On a weighted Laplace differential operator for the unit disc
(2011) In Master Thesis in Mathematical Science MATM01 20112Mathematics (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:
http://lup.lub.lu.se/student-papers/record/2493421
- author
- Teka, Yohannis Wubeshet LU
- supervisor
- organization
- course
- MATM01 20112
- year
- 2011
- 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}}, }