Higher integrability for measures satisfying a PDE constraint

Abstract

We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[ \mathcal{A} \mu = \sigma, \] where $\mu \in \mathcal{M}(\Omega;V)$ and $\sigma\in \mathcal{M}(\Omega;W)$ are vector measures and the polar $\frac{\mathrm{d} \mu}{\mathrm{d} |\mu|}$ is uniformly close to a convex cone of $V$ intersecting the wave cone of $\mathcal{A}$ only at the origin. More precisely, we prove local compensated compactness estimates of the form \[ \|\mu\|_{\mathrm{L}^p(\Omega')} \lesssim |\mu|(\Omega) + |\sigma|(\Omega), \qquad \Omega' \Subset \Omega. \] Here, the exponent $p$ belongs to the (optimal) range $1 \leq p < d/(d-k)$, $d$ is the dimension of $\Omega$, and $k$ is the order of $\mathcal{A}$. We also obtain the limiting case $p = d/(d-k)$ for canceling constant-rank operators. We consider applications to compensated compactness and {applications to the theory of} functions of bounded variation and bounded deformation.