On the Structure of $\infty$-Harmonic Maps

Abstract

Let H ∈ C 2(ℝN×n), H ≥ 0. The PDE system (Formula presented.) arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E ∞(u, Ω) = {norm of matrix}H(Du){norm of matrix}L ∞(Ω) defined on maps u: Ω ⊆ ℝn → ℝN. (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 ℝN×n, which we call the "∞-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 {pipe}Du{pipe} 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.