Integral representations in the complex plane and iterated boundary value problems

Abstract

Fundamental solutions to differential operators lead to integral operators providing integral representation formulas for solutions to related differential equations. Proper modifications of the fundamental solutions result in integral operators which are related to certain boundary value problems. For complex partial differential operators of arbitrary order in the plane, fundamental solutions are achievable by properly integrating the Cauchy kernel. Particular such complex model differential operators are the polyanalytic and the polyharmonic operators. A hierarchy of integral operators is available for these model operators leading to polyanalytic Cauchy–Schwarz and to polyharmonic Green, Neumann, Robin, and hybrid Green integral operators. The theory is supplemented here by constructing a new polyanalytic Pompeiu (Pompeiu–Vekua) integral operator of any order adjusted to (iterated) Neumann boundary conditions.