Hardy inequalities on Riemannian manifolds and applications

D'Ambrosio, Lorenzo; Dipierro, Serena

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 449-475 / Harvested from Numdam

Résumé

We prove a simple sufficient criterion to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second order differential operator $Δ_{p} u : = div ({| \nabla u |}^{p - 2} \nabla u)$ . Namely, if ρ is a nonnegative weight such that $- Δ_{p} ρ ⩾ 0$ , then the Hardy inequality $c \int_{M} \frac{{| u |}^{p}}{ρ^{p}} {| \nabla ρ |}^{p} d v_{g} ⩽ \int_{M} {| \nabla u |}^{p} d v_{g}, u \in C_{0}^{\infty} (M),$ holds. We show concrete examples specializing the function ρ.Our approach allows to obtain a characterization of p-hyperbolic manifolds as well as other inequalities related to Caccioppoli inequalities, weighted Gagliardo–Nirenberg inequalities, uncertain principle and first order Caffarelli–Kohn–Nirenberg interpolation inequality.

Publié le : 2014-01-01

MR 3208450 | Zbl 1317.46022

DOI : https://doi.org/10.1016/j.anihpc.2013.04.004
Classification: 58J05, 31C12, 26D10

@article{AIHPC_2014__31_3_449_0,
     author = {D'Ambrosio, Lorenzo and Dipierro, Serena},
     title = {Hardy inequalities on Riemannian manifolds and applications},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {31},
     year = {2014},
     pages = {449-475},
     doi = {10.1016/j.anihpc.2013.04.004},
     mrnumber = {3208450},
     zbl = {1317.46022},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_3_449_0}
}

D'Ambrosio, Lorenzo; Dipierro, Serena. Hardy inequalities on Riemannian manifolds and applications. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 449-475. doi : 10.1016/j.anihpc.2013.04.004. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_3_449_0/

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