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

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.

Journal: Comptes Rendus Mathematique, 6 pages