Plates with incompatible prestrain of higher order

We study the effective elastic behaviour of the incompatibly prestrained thin plates, characterized by a Riemann metric $G$ on the reference configuration. We assume that the prestrain is "weak", i.e. it induces scaling of the incompatible elastic energy $E^h$ of order less than $h^2$ in terms of the plate's thickness $h$. We essentially prove two results. First, we establish the $\Gamma$-limit of the scaled energies $h^{-4}E^h$ and show that it consists of a von K\'arm\'an-like energy, given in terms of the first order infinitesimal isometries and of the admissible strains on the surface isometrically immersing $G_{2\times 2}$ (i.e. the prestrain metric on the midplate) in $\mathbb{R}^3$. Second, we prove that in the scaling regime $E^h\sim h^\beta$ with $\beta>2$, there is no other limiting theory: if $\inf h^{-2} E^h \to 0$ then $\inf E^h\leq Ch^4$, and if $\inf h^{-4}E^h\to 0$ then $G$ is realizable and hence $\min E^h = 0$ for every $h$.