  We establish a $\Gamma$-convergence result for $h\to 0$ of a thin nonlinearly
elastic 3D-plate of thickness $h>0$ which is assumed to be glued to a support
region in the 2D-plane $x_3=0$ over the $h$-2D-neighborhood of a given closed
set $K$. In the regime of very small vertical forces we identify the
$\Gamma$-limit as being the bi-harmonic energy, with Dirichlet condition on the
gluing region $K$, following a general strategy by Friesecke, James, and
M\"uller that we have to adapt in presence of the glued region. Then we
introduce a shape optimization problem that we call "optimal support problem"
and which aims to find the best glued plate. In this problem the bi-harmonic
energy is optimized among all possible glued regions $K$ that we assume to be
connected and for which we penalize the length. By relating the dual problem
with Griffith almost-minimizers, we are able to prove that any minimizer is
$C^{1,\alpha}$ regular outside a set of Hausdorff dimension strictly less then
one.
