Boundary value problems on hypersurfaces and $\Gamma $-convergence

We consider two examples of boundary value problems (BVPs) on hypersurfaces: heat conduction by an "isotropic" media, governed by the Laplace equation and bending of elastic "isotropic" media governed by Láme equations. The boundary conditions are classical Dirichlet-Neumann mixed type. The domain $\Omega^{h }:=\mathcal{C}\times (-h,h )$ is of thickness $2h$. Here $\mathcal{C}\subset \mathcal{S}$ is a smooth subsurface of a closed hypersurface $\mathcal{S}$ with smooth nonempty boundary $\partial \mathcal{C}$.
The object of the investigation is what happens with the above mentioned mixed boundary value problems when the thickness of the layer converges to zero $h\to0$. It is shown that the corresponding BVPs converge in the sense of $\Gamma$-convergence to a certain BVPs on the mid surface $\mathcal{C}$: The BVP for the Laplace equation converges to the BVP for the Dirichlet BVP for the Laplace-Beltrami equation, while for the Láme equation we get a new form of BVP for the shell equation.
The suggested approach is based on the fact that the Laplace and Láme operators are represented in terms of Günter's tangential and normal (to the surface) derivatives. Namely, if $\nu$ is the unit normal vector field on the surface, extended in the domain $\Omega_h$, the Günter's derivatives read
$$ \mathcal{D}_j:=\partial_j-\nu_j\mathcal{D}_4, \qquad \mathcal{D}_4 =\partial_\nu=\displaystyle\sum\limits_{k=1}^3\nu_k\partial_k,\qquad j=1,\ldots,n $$
and the Laplace-Beltrami operator on the surface $\mathcal{C}$ is represented as follows
$$ \Delta_\mathcal{C}=\mathcal{D}_1^2+\mathcal{D}_2^2+\mathcal{D}_3^2. $$
Moreover, the Laplace and the Láme operators in the domain
$$ \Delta_{\Omega^h}=\partial _1^2+\partial_2^2+\partial _3^2,\qquad \mathcal{L}_{\Omega^h}=-2\mu\,\Delta-(\lambda+2\mu)\,\nabla{\rm div} $$
are represented as follows:
$$ \Delta _{\Omega ^h}= \displaystyle\sum\limits_{j=1}^{4} \mathcal{D}_{j}^{2}+2\mathcal{H}_\mathcal{C}\mathcal{D}_4,\qquad \mathcal{L}_{\Omega^h}=-2\mu\,\Delta_{\Omega^h} -(\lambda+2\mu)\,\nabla_{\Omega^h}{\rm div} _{\Omega^h}. $$
Here $\mathcal{H}_\mathcal{C}$ is the mean curvature of the surface $\mathcal{C}$ and
$$ \nabla_{\Omega^h}\varphi:=\Bigl\{\mathcal{D}_1\varphi,...,\mathcal{D}_4 \varphi\Bigr\}^\top,\qquad {\rm div}_{\Omega^h}{\mathbf U}:=\sum_{j=1}^4\mathcal{D}_jU^0_j+\mathcal{H}_\mathcal{C} U^0_4,\\ {\mathbf U}=(U_1,U_2,U_3)^\top,\qquad U^0_j:=U_j-U^0_4,\quad U_4^0:=\langle\nu,{\mathbf U}\rangle,\quad j=1,2,3 $$
are the gradient and divergence.
The work is carried out in collaboration with T. Buchukuri and G. Tephnadze (Tbilisi).