Quantitative Rigidity of Differential Inclusions in Two Dimensions

• Published in: International mathematics research notices 2024-04, Vol.2024 (8), p.6325-6349
• Main Authors: Lamy, Xavier, Lorent, Andrew, Peng, Guanying
• Format: Article
• Language: English
• Subjects: Analysis of PDEs; Mathematics
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rnad108

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 $n\geq 3$.