A Neumann problem for the polyanalytic operator in planar domains with harmonic Green function

Abstract

The higher order Pompeiu integral operator providing a particular solution to the inhomogeneous polyanalytic equation provides via the higher order Cauchy–Pompeiu representation an attempt for solving the respective higher order Dirichlet problem under certain solvability conditions. An alteration of this Pompeiu operator, the Schwarz–Pompeiu integral serves to solve the well-posed Schwarz problem for the polyanalytic operator via the Cauchy–Schwarz–Pompeiu representation. This alteration, explicitly known for the unit disc, is available for domains with the harmonic Green function. Another alteration of the Pompeiu operator with regard to the Neumann problem is introduced also for domains with the harmonic Green function. In fact, the novelty of the new integral operator is not the operator itself. For the unit disc, it is just the known higher order polyanalytic Pompeiu operator. For other domains however, an explicit evaluation of the integrals involved is not possible. The novelty is the combination with the effect of the harmonic Green function, which enables to handle the Neumann problem. As the higher order Neumann problem, however, is also over-determined for the polyanalytic operator, again solvability conditions on the data are required. They are worked out here in combination with the representation formula for the solution.

Keywords:

• Polyanalytic equation
• harmonic Green function
• integral representations
• Neumann problem

2020 Mathematics Subject Classifications:

• 30E25
• 30G20
• 31A10
• 31A30
• 35J40

Disclosure statement

No potential conflict of interest was reported by the author(s).