The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type
Partial differential equations of mixed elliptic-hyperbolic type arise in diverse areas of physics and geometry, including fluid and plasma dynamics, optics, cosmology, traffic engineering, projective geometry, geometric variational theory, and the theory
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2043
Thomas H. Otway
The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type
123
Thomas H. Otway Department of Mathematical Sciences Yeshiva University New York USA
ISBN 978-3-642-24414-8 e-ISBN 978-3-642-24415-5 DOI 10.1007/978-3-642-24415-5 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011943496 Mathematics Subject Classification (2010): 35-XX, 35M10 c Springer-Verlag Berlin Heidelberg 2012 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Partial differential equations of mixed elliptic–hyperbolic type arise in diverse areas of physics and geometry, including fluid and plasma dynamics, optics, cosmology, traffic engineering, projective geometry, geometric variational theory, and the theory of isometric embeddings. And yet even the linear theory of these equations is at a very early stage. This course examines various Dirichlet problems which can be formulated for equations of Keldysh type, one of the two main classes of linear elliptic–hyperbolic equations. Open boundary conditions (in which data are prescribed on only part of the boundary) and closed boundary conditions (in which data are prescribed on the entire boundary) are both considered. Emphasis is on the formulation of boundary conditions for which solutions can be shown to exist in an appropriate function space. Specific applications to plasma physics, optics, and analysis on projective spaces are discussed. These notes were written to supplement a series of ten lectures given at Henan University in the summer of 2010. They are intended for graduate students and researchers in pure or applied analysis. In particular, the reader is expected to have a background in functional analysis – including Sobolev spaces – and the basic theory of partial differential equations, but not necessarily any prior expertise in the theory of mixed elliptic–hyperbolic equations. A familiarity with the geometry of differential forms is assumed in Sect. 5.6, but that mater
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