On the numerical approximation of vectorial absolute minimisers in $L^\infty$
For researchers in Calculus of Variations in $L^\\infty$, this provides numerical evidence for a method addressing the open problem of existence of absolute minimisers in the vectorial case.
The paper presents numerical experiments for approximating vectorial absolute minimisers of the $L^\\infty$ norm of the gradient, a problem where existence under boundary data is open for $n,N\\geq 2$. The experiments are based on a new method from prior work.
Let $Ω$ be an open set. We consider the supremal functional \[ \tag{1} \label{1} \ \ \ \ \ \ \mathrm{E}_\infty (u,\mathcal{O})\, :=\, \| \mathrm D u \|_{L^\infty( \mathcal{O} )}, \ \ \ \mathcal{O} \subseteq Ω\text{ open}, \] applied to locally Lipschitz mappings $u : \mathbb R^n \supseteq Ω\longrightarrow \mathbb R^N$, 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].