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Weighted interpolation inequalities: a perturbation approach

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Abstract

We study optimal functions in a family of Caffarelli–Kohn–Nirenberg inequalities with a power-law weight, in a regime for which standard symmetrization techniques fail. We establish the existence of optimal functions, study their properties and prove that they are radial when the power in the weight is small enough. Radial symmetry up to translations is true for the limiting case where the weight vanishes, a case which corresponds to a well-known subfamily of Gagliardo–Nirenberg inequalities. Our approach is based on a concentration-compactness analysis and on a perturbation method which uses a spectral gap inequality. As a consequence, we prove that optimal functions are explicit and given by Barenblatt-type profiles in the perturbative regime.

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Acknowledgments

J.D. has been supported by the ANR projects NoNAP, STAB and Kibord. B.N. has been supported by the ANR project STAB. M.M. thanks Alessandro Zilio for a very helpful discussion concerning the regularity theory for the Euler–Lagrange equation. M.M. has been partially funded by the “Università Italo-Francese/Université Franco-Italienne” (Bando Vinci 2013). The authors thank Guillaume Carlier for pointing them Ref. [1]. They thank the referees for their careful reading which helped them to improve the manuscript. © 2016 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes

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Authors and Affiliations

  1. Ceremade, CNRS UMR No. 7534, Université Paris-Dauphine, PSL*, Place de Lattre de Tassigny, 75775, Paris Cédex 16, France

    Jean Dolbeault

  2. Dipartimento di Matematica Francesco Brioschi, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Matteo Muratori

  3. SAMM, Université Paris 1, 90, rue de Tolbiac, 75634, Paris Cédex 13, France

    Bruno Nazaret

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  1. Jean Dolbeault
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Correspondence to Jean Dolbeault.

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Dolbeault, J., Muratori, M. & Nazaret, B. Weighted interpolation inequalities: a perturbation approach. Math. Ann. 369, 1237–1270 (2017). https://doi.org/10.1007/s00208-016-1480-4

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