Higher integrability for measures satisfying a PDE constraint

Abstract

We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[ \mathcal{A} μ= σ, \] 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 \[ \|μ\|_{\mathrm{L}^p(Ω')} \lesssim |μ|(Ω) + |σ|(Ω), \qquad Ω' \Subset Ω. \] Here, the exponent $p$ belongs to the (optimal) range $1\le p<d/(d-k)$, $d$ 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.

29 pages