  We study the convex hull property for systems of partial differential
equations. This is a generalisation of the maximum principle for a single
equation. We show that the convex hull property holds for a class of elliptic
and parabolic systems of non-linear partial differential equations. In
particular, this includes the case of the parabolic $p$-Laplace system. The
coupling conditions for coefficients are demonstrated to be optimal by means of
respective counterexamples.
