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

TL;DR

This paper develops a comprehensive PDE system for Aronsson's absolute minimizers in $L^inity$ variational problems, revealing discontinuous coefficients, interfaces, and phase splitting, advancing understanding of vector-valued infinity harmonic maps.

Contribution

It derives the complete Aronsson PDE system for vector-valued maps, including discontinuous coefficients and varifold solutions, and analyzes their structural properties and singular solutions.

Findings

Derived the full Aronsson PDE system with discontinuous coefficients.

Identified interfaces where the gradient rank is discontinuous.

Constructed singular $C^1$ diffeomorphisms and Aronsson maps.

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.