Boundary value problems for bi-polyanalytic functions

Abstract

The theory of bi-analytic functions introduced by Hua, Lin and Wu in the 1980s in order to solve some second-order systems of two partial differential equations in two variables is the theory of the second-order complex partial differential equation for some constant real α. Here the equation is investigated for 1 ≤ m, n. In the case m = 1 the solutions are called bi-polyanalytic. Different kinds of boundary conditions are introduced for these equations. They are originating from the well-known Schwarz, Dirichlet and Neumann problems from complex analysis, see e.g. in 2,9. Some are well-posed, others are only solvable under certain solvability conditions. Basic tools are higher-order Cauchy Pompeiu representations and the respective boundary value problems for analytic functions. In order to be explicit, the problems are investigated in the unit disc.

§Dedicated to Professor Wei Lin on the occasion of his 70th birthday

Keywords:

• Bi-polyanalytic functions
• Cauchy–Pompeiu representation
• boundary value problems for analytic
• polyanalytic and bi-polyanalytic functions
• Schwarz kernel
• Poisson kernel
• Green and Neumann function

Keywords:

• AMS Classifications:
• 30G20
• 30E25
• 35J55
• 35J30
• 35J40

Acknowledgment

The second author was supported by DAAD (German Academic Exchange Service) reinvitation programme in summer 2003.

Notes

§Dedicated to Professor Wei Lin on the occasion of his 70th birthday