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arXiv · 1802.09892

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

Vladimir G. Tkachev
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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.

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Vladimir G. Tkachev. 2018-02-27. New explicit solutions to the $p$-Laplace equation based on isoparametric foliations. https://doi.org/10.1016/j.difgeo.2020.101629

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