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On the Symmetry of Minimizers

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Abstract

For a large class of variational problems we prove that minimizers are symmetric whenever they are C 1.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berestycki H., Lions P.-L.: Nonlinear scalar field equations, I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–345 (1983)

    MATH  Google Scholar 

  4. Brock F.: Positivity and radial symmetry of solutions to some variational problems in R N. J. Math. Anal. Appl. 296, 226–243 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Chen, Y.-Z., Wu, L.-C.: Second Order Elliptic Equations and Elliptic Systems. Translations of Mathematical Monographs, vol.174. AMS, Providence, 1998

  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)

    MATH  Google Scholar 

  7. Giaquinta M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)

    MATH  Google Scholar 

  8. Giaquinta M.: Introduction to the Regularity Theory for Nonlinear Elliptic Systems. Birkhäuser Verlag, Basel (1993)

    MATH  Google Scholar 

  9. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, New York (1983)

    Book  MATH  Google Scholar 

  10. Ladyzhenskaya O.A., Ural’tseva N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    MATH  Google Scholar 

  11. Lopes O.: Radial symmetry of minimizers for some translation and rotation invariant functionals. J. Differ. Equ. 124, 378–388 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Lopes O.: Radial and nonradial minimizers for some radially symmetric functionals. Eletr. J. Differ. Equ. 3, 1–14 (1996)

    MATH  Google Scholar 

  13. Lopes O., Mariş M.: Symmetry of minimizers for some nonlocal variational problems. J. Funct. Anal. 254(2), 535–592 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs. AMS, Providence, 1997

  15. Pacella F., Weth T.: Symmetry of solutions to semilinear elliptic equations via Morse index. Proc. Am. Math. Soc. 135(6), 1753–1762 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Smets D., Willem M.: Partial symmetry and asymptotic behavior for some elliptic variational problems. Calc. Var. PDE 18, 57–75 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Spanier, E.H.: Algebraic Topology. McGraw-Hill, New York, 1966

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Authors and Affiliations

  1. Département de Mathématiques UMR 6623, Université de Franche-Comté, 16, Route de Gray, 25030, Besançon, France

    Mihai Mariş

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  1. Mihai Mariş
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Correspondence to Mihai Mariş.

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Communicated by P. Rabinowitz

Dedicated to Dorel Miheţ, for his teaching, his friendship, and the inspiration he gave to me.

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Mariş, M. On the Symmetry of Minimizers. Arch Rational Mech Anal 192, 311–330 (2009). https://doi.org/10.1007/s00205-008-0136-2

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  • DOI: https://doi.org/10.1007/s00205-008-0136-2

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