Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Spherical harmonics generating bounded biharmonics

Authors: Bradley Beaver, Leo Sario and Cecilia Wang
Journal: Proc. Amer. Math. Soc. 84 (1982), 485-491
MSC: Primary 31C12; Secondary 31A30, 31B30
MathSciNet review: 643735
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Abstract: Let $ {H^2}B(R)$ be the family of bounded nonharmonic biharmonic functions on a Riemannian manifold $ R$. On the punctured Euclidean $ N$-space $ E_0^N = \{ x = ({x^1}, \ldots ,{x^N})\vert < \vert x\vert < \infty \} $, $ {H^2}B$ is void for $ N > 3$, whereas for $ N = 2,3$, it is generated by certain fundamental spherical harmonics. It is also known that $ {H^2}B$ remains void on the Riemannian manifold $ E_\alpha ^N$, $ N > 3$, obtained by endowing $ E_0^N$ with the non-Euclidean metric $ d{s_\alpha } = {r^\alpha }\vert dx\vert$, $ \alpha \in R$.

The purpose of the present paper is to show that the fundamental spherical harmonics continue generating $ {H^2}B(E_\alpha ^3)$, despite the distorting metric $ d{s_\alpha }$. An analogous result holds for $ E_\alpha ^2$.

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Article copyright: © Copyright 1982 American Mathematical Society