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

Abstract

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