A mixed boundary value problem for the L\’ame equation in
a thin layer
around a surface $\cC$ with the Lipshitz boundary is
investigated. The main goal is to find out what happens when the
thickness of the layer tends to zero $h\to0$. To this
end we reformulate BVP into an equivalent variational problem and prove
that the energy functional has the $\Gamma$-limit being
the energy functional on the mid-surface $\cC$. The
corresponding BVP on $\cC$, considered as the
$\Gamma$-limit of the initial BVP, is written in terms
of G\”unter’s tangential derivatives on
$\cC$ and represents a new form of the shell equation.
It is shown that the Neumann boundary condition from the initial BVP on
the upper and lower surfaces transforms into a right-hand side term of
the basic equation of the limit BVP.