Higher integrability for measures satisfying a PDE constraint
We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[ \mathcal{A} μ= σ, \] where $μ\in \mathcal{M}(Ω;V)$ and $σ\in \mathcal{M}(Ω;W)$ are vector measures and the polar $\frac{\mathrm{d} μ}{\mathrm{d} |μ|}$ 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 \[ \|μ\|_{\mathrm{L}^p(Ω')} \lesssim |μ|(Ω) + |σ|(Ω), \qquad Ω' \Subset Ω. \] Here, the exponent $p$ belongs to the (optimal) range $1 \leq p < d/(d-k)$, $d$ is the dimension of $Ω$, 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.