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Weighted interpolation inequalities: a perturbation approach

Dolbeault, Jean; Muratori, Matteo; Nazaret, Bruno (2017), Weighted interpolation inequalities: a perturbation approach, Mathematische Annalen, 369, 3-4, p. 1237-1270. 10.1007/s00208-016-1480-4

Type
Article accepté pour publication ou publié
External document link
http://arxiv.org/abs/1509.09127v1
Date
2017
Journal name
Mathematische Annalen
Volume
369
Number
3-4
Publisher
B. G. Teubner
Published in
Paris
Pages
1237-1270
Publication identifier
10.1007/s00208-016-1480-4
Metadata
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Author(s)
Dolbeault, Jean cc
Muratori, Matteo
Nazaret, Bruno
Abstract (EN)
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.
Subjects / Keywords
Functional inequalities; Weights; Optimal functions; Best constants; Symmetry; Concentration-compactness; Gamma-convergence

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