Spherical harmonics generating bounded biharmonics
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- by Bradley Beaver, Leo Sario and Cecilia Wang PDF
- Proc. Amer. Math. Soc. 84 (1982), 485-491 Request permission
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 $E_0^N = \{ x = ({x^1}, \ldots ,{x^N})|0 < |x| < \infty \}$, ${H^2}B$ 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 $d{s_\alpha } = {r^\alpha }|dx|$, $\alpha \in R$. 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 $E_\alpha ^2$.References
- Claus Müller, Spherical harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR 0199449
- Leo Sario and Cecilia Wang, Generators of the space of bounded biharmonic functions, Math. Z. 127 (1972), 273–280. MR 320349, DOI 10.1007/BF01114930
- Leo Sario and Cecilia Wang, Riemannian manifolds of dimension $N\geq 4$ without bounded biharmonic functions, J. London Math. Soc. (2) 7 (1974), 635–644. MR 425834, DOI 10.1112/jlms/s2-7.4.635
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 84 (1982), 485-491
- MSC: Primary 31C12; Secondary 31A30, 31B30
- DOI: https://doi.org/10.1090/S0002-9939-1982-0643735-7
- MathSciNet review: 643735