Optimal L p -Riemannian Gagliardo–Nirenberg inequalities

Abstract

Let (M, g) be a compact Riemannian manifold of dimension n ≥ 2 and 1 < p ≤ 2. In this work we prove the validity of the optimal Gagliardo-Nirenberg inequalityfor a family of parameters r, q and θ. Our proof relies strongly on a new distance lemma which holds for 1 < p ≤ 2.In particular, we obtain Riemannian versions of L p -Euclidean Gagliardo-Nirenberg inequalities of [8] and extend the optimal L 2 -Riemannian Gagliardo-Nirenberg inequality of [5] in a unified framework.

Sharp Lp-entropy inequalities on manifolds

“…Therefore, lim inf k→+∞ ν k ≥ A 0 (p) −1 and so the limit (19) follows from (13). Let x k ∈ M be a maximum point of u k , that is…”

“…Let 1 < p ≤ 2 and 1 ≤ q < p. In [12], [13] and [21], Ceccon-Montenegro and Humbert established, respectively for 1 < p < 2 and p = 2, the existence of a constant B ∈ R such that the sharp L p -Nash inequality…”

Sharp constants in RiemannianLp-Gagliardo–Nirenberg inequalities

“…Thereby, we will focus our attention only on the case q > 1. By (8), ũ α satisfies the Euler-Lagrange equation…”

“…Recently, the Riemannian Gagliardo-Nirenberg optimal constants studied in [8] where applied by [22] to obtain global existence theorems for Zakharov system in T 2 . A particularly important family of applications of optimal Gagliardo-Nirenberg inequalities is the transition to optimal Entropy inequalities, in the spirit of [10,15].…”

Construction of blow-up solutions for Zakharov system on \( T^{2} \)

“…Corollary 2 can be derived by using the same argument as Glangetas and Merle with a sharp Gagliardo-Nirenberg inequality on T 2 by Ceccon and Montenegro [9]. However, for the proof of Theorem 2, it is not sufficient by itself to replace the sharp Gagliardo-Nirenberg inequality on R 2 with that on T 2 .…”