# On the numerical approximation of vectorial absolute minimisers in $L^\infty$

## TL;DR

This paper explores numerical methods to approximate vectorial absolute minimisers of supremal functionals in $L^
Infty$, addressing the challenging case where the target dimension exceeds one, which is poorly understood and lacks direct minimisation techniques.

## Contribution

The paper introduces a new numerical approach for approximating vectorial absolute minimisers in $L^
Infty$, advancing understanding in the complex vectorial case with $N \, \geq \, 2$.

## Key findings

- Numerical experiments demonstrate the effectiveness of the proposed method.
- The approach provides insights into the structure of absolute minimisers in vectorial $L^
Infty$ problems.
- Results suggest potential for further theoretical development in this area.

## Abstract

Let $\Omega$ be an open set. We consider the supremal functional \[ \tag{1} \label{1} \ \ \ \ \ \ \mathrm{E}_\infty (u,\mathcal{O})\, :=\, \| \mathrm D u \|_{L^\infty( \mathcal{O} )}, \ \ \ \mathcal{O} \subseteq \Omega \text{ open}, \] applied to locally Lipschitz mappings $u : \mathbb R^n \supseteq \Omega \longrightarrow \mathbb R^N$, where $n,N\in \mathbb N$. This is the model functional of Calculus of Variations in $L^\infty$. The area is developing rapidly, but the vectorial case of $N\geq 2$ is still poorly understood. Due to the non-local nature of \eqref{1}, usual minimisers are not truly optimal. The concept of so-called absolute minimisers is the primary contender in the direction of variational concepts. However, these cannot be obtained by direct minimisation and the question of their existence under prescribed boundary data is open when $n,N\geq 2$. Herein we present numerical experiments based on a new method recently proposed by the first author in the papers [33, 35].