Quantitative rigidity of differential inclusions in two dimensions

2022/08/17 by Xavier Lamy, Lamy, Xavier, Andrew Lorent +3
Computer Science · Mathematics · #Contact Mechanics and Variational Inequalities #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2208.08526

Abstract

For any compact connected one-dimensional submanifold K⊂ \mathbb R^{2× 2} which has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate inf_{M∈ K}∫_B_1/2| Du -M |² dx ≤ C ∫_B1 dist²(Du, K) dx, ∀ u∈ H¹(B₁;\mathbb R²). This is an optimal generalization, for compact connected submanifolds of \mathbb R^{2× 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⊂\mathbb R^{2× 2} with respect to conformal-anticonformal decomposition, which provide a quasilinear elliptic PDE satisfied by solutions of the exact differential inclusion Du∈ K. We also give an example showing that no analogous result can hold true in \mathbb R^{n× n} for n≥ 3.

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