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Archive for Rational Mechanics and Analysis

, Volume 192, Issue 2, pp 311–330 | Cite as

On the Symmetry of Minimizers

  • Mihai MarişEmail author
Article

Abstract

For a large class of variational problems we prove that minimizers are symmetric whenever they are C1.

Keywords

Orthonormal Basis Variational Problem Lagrange Equation Radial Symmetry Quasilinear Elliptic Equation 

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References

  1. 1.
    Ball J.M., Mizel V.J.: One-dimensional variational problems whose minimizers do not satisfy the Euler–Lagrange equation. Arch. Ration. Mech. Anal. 90(4), 325–388 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bartsch T., Weth T., Willem M.: Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1–18 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berestycki H., Lions P.-L.: Nonlinear scalar field equations, I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–345 (1983)zbMATHGoogle Scholar
  4. 4.
    Brock F.: Positivity and radial symmetry of solutions to some variational problems in R N. J. Math. Anal. Appl. 296, 226–243 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Y.-Z., Wu, L.-C.: Second Order Elliptic Equations and Elliptic Systems. Translations of Mathematical Monographs, vol.174. AMS, Providence, 1998Google Scholar
  6. 6.
    Ferrero A., Gazzola F.: On subcriticality assumptions for the existence of ground states of quasilinear elliptic equations. Adv. Differ. Equ. 8(9), 1081–1106 (2003)zbMATHGoogle Scholar
  7. 7.
    Giaquinta M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)zbMATHGoogle Scholar
  8. 8.
    Giaquinta M.: Introduction to the Regularity Theory for Nonlinear Elliptic Systems. Birkhäuser Verlag, Basel (1993)zbMATHGoogle Scholar
  9. 9.
    Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ladyzhenskaya O.A., Ural’tseva N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)zbMATHGoogle Scholar
  11. 11.
    Lopes O.: Radial symmetry of minimizers for some translation and rotation invariant functionals. J. Differ. Equ. 124, 378–388 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lopes O.: Radial and nonradial minimizers for some radially symmetric functionals. Eletr. J. Differ. Equ. 3, 1–14 (1996)zbMATHGoogle Scholar
  13. 13.
    Lopes O., Mariş M.: Symmetry of minimizers for some nonlocal variational problems. J. Funct. Anal. 254(2), 535–592 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs. AMS, Providence, 1997Google Scholar
  15. 15.
    Pacella F., Weth T.: Symmetry of solutions to semilinear elliptic equations via Morse index. Proc. Am. Math. Soc. 135(6), 1753–1762 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pucci P., Serrin J., Zou H.: A strong maximum principle and a compact support principle for singular elliptic inequalities. J. Math. Pures Appl. 78, 769–789 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Smets D., Willem M.: Partial symmetry and asymptotic behavior for some elliptic variational problems. Calc. Var. PDE 18, 57–75 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Spanier, E.H.: Algebraic Topology. McGraw-Hill, New York, 1966Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Département de Mathématiques UMR 6623Université de Franche-ComtéBesançonFrance

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