  We study minimisation problems in $L^\infty$ for general quasiconvex first
order functionals, where the class of admissible mappings is constrained by the
sublevel sets of another supremal functional and by the zero set of a nonlinear
operator. Examples of admissible operators include those expressing pointwise,
unilateral, integral isoperimetric, elliptic quasilinear differential, jacobian
and null Lagrangian constraints. Via the method of $L^p$ approximations as
$p\to \infty$, we illustrate the existence of a special $L^\infty$ minimiser
which solves a divergence PDE system involving certain auxiliary measures as
coefficients. This system can be seen as a divergence form counterpart of the
Aronsson PDE system which is associated with the constrained $L^\infty$
variational problem.
