Weighted interpolation inequalities: a perturbation approach

Dolbeault, Jean; Muratori, Matteo; Nazaret, Bruno

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

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