The asymptotically sharp Korn interpolation and second inequalities for shells

TL;DR

This paper establishes optimal asymptotic Korn inequalities for shells with bounded curvatures, linking Korn's interpolation and second inequalities to Poincaré estimates, without boundary conditions.

Contribution

It introduces asymptotically sharp Korn interpolation and second inequalities for curved shells, with optimal constants and no boundary restrictions.

Findings

Constants are optimal in shell thickness asymptotics.

Korn inequalities reduce to Poincaré estimates for shells.

Results apply to linear geometric rigidity estimates without boundary conditions.

Abstract

We consider shells in three dimensional Euclidean space which have bounded principal curvatures. We prove Korn's interpolation (or the so called first and a half\footnote{The inequality first introduced in [6]}) and second inequalities on that kind of shells for $\Bu\in W^{1,2}$ vector fields, imposing no boundary or normalization conditions on $\Bu.$ The constants in the estimates are optimal in terms of the asymptotics in the shell thickness $h,$ having the scalings $h$ or $O(1).$ The Korn interpolation inequality reduces the problem of deriving any linear Korn type estimate for shells to simply proving a Poincar\'e type estimate with the symmetrized gradient on the right hand side. In particular this applies to linear geometric rigidity estimates for shells, i.e., Korn's fist inequality without boundary conditions.