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 On the Structure of Infinity-Harmonic Maps

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Let $H \in C^2\R^N \ot \R^n$ be a Hamiltonian. Here we study \emph{Aronsson Maps}, that is solutions $u : \Om \sub \R^n \larrow \R^N$ to the system \\label{1} \A \infty u \ :=\ \BigH P \ot H P, +\, H H P^\bot H {PP} \BigDu: D^2 u\ = \ 0 \tag{1} with emphasis on the special case of 2-d \emph{$\infty$-Harmonic maps} for $n=2\leq N$ with $HP=1-2|P|^2$ the Euclidean norm. We also consider the 1-d case of Aronsson ODEs when $Hx,ux,ux$ depends on all arguments. \eqref{1} was first derived in the authors recent work \cite{K3}. By establishing a general Rigidity Theorem for Rank-One Maps of independent interest, we analyse the phase separation of 2-d $\infty$-Harmonic maps and their interfaces whereon the coefficients of the system become discontinuous. As a corollary, we extend the Aronsson-Evans-Yu theorem on the non-existence of zeros of $|Du|$ for solutions to all $N \geq 2$ and establish a Maximum Principle Convex Hull Property for N=2. We further classify all $H$ for which \eqref{1} is elliptic: they are the -geometric- ones, which depend on $Du$ via the Riemannian metric $Du^\top Du$. We also study existence, uniqueness and regularity of the initial value problem for Aronsson ODEs.

Author: Nikolaos I. Katzourakis

Source: https://archive.org/

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