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arXiv · 1801.09756

Counterexamples in Calculus of Variations in $L^\infty$ through the vectorial Eikonal equation

Nikos Katzourakis·Giles Shaw
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Abstract

We show that for any regular bounded domain $\Omega\subseteq \mathbb R^n$, $n=2,3$, there exist infinitely many global diffeomorphisms equal to the identity on $\partial \Omega$ which solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the $\infty$-Laplace system arising in vectorial Calculus of Variations in $L^\infty$ does not suffice to characterise either limits of $p$-Harmonic maps as $p\to \infty$, or absolute minimisers in the sense of Aronsson.

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Nikos Katzourakis, Giles Shaw. 2018-01-29. Counterexamples in Calculus of Variations in $L^\infty$ through the vectorial Eikonal equation. https://arxiv.org/abs/1801.09756

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