Boundary value problems for bi-polyanalytic functions

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 Begehr, H. 1994. Complex Analytic Methods for Partial Differential Equations. An Introductory Text, Singapore: World Scientific. [Crossref] , [Google Scholar],9 Gakhov, FD. 1966. Boundary Value Problems, Oxford: Pergamon. [Crossref] , [Google Scholar]. 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.

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