Explicit Infinity-Harmonic Maps whose Interfaces have Junctions and Corners

الأصل · EN

Given a map u: ⁿ ⁿ, the ∞-Laplacian is the system 1 ∞ u:= (Du Du + |Du|² [Du] I): D² u = 0 1 and arises as the "Euler-Lagrange PDE" of the supremal functional E∞(u,)= DuL∞(). 1 is the model PDE of vector-valued Calculus of Variations in L∞ and first appeared in the author's recent work K1,K2,K3. 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 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 1 and shows there can be no regularity theory of interfaces.