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Iterations of Pompeiu operators. (English) Zbl 0908.47048

Summary: The Pompeiu operator \(T\) was extensively used by I. N. Vekua in his treatment of generalized Cauchy-Riemann systems. In the case of several complex variables when polydomains are considered, proper combinations of different \(T\)-operators for different components of the variable lead to a particular solution of the inhomogeneous Cauchy-Riemann system. This is applied to solve explicitely the Dirichlet problem in the unit polydisc for the inhomogeneous pluriharmonic system in the case of two complex variables.

MSC:

47G10 Integral operators
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35C15 Integral representations of solutions to PDEs
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