Mathematical Sciences

Lund University

  • Title: On a weighted Laplace differential operator for the unit disc.
  • Description: 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.
  • Start Date: Dec. 9, 2011
  • Finished Date: Dec. 9, 2011
  • Supervisor: Anders Olofsson
  • Student: Yohannis Wubeshet Teka