On isosupremic vectorial minimisation problems in $L^\infty$ with general nonlinear constraints

Abstract

We study minimisation problems in L ∞ 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 → ∞, we illustrate the existence of a special L ∞ 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 ∞ variational problem. Contents 1. Introduction and main results 1 2. Minimisers of L p problems and convergence as p → ∞ 9 3. The equations for constrained minimisers in L p and in L ∞ 12 4. Explicit classes of nonlinear operators 20 4.1. Pointwise constraints, unilateral constraints and inclusions 20 4.2. Integral and isoperimetric constraints 22 4.3. Quasilinear second order differential constraints 23 4.4. Null Lagrangians and determinant constraints 24 References 25