Higher integrability for measures satisfying a PDE constraint

TL;DR

This paper proves higher integrability estimates for measures satisfying linear PDE constraints, extending the understanding of regularity and compactness in measure solutions with applications to BV and BD functions.

Contribution

It establishes optimal higher integrability results for PDE-constrained measures, including the limiting case for canceling operators, advancing the theory of compensated compactness.

Findings

Proves local higher integrability estimates for PDE measures.

Identifies the optimal range for the integrability exponent p.

Extends results to the limiting case for canceling operators.

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.