On Korn’s constant for thin cylindrical domains

Abstract

We consider an ε-parametrized collection of cylinders of cross section εω, where $\omega \subset {ℝ}^2$ , and of fixed length ℓ. By Korn’s inequality, there exists a positive constant K^ε such that ${\int }_{{\Omega }^{\epsilon }}|\text{sym}\nabla \mathit{u}{|}^2{\text{d}}^3\mathit{x}\ge K^{\epsilon }{\int }_{{\Omega }^{\epsilon }}|\nabla \mathit{u}{|}^2{\text{d}}^3\mathit{x}$ provided that $\mathit{u}\in H^1(\Omega ;{ℝ}^3)$ satisfies a condition that rules out infinitesimal rotations. We show that K^ε∕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.