The biharmonic optimal support problem

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.