Topological Methods for Variational Problems with Symmetries
Symmetry has a strong impact on the number and shape of solutions to variational problems. This has been observed, for instance, in the search for periodic solutions of Hamiltonian systems or of the nonlinear wave equation; when one is interested in ellip
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ziirich F. Takens, Groningen
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Thomas Bartsch
Topological Methods for Variational Problems with Symmetries
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Author Thomas Bartsch Mathematisches Institut Universitat Heidelberg Im Neuenheimer Feld 288 D-69120 Heidelberg, Germany
Mathematics Subject Classification (1991): 34Cxx, 35120, 47H15, 49R05, 55M30, 55P91,58Exx, 58Fxx ISBN 3-540-57378-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57378-X 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
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meinen Eltern
Preface Symmetry has a strong impact on the number and shape of solutions to variational problems. This can be observed, for instance, when one looks for periodic solutions of autonomous differential equations and exploits the invariance under time shifts or when one is interested in elliptic equations on symmetric domains and wants to find special solutions. Two topological methods have been devised in order to find critical points that are not minima or maxima of variational integrals: the theories of LusternikSchnirelmann and of Morse. In these notes we want to present recent techniques and results in critical point theory for functionals invariant under a symmetry group. We develop LusternikSchnirelmann (minimax) theory and a generalization of Morse theory, the MorseConley theory, in some detail. Both theories are based on topological notions: the LusternikSchnirelmann category and geometrical index theories like the genus on the one hand and the Conley index (generalizing the Morse index) on the other hand. These notions belong to the realm of homotopy theory with a typical consequence: They are easy to define but difficult to compute. We present a variety of new computations of the category where very general classes of symmetry groups are involved, and we give examples showing that our results cannot be extended further without serious restrictions. In order to do this we prove new generalizations of the BorsukUlam theorem and give counterexamples to more general versions. It is here that we need to use some algebraic topology, namely cohomology theory, and an equivariant version of the cuplength. A variation of the equivariant cuplength, the "length", turns out to be very useful
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