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

L. Desvillettes; Cédric Villani

ESAIM: Control, Optimisation and Calculus of Variations (2002)

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

Abstract

top
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

top

Desvillettes, L., and Villani, Cédric. "On a variant of Korn’s inequality arising in statistical mechanics." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 603-619. <http://eudml.org/doc/244703>.

@article{Desvillettes2002,
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; Korn's inequality; Monge-Kantorovich mass transportation problem},
language = {eng},
pages = {603-619},
publisher = {EDP-Sciences},
title = {On a variant of Korn’s inequality arising in statistical mechanics},
url = {http://eudml.org/doc/244703},
volume = {8},
year = {2002},
}

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
PY - 2002
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; Korn's inequality; Monge-Kantorovich mass transportation problem
UR - http://eudml.org/doc/244703
ER -

References

top
  1. [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). Zbl0648.73014MR936420
  2. [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. Zbl0937.35502
  3. [3] L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and BGK equations. Arch. Rational Mech. Anal. 110 (1990) 73-91. Zbl0705.76070MR1031086
  4. [4] L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The Boltzmann equation. Work in progress. Zbl1029.82032
  5. [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. Zbl0331.35002MR521262
  6. [6] K.O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn’s inequality. Ann. Math. 48 (1947) 441-471. Zbl0029.17002
  7. [7] S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry, Second Edition. Springer-Verlag, Berlin (1990). Zbl0716.53001MR1083149
  8. [8] J. Gobert, Une inégalité fondamentale de la théorie de l’élasticité. Bull. Soc. Roy. Sci. Liège 31 (1962) 182-191. Zbl0112.38902
  9. [9] H. Grad, On Boltzmann’s H -theorem. J. Soc. Indust. Appl. Math. 13 (1965) 259-277. 
  10. [10] C.O. Horgan, Korn’s inequalities and their applications in continuum mechanics. SIAM Rev. 37 (1995) 491-511. Zbl0840.73010
  11. [11] C.O. Horgan and L.E. Payne, On inequalities of Korn, Friedrichs and Babuska–Aziz. Arch. Rational Mech. Anal. 82 (1983) 165-179. Zbl0512.73017
  12. [12] R.V. Kohn, New integral estimates for deformations in terms of their nonlinear strains. Arch. Rational Mech. Anal. 78 (1982) 131-172. Zbl0491.73023MR648942
  13. [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. Zbl39.0853.03JFM39.0853.03
  14. [14] J.A. Nitsche, On Korn’s second inequality. RAIRO: Anal. Numér. 15 (1981) 237-248. Zbl0467.35019
  15. [15] V.A. Kondratiev and O.A. Oleinik, On Korn’s inequalities. C. R. Acad. Sci. Paris Sér. I Math. 308 (1989) 483-487. Zbl0698.35067
  16. [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. Zbl1001.74560
  17. [17] C. Villani, Topics in mass transportation. Preprint (2002). 
  18. [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. Zbl0382.35047MR466988

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.