Abstract:
we prove general optimal euclidean sobolev and gagliardo-nirenberg inequalities by using mass transportation and convex analysis results. explicit extremals and the computation of some optimal constants are also provided. in particular we extend the optimal gagliardo-nirenberg inequality proved by del pino and dolbeault 2003 and the optimal inequalities proved by cordero-erausquin et al. 2004.

Abstract:
We prove general optimal euclidean Sobolev and Gagliardo-Nirenberg inequalities by using mass transportation and convex analysis results. Explicit extremals and the computation of some optimal constants are also provided. In particular we extend the optimal Gagliardo-Nirenberg inequality proved by Del Pino and Dolbeault 2003 and the optimal inequalities proved by Cordero-Erausquin et al. 2004.

Abstract:
Let (M,g) be a compact Riemannian manifold of dimension n \geq 2. In this work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg inequality for 1 < p \leq 2. Our proof relies strongly on a new distance lemma which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities due to Del Pino and Dolbeault and the optimal L^2-Riemannian Gagliardo-Nirenberg inequality due to Broutteland in a unified framework.

Abstract:
We discuss the existence, nonexistence and uniqueness of positive viscosity solutions for Lane-Emden systems involving the fractional Laplacian on bounded domains. As a byproduct, we obtain the critical hyperbole associated to the these systems.

Abstract:
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.

Abstract:
A classical open problem involving the Laplace operator on symmetric domains in Rn is whether all its Dirichlet eigenvalues vary smoothly upon one-parameter C1 perturbations of the domain. We provide a fairly complete answer to this question in dimension n = 2 on disks and squares and also for the second eigenvalue on balls in Rn for any n >= 3. Our approach bases on a suitable setting of the problem and uses an appropriate degenerate version of the implicit function theorem on Banach spaces as central tool.

Abstract:
\small{In 2004, Del Pino and Dolbeault \cite{DPDo} and Gentil \cite{G} investigated, independently, best constants and extremals associated to sharp Euclidean $L^p$-entropy inequalities. In this work, we present some important advances in the Riemannian context. Namely, let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 3$. For $1 < p \leq 2$, we prove that the sharp Riemannian $L^p$-entropy inequality \[\int_M |u|^p \log(|u|^p) dv_g \leq \frac{n}{p} \log ({\cal A}_{opt} \int_M |\nabla u|_g^p dv_g + {\cal B}) \] \n holds on all functions $u \in H^{1,p}(M)$ such that $||u||_{L^p(M)} = 1$. Moreover, we show that the first best Riemannian constant ${\cal A}_{opt}$ is equal to the corresponding Euclidean one. Our approach is inspired on the Bakry, Coulhon, Ledoux and Sallof-Coste's idea \cite{Ba} of getting Euclidean entropy inequalities as a limit case of suitable Gagliardo-Nirenberg inequalities. It is conjectured that the above inequality sometimes fails for $p > 2$.}

Abstract:
In this note we prove that any compact Riemannian manifold of dimension $ngeq 4$ which is non-conformal to the standard n-sphere and has positive Yamabe invariant admits infinitely many conformal metrics with nonconstant positive scalar curvature on which the classical sharp Sobolev inequalities admit extremal functions. In particular we show the existence of compact Riemannian manifolds with nonconstant positive scalar curvature for which extremal functions exist. Our proof is simple and bases on results of the best constants theory and Yamabe problem.

Abstract:
We concerns here with the continuity on the geometry of the second Riemannian L^p-Sobolev best constant B_0(p,g) associated to the AB program. Precisely, for 1 <= p <= 2, we prove that B_0(p,g) depends continuously on g in the C^2-topology. Moreover, this topology is sharp for p = 2. From this discussion, we deduce some existence and C^0-compactness results on extremal functions.