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

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