Biharmonic functions and bi-eigenfunctions on some model spaces

2024/07/18 by Ye‐Lin Ou, Ou, Ye-Lin
Mathematics · #53C12 #58E20 #Advanced Differential Equations and Dynamical Systems #Differential Equations and Boundary Problems #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2407.13560

openalex publication_date 2024/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ( ℝ⁺× S^m-1, dr² + σ²(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.

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