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

L. Desvillettes; Cédric Villani

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 603-619
  • ISSN: 1292-8119

Abstract

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We state and prove a Korn-like inequality for a vector field in a bounded open set of 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.

How to cite

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Desvillettes, L., and Villani, Cédric. "On a variant of Korn's inequality arising in statistical mechanics." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 603-619. <http://eudml.org/doc/90662>.

@article{Desvillettes2010,
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. },
author = {Desvillettes, L., Villani, Cédric},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Korn's inequality; Boltzmann equation; Monge–Kantorovich mass transportation problem.; Monge-Kantorovich mass transportation problem},
language = {eng},
month = {3},
pages = {603-619},
publisher = {EDP Sciences},
title = {On a variant of Korn's inequality arising in statistical mechanics},
url = {http://eudml.org/doc/90662},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Desvillettes, L.
AU - Villani, Cédric
TI - On a variant of Korn's inequality arising in statistical mechanics
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 603
EP - 619
AB - 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.
LA - eng
KW - Korn's inequality; Boltzmann equation; Monge–Kantorovich mass transportation problem.; Monge-Kantorovich mass transportation problem
UR - http://eudml.org/doc/90662
ER -

References

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  16. E.I. Ryzhak, Korn's constant for a parallelepiped with a free face or pair of faces. Math. Mech. Solids4 (1999) 35-55.  
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