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 $E_{\infty }(u,{\Omega }^{\prime })=\Vert \mathcal{L}(\cdot ,u,Du){\Vert }_{L^{\infty }({\Omega }^{\prime })}$, ${\Omega }^{\prime }⋐\Omega$, applied to $W^{1,\infty }$ maps $u:\Omega \subseteq \mathbb{R}⟶{\mathbb{R}}^N$ with given boundary values. The assumptions on $\mathcal{L}($ are minimal, improving earlier existence results previously established by Barron-Jensen-Wang and by the second author.

arXiv comment9 pages, Journal: Proceedings of the AMS