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Detailed view Article in peer-reviewed journal
Nonlinear Analysis: Theory, Methods and Applications 75, 16 (2012) 5985 - 6001
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Improved Poincaré inequalities
Jean Dolbeault1, Bruno Volzone2

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.
1:  CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2:  Dipartimento per le Tecnologie
Hardy inequality – Poincaré inequality – Best constant – Remainder terms – Weighted norms