On the Structure of Infinity-Harmonic Maps

Details On the Structure of Infinity-Harmonic Maps On the Structure of Infinity-Harmonic Maps

2012-04-24

arxiv.org/abs/1204.5374v1

Mathematics

Analysis of PDEs

31 pages, 8 figures

Scientific paper

In this paper we primarily study 2-dimensional $\infty$-Harmonic maps $u : \Om \sub \R^2 \longrightarrow \R^N$, $N\geq 2$, that is solutions to \[\label{1} \Delta_\infty u \ :=\ \Big(Du \otimes Du \, +\, |Du|^2 [Du]^\bot \otimes I \Big) : D^2 u\ = \ 0. \tag{1}\] PDE \eqref{1} and the Aronsson system with respect to general Hamiltonians were derived in recent work \cite{K3}. By establishing a general Rigidity Theorem for Rank-One Lipschitz Maps of independent interest, we analyse the structure of phase separation of solutions and of their interfaces whereon the coefficients of \eqref{1} become discontinuous. As a corollary, we extend the Aronsson-Evans-Yu theorem on the non-existence of zeros of $|Du|$ for solutions to \eqref{1} to all $N \geq 2$ and establish a Maximum Principle (Convex Hull Property) for N=2. We further classify all Hamiltonians $H \in C^2(\R^N \ot \R^n)$ which lead to elliptic Aronsson systems: they are the "geometric" ones, which depend on $Du$ via the Riemannian metric $Du^\top Du$. We finally study existence, uniqueness and regularity of solutions to the initial value problem for Aronsson ODE systems.

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