Quantitative Rigidity of Differential Inclusions in Two Dimensions.

For any compact connected one-dimensional submanifold |$K\subset \mathbb R^{2\times 2}$| without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$ 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 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 partial differential equation 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 |$n\geq 3$|⁠. [ABSTRACT FROM AUTHOR]