On the Korn Interpolation and Second Inequalities in Thin Domains
Davit Harutyunyan · SIAM Journal on Mathematical Analysis · 2018
We consider shells of nonconstant thickness in three dimensional Euclidean space around surfaces which have bounded principal curvatures. We derive Korn's interpolation inequality (or the so-called first (and a half) inequality introduced in [Y. Grabovsky and D. Harutyunyan, SIAM J. Math. Anal., 46 (2014), pp. 3277--3295]) and Korn's second inequality on such domains for $\bm{u}\in H^1$ vector fields, imposing no boundary or normalization conditions on $\bm{u}$. The constants in the estimates are asymptotically optimal in terms of the domain thickness $h$, with the leading order constant having the scaling $h$ as $h\to0$. This is the first work that determines the asymptotics of the optimal constant in the classical Korn second inequality for shells in terms of the domain thickness in almost full generality, the inequality being fulfilled for practically all thin domains $\Omega\in\mathbb{R}^3$ and all vector fields $\bm{u}\in H^1(\Omega)$. Moreover, Korn's interpolation inequality is stronger than Korn's second inequality, and it reduces the problem of estimating the gradient $ abla\bm{u}$ in terms of the symmetrized gradient $e(\bm{u})$, in particular, any linear geometric rigidity estimates for thin domains, to the easier problem of proving the corresponding Poincaré-like estimates on the field $\bm{u}$ itself.