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

Desvillettes, L.; Villani, Cédric

doi:10.1051/cocv:2002036

Desvillettes, L. ; Villani, Cédric

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 603-619.

Résumé

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.

Zbl 1092.82032 | 1 citation dans Numdam

DOI : https://doi.org/10.1051/cocv:2002036
Classification : 49J40, 82C40, 76P05
Mots clés : Korn's inequality, Boltzmann equation, Monge-Kantorovich mass transportation problem

@article{COCV_2002__8__603_0,
     author = {Desvillettes, Laurent and Villani, C\'edric},
     title = {On a variant of Korn's inequality arising in statistical mechanics},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {603--619},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002036},
     zbl = {1092.82032},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2002__8__603_0/}
}

Desvillettes, L.; Villani, Cédric. On a variant of Korn's inequality arising in statistical mechanics. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 603-619. doi : 10.1051/cocv:2002036. http://www.numdam.org/item/COCV_2002__8__603_0/

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