HIGHER INTEGRABILITY FOR MEASURES SATISFYING A PDE CONSTRAINT

We establish higher integrability estimates for constant-coefficient systems of linear PDEs Aμ = σ, where μ ∈ M(Ω; V ) and σ ∈ M(Ω; W) are vector measures and the polar _d^d_⃒^μ_μ⃒ is uniformly close to a convex cone of V intersecting the wave cone of A only at the origin. More precisely, we prove local compensated compactness estimates of the form ∣μ∣_Lp_(Ώ₎ ≲ |μ|(Ω) + |σ|(Ω), Ω^́ ∈ Ω. Here, the exponent p belongs to the (optimal) range 1 ≤ p < d/(d − k), d is the dimension of Ω, and k is the order of 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.