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.

中文翻译：

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基于等参叶面的p -Laplace方程的新显式解

与80年代后期G. Aronsson获得的无穷多个二维p调和函数的显式实例相反，对高维情况的了解很少。在本文中，我们展示了如何使用等参多项式生成p调和和双调和函数的各种示例。值得注意的是，对于p和环境维数n的一些不同的值，这产生了有理和代数p谐波函数的第一个示例。此外，我们表明不存在等参类型的p调和多项式。这支持了J. Lewis在1980年提出的一个问题的否定答案。