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

Abstract

Given a map $u : Ω\subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$, the $\infty$-Laplacian is the system \[ \label{1} Δ_\infty u \, :=\, \Big(\text{D}u \otimes \text{D}u + |\text{D}u|^2 [\text{D}u]^\bot \! \otimes I \Big) : \text{D}^2 u\, = \, 0. \tag{1} \] \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 \[ \label{2} E_\infty(u,Ω)\, :=\, \| \text{D}u \|_{L^\infty(Ω)}. \tag{2} \] 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.