New explicit solutions to the $p$-Laplace equation based on isoparametric foliations

TL;DR

This paper introduces new explicit solutions to the higher-dimensional p-Laplace equation using isoparametric polynomials, including rational and algebraic examples, and demonstrates the non-existence of certain p-harmonic polynomials.

Contribution

It develops a novel method using isoparametric polynomials to generate diverse p-harmonic functions in higher dimensions, including the first rational and algebraic solutions.

Findings

Constructs new p-harmonic solutions using isoparametric polynomials.

Shows non-existence of p-harmonic polynomials of isoparametric type.

Provides examples of rational and algebraic p-harmonic functions for specific p and n.

Abstract

In contrast to an infinite family of explicit examples of two-dimensional $p$-harmonic functions obtained by G.Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of $p$-harmonic and biharmonic functions. Remarkably, for some distinguished values of $p$ and the ambient dimension $n$ this yields first examples of rational and algebraic $p$-harmonic functions. Moreover, we show that there are no $p$-harmonic polynomials of the isoparametric type. This supports a negative answer to a question proposed in 1980 by J. Lewis.