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Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

Bonforte, Matteo; Dolbeault, Jean; Muratori, Matteo; Nazaret, Bruno (2017), Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities, Kinetic & Related Models, 10, 1, p. 33-59. 10.3934/krm.2017002

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Type
Article accepté pour publication ou publié
Date
2017
Journal name
Kinetic & Related Models
Volume
10
Number
1
Publisher
American Institute of Mathematical Sciences
Pages
33-59
Publication identifier
10.3934/krm.2017002
Metadata
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Author(s)
Bonforte, Matteo
Dolbeault, Jean cc
Muratori, Matteo
Nazaret, Bruno
Abstract (EN)
In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincaré inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.
Subjects / Keywords
symmetry breaking; symmetry; linearization; functional inequalities; flows; Interpolation; fast diffusion equation; semilinear elliptic equations; Caffarelli-Kohn-Nirenberg inequalities; weights; optimal functions; best constants; spectrum; Hardy-Poincaré inequality; spectral gap; entropy methods

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