Jurandir Ceccon

Sharp L p -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 compact Riemannian manifold of dimension n≥3. For 1<p≤2, we establish the validity of the sharp Riemannian Lp-entropy inequality ∫M|u|plog⁡(|u|p)dvg≤nplog⁡(Aopt∫M|∇gu|pdvg+B) on all functions u∈H1,p(M) such that ‖u‖Lp(M)=1 for some constant B. Moreover, we prove that the first best constant Aopt is equal to the corresponding Euclidean one. Our approach is inspired on the Bakry, Coulhon, Ledoux and Saloff-Costes 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.

Sharp constants in Riemannian Lp-Gagliardo–Nirenberg inequalities

Abstract Let ( M , g ) be a smooth compact Riemannian manifold of dimension n ≥ 2 , 1 p n and 1 ≤ q r p ⁎ = n p n − p be real parameters. This paper concerns the validity of the optimal Gagliardo–Nirenberg inequality ( ∫ M | u | r d v g ) τ r θ ≤ ( A opt ( ∫ M | ∇ g u | p d v g ) τ p + B opt ( ∫ M | u | p d v g ) τ p ) ( ∫ M | u | q d v g ) τ ( 1 − θ ) θ q . This kind of inequality is studied in Chen and Sun (2010) [12] where the authors established its validity when 2 p r p ⁎ and (implicitly) τ = 1 . Here we solve the case p ≥ r and introduce one more parameter 1 ≤ τ ≤ min ⁡ { p , 2 } . Moreover, we prove the existence of extremal functions for the optimal inequality above.

Equivalence of optimal L 1 -inequalities on Riemannian manifolds

Let

Equivalence of optimal L1-inequalities on Riemannian manifolds

(M,g)

Equivalence of optimal L1L1-inequalities on Riemannian manifolds

be a smooth compact Riemannian manifold of dimension

Optimal Lp-Riemannian Gagliardo–Nirenberg inequalities

n \geq 2

Optimal Riemannian Lp-Gagliardo–Nirenberg inequalities revisited

. This paper concerns to the validity of the optimal Riemannian

Extremals for sharp GNS inequalities on compact manifolds

L^1

Optimal L p -Riemannian GagliardoNirenberg inequalities

-Entropy inequality \[ {\bf Ent}_{dv_g}(u) \leq n \log \left(A_{opt} \|D u\|_{BV(M)} + B_{opt}\right) \] for all

Compactness results for divergence type nonlinear elliptic equations

u \in BV(M)