| Abstract: |
| For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ without boundary which has no rank-one connection and is elliptic,
we prove the quantitative rigidity estimate
\begin{align*}
\inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\,dx \leq C \int_{B_1} \mathrm{dist}^2(Du, K)\, dx,
\qquad\forall u\in H^1(B_1;\mathbb R^2).
\end{align*}
This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$ without boundary, of the celebrated quantitative rigidity estimate
of Friesecke, James and M\uller for the approximate differential inclusion into $SO(n)$. Furthermore, no analogous result can hold true in $\mathbb R^{n\times n}$ for $n\geq 3$. This is joint work with Xavier Lamy and Andrew Lorent. |
|