Explicit Infinity-Harmonic Maps whose Interfaces have Junctions and Corners

TL;DR

This paper constructs explicit smooth solutions to the vector-valued infinity-Laplacian PDE in two dimensions, revealing complex interface junctions and corners, and demonstrating the inherent irregularity of solutions' interfaces.

Contribution

It introduces new explicit solutions with junctions and corners for the infinity-Laplacian, illustrating the complexity and irregularity of interfaces in vector-valued cases.

Findings

Solutions exhibit triple junctions and nonsmooth corners.

Interfaces can be highly irregular with discontinuous coefficients.

Regularity theory for interfaces cannot be established due to complexity.

Abstract

Given a map $u : \Om \sub \R^n \larrow \R^N$, the $\infty$-Laplacian is the system \[ \label{1} \De_\infty u \, :=\, \Big(Du \ot Du + |Du|^2 [Du]^\bot \ \ot I \Big) : D^2 u\, = \, 0 \tag{1} \] and arises as the "Euler-Lagrange PDE" of the supremal functional $E_\infty(u,\Om)= \|Du\|_{L^\infty(\Om)}.$ \eqref{1} is the model PDE of vector-valued Calculus of Variations in $L^\infty$ and first appeared in the author's recent work \cite{K1,K2,K3}. Solutions to \eqref{1} present a natural phase separation with qualitatively different behaviour on each phase. Moreover, on the interfaces the coefficients of \eqref{1} are discontinuous. Herein we constuct new explicit smooth solutions for $n=N=2$ for which the interfaces have triple junctions and nonsmooth corners. The high complexity of these solutions provides further understanding of the PDE \eqref{1} and shows there can be no regularity theory of interfaces.