Higher integrability for measures satisfying a PDE constraint

We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[ A μ = σ , A μ = σ , \] where μ ∈ M ( Ω ; V ) μ ∈ M(Ω ;V) and σ ∈ M ( Ω ; W ) σ ∈ M(Ω ;W) are vector measures and the polar d μ d | μ | dμ /d|μ | is uniformly close to a convex cone of V V intersecting the wave cone of A A only at the origin. More precisely, we prove local compensated compactness estimates of the form \[ ‖ μ ‖ L p ( Ω ′ ) ≲ | μ | ( Ω ) + | σ | ( Ω ) , Ω ′ ⋐ Ω . \|μ \|L^p(Ω ’) |μ |(Ω ) + |σ |(Ω ), Ω ’ Ω . \] Here, the exponent p p belongs to the (optimal) range 1 ≤ p > d / ( d − k ) 1 ≤ p > d/(d-k) , d d is the dimension of Ω Ω , and k k is the order of A A . We also obtain the limiting case p = d / ( d − k ) 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.