Existence of 1𝐷 vectorial Absolute Minimisers in 𝐿^{∞} under minimal assumptions

Abugirda, Hussien; Katzourakis, Nikos

doi:10.1090/proc/13421

Existence of $1D$ vectorial Absolute Minimisers in $L^\infty$ under minimal assumptions
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by Hussien Abugirda and Nikos Katzourakis

Proc. Amer. Math. Soc. 145 (2017), 2567-2575

DOI: https://doi.org/10.1090/proc/13421

Published electronically: December 27, 2016

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Abstract:

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

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