Sharp Lp-entropy inequalities on manifolds

In 2003, Del Pino and Dolbeault [14] and Gentil [19] investigated, independently, best constants and extremals associated to Euclidean Lp-entropy inequalities for p > 1. In this work, we present some contributions in the Riemannian context. Namely, let (M,g) be a closed Riemannian manifold of dimension n >= 3. For 1 < p <= 2, we establish the validity of the sharp Riemannian Lp-entropy inequality int_M |u|^p log(|u|^p) dv_g <= n/p log ( A_{opt} int_M |Grad_g u|^p dv_g + B ) on all functions u em H^{1,p}(M) such that ||u||_{Lp(M)} = 1 for some constant B. Moreover, we prove that the first best constant A_{opt} is equal to the corresponding Euclidean one. Our approach is inspired on the Bakry, Coulhon, Ledoux and Sallof-Coste's idea [3] of getting Euclidean entropy inequalities as a limit case of suitable subcritical interpolation inequalities. It is conjectured that the inequality sometimes fails for p > 2.