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

TL;DR

This paper demonstrates the existence of infinitely many solutions to the vectorial Eikonal equation in bounded domains, showing limitations in characterizing limits of p-harmonic maps and absolute minimizers in vectorial Calculus of Variations in $L^ Infty$.

Contribution

It constructs explicit solutions to the vectorial Eikonal equation, revealing that the $ Infty$-Laplace system alone cannot fully characterize certain variational limits.

Findings

Existence of infinitely many solutions to the vectorial Eikonal equation.

Explicit examples of solutions on annular domains.

Limitations of the $ Infty$-Laplace system in characterizing variational limits.

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.