The asymptotically sharp Korn interpolation and second inequalities for shells

摘要

We consider shells in three-dimensional Euclidean space that have bounded principal curvatures. We prove Korn's interpolation (or the so-called first and a half) and the second inequalities on that kind of shells foru∈W1,2vector fields, imposing no boundary or normalization conditions onu. The constants in the estimates are optimal in terms of the asymptotics in the shell thicknessh, having the scalingshorO(1). The Korn interpolation inequality reduces the problem of deriving any linear Korn type estimate for shells to simply proving a Poincaré-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.