S\right\};$$ and $$V=\left\{
v=(v^1;v^2;v^3)\in [H^1({{\cal O}})]^3\, :\;
 {\rm div}\, v=0,\;
v=0~\mbox{on}~ S
  \right\}.$$ We equip $X$ with $L_2$-type norm $\|\cdot\|_{{\cal O}}$ and denote by $(\cdot,\cdot)_{{\cal O}}$ the corresponding inner product. The space $V$ is endowed with the norm $\|\cdot\|_V= \|\nabla\cdot\|_{{\cal O}}$. For some details concerning this type spaces we refer to [@temam-NS], for instance.

We also need the Sobolev spaces consisting of functions with zero average on the domain ${{\Omega}}$, namely we consider the space $$\widehat{L}_2({{\Omega}})=\left\{u\in L_2({{\Omega}}): \int_{{\Omega}}u(x') dx' =0 \right\}$$ and also $\widehat H^s({{\Omega}})=H^s({{\Omega}})\cap\widehat L_2({{\Omega}})$ for $s>0$ with the standard $H^s({{\Omega}})$-norm. The notations $\widehat H^s_*({{\Omega}})$ and $\widehat H^s_0({{\Omega}})$ have a similar meaning.

To describe shell displacement we use the spaces $$\label{WY-space}
W=H^1_0({{\Omega}}) \times H^1_0({{\Omega}}) \times H^2_0({{\Omega}}), \quad Y=L_2({{\Omega}}) \times L_2({{\Omega}}) \times H_\alpha,$$ where $H_\alpha$ equals to $\widehat{H}^1_0({{\Omega}})$ with the equivalent norm $||\cdot||^2_{{H_\alpha}}=||\cdot||^2_{{\Omega}}+\alpha||{{\nabla}}\cdot||^2_{{\Omega}}$ and corresponding inner product.

\[re:hat-space\] [Below we also use $\widehat{H}^2_0({{\Omega}})$ as a state space for the displacement of the plate. It is clear that $\widehat{H}^2_0({{\Omega}})$ is a closed subspace of $H^2_0({{\Omega}})$. We denote by $\widehat{P}$ the projection on $\widehat{H}^2_0({{\Omega}})$ in $H^2_0({{\Omega}})$ which is orthogonal with respect to the inner product $(\Delta\cdot, \Delta\cdot)_{{\Omega}}$. One can see that $(I-\widehat{P})H^2_0({{\Omega}})$ consists of functions $u\in H^2_0({{\Omega}})$ such that $\Delta^2u=const$ and thus 