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

TL;DR

This paper investigates $L^ Infty$ minimisation problems with nonlinear constraints, establishing existence of special minimisers via $L^p$ approximation and linking to divergence PDE systems related to Aronsson equations.

Contribution

It introduces a novel approach to constrained $L^ Infty$ minimisation problems using $L^p$ approximation, connecting solutions to divergence PDE systems with auxiliary measures.

Findings

Existence of a special $L^ Infty$ minimiser solving a divergence PDE system.

Connection between the minimiser and a divergence form Aronsson PDE system.

Applicability to various nonlinear operator constraints.

Abstract

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.