Year: 2024
Type: preprint
Abstract: In this paper, we first give a convenient formula for bi-Laplacian on a sphere and the complete description of its eigenvalues, buckling eigenvalues, and their corresponding eigenfunctions. We then show that the radial (or rotationally symmetric) solutions for biharmonic equation on the model space $( \mathbb{R}^+\times S^{m-1}, dr^2 + σ^2(r)\, g^{S^{m-1}})$ can be given by an integral formula. We also prove that the model space always admits proper biharmonic functions as the products of any eigenfunctions of the factor sphere with certain radial functions. Many explicit examples of proper biharmonic functions on space forms are given. Finally, we give a complete classification of proper biharmonic functions with positive Laplacian on the punctured Euclidean space.
Source: arXiv (Cornell University)
Author: Ou, Ye-Lin
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403321634