Heinrich Begehr
abstract:
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.
Mathematics Subject Classification: 47G10, 35J05, 35C15.
Key words and phrases: Pompeiu operator, Cauchy Pompeiu representation, higher order equations, one and several complex variables, boundary value problems, polydomains.