$\mathscr{A}$-free truncation and higher integrability of minimisers

We show higher integrability of minimisers of functionals $$I(u) = \int_{\Omega} f(x,u(x)) ~\mathrm{d}x$$ subject to a differential constraint $\mathscr{A} u=0$ under natural $p$-growth and $p$-coercivity conditions for $f$ and regularity assumptions on $\Omega$. For the differential operator $\mathscr{A}$ we asssume a rather abstract truncation property that, for instance, holds for operators $\mathscr{A}=\mathrm{curl}$ and $\mathscr{A}=\mathrm{div}$. The proofs are based on the comparison of the minimiser to the truncated version of the minimiser.