Explicit 2D ∞-harmonic maps whose interfaces have junctions and corners

Abstract

Given a map u:Ω⊆Rn→RN, the ∞-Laplacian is the system:(1)δ∞u:=(Du⊗Du+|Du|2[Du]⊥⊗I):D2u=0 and arises as the "Euler-Lagrange PDE" of the supremal functional E∞(u,Ω)={norm of matrix}Du{norm of matrix}L∞(Ω). (1) is the model PDE of the vector-valued Calculus of Variations in L∞ and first appeared in the author's recent work [10-14]. Solutions to (1) present a natural phase separation with qualitatively different behaviour on each phase. Moreover, on the interfaces the coefficients of (1) are discontinuous. Herein we construct new explicit smooth solutions for n=N=2, for which the interfaces have triple junctions and non-smooth corners. The high complexity of these solutions provides further understanding of the PDE (1) and limits what might be true in future regularity considerations of the interfaces.