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A geometric lower bound on Grad’s number. (English) Zbl 1167.49040
Summary: We provide a new geometric lower bound on the so-called Grad’s number of a domain $$\Omega$$ in terms of how far $$\Omega$$ is from being axisymmetric. Such an estimate is important in the study of the trend to equilibrium for the Boltzmann equation for dilute gases.
##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 49J40 Variational inequalities 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. The Clarendon Press, Oxford University Press, New York (2000). Zbl0957.49001 MR1857292 · Zbl 0957.49001 [2] L. Desvillettes and C. Villani, On a variant of Korn’s inequality arising in statistical mechanics. ESAIM: COCV 8 (2002) 603-619. Zbl1092.82032 MR1932965 · Zbl 1092.82032 · doi:10.1051/cocv:2002036 · numdam:COCV_2002__8__603_0 · eudml:244703 [3] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159 (2005) 245-316. Zbl1162.82316 MR2116276 · Zbl 1162.82316 · doi:10.1007/s00222-004-0389-9 [4] A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities. Preprint (2007). · Zbl 1196.49033 [5] C. Villani, Hypocoercivity. Memoirs Amer. Math. Soc. (to appear). Zblpre05657453 MR2562709 · Zbl 1197.35004 [6] W.P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics 120. Springer-Verlag, New York (1989). Zbl0692.46022 MR1014685 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
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