Quantitative rigidity of differential inclusions in two dimensions

For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ which has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\,dx \leq C \int_{B_1} \mathrm{dist}^2(Du, K)\, dx, \qquad\forall u\in H^1(B_1;\mathbb R^2).$$ This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$, of the celebrated quantitative rigidity estimate of Friesecke, James and M\"uller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $K\subset\mathbb R^{2\times 2}$ with respect to conformal-anticonformal decomposition, which provide a quasilinear elliptic PDE satisfied by solutions of the exact differential inclusion $Du\in K$. We also give an example showing that no analogous result can hold true in $\mathbb R^{n\times n}$ for $n\geq 3$.