Polyanalytic boundary value problems for planar domains with harmonic Green function

There are three basic boundary value problems for the inhomogeneous polyanalytic equation in planar domains, the well-posed iterated Schwarz problem, and further two over-determined iterated problems of Dirichlet and Neumann type. These problems are investigated in planar domains having a harmonic Green function. For the Schwarz problem, treated earlier [Ü. Aksoy, H. Begehr, A.O. Çelebi, AV Bitsadze’s observation on bianalytic functions and the Schwarz problem. Complex Var Elliptic Equ 64(8): 1257–1274 (2019)], just a modification is mentioned. While the Dirichlet problem is completely discussed for arbitrary order, the Neumann problem is just handled for order up to three. But a generalization to arbitrary order is likely.

The Neumann problem for degenerate Hessian quotient equations

In this paper, we prove the existence of the [Formula: see text]-solution to the classical Neumann problem for the degenerate elliptic Hessian quotient equation [Formula: see text] under the condition that [Formula: see text].