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.