . . . . . . . . . . . 56
2.3 Parameter-dependent operators on an infinite cylinder . . . . . . . . . 70
2.3.1 Weighted cylindrical spaces . . . . . . . . . . . . . . . . . . . . 70
2.3.2 Elements of the calculus . . . . . . . . . . . . . . . . . . . . . . 71
3 Axiomatic approach with corner-degenerate symbols 75
3.1 Symbols associated with order reductions . . . . . . . . . . . . . . . . 75
3.1.1 Scales and order reducing families . . . . . . . . . . . . . . . . 75
3.1.2 Symbols based on order reductions . . . . . . . . . . . . . . . . 80
3.1.3 An example from the parameter-dependent cone calculus . . . 84
3.2 Operators referring to a corner point . . . . . . . . . . . . . . . . . . . 97
3.2.1 Weighted spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2.2 Mellin quantisation and kernel cut-off . . . . . . . . . . . . . . 100
3.2.3 Meromorphic Mellin symbols and operators with asymptotics . 105
Bibliography 115
Index 118Introduction
Partial differential equations arise in various branches of mathematics, physics, and
engineering in a natural way. They describe a large variety of different situations.
Elliptic equations are an important subclass with applications in almost all areas of
mathematics, from harmonic analysis to geometry, as well as in numerous fields of
physics. The standard example of an elliptic equation is Laplace’s equation, Δu=0,
its solutions describe the behaviour of electric, gravitational, and fluid potentials,
and are therefore significant in many applications, especially in electromagnetism,
astronomy, and fluid dynamics. An important method to express the solutions of
elliptic partial differential equations is to extend the class of the operators to the
so-called pseudo-differential operators. A basic reference is the work of Kohn and
Nirenberg[15]wherepseudo-differentialoperatorshavebeenestablishedasacalculus,
see also H¨ormander [12], [11], Kumano-go [19], Shubin [46].
The analysis on manifolds with geometric singularities (such as conical points,
edges, or corners) is motivated by models of the applied sciences, especially of me-
chanics, elasticity theory, particle physics, and astronomy, as well as by pure mathe-
matics, such as geometry and topology. More information on the general role of the
singular analysis for models in mechanics may be found in [9]. The singularities can
arise either from the geometry of the underlying configuration or from the operator
itself. For example, the standard Laplacian in polar coordinates takes the form of a
singular operator, an example of a special class of differential operators, the so-called
Fuchs type operators.
The “traditional” analysis is based on adequate algebras of pseudo-differential oper-
ators that contain geometric differential operators, e.g., Laplacians, associated with
correspondingsingularRiemannianmetrics,togetherwiththeparametricesofelliptic
elements. This paper is aimed at studying pseudo-differential operators on configura-
tions with such singularities.
Our investigations are focused on new elements of the analysis on configurations
with higher singularities, especially on problems appearing on infinite cones which
require the development of pseudo-differential structures from the point of view of
conical exits to infinity. The new difficulty in the case of higher singularities comes
from singularities on cross sections of cones that generate non-compact edges going
to infinity with the new corner axis variable. To illustrate the idea, let us first con-
sider,forexample,theLaplacianonamanifoldwithconicalsingularities(say,without
boundary). In this case the ellipticity does not only refer to the “standard” princi-
pal homogeneous symbol but also to the so-called conormal symbol. The latter one,
iiiiv INTRODUCTION
contributed by the conical point, is operator-valued and singles out the weights in
Sobolev spaces, where the operator has the Fredholm property.
Another example of ellipticity with different principal symbolic components is the
case of boundary value problems. The boundary, say smooth, interpreted as an edge,
contributes the operator-valued boundary (or edge) symbol which is responsible for
the nature of boundary conditions (for instance, of Dirichlet or Neumann type in the
case of the Laplacian). In general, if the configuration has polyhedral singularities
of order k, we have to expect a principal symbolic hierarchy of length k + 1, with
components contributed by the various strata. In order to characterise the solvability
of elliptic equations, especially, the regularity of solutions in suitable scales of spaces,
it is natural to embed the problem in a pseudo-differential calculus, and to construct
a parametrix. For higher singularities this is a program of tremendous complexity.
It is therefore advisable to organise the general elements of the calculus by means
of an axiomatic framework which contains the typical features, such as the cone- or
edge-degenerate behaviour of symbols but ignores the (in general) huge tail of k−1
iterative steps to reach the singularity level k.
AtpresenttheanalysisofPDEsonmanifolds(or,moregenerally,stratifiedspaces)
withregularsingularitiesisanimportantresearchfieldwithmanyopenproblemsand
new challenges. Moreover, there are traditional aspects witha long history,motivated
by applications to models in physics and other sciences. Let us give some references
on crucial results and recent development of the calculus.
The“concrete”(pseudo-differential)calculusofoperatorsonmanifoldswithconicalor
edge singularities may be found in several papers and monographs, see, for instance,
[32], [36], [35], [5]. Operators on manifolds of singularity order 2 are studied in [37],
[41],[20],[7].Theoriesofthatkindarealsopossibleforboundaryvalueproblemswith
thetransmissionpropertyatthe(smoothpartofthe)boundary,see,forinstance,[31],
[14], [9]. This is useful in numerous applications, for instance, to models of elasticity
or crack theory, see [14], [10], [8]. Elements of operator structures on manifolds with
highersingularitiesaredeveloped,forinstance,in[40],[1].Thenatureofsuch