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Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities

Dolbeault, Jean; Esteban, Maria J.; Loss, Michael; Muratori, Matteo (2017), Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities, Comptes rendus. Mathématique, 355, 2, p. 133-154. 10.1016/j.crma.2017.01.004

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Type
Article accepté pour publication ou publié
Date
2017
Journal name
Comptes rendus. Mathématique
Volume
355
Number
2
Publisher
CNRS
Pages
133-154
Publication identifier
10.1016/j.crma.2017.01.004
Metadata
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Author(s)
Dolbeault, Jean cc
Esteban, Maria J. cc
Loss, Michael
Muratori, Matteo
Abstract (EN)
We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition which determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden-Fowler transformation have to be modified.
Subjects / Keywords
Emden-Fowler transformation; carré du champ; fast diffusion equation; flows; semilinear elliptic equations; symmetry breaking; symmetry; best constants; optimal functions; uniqueness; rigidity results; Functional inequalities; interpolation; Caffarelli-Kohn-Nirenberg inequalities; weights

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