We consider a homogeneous, constant rank differential operator $\mathcal{ A}$ and prove a characterisation theorem for $\mathcal{ A}$-quasiaffine functions in the spirit of Ball, Currie and Olver (1981); i.e. functions such that \[ f(v) = \int_{ T_N} f(v + \psi(y))~\mathrm{ d}y \] for all $v$ and all $\mathcal{ A}$-free test functions $\psi$ with zero mean. This result is used to get a sufficient, but not necessary condition for the differential operator $\mathcal{ A}$, such that linearity along the characteristic cone of $\mathcal{ A}$ implies $\mathcal{ A}$-quasiaffinity. We show that this implication is true if $\mathcal{ A}$ admits a first order potential.
Categories: math.AP