The biharmonic optimal support problem
Résumé
We establish a Γ-convergence result for h → 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 x3 = 0 over the h-2D-neighborhood of a given closed set K. In the regime of very small vertical forces we identify the Γ-limit as being the bi-harmonic energy, with Dirichlet condition on the gluing region K, following a general strategy by Friesecke, James, and Müller 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 C1,α regular outside a set of Hausdorff dimension strictly less then one.
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