Variational models for prestrained plates with Monge-Ampère constraint

Abstract

We derive a new variational model in the description of prestrained elastic thin films. The model consists of minimizing a biharmonic energy of the out-of plane displacements $v\in W^{2,2}(\Omega, \mathbb{R})$, satisfying the Monge-Ampèere constraint: $$ \det\nabla^2v = f . $$ Here, $f=-\mbox{curl}^T\mbox{curl} (S_g)_{2\times 2}$ is the linearized Gauss curvature of the incompatibility (prestrain) family of Riemannian metrics $G^h= \mbox{Id}_3 + 2 h^\gamma S_g+ h.o.t.$, imposed on the referential configurations of the thin films with midplate $\Omega$ and small thickness $h$. We further discuss multiplicity properties of the minimizers of this model in some special cases.