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

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.