# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### On a weighted Laplace differential operator for the unit disc

(2011) In Master Thesis in Mathematical Science MATM01 20112
Mathematics (Faculty of Sciences)
Abstract
It is well-known that the classical Poisson kernel for the unit disc $\D$ in the complex plane is naturally associated to the Laplacian. In a recent paper Duman has shown that Poisson integrals with respect to the kernel $$K_2(z)=\frac{1}{2}\frac{(1-\lvert z\rvert^2)^3}{\lvert 1-z\rvert^4}, \quad z\in\D,$$ solve the Dirichlet problem for the unit disc for a certain second order differential operator $D_2(z,\partial)$. In this paper we calculate the differential operator $D_2(z,\partial)$ explicitly. We also prove uniqueness of solutions for the above mentioned Dirichlet problem for the differential operator $D_2(z,\partial)$. The analysis of the uniqueness problem makes use of an interesting connection to the classical hypergeometric... (More)
It is well-known that the classical Poisson kernel for the unit disc $\D$ in the complex plane is naturally associated to the Laplacian. In a recent paper Duman has shown that Poisson integrals with respect to the kernel $$K_2(z)=\frac{1}{2}\frac{(1-\lvert z\rvert^2)^3}{\lvert 1-z\rvert^4}, \quad z\in\D,$$ solve the Dirichlet problem for the unit disc for a certain second order differential operator $D_2(z,\partial)$. In this paper we calculate the differential operator $D_2(z,\partial)$ explicitly. We also prove uniqueness of solutions for the above mentioned Dirichlet problem for the differential operator $D_2(z,\partial)$. The analysis of the uniqueness problem makes use of an interesting connection to the classical hypergeometric differential equation. (Less)
Please use this url to cite or link to this publication:
author
supervisor
organization
course
MATM01 20112
year
type
H2 - Master's Degree (Two Years)
subject
publication/series
Master Thesis in Mathematical Science
report number
LUNFMA-3067-2011
ISSN
1404-6342
other publication id
2011:E54
language
English
id
2493421
date added to LUP
2014-12-15 14:28:49
date last changed
2014-12-15 14:28:49
@misc{2493421,
abstract     = {It is well-known that the classical Poisson kernel for the unit disc $\D$ in the complex plane is naturally associated to the Laplacian. In a recent paper Duman has shown that Poisson integrals with respect to the kernel $$K_2(z)=\frac{1}{2}\frac{(1-\lvert z\rvert^2)^3}{\lvert 1-z\rvert^4}, \quad z\in\D,$$ solve the Dirichlet problem for the unit disc for a certain second order differential operator $D_2(z,\partial)$. In this paper we calculate the differential operator $D_2(z,\partial)$ explicitly. We also prove uniqueness of solutions for the above mentioned Dirichlet problem for the differential operator $D_2(z,\partial)$. The analysis of the uniqueness problem makes use of an interesting connection to the classical hypergeometric differential equation.},
author       = {Teka, Yohannis Wubeshet},
issn         = {1404-6342},
language     = {eng},
note         = {Student Paper},
series       = {Master Thesis in Mathematical Science},
title        = {On a weighted Laplace differential operator for the unit disc},
year         = {2011},
}