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Article Dans Une Revue Mathematische Annalen Année : 2017

Weighted interpolation inequalities: a perturbation approach

Résumé

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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Dates et versions

hal-01207009 , version 1 (29-09-2015)
hal-01207009 , version 2 (17-03-2016)
hal-01207009 , version 3 (17-09-2016)

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Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted interpolation inequalities: a perturbation approach. Mathematische Annalen, 2017, ⟨10.1007/s00208-016-1480-4⟩. ⟨hal-01207009v3⟩
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