Korn and Poincaré-Korn inequalities for thin domains in $GSB{D}^p$

We prove Korn’s first inequalities with and without boundary conditions in $GSB{D}^p$ ($1<p<\mathrm{\infty }$) in $C^{1,1}$-regular thin domains with Lipschitz thickness boundaries with a constant $C{h}^{-p},$ that is optimal if the domain mid-surface contains a flat region, e.g., if the domain contains a piece of plate. For the one with boundary condition, we impose zero Dirichlet boundary conditions on the thin lateral faces of the shell, where we prove that the same estimate holds with no subtracted rigid motion like in the classical case for Sobolev vector fields.