Biharmonic Functions On S^2 And H^2
Document Type
Thesis
Degree Name
Master of Science (MS)
Department
Mathematics
Date of Award
5-22-2024
Abstract
The study herein aims to understand bi-harmonic functions on 2-dimensional sphere S² and hyperbolic space H² in relation with harmonic and bi-harmonic functions on Eu- clidean space R². For this purpose the properties of harmonic functions and the conceptual construc- tion of bi-harmonic functions from general functions using these properties are reviewed. The Poisson’s equations on R² are applied to construct and/or classify bi-harmonic functions on the warped product spaces as well as the conformally flat surfaces of S² and H² . The construction of bi-harmonic functions on R² by using analytic complex functions and some applications in elasticity are reviewed. We are able to describe all rotationally symmetric bi-harmonic functions on S² and H² using conformal models and polar coordinates in R². It is concluded from these studies that although the bi-harmonic functions on R² can be abundantly constructed from the arbitrary analytic complex functions, explicit solu- tions of bi-harmonic equations on 2-dimensional sphere S² and hyperbolic space H² could be a challenge due to the difficulty in solving the corresponding nonlinear Poisson’s equations.
Advisor
Ye-Lin Ou
Subject Categories
Mathematics | Physical Sciences and Mathematics
Recommended Citation
Lee, Han Seon, "Biharmonic Functions On S^2 And H^2" (2024). Electronic Theses & Dissertations. 1162.
https://digitalcommons.tamuc.edu/etd/1162
