Propagation of Singularities for Fuchsian Operators
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984 Antonio Sove Jeff E. Lewis Cesare Parenti
Propagation of Singularities for Fuchsian Operators
Spri nger-Verlag Berlin Heidelberg New York Tokyo 1983
Authors Antonio Bove Department of Mathematics, University of Trento 38050 Povo (Trento), Italy Jeff E. Lewis Department of Mathematics, University of Illinois at Chicago P.O.Box 4348, Chicago, IL 60680, USA Cesare Parenti Department of Mathematics, University of Bologna Piazza di Porta S. Donato, 5, 40127 Bologna, Italy
AMS Subject Classifications (l980): 58G16, 58G17, 35L40 ISBN 3-540-12285-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12285-0 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
TABLE OF CONTENTS
Introduction
1
1. Preliminaries and Review of Results of N. Hanges
10
2. General Fuchsian Systems
24
3. Applications to Fuchsian Hyperbolic P.D.E.
96
4. Operators with Multiple Non-Involutive Characteristics
133
5.
References
158
6.
Subject Index
161
ACKNOWLEDGEMENTS One of us (J.E.L.) would like to thank the Italian Research Council for supporting his staying at the University of Bologna during the completion of this work. A.B. and C.P. were partially supported by the C.N.R., gruppo G.N.A.F.A. The authors would like to thank Mrs S. Serra and Mrs. M. Stettermajer for their excellent typing work.
INTRODUCTION The main purpose of this monograph is the study of Fuchsian systems of the form
(0.1 )
Pu
(td I - A(t,x,D ,D »)u(t,x) tNt x
where
A
is an
order
o
defined on
describing
matrix of classical pseudodifferential operators (p d 0) of
n+l lR
n
lR x lR t x
More precisely, we are interested in
00
,
where
WF(v)
denotes the wave front set of the distribution
L. H6rmander [14] N
we put
WF(v)
jLJ1
WF (V
(for a vector-valued distribution
v
as
v=
j» .
It is well known that the structure of the set the characteristics of the operator
WF(u) 'WF(Pu) c
(0.2)
r
C -singularities of the solutions of system (0.1) i.e. the set
WF(u) , WF(Pu) defined in
Nx N
f(t,x)
WF(u) 'WF (Pu)
depends on
P, Le.
{ (t,x,T,i',;)
* n+l ' 0 ET lR
Itt
o}
Char P
Near a point i',;0
to,
a complete description of
WF (u) 'WF (Pu)
follows from the general results
2 of J.J. Duistermaat - L. Hormander [10]; in particular
WF(u) 'WF(Pu)
under the action of the hamiltonian vector fields
a at
H
r
and
H
is invariant respecti-
t
vely. Therefore we concentrate our analysis of
WF (u) ,WF (Pu)
near the points
of the two following disjoint subsets
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