Quantitative Immersability of Riemann Metrics and the Infinite Hierarchy of Prestrained Shell Models

Abstract

We propose results that relate the following two contexts:

• (i) Given a Riemann metric G on $\Omega ^1=\omega \times (-\frac{1}{2}, \frac{1}{2})$, we find the infimum of the averaged pointwise deficit of an immersion from attaining the orientation-preserving isometric immersion of $G_{\mid \Omega ^h}$ on $\Omega ^h=\omega \times (-\frac{h}{2}, \frac{h}{2})$, over all weakly regular immersions. This deficit is measured by the non-Euclidean energies ${\mathcal {E}}^h$, which can be seen as modifications of the classical nonlinear three-dimensional elasticity.

• (ii) We complete the scaling analysis of ${\mathcal {E}}^h$ in the context of dimension reduction as $h\rightarrow 0$, and the derivation of $\Gamma$-limits of the scaled energies $h^{-2n}{\mathcal {E}}^h$ for all $n\geqq 1$. We show the energy quantisation, in the sense that the even powers 2n of h are indeed the only possible ones (all of them are also attained).

For each n, we identify conditions for the validity of the scaling $h^{2n}$, in terms of the vanishing of Riemann curvatures of G up to appropriate orders, and in terms of the matched isometry expansions. We also establish the asymptotic behaviour of the minimizing immersions as $h\rightarrow 0$.