New explicit solutions to the p-Laplace equation based on isoparametric foliations |
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Institution: | Linköping University, Department of Mathematics, SE-581 83, Sweden |
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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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Keywords: | Isoparametric polynomials ∞-Laplacian Biharmonic functions Minimal submanifold Focal varieties |
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