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

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