The sharp $L^p$ Korn interpolation and second inequalities in thin domains

In the present paper we extend the $L^2$ Korn interpolation and second inequalities in thin domains, proven in [\ref{bib:Harutyunyan.4}], to the space $L^p$ for any $1<p<\infty.$ A thin domain in space is roughly speaking a shell with non-constant thickness around a smooth enough two dimensional surface. The inequality that we prove in $L^p$ holds for practically any thin domain $\Omega\subset\mathbb R^3$ and any vector field $\Bu\in W^{1,p}(\Omega).$ The constants in the estimate are asymptotically optimal in terms of the domain thickness $h.$ This in particular solves the problem of finding the asymptotics of the optimal constant in the classical Korn second inequality in $L^p$ for thin domains in terms of the domain thickness in almost full generality. The remarkable fact is that the interpolation inequality reduces the problem of estimating the gradient $\nabla\Bu$ in terms of the strain $e(\Bu)$ to the easier problem of estimating only the vector field $\Bu$, which is a Korn-Poincar\'e inequality.