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

TL;DR

This paper proves higher integrability of minimisers for certain functionals with differential constraints, using a novel truncation property applicable to operators like curl and div, under natural growth and regularity conditions.

Contribution

It introduces an abstract truncation property for differential operators and demonstrates higher integrability of minimisers under this framework.

Findings

Higher integrability of minimisers established.

Applicable to operators like curl and div.

Uses comparison with truncated minimisers.

Abstract

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.