On Korn’s constant for thin cylindrical domains

Abstract

We consider an ε-parametrized collection of cylinders of cross section εω, where [Formula: see text], and of fixed length ℓ. By Korn’s inequality, there exists a positive constant Kε such that [Formula: see text] provided that [Formula: see text] satisfies a condition that rules out infinitesimal rotations. We show that Kε∕ε2 converges to a strictly positive limit, and we characterize this limit in terms of certain parameters that depend on the geometry of ω and on ℓ.