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On a variant of Korn's inequality arising in statistical mechanics

Published online by Cambridge University Press:  15 August 2002

L. Desvillettes  and
Cédric Villani
L. Desvillettes
Affiliation:
Centre de Mathématiques et Leurs Applications, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, 94235 Cachan, France; desville@cmla.ens-cachan.fr.
Cédric Villani
Affiliation:
UMPA, École Normale Supérieure de Lyon, 69364 Lyon Cedex 07, France; cvillani@umpa.ens-lyon.fr.
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Abstract

We state and prove a Korn-like inequality for a vector field in a bounded open set of $\mathbb{R}^N$, satisfying a tangency boundary condition. This inequality, which is crucial in our study of the trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case are briefly discussed.

Keywords

Korn's inequalityBoltzmann equationMonge–Kantorovich mass transportation problem.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 603 - 619
Copyright
© EDP Sciences, SMAI, 2002

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