On the Numerical Approximation of $\infty$-Harmonic Mappings

TL;DR

This paper develops numerical methods to approximate solutions of the vectorial $ ext{infty}$-Laplacian system in two dimensions, revealing new phenomena and insights into the structure of solutions in the context of $L^ty$ calculus of variations.

Contribution

It introduces numerical approximation techniques for the $ ext{infty}$-Laplacian system in 2D and explores the solution structure and phenomena in $L^ty$ calculus of variations.

Findings

Demonstrates interesting phenomena in $L^ty$ solutions

Provides insights into phase separation in solutions

Reveals structure of solutions for specific boundary data

Abstract

Given a map $u : \Omega \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$, the $\infty$-Laplacian is the system \[ \label{1} \Delta_\infty u \, :=\, \Big(\text{D}u \otimes \text{D}u + |\text{D}u|^2 [\text{D}u]^\bot \! \otimes I \Big) : \text{D}^2 u\, = \, 0. \tag{1} \] \eqref{1} is the model system of vectorial Calculus of Variations in $L^\infty$ and arises as the "Euler-Lagrange" equation in relation to the supremal functional \[ \label{2} E_\infty(u,\Omega)\, :=\, \| \text{D}u \|_{L^\infty(\Omega)}. \tag{2} \] The scalar case of \eqref{1} has been introduced by Aronsson in the 1960s and by now is relatively classical and well understood. The general system \eqref{1} has been discovered and studied by the first author in a series of recent papers. Supremal functionals are fundamental for applications because they provide more realistic models as opposed to conventional integral models. Herein we provide numerical approximations of solutions to the Dirichlet problem when $n=2$ and $N=2,3$ for certain carefully selected boundary data on the unit square. Our experiments demonstrate interesting and unexpected phenomena occurring in $L^\infty$ and provide insights on the structure of general solutions and the natural separation to phases they present.