  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.
