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

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.