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


Authors: Hussien Abugirda and Nikos Katzourakis
Journal: Proc. Amer. Math. Soc. 145 (2017), 2567-2575
MSC (2010): Primary 35J47, 35J62, 53C24; Secondary 49J99
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 ')\!=\!\Vert\mathscr {L}(\cdot ,u,\mathrm {D} u)\Vert _{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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Additional Information

Hussien Abugirda
Affiliation: Department of Mathematics, College of Science, University of Basra, Basra, Iraq — and — Department of Mathematics and Statistics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, United Kingdom
Email: h.a.h.abugirda@student.reading.ac.uk

Nikos Katzourakis
Affiliation: Department of Mathematics and Statistics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, United Kingdom
Email: n.katzourakis@reading.ac.uk

DOI: https://doi.org/10.1090/proc/13421
Keywords: Vectorial calculus of variations in $L^\infty$, vectorial Absolute Minimisers, $\infty$-Laplacian
Received by editor(s): April 19, 2016
Received by editor(s) in revised form: July 22, 2016
Published electronically: December 27, 2016
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society