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 wellknown 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 $$ K_2(z)=\frac{1}{2}\frac{(1\lvert z\rvert^2)^3}{\lvert 1z\rvert^4}, \quad z\in\D, $$ 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 wellknown 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 $$ K_2(z)=\frac{1}{2}\frac{(1\lvert z\rvert^2)^3}{\lvert 1z\rvert^4}, \quad z\in\D, $$ 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/studentpapers/record/2493421
 author
 Teka, Yohannis Wubeshet ^{LU}
 supervisor

 Anders Olofsson ^{LU}
 organization
 course
 MATM01 20112
 year
 2011
 type
 H2  Master's Degree (Two Years)
 subject
 publication/series
 Master Thesis in Mathematical Science
 report number
 LUNFMA30672011
 ISSN
 14046342
 other publication id
 2011:E54
 language
 English
 id
 2493421
 date added to LUP
 20141215 14:28:49
 date last changed
 20141215 14:28:49
@misc{2493421, abstract = {It is wellknown 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 $$ K_2(z)=\frac{1}{2}\frac{(1\lvert z\rvert^2)^3}{\lvert 1z\rvert^4}, \quad z\in\D, $$ 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 = {14046342}, language = {eng}, note = {Student Paper}, series = {Master Thesis in Mathematical Science}, title = {On a weighted Laplace differential operator for the unit disc}, year = {2011}, }