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 $$\underset{M\in K}{\inf }{\int }_{B_{1\mathrm{/}2}}\mathrm{\mid }Du-M{\mathrm{\mid }}^{2}dx\le C{\int }_{B_1}{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}^2(Du,K)dx,\mathrm{\forall }u\in H^1(B_1;{\mathbb{R}}^2)\mathrm{.}$$ 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 Müller 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 $n\ge 3$.