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

TL;DR

This paper establishes geometric rigidity inequalities for incompatible fields in dimensions three and higher, providing strong $L^p$ estimates in the supercritical regime and weak-$L^1$ estimates at the critical exponent, with applications to $BV$ bounds.

Contribution

It extends geometric rigidity estimates to higher dimensions for incompatible fields, introducing new $L^p$ and weak-$L^1$ bounds relevant for fields with bounded Curl.

Findings

Strong scaling-invariant $L^p$ estimates in supercritical regime

Weak-$L^1$ inequality at critical exponent

Derived $BV$ bounds for $SO(n)$-valued fields with bounded Curl

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 $1^* = \frac{n}{n-1}$ 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.