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An Introduction to Maximum Principles and Symmetry in Elliptic Problems

An Introduction to Maximum Principles and Symmetry in Elliptic Problems

Part of Cambridge Tracts in Mathematics

  • Date Published: April 2011
  • availability: Available
  • format: Paperback
  • isbn: 9780521172783


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About the Authors
  • Originally published in 2000, this was the first book to present the basic theory of the symmetry of solutions to second-order elliptic partial differential equations by means of the maximum principle. It proceeds from elementary facts about the linear case to recent results about positive solutions of non-linear elliptic equations. Gidas, Ni and Nirenberg, building on work of Alexandrov and of Serrin, have shown that the shape of the set on which such elliptic equations are solved has a strong effect on the form of positive solutions. In particular, if the equation and its boundary condition allow spherically symmetric solutions, then, remarkably, all positive solutions are spherically symmetric. Results are presented with minimal prerequisites in a style suited to graduate students. Two long and leisurely appendices give basic facts about the Laplace and Poisson equations. There is a plentiful supply of exercises, with detailed hints.

    • Was the first book to present the basic theory of symmetry by way of the maximum principle
    • Proofs spelled out in exceptional detail, and are elementary as can be
    • Exercises include results, made accessible by detailed hints
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    Reviews & endorsements

    Review of the hardback: 'The originality of this book mainly consists in new proofs and new extensions of quite well known results concerning maximum principles and symmetry properties related to semilinear elliptic equations.' Mathematical Reviews

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    Product details

    • Date Published: April 2011
    • format: Paperback
    • isbn: 9780521172783
    • length: 352 pages
    • dimensions: 229 x 152 x 20 mm
    • weight: 0.52kg
    • availability: Available
  • Table of Contents

    0. Some notation, terminology and basic calculus
    1. Introduction
    2. Some maximum principles for elliptic equations
    3. Symmetry for a non-linear Poisson equation
    4. Symmetry for the non-linear Poisson equation in RN
    5. Monotonicity of positive solutions in a bounded set O. Appendix A. On the Newtonian potential
    Appendix B. Rudimentary facts about harmonic functions and the Poisson equation
    Appendix C. Construction of the primary function of Siegel type
    Appendix D. On the divergence theorem and related matters
    Appendix E. The edge-point lemma
    Notes on sources

  • Author

    L. E. Fraenkel, University of Bath

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