On a variant of Korn's inequality arising in statistical mechanics
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 603-619.

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.

DOI : 10.1051/cocv:2002036
Classification : 49J40, 82C40, 76P05
Mots clés : Korn's inequality, Boltzmann equation, Monge-Kantorovich mass transportation problem 
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     title = {On a variant of {Korn's} inequality arising in statistical mechanics},
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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://archive.numdam.org/articles/10.1051/cocv:2002036/

[1] P.G. Ciarlet, Mathematical elasticity. Vol. I. Three-dimensional elasticity. Vol. II: Theory of plates. Vol. III: Theory of shells. North-Holland Publishing Co., Amsterdam (1988, 1997, 2000). | MR | Zbl

[2] D. Cioranescu, O.A. Oleinik and G. Tronel, On Korn's inequalities for frame type structures and junctions. C. R. Acad. Sci. Paris Sér. I Math. 309 (1989) 591-596. | Zbl

[3] L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and BGK equations. Arch. Rational Mech. Anal. 110 (1990) 73-91. | MR | Zbl

[4] L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The Boltzmann equation. Work in progress.

[5] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics. Springer-Verlag, Berlin (1976). Translated from the French by C.W. John, Grundlehren der Mathematischen Wissenschaften, 219. | MR | Zbl

[6] K.O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality. Ann. Math. 48 (1947) 441-471. | Zbl

[7] S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry, Second Edition. Springer-Verlag, Berlin (1990). | MR | Zbl

[8] J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité. Bull. Soc. Roy. Sci. Liège 31 (1962) 182-191. | Zbl

[9] H. Grad, On Boltzmann’s H-theorem. J. Soc. Indust. Appl. Math. 13 (1965) 259-277.

[10] C.O. Horgan, Korn's inequalities and their applications in continuum mechanics. SIAM Rev. 37 (1995) 491-511. | Zbl

[11] C.O. Horgan and L.E. Payne, On inequalities of Korn, Friedrichs and Babuska-Aziz. Arch. Rational Mech. Anal. 82 (1983) 165-179. | Zbl

[12] R.V. Kohn, New integral estimates for deformations in terms of their nonlinear strains. Arch. Rational Mech. Anal. 78 (1982) 131-172. | MR | Zbl

[13] A. Korn, Solution générale du problème d'équilibre dans la théorie de l'élasticité, dans le cas où les effets sont donnés à la surface. Ann. Fac. Sci. Univ. Toulouse 10 (1908) 165-269. | JFM | Numdam

[14] J.A. Nitsche, On Korn's second inequality. RAIRO: Anal. Numér. 15 (1981) 237-248. | Numdam | Zbl

[15] V.A. Kondratiev and O.A. Oleinik, On Korn's inequalities. C. R. Acad. Sci. Paris Sér. I Math. 308 (1989) 483-487. | Zbl

[16] E.I. Ryzhak, Korn's constant for a parallelepiped with a free face or pair of faces. Math. Mech. Solids 4 (1999) 35-55. | Zbl

[17] C. Villani, Topics in mass transportation. Preprint (2002).

[18] Y. Shizuta and K. Asano, Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Japan Acad. Ser. A Math. Sci. 53 (1977) 3-5. | MR | Zbl

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