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

Abstract

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

16 pages