We state and prove a Korn-like inequality for a vector field in a
bounded open set of
, 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.