Improved Poincaré inequalities

Volzone, Bruno; Dolbeault, Jean

Volzone, Bruno; Dolbeault, Jean (2012), Improved Poincaré inequalities, Nonlinear Analysis: Theory, Methods & Applications, 75, 16, p. 5985–6001. http://dx.doi.org/10.1016/j.na.2012.05.008

Type

Article accepté pour publication ou publié

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http://hal.archives-ouvertes.fr/hal-00638281/fr/

Date

2012

Nom de la revue

Nonlinear Analysis: Theory, Methods & Applications

Volume

Numéro

Éditeur

Elsevier

Pages

5985–6001

Résumé (EN)

Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be build, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to identify the optimal constants in the asymptotic expansion, order by order. We also apply the same strategy to a family of Hardy-Poincaré inequalities which interpolate between Hardy and gaussian Poincaré inequalities.

Mots-clés

Weighted norms; Remainder terms; Best constant; Poincaré inequality; Hardy inequality