Existence of $1D$ vectorial Absolute Minimisers in $L^\infty $ under minimal assumptions
Existence of $1D$ vectorial Absolute Minimisers in $L^\infty $ under minimal assumptions
We prove the existence of vectorial Absolute Minimisers in the sense of Aronsson to the supremal functional $E_\infty (u,\Omega â)\!=\!\|\mathscr {L}(\cdot ,u,\mathrm {D} u)\|_{L^\infty (\Omega â)}$, $\Omega â\Subset \Omega$, 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 …