Quantitative rigidity of differential inclusions in two dimensions

TL;DR

This paper establishes a quantitative rigidity estimate for differential inclusions in two dimensions, generalizing a classical result and relying on elliptic properties of certain submanifolds in matrix space.

Contribution

It proves a new rigidity estimate for elliptic submanifolds of matrices in 2D, extending the classical Friesecke-James-Müller result to a broader class.

Findings

Optimal rigidity estimate for 2D elliptic submanifolds

Uses conformal-anticonformal decomposition for PDE analysis

Shows no similar result in higher dimensions n≥3

Abstract

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 $n\geq 3$.