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Weighted Korn and Poincaré-Korn inequalities in the Euclidean space and associated operators

Carrapatoso, Kleber; Dolbeault, Jean; Hérau, Frédéric; Mischler, Stéphane; Mouhot, Clément (2020), Weighted Korn and Poincaré-Korn inequalities in the Euclidean space and associated operators. https://basepub.dauphine.psl.eu/handle/123456789/22119

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CDHMM.pdf (417.1Kb)
Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-03059166
Date
2020
Series title
Cahier de recherche du CEREMADE
Published in
Paris
Pages
20
Metadata
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Author(s)
Carrapatoso, Kleber
Centre de Mathématiques Laurent Schwartz [CMLS]
Dolbeault, Jean cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Hérau, Frédéric cc
Laboratoire de Mathématiques Jean Leray [LMJL]
Mischler, Stéphane
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Mouhot, Clément
Department of Pure Mathematics and Mathematical Statistics [DPMMS]
Abstract (EN)
We prove functional inequalities on vector fields on the Euclidean space when it is equipped with a bounded measure that satisfies a Poincaré inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part and, in an improved form of the inequality, an additional term. We also consider Poincaré-Korn inequalities for estimating a projection of the vector field by the symmetric part of the differential matrix and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, on a bounded domain.
Subjects / Keywords
Korn inequality; weighted Poincaré inequality; Poincaré-Korn inequality; Lions' lemma; Witten-Laplace operator; Grad's number

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