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

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