Advertisement

Communications in Mathematical Physics

, Volume 68, Issue 3, pp 209–243 | Cite as

Symmetry and related properties via the maximum principle

  • B. Gidas
  • Wei-Ming Ni
  • L. Nirenberg
Article

Abstract

We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting plane. We treat solutions in bounded domains and in the entire space.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Maximum Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Corrigan, F., Fairlie, D.: Scalar field theory and exact solutions in a classicalSU(2) gauge theory. Phys. Lett.67B, 69 (1977)Google Scholar
  2. 2.
    Courant, R., Hilbert, D.: Methods of mathematical physics. Interscience-Wiley, Vol. II (1962)Google Scholar
  3. 3.
    Hopf, H.: Lectures on differential geometry in the large. Stanford University, 1956Google Scholar
  4. 4.
    Jackiw, R., Rebbi, C.: Conformal properties of pseudoparticle configurations. Phys. Rev. D16, 1052 (1976)Google Scholar
  5. 5.
    Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal and projective transformations. In: Contributions to Analysis, pp. 245–272. Academic Press 1974Google Scholar
  6. 6.
    Obata, M.: The conjectures on conformal transformations of Riemannian manifolds. J. Diff. Geom.6, 247–258 (1971)Google Scholar
  7. 7.
    Protter, M., Weinberger, H.: Maximum principles in differential equations. Prentice-Hall 1967Google Scholar
  8. 8.
    Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech.43, 304–318 (1971)Google Scholar
  9. 9.
    Wilczek, F.: Geometry and interactions of instantons. In: Quark confinement and field theory. Stump, D., Weingarten, D. (eds.). New York: Wiley 1977Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • B. Gidas
    • 1
  • Wei-Ming Ni
    • 2
  • L. Nirenberg
    • 2
  1. 1.Rockefeller UniversityUSA
  2. 2.Courant Institute of Math. Sci.New York UniversityNew YorkUSA

Personalised recommendations