Conical Refraction and Higher Microlocalization
The main topic of the book is higher analytic microlocalization and its application to problems of propagation of singularities. The part on higher microlocalization could serve as an introduction to the subject. The results on propagation refer to soluti
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Otto Liess
Conical Refraction and Higher Microlocalization
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Otto Liess Department of Mathematics University of Bologna Piazza Porta di San Donato 5 Bologna, Italy
Mathematics Subject Classification (1991): 35-02, 35A27, 58G 17 ISBN 3-540-57105-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57105-1 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany
214613140-543210 - Printed on acid-free paper
Preface These notes focus on some results concerning propagation of analytic microlocal singularities for solutions of partial differential equations with characteristics of variable multiplicity, and on the tools from the theory of higher involutive microlocalization needed in the proofs. The simplest model to which the results apply is Maxwell's system for homogeneous anisotropic optical media (typical examples of which are crystals); then the underlying physical phenomenon is that of conical refraction. The main difficulty in the study of operators with characteristics of variable multiplicity stems from the fact that the characteristic variety of such operators is not smooth. Indeed, near a singular point, a number of constructions usually performed in the study of the propagation of singularities will degenerate or break down. In the analytic category, these difficulties can be best investigated from the point of view of higher microlocalization. Unfortunately, none of the theories on higher analytic microlocalization in use nowadays completely covers the situation that we encounter later in the notes. Rather than adapting or extending the existing theories to the present needs, we have chosen to build up a new theory from a uniform point of view. Actually the results on higher microlocalization are sufficiently well delimited from the other results of the text and could in principle be read independently of the rest. Special emphasis is put, on the other hand, upon the relation and interplay between the results on propagation of microlocal singularities and similar results and constructions in geometrical optics. All microlocalization processes seem to follow some underlying common pattern. Therefore some overlap with other articles and books on higher microlocalization has been inevitable. It is also clear in this situation that we have been greatly influenced by the
