Existence of 1D vectorial Absolute Minimisers in $L^\infty$ under minimal assumptions

TL;DR

This paper proves the existence of vectorial absolute minimizers for a supremal functional in one dimension under minimal assumptions, extending previous results and broadening the scope of such minimizers.

Contribution

It establishes the existence of vectorial absolute minimizers in 1D for $L^ abla$ functionals with minimal assumptions, improving prior results.

Findings

Existence of minimizers proven under minimal conditions.

Extends previous results by Barron-Jensen-Wang.

Applicable to $W^{1, abla}$ maps with boundary values.

Abstract

We prove the existence of vectorial Absolute Minimisers in the sense of Aronsson to the supremal functional $E_\infty(u,\Omega') = \|\mathscr{L}(\cdot,u,D u)\|_{L^\infty(\Omega')}$, $\Omega'\Subset \Omega$, applied to $W^{1,\infty}$ maps $u:\Omega\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$ with given boundary values. The assumptions on $\mathscr{L}($ are minimal, improving earlier existence results previously established by Barron-Jensen-Wang and by the second author.