Higher-order constraint conjecture for elastic shell models

Let $S$ be a midsurface, let $E^h(u^h)$ denote the elastic energy of a shell of thickness $h$, and suppose

$$E^h(u^h)\approx h^\beta,\qquad \beta>2.$$

For $i\geq 1$, write

$$\beta_i=2+\frac{2}{i-1}$$

when $i>1$, and let $\mathcal{V}_N$ denote the space of $N$-tuples $(V_1,\ldots,V_N)$ of displacements whose formal deformation

$$u_\epsilon=\operatorname{id}+\sum_{i=1}^N\epsilon^iV_i$$

preserves the metric up to order $\epsilon^N$. Higher-order constraint conjecture. For values of $\beta$ closer to $2$, the limiting theory of thin shells should involve additional constraints beyond the known inclusion $V\in\mathcal{\tilde V}_2$, with the relevant constraints arising from higher-order infinitesimal isometries and, at the successive scaling thresholds, from the spaces $\mathcal{V}_N$. In particular, for $\beta\in[\beta_{N+1},\beta_N)$, the limiting functional should be defined on $\mathcal{V}_N$ and should have the form stated in the source: at $\beta=\beta_{N+1}$ it contains both the $(N+1)$-st order metric-change term and the first-order change in the second fundamental form, while for $\beta\in(\beta_{N+1},\beta_N)$ it contains only the latter; the $\mathcal{V}_N$ constraint may be relaxed to $\mathcal{V}_M$, $M<N$, when the midsurface has the stated matching property. The range $2<\beta<4$ remains open for general shells, and the conjectured additional constraints are intended to describe the limiting theories at scalings progressively closer to $2$.