L-Infininity Variational Problems for Maps and the Aronsson PDE System

Abstract

By employing Aronsson's Absolute Minimizers of $L^\infty$ functionals, we prove that Absolutely Minimizing Maps $u:\R^n \larrow \R^N$ solve a "tangential" Aronsson PDE system. By following Sheffield-Smart \cite{SS}, we derive $\De_\infty$ with respect to the dual operator norm and show that such maps miss information along a hyperplane when compared to Tight Maps. We recover the lost term which causes non-uniqueness and derive the complete Aronsson system which has \emph{discontinuous coefficients}. In particular, the Euclidean $\infty$-Laplacian is $\De_\infty u = Du \ot Du : D^2u\, +\, |Du|^2[Du]^\bot \De u$ where $[Du]^\bot$ is the projection on the null space of $Du^\top$. We exibit $C^\infty$ solutions having interfaces along which the rank of their gradient is discontinuous and propose a modification with $C^0$ coefficients which admits \emph{varifold solutions}. Away from the interfaces, Aronsson Maps satisfy a structural property of local splitting to 2 phases, an horizontal and a vertical; horizontally they possess gradient flows similar to the scalar case and vertically solve a linear system coupled by a scalar Hamilton Jacobi PDE. We also construct singular $\infty$-Harmonic local $C^1$ diffeomorphisms and singular Aronsson Maps.