On isosupremic vectorial minimisation problems in L ^∞ with general nonlinear constraints

Abstract

We study minimisation problems in L ∞ {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 {L^{p}} approximations as p → ∞ {p\to\infty} , we illustrate the existence of a special L ∞ {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 ∞ {L^{\infty}} variational problem.