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

By employing Aronsson’s Absolute Minimizers of $L^{\mathrm{\infty }}$ functionals, we prove that Absolutely Minimizing Maps $u:{\mathbb{R}}^n\text{larrow}{\mathbb{R}}^N$ solve a “tangential” Aronsson PDE system. By following Sheffield-Smart \cite{SS}, we derive ${\text{De}}_{\mathrm{\infty }}$ 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 $\mathrm{\infty }$-Laplacian is $\De_\infty u = Du \ot Du : D^2u, +, |Du|^2[Du]^\bot \De u$where$[Du]^\bot$istheprojectiononthenullspaceof$Du^\top$\mathrm{.}Weexibit$C^\infty$ 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 $\mathrm{\infty }$-Harmonic local $C^1$ diffeomorphisms and singular Aronsson Maps.