# Vectorial variational principles in $L^\infty$ and their characterisation through PDE systems

## TL;DR

This paper explores two minimality principles for supremal functionals in $L^
Infty$, characterising them via PDE systems and showing their equivalence in the scalar case, linking divergence and non-divergence equations.

## Contribution

It introduces a PDE-based characterisation of absolute minimisers and $
Infty$-minimal maps, connecting divergence systems with Aronsson's PDEs.

## Key findings

- Absolute minimisers characterise a divergence system with probability measures.
- $
Infty$-minimal maps characterise Aronsson's PDE system.
- In the scalar case, the variational concepts coincide, linking divergence and non-divergence equations.

## Abstract

We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson's standard notion of absolute minimisers and the concept of $\infty$-minimal maps introduced more recently by the second author. We prove that $C^1$ absolute minimisers characterise a divergence system with parameters probability measures and that $C^2$ $\infty$-minimal maps characterise Aronsson's PDE system. Since in the scalar case these different variational concepts coincide, it follows that the non-divergence Aronsson's equation has an equivalent divergence counterpart.