Marcos Montenegro

Best constants in second-order Sobolev inequalities on Riemannian manifolds and applications

Abstract Let (M,g) be a smooth compact Riemannian manifold, with or without boundary, of dimension n⩾3 and 1 ‖u‖= ‖ Δ g u‖ L p (M) p +‖u‖ L p (M) p 1/p on each of the spaces H2,p(M), H02,p(M) and H2,p(M)∩H01,p(M), we study an asymptotically sharp inequality associated to the critical Sobolev embedding of these spaces. As an application, we investigate the influence of the geometry in the existence of solutions for some fourth-order problems involving critical exponents on manifolds. In particular, new phenomena arise in Brezis–Nirenberg type problems on manifolds with positive scalar curvature somewhere, in contrast with the Euclidean case. We also show that on such manifolds the corresponding optimal inequality for p=2 is not valid.

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.

The effect of diffusion on critical quasilinear elliptic problems

We discuss the role of the diffusion coefficient

General optimal euclidean Sobolev and Gagliardo-Nirenberg inequalities

a(x)

Uniform bounds of minimizers of non-smooth constrained functionals on maps spaces

on the existence of a positive solution for the quasilinear elliptic problem involving critical exponent

Optimal Lp-Riemannian Gagliardo–Nirenberg inequalities

\cases - \text{div}( a(x) |\nabla u|^{p-2} \nabla u) = u^{p^* - 1} + \lambda u^{p-1} & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega,\ \endcases

On nontrivial solutions of critical polyharmonic elliptic systems

where

Sharp affine Sobolev type inequalities via the Lp Busemann–Petty centroid inequality

\Omega