On the Structure of ∞-Harmonic Maps

Let H is an element of C-2 (IRNxn), H >= 0. The PDE system A(infinity)u := (H-P circle times H-P + H[H-P]H-perpendicular to(PP))(Du) : D(2)u = 0 (1) arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E-infinity(u, Omega) = parallel to H(Du)parallel to(L infinity(Omega)) defined on maps u : Omega subset of IRn -> IRN. (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 <= N with H the Euclidean norm on IRNxn, which we call the "infinity-Laplacian". By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N >= 2 the Aronsson-Evans- Yu theorem regarding non existence of zeros of vertical bar Du vertical bar and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(., u, u') depending on all the arguments.