An introduction to maximum principles and symmetry in elliptic problems
著者
書誌事項
An introduction to maximum principles and symmetry in elliptic problems
（Cambridge tracts in mathematics, 128）
Cambridge University Press, 2000
- タイトル別名
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Introduction to maximum principles and symmetry in elliptic problems
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注記
Includes bibliographical references (p. 332-336) and index
内容説明・目次
内容説明
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.
目次
- Preface
- 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
- References
- Index.
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