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Springer, Mathematische Annalen, 3-4(369), p. 1237-1270, 2016

DOI: 10.1007/s00208-016-1480-4

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Weighted interpolation inequalities: a perturbation approach

Journal article published in 2015 by Jean Dolbeault ORCID, Muratori Matteo, Bruno Nazaret
This paper is available in a repository.
This paper is available in a repository.

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Abstract

We study optimal functions in a family of Caffarelli-Kohn-Niren-berg inequalities with a power-law weight, in a regime for which standardsymmetrization techniques fail. We establish the existence of optimal func-tions, study their properties and prove that they are radialwhen the powerin the weight is small enough. Radial symmetry up to translations is truefor the limiting case where the weight vanishes, a case whichcorresponds toa well-known subfamily of Gagliardo-Nirenberg inequalities. Our approach isbased on a concentration-compactness analysis and on a perturbation methodwhich uses a spectral gap inequality. As a consequence, we prove that optimalfunctions are explicit and given by Barenblatt-type profiles in the perturbativeregime. ; non ; non ; recherche ; International