Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Abstract

Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every n ≥ 3 and 1 < p ≤ n+2 n−2 , there exists a best constant S n,p,λ (B n ) > 0 such that, where (n−1) 2 4is the bottom of the L 2 -spectrum of −∆ B n . It is known from the results of Mancini and Sandeep [56] that under appropriate assumptions on n, p and λ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant S n,p,λ (B n ). In this article we investigate the quantitative gradient stability of the above inequality and the associated Euler-Lagrange equation locally around a bubble. In our first result, we prove a sharp quantitative stability of the above Poincaré-Sobolev inequality: if u ∈ H 1 (B n ) almost optimizes the above inequality then u is close to the manifold of optimizers in a quantitative way. Secondly, we prove the quantitative stability of its critical points: if u ∈ H 1 (B n ) almost solves the Euler-Lagrange equation corresponding to the above Poincaré-Sobolev inequality and the energy of u is close to the energy of an extremizer, then we derive the following quantitative bound dist (u, Z) ≤ C(n, p, λ), where Z denotes the manifold of non-negative finite energy solutions of −∆ B n w−λw = |w| p−1 w. Our result generalizes the sharp quantitative stability of Sobolev inequality in R n of Bianchi-Egnell [13] and to the Poincaré-Sobolev inequality on the hyperbolic space.Furthermore, combining our stability results and implementing a refined smoothing estimates, we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity.