Date
2017
Dewey
Analyse
Sujet
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
Journal issue
Kinetic & Related Models
Volume
10
Number
1
Publication date
2017
Article pages
33-59
Publisher
American Institute of Mathematical Sciences
Author
Bonforte, Matteo
Dolbeault, Jean
Muratori, Matteo
Nazaret, Bruno
Type
Article accepté pour publication ou publié
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.