Explicit $\infty$-harmonic functions in high dimensions

TL;DR

This paper derives new explicit solutions to the high-dimensional $ abla$-Laplace equation, expanding the class of known $ abla$-harmonic functions with symmetry conditions in arbitrary dimensions.

Contribution

It introduces novel explicit solutions to the $ abla$-Laplace equation in high dimensions, generalizing known two-dimensional cases.

Findings

New explicit $ abla$-harmonic functions in arbitrary dimensions

Solutions obey specific symmetry conditions

Contains known 2D solutions as special cases

Abstract

The aim of this work is to derive new explicit solutions to the $\infty$-Laplace equation, the fundamental PDE arising in Calculus of Variations in the space $L^\infty$. These solutions obey certain symmetry conditions and are derived in arbitrary dimensions, containing as particular sub-cases the already known classes two-dimensional infinity-harmonic functions.