Geometric Rigidity Estimates for Incompatible Fields in Dimension $\ge$ 3

Abstract

We prove geometric rigidity inequalities for incompatible fields in dimension higher than 2. We are able to obtain strong scaling-invariant $L^p$ estimates in the supercritical regime, while for critical exponent $1^{\ast }=\frac{n}{n-1}$ we have a scaling invariant inequality only for the weak-$L^1$ norm. Although not optimal, such an estimate in $L^{1,\infty }$ is enough in order to infer a useful lemma which gives $BV$ bounds for $SO(n)$-valued fields with bounded Curl.