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arXiv: 1111.1516

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Nonlinear Analysis: Theory, Methods and Applications 75, 16 (2012) 5985 - 6001

Improved Poincaré inequalities

Jean Dolbeault 1, Bruno Volzone 2

(2012-12-31)

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:	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris IX - Paris Dauphine
2:	Dipartimento per le Tecnologie
	Università degli Studi di Napoli ''Parthenope''

Subject

:

Mathematics/Analysis of PDEs

Keyword(s): Hardy inequality – Poincaré inequality – Best constant – Remainder terms – Weighted norms

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From: Jean Dolbeault <>

Submitted on: Friday, 4 November 2011 14:39:51

Updated on: Wednesday, 5 December 2012 19:24:55