Explicit Infinity-Harmonic Maps whose Interfaces have Junctions and Corners

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.