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Rigidity results with applications to best constants and symmetry of Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities

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Abstract

We take advantage of a rigidity result for the equation satisfied by an extremal function associated with a special case of the Caffarelli-Kohn-Nirenberg inequalities to get a symmetry result for a larger set of inequalities. The main ingredient is a reparametrization of the solutions to the Euler-Lagrange equations and estimates based on the rigidity result. The symmetry results cover a range of parameters which go well beyond the one that can be achieved by symmetrization methods or comparison techniques so far.

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Acknowledgments

J. D. thanks S. F. and A. T. for welcoming him in Heraklion. J. D. and M. J. E. have been supported by the ANR project NoNAP. J. D. has also been supported by the ANR projects STAB and Kibord.

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Authors and Affiliations

  1. CEREMADE, UMR n° 7534, CNRS & Université Paris-Dauphine, Paris, France

    Jean Dolbeault & Maria J. Esteban

  2. Department of Mathematics and Applied Mathematics, University of Crete, University Campus, Voutes, 700 13, Heraklion, Crete, Greece

    Stathis Filippas & Achilles Tertikas

  3. Institute of Applied and Computational Mathematics, FORTH, 711 10, Heraklion, Crete, Greece

    Stathis Filippas & Achilles Tertikas

Authors
  1. Jean Dolbeault
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  2. Maria J. Esteban
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  3. Stathis Filippas
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  4. Achilles Tertikas
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Corresponding author

Correspondence to Maria J. Esteban.

Additional information

Communicated by A. Malchiodi.

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Dolbeault, J., Esteban, M.J., Filippas, S. et al. Rigidity results with applications to best constants and symmetry of Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities. Calc. Var. 54, 2465–2481 (2015). https://doi.org/10.1007/s00526-015-0871-9

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  • DOI: https://doi.org/10.1007/s00526-015-0871-9

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