Scale-invariant bounds on thin sets for measures satisfying a first-order PDE and the anisotropic Michael-Simon inequality

Abstract

We prove mass estimates for PDE-constrained measures on sets of $\sigma$-finite ${\mathcal H}^m$-measure, assuming an appropriate rank condition. As an application, we derive a Michael--Simon inequality for varifolds with bounded anisotropic first variation with respect to an integrand that satisfies the natural atomic condition (AC1).